Accommodating Wind Energy Characteristics in Power Transmission Planning Applications. Mr. Daniel Burke, (BE)

Transcription

1 Accommodating Wind Energy Characteristics in Power Transmission Planning Applications Mr. Daniel Burke, (BE) Thesis submitted to University College Dublin in fulfilment of the requirements for the degree of Doctor of Philosophy, (PhD) at the School of Electrical, Electronic and Mechanical Engineering Head of School: Prof. David Fitzpatrick Principal Supervisor: Prof. Mark O Malley August 2010

2 Table of Contents Table of Contents. ii Abstract. vi Statement of Original Authorship.. viii Clarification of Collaborative Research Responsibilities. viii Acknowledgements ix Publications, Submissions & Working Papers Arising From Thesis x Nomenclature & List of Acronyms. xi List of Figures xii List of Tables. xiv Chapter 1 Introduction I. Power system industry overview 1 II. Wind energy integration overview 4 III. Transmission system context of wind energy deployment.. 6 IV. Transmission planning methodology implications of wind characteristics 7 V. Context of research outlined in thesis. 8 VI. References.. 10 Chapter 2 Maximising Firm Wind Connection to Security Constrained Transmission Networks, journal paper published in the IEEE Transactions on Power Systems, Vol.25, No.2, May Abstract 13 I. Nomenclature and units 13 II. Introduction 14 III. Methodology. 16 A. Optimisation methodology overview 16 B. Optimisation methodology sub-tasks ) Approximating system wind power output time series 18 2) (MIP) Unit commitment and economic dispatch ) Formulating the load flow inequality constraints. 20 4) Removing redundant linear constraints ) Linear programming firm wind energy feasibility test ) Updating the system-total wind power time series 26 7) Incrementing the firm wind energy penetration target 26 ii

3 IV. Methodology application to a test-system.. 27 V. Results 28 A. Maximising the firm wind energy penetration level.. 28 B. Convergence of the system-total wind power time series 28 C. Investigating the solution load flow results. 29 D. Computational requirements 30 VI. Discussion 32 VII. Conclusion.. 34 VIII. References.. 34 Chapter 3 Considering Operational Wind Management Strategy Impacts on Transmission Planning, working paper in preparation for submission. Abstract. 37 I. Introduction 37 II. System power flow modelling.. 39 A. Impact of operational wind issues in planning timeframes.. 39 B. Scheduling model power flow investigations. 40 C. Long-term uncertainty sensitivity analysis 41 III. Test system. 42 IV. Results 43 A. Scheduling model plant capacity factor impact results 43 B. Scheduling model power flow impact results. 44 C. Long-term planning model uncertainty sensitivities. 46 1) Load profile uncertainty. 46 2) Conventional plant fuel price uncertainty V. Discussion and conclusions. 50 VI. References.. 51 Chapter 4 A Study of Optimal Non-Firm Wind Capacity Connection to Congested Transmission Networks, journal paper in review with the IEEE Transactions on Sustainable Energy. Abstract. 54 I. Nomenclature.. 54 II. Introduction 55 III. Test power system 57 IV. Optimal non-firm wind allocation model.. 58 A. Linear programming model formulation 58 B. Model decomposition and solution 60 V. Model investigation and sensitivity analyses. 61 A. Incrementing wind capacity targets.. 61 B. Wind resource profile sensitivities. 61 C. Load profile uncertainty sensitivities 62 D. Fuel-price uncertainty sensitivities. 63 E. Contingency management policy sensitivities 63 VI. Optimal non-firm capacity model results. 64 A. Increasing wind capacity targets.. 64 B. Wind resource profile sensitivities. 66 C. Load profile uncertainty sensitivities 67 iii

4 D. Fuel price uncertainty sensitivities. 67 E. Contingency management policy sensitivity 68 VII. Discussion and conclusions. 68 VIII. References.. 70 Chapter 5 Factors Influencing Wind Energy Curtailment, journal paper in review with the IEEE Transactions on Sustainable Energy. Abstract. 72 I. Introduction 72 II. Test power system details 74 III. Wind curtailment study implementation.. 75 A. Historical data timeframe modelling case study.. 76 B. Inter-locational curtailment risk-dependency case study.. 76 C. Inertial/congestion curtailment dependency case study 78 IV. Wind curtailment results.. 79 A. Historical data timeframe considerations. 79 B. Inter-locational curtailment risk dependency case study.. 83 C. Inertial/congestion curtailment dependency case study 86 V. Discussion 86 VI. Conclusions. 88 VII. References.. 89 Chapter 6 A Study of Multivariate Component Analysis Applied to Spatially Distributed Wind Power, journal paper in review with the IEEE Transactions on Power Systems. Abstract. 90 I. Nomenclature.. 90 II. Introduction 91 III. Wind power component analysis study. 93 A. Principal component analysis of wind power data. 93 B. Wind power data reconstruction residual effects 94 C. Independent component analysis of wind power data 94 D. Measuring independence using mutual information 96 IV. Multivariate wind data and test power system 96 A. Irish wind power time series data 96 B. Test power system information 97 C. Residual error system power flow case studies. 98 V. Results 99 A. Multivariate wind data 99 B. Principal component analysis results 100 C. Independent component analysis results. 105 VI. Discussion and conclusion 105 VII. References Chapter 7 Scenario Reduction Applied to Distributed Wind Power Transmission Studies, working paper in preparation for submission. Abstract. 109 I. Introduction 109 iv

5 II. Transmission studies and scenario reduction 111 A. Transmission planning problems investigated 111 1) Optimal non-firm wind capacity connection 111 2) Distributed wind curtailment analysis B. Application of scenario reduction 112 III. Test power system 113 IV. Results 115 A. Retained multivariate wind samples. 115 B. Optimal non-firm wind capacity solution variations C. Wind energy curtailment study variations 121 V. Conclusion VI. References Chapter 8 Considering the Benefits of Probabilistic Wind Characteristics in Transmission Planning, letter in preparation for submission. Abstract. 124 I. Introduction 124 II. Wind profile modelling III. Discussion 126 IV. References Chapter 9 Discussion, Conclusions and Recommendations for Future Research Work Appendix I - Test System Parameters for Chapter Appendix II - Test System Parameters for Chapter 3, 4, 5, 6, Appendix III - Interfacing MATLAB and GAMS Appendix IV - Benders Decomposition v

6 Abstract As an indigenous and renewable form of generation, wind energy integration has significant potential with regard to environmental and security of supply concerns in many electrical power systems. In contrast to most conventional power generation sources though, wind has low capacity factor, low capacity credit, appreciable variability and uncertainty, and spatially interdependent power generation characteristics. Significant reappraisal of traditional system modelling, planning, and market-design mechanisms may be necessary therefore. Present transmission capacity limitations and expansion delays only serve to further complicate this challenge. The research methods and results presented in this thesis pertain to concise and pragmatic representation of wind power s distinctive characteristics in transmission studies, proposing computationally efficient approaches to the resultant large-scale optimisation models, and giving due consideration to related issues within a deregulated power system context. Physically-firm wind power connection feasibility study requires power flow investigations over a large number of diverse study cases to ensure transmission network security is preserved. Most of these individual cases will be dominated by a very small number of critically onerous conditions however. An efficient redundancy removal scheme is proposed to exploit this property for computationally tractable optimal firm wind connection schemes. Wind generation rarely reaches maximum rated capacity in most power system locations though. Post-optimal analysis of such optimal firm capacity connection models indicates that a development of transmission networks to accommodate all available wind power production over extended timeframes will be unnecessarily costly. Non-physically-firm wind power connections will allow the connection of much greater wind capacity if a small amount of the available wind energy is allowed to be curtailed during onerous but low-probability transmission system overloading instances. The wind energy curtailment associated with such non-physically-firm network connections may have significant commercial investment implications in deregulated power systems however. Future power system uncertainties such as load profile and fossil fuel prices will result in non-negligible curtailment risk at individual locations. An analysis of inter-locational risk dependencies suggests that effective risk management schemes may be possible, though precise modelling choices for the various uncertainties will be quite influential. Further investigations suggest that wind curtailment caused by other system operational constraints may be a subset of the wind curtailment already identified by network congestion studies. An important issue of consideration for optimisation model formulation in wind capacity investment problems is the impact of model dimensionality and complexity. A study of stochastic unit-commitment influences on power flow patterns is conducted to determine the relevance of techniques managing wind variability and forecast uncertainty in longer-term transmission planning applications. For many power systems without large scale energy-limited storage, it is vi

7 shown that the inclusion of these operational management issues may be unnecessary from a computational-burden/accuracy trade-off viewpoint, when a pragmatic consideration of long term load profile and/or fuel and carbon price uncertainties must also be accounted for. If sequential dependence from one power system operational time-step to the next can justifiably be relaxed, there are significant modelling and computational implications. For example linear programming optimisation constraint matrices modelling multi-periodicity of wind are then blockdiagonally separable, and are thus easily exploited by standard decomposition schemes. Furthermore, scenario-reduction schemes are shown to be very promising for wind/transmission optimisation problems, combining similar but sequentially un-related hourly wind/load time series cases to give a much lower number of probability-weighted case-study samples. Considerable computational advantage results for relatively little accuracy degradation. Multivariate wind time series model compression efficiencies may also be possible in the spatial sense for wind and transmission study applications, as distributed wind power production patterns at nearby but geographically distinct locations will have significant interdependency due to common weather influences. Dimension reduction schemes using multivariate techniques such as principal and independent component analysis are investigated to determine if this spatial dependency can be efficiently characterised with a lower number of alternative transformed random variables. Results indicate that while significant dimensionality reduction will most likely be possible, the non-parametric statistical dependency of multivariate wind power output means that even though the alternative wind components will be uncorrelated, they are not necessarily statistically independent if a linear separation model is applied. In a power and energy industry experiencing great change as at present, the transmission planning challenge is most astutely considered as a decision-making-under-uncertainty task. Analysis of historical data suggests that wind at distinct locations is characterised by relatively stable medium-term probability distributions. In comparison to the effect of more generally uncertain parameters that must also be somehow factored into decision making processes, this probabilistic certainty wind characteristic may have associated decision-making benefits. In summary, the analysis and results presented in this thesis have important relevance for economic wind integration to constrained transmission networks. Even with relatively concise representation of wind power s multi-periodicity and spatial diversity characteristics, optimisation models in power systems will still be of large scale. The importance of effective decomposition algorithms will likely become prevalent as the scope for wind/transmission optimisation increases with other applications. Even though some level of wind energy curtailment is most likely optimal from a least-cost wind integration point of view, managing curtailment from system operational real-time security and longer-term market risk perspectives, particularly in systems presently without significant congestion or locational pricing experience, will be a considerable challenge. vii

8 Statement of Original Authorship I hereby certify that the submitted work is my own work, was completed while registered as a candidate for the degree stated on the Title Page, and I have not obtained a degree elsewhere on the basis of the research presented in this submitted work. Clarification of Collaborative Research Responsibilities The conception, implementation and composition of the original research outlined in Chapters 2 to 8 of this thesis was principally the contribution of the PhD candidate, Mr. Daniel Burke, under the oversight of Prof. Mark O Malley at the Electricity Research Centre, School of Electrical, Electronic and Mechanical Engineering, University College Dublin. The working paper entitled Considering Operational Wind Management Strategy Impacts on Transmission Planning, as given in Chapter 3 of this thesis, was conducted in collaboration with Dr. Aidan Tuohy, a colleague at the Electricity Research Centre. Dr. Tuohy s contribution to this work related to implementation of the stochastic unit commitment case studies as described therein. viii

9 Acknowledgements Most sincere thanks to everyone at the Electricity Research Centre (ERC), and indeed all the other assorted past inhabitants of Room 157, for the great help, advice and comradeship I ve enjoyed over the last four years - putting up with my various personal oddities has no doubt tested your tolerance levels on a number of occasions! Dr. Aidan Tuohy deserves particular mention for his collaboration with one of the research papers, as does Mr. Paul Smith for his help in proof-reading this thesis. The administrative and technical assistance of the other staff at the School of Electrical, Electronic and Mechanical Engineering is noteworthy as well. Sincere thanks to my parents, extended family and friends for all the personal support over the last few years without which I wouldn t have been able to attempt this challenge. The teaching and guidance of various previous mentors at Inchigeela National School, De La Salle College Macroom, and indeed the Department of Electrical and Electronic Engineering at University College Cork, is very much appreciated as a positive formative influence as well. It is important to express due recognition and gratitude for the financial support of the Sustainable Energy Authority of Ireland, administered through the Irish Research Council for Science Engineering and Technology, that along with similar support from the ERC industrial members and sponsors has made this postgraduate research work possible. Finally, the greatest acknowledgement of all must go to Prof. Mark O Malley. Not only for giving me the initial academic opportunity and direct research guidance over this extended time, but equally importantly for the less tangible personal qualities of disciplined yet creative application that his leadership of the ERC inspires in all associated staff. ix

10 Publications, Submissions & Working Papers Arising From Thesis Journal Paper Contributions: D.J. Burke, M.J. O Malley Maximizing Firm Wind Connection to Security Constrained Transmission Networks, IEEE Transactions on Power Systems, Vol.25, No.2, May D.J. Burke, A. Tuohy and M.J. O Malley Considering Operational Wind Management Strategy Impacts on Transmission Planning, (Working paper in preparation for submission). D.J. Burke, M.J. O Malley A Study of Optimal Non-Firm Wind Capacity Connection to Congested Transmission Systems, IEEE Transactions on Sustainable Energy, (In Review). D.J. Burke, M.J. O Malley Factors Influencing Wind Energy Curtailment, IEEE Transactions on Sustainable Energy, (In Review). D.J. Burke, M.J. O Malley A Study of Multivariate Component Analysis Applied to Spatially Distributed Wind Power, IEEE Transactions on Power Systems, (In Review). D.J. Burke, M.J. O Malley Scenario Reduction Applied to Distributed Wind Power Transmission Studies, (Working paper in preparation for submission). Letters: D.J. Burke, M.J. O Malley Considering the Benefits of Probabilistic Wind Characteristics in Transmission Planning, (Working letter in preparation for submission). Conference Paper Contributions: D. Burke and M.J. O Malley Optimal Wind Power Location on Transmission Systems A Probabilistic Approach, presented at the IEEE Probabilistic Methods Applied to Power Systems Conference, Rincon, Puerto Rico, May D. Burke and M.J. O Malley Optimal Firm Wind Capacity Allocation To Power Systems With Security Constraints, presented at the IEEE Power Systems Conference and Exhibition, Seattle, USA, March D. Burke and M.J. O Malley Transmission Connected Wind Curtailment With Increasing Wind Capacity Connection presented at the IEEE Power and Energy Society General Meeting, Calgary, Canada, July x

11 Nomenclature & Acronyms Detailed nomenclature and acronym information is given as appropriate in each separate chapter of this thesis. xi

12 List of Figures Chapter 2 Maximising Firm Wind Connection to Security Constrained Transmission Networks Fig. 1. The firm wind energy maximisation methodology flowchart.. 17 Fig. 2. Linear program constraint redundancy no feasible space intersection Fig. 3. Maximal vector subset of a 2-D dataset Fig. 4. The test system transmission network schematic.. 27 Fig. 5. Histogram of power imbalance between methodology re-iterations 29 Fig. 6. Load flow distribution for branch 6-12 under branch failure 30 Fig. 7. A typical graph of ordered rank-product values for pre-processing Chapter 3 Considering Operational Wind Management Strategy Impacts on Transmission Planning Fig. 1. The test power system under investigation Fig. 2. Power flow histograms for line adjacent to HVDC interconnection (line 12-19) Fig. 3. Power flow histograms for line adjacent to flexible ADGT plant location (line 6-11) Fig. 4. Sample power flow histogram from elsewhere in the system (line 17-21) Fig. 5. Sample pdf of line flows under the influence of peak customer demand uncertainty (Case I, line 1-3) Fig. 6. Sample pdf of line flows under the influence of spatial customer demand uncertainty (Case II, line 10-16) 47 Fig. 7. Seasonal base-load coal and CCGT average unit costs (Case III) Fig. 8. Sample pdf of line flows under the influence of fossil fuel price uncertainty (Case III, line 19-22) 49 Fig. 9. Sample pdf of line flows under the influence of fossil fuel price uncertainty (Case III, line 9-13).. 49 Chapter 4 A Study of Optimal Non-Firm Wind Capacity Connection to Congested Transmission Networks Fig. 1. The test power system under investigation Fig. 2. Block diagonal constraint matrix structure Fig. 3. Master/sub-problem Benders decomposition scheme.. 61 Fig. 4. Irish wind farm inter-yearly capacity factor variation (specific y-axis info not given for commercial sensitivity reasons).. 62 Fig. 5. Wind energy curtailments for increasing total wind capacity targets. 65 Fig. 6. Value of optimal non-firm wind capacity allocation, as % of Area 1 total cost, at each progressive Benders iteration Chapter 5 Factors Influencing Wind Energy Curtailment Fig. 1. The test power system under investigation Fig. 2. Variations in wind energy curtailment at Farm-9 with respect to number of years of data (15-minute sample resolution). 79 Fig. 3. Variations in wind energy curtailment at Farm-11 with respect to number of years of data (15-minute sample resolution). 79 Fig. 4. System-averaged mean absolute wind energy curtailment error with respect to number of years of data and data sampling resolution. 80 Fig. 5. System-averaged mean absolute wind capacity factor error with respect to number of years of data and data sampling resolution.. 81 Fig. 6. A comparison of capacity factor and wind energy curtailment variations over the 8 year period for Farm 9 (15-min sampling). 81 Fig. 7. A comparison of capacity factor and wind energy curtailment variations over the 8 year period xii

13 for Farm11 (15-min sampling).. 81 Fig. 8. Wind curtailment risk dependency for Farms 5 and 11 (Case I).. 84 Fig. 9. Wind curtailment risk dependency for Farms 13 and 15 (Case II) Fig. 10. Scatter plot of inertial/network-congestion curtailment 8 GW wind. 87 Fig. 11. Time series of inertial/network-congestion curtailments 8 GW wind 87 Chapter 6 A Study of Multivariate Component Analysis Applied to Spatially Distributed Wind Power Fig. 1. Irish wind power zones used in this study (each with 2-5 wind farms).. 97 Fig. 2. Test power system network schematic and wind zone locations Fig. 3. Marginal probability density function histograms for Zones X, Y Fig. 4. Bi-variate dependency scatter plot for Zones X, Y. 99 Fig. 5. Ordered eigenvalue plot associated with each principal component Fig. 6. The resulting principal component time series Fig. 7. Zone A residual error for 1, 3, 5, 7, 9 retained principal components Fig. 8. Reduction in zonal rms reconstruction error for additional components Fig. 9. Case-I : Transmission network power flow modeling residual error (line 15-25) Fig. 10. Case-II : Wind energy curtailment % with respect to number of retained components Fig. 11. Scatter plot of retained principal components 1 and Fig. 12. The ICA estimates for 5 retained components Fig. 13. Scatter plot of independent components 2 and Fig. 14. Percentage reduction in mutual information from PCA to ICA results Chapter 7 Scenario Reduction Applied to Distributed Wind Power Transmission Studies Fig. 1. The test power system under investigation. 115 Fig. 2. Bi-variate scatter plot of 500 hourly wind power samples Fig. 3. Effect of scenario reduction 250 samples retained Fig. 4. Effect of scenario reduction 100 samples retained Fig. 5. Corresponding number of Benders decomposition iterations required Fig. 6. Optimal wind capacity allocations solution variations. 119 Fig. 7. Estimated cost function values for different numbers of retained scenarios Fig. 8. True optimal cost function values for different numbers of retained scenarios expressed as percentage deviations of Area-1 total cost Fig. 9. Estimated wind energy curtailment percentage variations Chapter 8 Considering the Benefits of Probabilistic Wind Characteristics in Transmission Planning Fig. 1. Wind profile variation from year to year for Irish wind plant A Fig. 2. Medium-term wind profile variation for Irish wind plant A Fig. 3. Wind profile variation from year to year for Irish wind plant B Fig. 4. Medium-term wind profile variation for Irish wind plant B xiii

14 List of Tables Chapter 2 Maximising Firm Wind Connection to Security Constrained Transmission Networks Table-I. Optimised Wind Energy Penetration Capacity Allocations (MW). 28 Table-II. 9.5% Wind Energy Re-iterated Wind Capacity Allocations (MW) 29 Chapter 3 Considering Operational Wind Management Strategy Impacts on Transmission Planning Table-I. Plant Capacity Factors For Different Scheduling Approaches 6 GW Installed Wind. 44 Chapter 4 A Study of Optimal Non-Firm Wind Capacity Connection to Congested Transmission Networks Table-I. Test System Wind Capacity Factor Information (%) Table-II. Optimal Non-Firm Wind Capacity Allocation, (MW). 65 Table-III. 6 GW Wind Capacity Allocation Solution Progression, (MW).. 66 Table-IV. Effect of Limited Wind Profile Model- 6 GW Capacity Solution (MW).. 67 Table-V. Effect of Load Profile Uncertainty - 6 GW Capacity Solution (MW). 67 Table-VI. Effect of Fuel Price Uncertainty - 6 GW Capacity Solution (MW). 68 Table-VII. Effect of Contingency Management - 6 GW Capacity Solution (MW).. 68 Chapter 5 Factors Influencing Wind Energy Curtailment Table-I. Optimal Non-Firm Wind Capacity Allocations, (MW). 77 Table-II. Wind Energy Curtailment %- Effect of Sensitivity Analysis. 82 Table-III. Mean Wind Energy Curtailments, (%) Table-IV. Distributed Wind Energy Curtailment Risk Correlations Case I 84 Table-V. Distributed Wind Energy Curtailment Risk Correlations Case II 85 Table-VI. Wind Energy Network/Inertial Curtailment Values, (%) 87 Chapter 6 A Study of Multivariate Component Analysis Applied to Spatially Distributed Wind Power Table-I. Zonal Wind Capacity Allocation Used For Scopf Analysis, (MW) 98 Table-II. Zonal Correlation Coefficients Before PCA Transformation. 100 Table-III. Principal Component Covariance After PCA Transformation Table-IV. Mutual Information of PCA and ICA Estimates, (Bits). 106 Chapter 7 Scenario Reduction Applied to Distributed Wind Power Transmission Studies Table-I. Test System Wind Farm Capacity Factors (%) Table-II. Optimal Non-Firm Wind Capacity Allocation Solution Vectors MW) Chapter 8 Considering the Benefits of Probabilistic Wind Characteristics in Transmission Planning Table-I. Euclidean Histogram Probability Distance Results APPENDIX 1 Table-A-I. Wind Farm Time Series Capacity Factors Table-A-II. Branch Reactance Parameters and Capacity Limits. 133 Table-A-III. Maximum Bus Load Values Table-A-IV. Conventional Plant Information 134 xiv

15 APPENDIX 2 Table-A-I. Test Power System Network Branch Information Table-A-II. Maximum Bus Load Values Table-A-III. Typical Unit Commitment Inter-Temporal Constraints Applied. 136 Table-A-IV. Conventional Generation Portfolio Information xv

16 CHAPTER 1 - INTRODUCTION I. POWER SYSTEM INDUSTRY OVERVIEW The electric power system is arguably one of the most critically important, technologically sophisticated and economically valuable infrastructural investments in the history of human civilisation. The contribution of a secure, reliable and cost effective supply of electric power is fundamental to a well functioning society and a productive modern economy, particularly as electrically powered technology applications evolve to play an increasingly indispensable role in daily life. The traditional power system architecture which essentially developed around large scale centralised conventional generation, with long distance high-voltage transmission to distribution connected customer demand, has generally served us well over the past century or so. However, there are presently a number of coincident factors that suggest this power system architecture, its generation portfolio composition, and its system operational principles must undergo significant reformation and redesign in order to meet the more complex challenges and responsibilities of the next few decades and beyond. With their well established generation technology, the economics of traditional coal, gas, and oil power generation sources are primarily reliant on a low cost and stable supply of their respective fossil fuels. The finite nature of hydrocarbon-based fuels is a geological certainty however, and while there are still abundant coal resources around the world, significant concerns have been recently voiced with regard to the true medium to long-term availability of oil and gas reserves that can be extracted at current operational costs [1]. Inherent political instability in the dominant oil and gas producing parts of the world, as well as continually rising energy demand from the economies of large developing countries, has greatly added to energy price volatility patterns observed in recent times [2]. There is thus great concern in the many nations with relatively few indigenous energy resources with regard to the medium and long-term security of their energy supply. An over-dependence on foreign oil and gas fossil fuel sources suggests possible exposure to future price spikes and/or supply disruption, if and when worldwide economic growth and its related energy demand recovers. The overwhelming evidence that human-activity related carbon dioxide emission is one of the most significant drivers of global climate change [3] also has considerable implications for the design of future electric power systems. Fossil-fuel based electric power generation contributes to a significant proportion of overall carbon dioxide emissions [4], and thus a transition of electricity generation portfolios to less carbon-polluting technologies is essential. It is generally agreed that 1

17 directly applying an economic cost to carbon dioxide emission is the most efficient and transparent means to gradually bring about the structural changes necessary to reduce carbon intensity in developed economies [5]. Whether this carbon dioxide cost should be implemented either by a direct taxation levy or the legal requirement to purchase tradable emissions permits [6], and what the optimal carbon price or total number of allocated permits should respectively be [7],[8], are of course very hotly debated economic topics with many issues of contention. In any case, this carbon price and/or regulatory uncertainty has significant economic cost and risk implications for existing and planned new fossil-fuel fired conventional generation investment. Coal generation investment is particularly affected, as coal is a relatively carbon-intensive fuel source. While conceptual carbon-capture and sequestration schemes have been proposed for large conventional generators [9], the technology is as of yet generally unproven, and its installation will no doubt add to conventional generation investment and operational costs. As a result of these fuel and carbon price volatility uncertainties, interest in nuclear power has experienced a renaissance in recent times due to both its low carbon intensity and its use of a fuel with a relatively stable supply chain [10]. Even accounting for recent developments in modularised reactor technology [11], the true lifetime cost effectiveness of nuclear power is somewhat disputed however, when all the construction costs, operational costs, decommissioning costs and long-term storage repository costs for high-grade nuclear waste are estimated [12]. For countries presently with a small or even a non-existent nuclear industrial complex, the knowledge and expertise gap required to develop and manage a large-scale nuclear generation fleet from scratch can be equally daunting. The perception of possible future energy price spikes due to carbon and fuel price volatilities will have an impact on future power system load profile shape and composition. Many schemes are promoted to focus widespread consumer efforts in relation to improving electric energy efficiency [13]. Reducing demand is one of the most direct and most cost-effective means for an economy to reduce carbon footprint and exposure to energy price volatilities [14]. It is proposed that a combination of time-of-use pricing, smart-metering technologies as well as other demand-side management schemes will have widespread application in the next decade and onwards for both power system load profile shaping and overall electrical energy demand reduction [15]. With the evolution of such changes along with well-functioning electricity markets, electric power demand will most likely not simply be fixed and price-unresponsive, but will increasingly be considered with inter-related price/quantity elasticity characteristics, possibly linked over multiple market-clearing time steps. This change will become especially relevant with additional load profile influences from increased home-heating and transport applications. The 2

18 projected transition of transport vehicle fleets from petrochemical fuel sources to hybrid electric and/or fully electric propulsion systems [16] will have significant impact on electricity demand and load-profile-shaping, depending on when and how it is both possible and cost-effective for batteries to be re-charged etc. In short, the convergence of these traditionally separate energy requirements towards the electric power system, in part due to its versatility as an energy transfer hub with power contributions from many diverse primary energy sources, will have significant impacts on composition and shape of future power system demand profiles. Power system planning and operational procedures will thus have to adapt accordingly. Advanced power system communications and control systems are envisaged as a key facilitator of this transition from the present electric power system, with its strong association to the traditional passive top-down architecture, to a more interactive and horizontally configured architecture in the near future. The revolution of information technology capability over the last few decades is expected to provide the means to ensure secure and reliable management of the power system in a configuration and operating context very different from its originally intended design function [17]. Wide-area and high-frequency power system parameter observation combined with swift remedial action control functionality may be an important application for such information technology capabilities, achieving greater performance from proportionally less infrastructural investment in the near future [18],[19]. The recent and projected growth in renewable energy will also have a revolutionary impact on the composition of electricity generation portfolios within this new power system industry context over the decades to come [20],[21]. Renewable energy sources such as traditional hydroelectricity and wind, wave, tidal, biomass, geothermal and solar power technologies, amongst others, have clear advantages over fossil-fuel based generation sources in a number of key ways. First of all, renewable energy developments generally have very low carbon intensity, and are therefore considered essential in the response to the global climate change problem. Secondly, renewable energy sources are typically indigenous to the power system areas where they are deployed, which is useful from a general security of supply perspective. Furthermore, in most cases, the major component of overall renewable energy costs relates to the initial infrastructural investment at the development stage of the project, with the natural energy itself subsequently harnessed almost cost-free. From a risk-adversity perspective, the relatively low uncertainty of most renewable electricity generation costs subsequent to initial construction is very useful for overall energy portfolio volatility reduction [22]. While many of the renewable technologies are still very much at the early stages of their development, particularly with regard to competitiveness in a market setting without strong carbon emissions regulation, 3

19 other more mature technologies such as wind power are presently playing an important role in this overall generation portfolio cost/risk trade-off in addition to any carbon-neutral advantages. As a result, there is significant political and regulatory emphasis on increasing the proportion of wind energy in electric power systems in the world today, with many nations considering very ambitious overall wind energy penetration targets [23], [24]. There is an Irish government target for a 40% renewable energy penetration level on the Irish power system for example [25], the vast majority of which will be achieved from large-scale wind power investment. In short, wind energy is a very promising renewable energy source, so a study of the implications of increasing its share in future power system generation portfolios is timely and important, both from an applied industrial and an inter-disciplinary research perspective. II. WIND ENERGY INTEGRATION OVERVIEW Wind energy is a fundamentally different type of power generation resource to the traditional composition of electricity generation portfolios, mainly due to either the latent properties of the available wind speed resource itself, or indeed the characteristics of the wind turbine technology used to harness it. As a result therefore, there are a number of power systems operational and planning challenges related to the integration of this alternative type of generation [26]. Wind generators typically tend to be induction machine based - in the past, squirrel-cage machines were preferred for their reliability and robustness, but new installed capacity at present is primarily of the doubly-fed induction machine class. Addition of large amounts of asynchronous induction machine based wind generation to power systems initially lead to concerns for inertial stability [27], fault-ride-through and network voltage stability performance [28]. Advances in doubly-fed induction machine power electronic control functionality have alleviated some of these concerns somewhat [29], [30], even if much of this technological capability is yet to be widely applied or implemented in practice. The dependence of wind generation on underlying weather trend variations thereby gives it an a-periodic and irregular production pattern. Wind power production has a low capacity factor (approximately 30-35% on the Irish power system for example), with power output often below half of the rated installed capacity for extended periods of time. Its coincidence with system load demand is generally quite statistically weak in most power systems (e.g. slightly higher long-run average wind power output in the colder Northern-European winter months for example). This partially dispatchable nature and poor load-following capability leads to a much lower capacity credit reliability contribution than that provided by other generation forms [31]. 4

20 Wind power not only exhibits significant sequential variability [32], but also significant related forecast uncertainty from a power system operational timeframe perspective [33]. Advanced wind forecasting techniques have focused on developing probabilistic models to appropriately represent the spread of forecast uncertainty, and its dynamic evolution for a given atmospheric stability situation and operational time horizon [34], [35]. Wind variability and forecast uncertainty has significant implications for the unit-commitment of other conventional generators in the power system. Large conventional units generally have inter-temporal start-up, ramp-rate, minimum-up-time, minimum-down-time constraints, as well as start-up fuel costs that must be considered in the overall least-cost system scheduling procedure. Advanced scheduling models have been developed using stochastic optimisation techniques that integrate wind forecast uncertainty to the decision-making process [36], [37]. Forecast uncertainty may require higher levels of primary, secondary and replacement reserve to be carried over different system operational time horizons [38], [39]. Wind forecast variability and uncertainty therefore has important market operation and design implications for most systems significant installed wind capacity generally leads to greater demand for regulating or balancing power, may require more frequent gate closures to reflect real-time operational issues, and more coordinated market implementation with neighbouring systems [40]. Wind capacity is typically installed in smaller and more geographically distributed individual quantities than comparable levels of conventional generation. Due to common weather influences, these many separate yet spatially inter-related wind generation sites will have appreciable power production statistical inter-dependency. The distinctive shape of each individual wind farm s power output statistical distribution results from passing the Weibull wind speed distribution through the turbines piece-wise non-linear wind-speed/wind-power transformation curve. The multivariate statistical dependency between power outputs at separate sites generally reduces with increasing geographical separation by distance [32], and is equally important to retain for power system planning and operational studies. Due to the weather and climate dependent nature of wind power production, strong seasonal and diurnal dependency patterns are often observed in wind power data. In short, the non-parametric probability distribution shapes, their multivariate dependencies of intermediate strength and possible non-stationary auto-correlative properties, ensure that advanced mathematical techniques are required for accurate statistical treatment of wind power data in a variety of important power system studies [41]. Alternatively, the availability of a recorded historical multivariate wind power time series dataset, with reasonable spatial distribution and appropriate length of timeframe, is quite useful in that any statistical and auto-correlative sequential 5

21 dependencies are inherently contained in the data without the need for artificial synthesis methods. III. TRANSMISSION SYSTEM CONTEXT OF WIND ENERGY DEPLOYMENT From a transmission planning perspective, the best wind resource sites typically tend to correspond to less-populated and therefore the less-developed regions of existing power system networks. As a result, a large proportion of installed wind capacity is presently connected at weaker low-voltage parts of the transmission network or indeed at distribution level in many power systems, with related technical challenges in transporting the available wind energy to the distant urban load centres. Connecting the much greater levels of wind capacity anticipated in the medium to long term will require significant transmission investment and network expansion in many power systems therefore. A number of detailed transmission studies have been carried out in many power systems to investigate the feasibility and technical requirements of large scale wind capacity connection. Many studies of the pan-european transmission system have identified specific regional interconnection bottlenecks as significant impediments to transporting the available wind energy from the lowest-development-cost windy areas on the western and northern peripheries to the more central demand concentrations [42]. Inter-regional transmission constraints also hinder the management of operational wind variability/forecast uncertainty in localised areas with high wind installation [40], as well as reducing the effective capacity credit contribution of wind energy in Europe as a whole. Similar conclusions have been formed from wind/transmission integration studies in North America [43],[44], and on the smaller and relatively more isolated Irish All-Island power system [45]. With the identification of the main system congestion points, conceptual expansion plans have been developed, often proposing very high voltage transmission and/or HVDC link solutions to expedite the concentrated development of wind capacity installations in the most attractive wind resource areas [46],[47]. Even though the levelised cost of transmission is a relatively small component of the overall long-term cost of electricity [45], the scale of the upfront capital investment required within a reasonably short time horizon in many power systems means that even small efficiencies achievable are often valued in the hundreds of millions or even billions of euro, depending on the system size. There is thus great scope for effective optimisation techniques applied to the challenges of planning and operating modern transmission systems with very high wind penetrations. 6

22 Implementing large-scale redesign of the transmission system network, with either capacity upgrades for existing facilities or the development of entirely new network paths, is an unavoidably time-consuming process. Construction lead-time delay in itself is also increasingly exacerbated in some areas by local opposition to new transmission network infrastructure development. Innovative use of any existing facilities may be required to accelerate the connection of wind capacity in the short to medium term, while effective technical and social solutions are found to other system development challenges in the long term. IV. TRANSMISSION PLANNING METHODOLOGY IMPLICATIONS OF WIND CHARACTERISTICS In the past, the rather predictable shape of the customer load demand profiles, with well-known daily, week-end and seasonal characteristics, as well as very similar demand patters in each part of the network, allowed transmission planning engineers to justifiably focus on a very limited number of deterministic snapshot power flow case-studies when assessing the network performance implications of new generation connections and new infrastructure developments. In northern European systems for example, judiciously chosen deterministic cases such as the winter-day-peak maximum loading scenario, summer-night-valley minimum loading scenario, and summer-day-peak typical loading scenario would have generally encapsulated the expected variation in transmission network flows observed over a much longer yearly timeframe [48]. Facilitating the stochastic and multivariate characteristics of wind power within such a limited deterministic study context is not possible however. Specific assumptions with regard to the level of total instantaneous wind power output, and equally importantly what individual contributions the spatial spread of wind generation sites make to this total wind power level, are difficult to justify for each of these deterministic case study examples. To facilitate the connection of wind generation, a re-appraisal of traditional transmission planning methodologies is therefore prudent. Assessment of transmission system performance over a much wider number of case studies has become necessary in order to properly characterise the contribution of distributed wind capacity investments to system power flow patterns. Year-long wind time series power flow studies are now common for detailed wind integration investigation [42],[43], with either multiperiod historical data or synthesised wind power patterns used to recreate the statistical nature of wind capacity s contribution to transmission system power flows. The computational implications of such time series power flow assessments in large-scale networks are significant however, particularly when many alternative transmission planning candidate solutions must be 7

23 investigated. Furthermore, the anticipation of more general future power system changes requires an effective consideration of the background uncertainty related to load profile and conventional plant fuel price parameters. The vast combination of possible power flow conditions to be assessed therefore has significant computational dimensionality implications for practical transmission planning decision-making in large scale power systems. Techniques to effectively limit transmission system study dimensionality while maintaining an appropriate degree of solution accuracy will have direct relevance to the wind integration challenges of today s electricity industry. The relatively low capacity factor of wind capacity installations also has some important implications for the design of future transmission networks. The very low probability of rated nameplate wind power output might suggest that a network judiciously designed to accommodate the available wind resource for most, but not all, of the time may be most cost effective. Non-physically firm transmission connections for wind power may be facilitated by occasional wind curtailment during system operation. The low capacity credit of wind power would furthermore imply that such transmission congestion related curtailment will have relatively low impact on overall power system reliability as compared to transmission congestion limitations placed on conventional generation export. V. CONTEXT OF RESEARCH OUTLINED IN THESIS The research methods, concepts and results presented in this thesis relate to the accommodation of wind energy characteristics in a number of power transmission planning applications. Significant emphasis is attributed to the task of concise representation of variable wind power output and multivariate wind power spatial inter-dependency, with a view to appropriate model dimensionality and complexity reduction wherever possible. Investigation of the large scale models that nevertheless still result is applied to take advantage of any special problem structure, so that effective solutions can be obtained within practical computational resource capabilities and timeframes. Pragmatic consideration of the effects of inherent power system parameter uncertainty is included where relevant, and some possible concepts proposed to mitigate it in particular problems. Chapter 2 presents a novel methodology to determine solutions to the optimal firm wind capacity allocation problem, with a focus on the development of an efficient linear programming redundancy removal pre-processing scheme. An approximate geographical smoothing assumption of system-total wind power output is used to include wind variability effects, while separating the 8

24 integer programming unit commitment procedure from the linear programming wind capacity placement task. Chapter 3 investigates in more detail the relevance of such operational timeframe wind variability and forecast uncertainty management issues to longer-term transmission planning studies, with the power flow modelling influences of deterministic and stochastic unit commitment techniques compared to a pragmatic consideration of the inherent long-term power system model uncertainty. Chapter 4 investigates the optimal non-firm wind capacity allocation problem for congested transmission networks, building on the conclusions of Chapter 3 to justify a block-diagonally separable constraint matrix model and a Benders decomposition solution approach. Non-physically-firm wind connections imply occasional wind curtailment for system congestion reasons however. A detailed consideration of a number of factors influencing such wind curtailment is presented in Chapter 5, investigating natural interyear wind profile variation, network congestion and inertial stability constraint effects. The effect of power system parameter uncertainty on precise wind curtailment volume estimation is highlighted in detail, with an investigation of inter-locational wind curtailment risk dependency also outlined to highlight the possible scope for curtailment risk management schemes. A multivariate component analysis study is investigated in Chapter 6, using principal and independent component analysis to reduce the multivariate spatial wind power representation to a much more effective lower number of random statistical variables. Chapter 7 furthermore investigates the utility of scenario reduction techniques to reduce the computational burden associated with multi-period wind power flow representation, with a specific focus on the solution degradation of two common wind/transmission planning studies for much lower numbers of discrete probability-weighted scenarios retained. Finally, Chapter 8 uses multi-year historical wind power output data to investigate the medium-term stability of wind power probability density functions, thus considering the benefits of wind power s probability characteristic in a transmission planning context that is significantly impacted by more general uncertainties in many other parameters. A critical analysis of the main research contributions and wider context of the material outlined in this thesis, along with recommendations for future research works, is subsequently outlined in Chapter 9. In keeping with the contemporary thesis submission procedures at University College Dublin, the chapters of this thesis are written in the style of self-contained research papers that are either already published, in peer-review, or are working papers in preparation for submission to various electrical power engineering journals. Nomenclature and references are given in each chapter as appropriate. 9

25 VI. REFERENCES [1] The Association for the Study of Peak-Oil and Gas, website - [2] S.P.A. Brown, M.K.Yucel What Drives Natural Gas Prices?, Energy Journal, 2008, Vol. 29 Issue 2, p [3] Intergovernmental Panel on Climate Change Fourth Assessment Report: Climate Change 2007 (AR4), available at website - [4] CO2 Emissions From Fuel Combustion, Highlights 2009 Edition, International Energy Agency Statistics, available at website - [5] W.D. Nordhaus Rolling the DICE An Optimal Transition Path for Controlling Greenhouse Gases, Resource and Energy Economics, Vol.15, Issue 1, March [6] Carbon Markets An International Business Guide, A. Brohe, N. Eyre and N. Howarth, Earthscan London, [7] The Economics of Climate Change The Stern Review, N. Stern, Cambridge University Press UK, [8] R.S.J. Tol The Social Cost of Carbon Trends, Outliers and Catastrophes, Economics, Vol.2, , August 12, [9] Where Europe Buries Carbon, IEEE Spectrum Magazine, August 2009, available at website - [10] Nuclear Reactor Renaissance, IEEE Spectrum Magazine, August 2010, available at website - [11] Downsizing Nuclear Plants, IEEE Spectrum Magazine, May 2010, available at website - [12] I.N. Kessides Nuclear Power Understanding the Economic Risks and Uncertainties, Energy Policy, Volume 38, Issue 8, August [13] Power Of One Campaign, Sustainable Energy Authority of Ireland, website - [14] A Cost Curve for Greenhouse Gas Reduction, McKinsey Quarterly, available at website - [15] S. Braithwait, Behaviour Modification, IEEE Power and Energy Magazine, May/June Edition 2010, available at website - [16] E. Ungar, K. Fell Plug-in, Turn-on, Load-up, IEEE Power and Energy Magazine, May/June Edition 2010, available at website - [17] J. McDonald IEEE Smart-Grid A Strategic Roadmap to Smart-Grid Success, available at website - [18] A.G. Phadke The Wide World of Wide Area Measurement, IEEE Power and Energy Magazine, September/October Edition 2008, available at [19] J. Wen, P. Arons, W.H. Edwin Liu The Role of Remedial Action Schemes in Renewable Generation Integrations, presented at the IEEE PES Innovative Smart Grid Technologies Conference, Maryland, January [20] European Renewable Energy Road Map, Commission of the European Communities, available at website - [21] The American Recovery and Reinvestment Act of 2009, available at website - [22] R. Doherty, H. Outhred, M.J. O Malley Establishing the Role That Wind Generation May Have in Future Generation Portfolios, IEEE Transactions on Power Systems, Vol. 21, No.3, August [23] 20% Wind Energy by 2030 Increasing Wind Energy s Contribution to U.S. Electricity Supply, U.S. Department of Energy, May 2008 available at website 10

26 [24] A Breath of Fresh Air EWEA 2009 Annual Report", European Wind Energy Association (EWEA), April Available online - pdf [25] Statement by Minister for the Environment, Heritage and Local Government, Parliamentary Debates, Dáil Éireann, Vol. 663, No. 4, 15th October 2008, P791 [26] Impact of Large Amounts of Wind Power on Design and Operation of Power Systems - Results of IEA Collaboration, IEA Task 25 Description, available at website - [27] Lalor, G., Mullane, A., and O Malley, M.J., Frequency Control and Wind Turbine Technologies, IEEE Transactions on Power Systems, Vol. 20, pp , [28] J. Kabouris, F.D. Kanellos Impacts of Large Scale Wind Penetration on Designing and Operation of Electric Power Systems, IEEE Transactions on Sustainable Energy, Vol.1, No.2, July [29] Mullane, A. and O Malley, M.J., The inertial-response of induction-machine based wind-turbines, IEEE Transactions on Power Systems, Vol. 20, pp , [30] E.Vittal, M.J. O Malley, A. Keane A Steady-State Voltage Stability Analysis of Power Systems with High Penetrations of Wind, IEEE Transactions on Power Systems, February [31] M. Milligan, K. Porter Wind Capacity Credit in the United States, presented at the IEEE Power and Energy Society General Meeting, Pittsburgh USA, July [32] T. Ackermann, (Editor) Wind Power in Power Systems, Wiley, [33] B. Ernst, B. Oakleaf, M. Ahlstrom, M. Lange, C. Moehrlen, B. Lange, U. Focken, K. Rohrig, Predicting the Wind, IEEE Power and Energy Magazine, November/December Edition 2007, available at website - [34] The Anemos Plus Project website at [35] P. Pinson, G. Papaefthymiou, B. Klockl, H.A. Nielsen, H. Madsen From Probabilistic Forecasts to Statistical Scenarios of Short-Term Wind Power Production, Wind Energy 12(1) 2009, pp [36] C. Weber, P. Meibom, R. Barth, H. Brand, WILMAR: A Stochastic Programming Tool to Analyze the Large-Scale Integration of Wind Energy Chapter-19, pp , Optimization in the Energy Industry, Springer Berlin Heidelberg [37] A.Tuohy, P.Meibom, E. Denny and M.J. O Malley Unit Commitment for Systems with Significant Wind Penetration, IEEE Trans. Power Systems Vol. 24, No. 2, May [38] R. Doherty, M.J. O Malley A New Approach to Quantify Reserve Demand in Systems With Significant Installed Wind Capacity, IEEE Transactions on Power Systems, Vol.20, May [39] J.M. Morales, A.J. Conejo, J. Perez Ruiz Economic Valuation of Reserves in Power Systems with High Penetration of Wind Power, IEEE Transactions on Power Systems, Vol.24, No.2, May [40] Integrating Wind, Developing Europe s Power Market for the Large Scale Integration of Wind Power, TRADEWIND Final Report, available online - [41] A. Lojowska, D. Kurowicka, G. Papaefthymiou and L. Van Der Sluis Advantages of ARMA-GARCH Wind Speed Time Series Modeling, presented at the IEEE PMAPS Conference, Singapore, June [42] European Wind Integration Study Final Report, available at [43] Eastern Wind Integration and Transmission Study, Final Report, prepared for the National Renewable Energy Laboratory USA, by Enernex Corporation, available online

27 [44] J.C. Smith, M.R. Milligan, E.A. De Meo, B. Parsons Utility Wind Integration and Operating Impact State-of-the-Art, IEEE Transactions on Power Systems, Vo.22, No.3, August [45] All Island Grid Study, Workstream 4 Analysis of Impacts and Benefits, Irish Government Department of Communications, Energy and Natural Resources/United Kingdom Department of Enterprise, Trade and Investment, Jan Available online - [46] North American Joint Coordinated System Plan, available at website - [47] Grid 25 A Strategy for the Development of Ireland s Electricity Grid for a Sustainable and Competitive Future, Eirgrid, available at website - [48] Transmission Forecast Statement , Eirgrid, available at website

28 CHAPTER 2 MAXIMISING FIRM WIND CONNECTION TO SECURITY CONSTRAINED TRANSMISSION NETWORKS Journal paper published in the IEEE Transactions on Power Systems, Vol.25, No.2, May Mr. Daniel J. Burke, and Prof. Mark J. O Malley. Abstract Prudent use of existing transmission capacity could be achieved by an optimal allocation of wind capacity to distinct transmission nodes. The statistical interdependency of geographically separate wind sites and the partially-dispatchable nature of wind power require a collective analysis of all potential wind farms over an extended time-frame in any optimised transmission planning study. The methodology presented in this paper separates this large optimisation problem into smaller sub-tasks, including a year-long sequential time series hourly integer unit commitment, a linear DC load-flow network model with hourly security constraints, and a linear programming optimisation model to estimate the maximum firm wind energy penetration for a given network. A novel maximal-vector based constraint redundancy analysis is employed to significantly reduce the linear programming optimisation dimensionality. Firm wind capacity connections are facilitated in this paper - i.e. those to which wind curtailment to manage congestion is not applicable within a typical system planning timeframe analysis. Each bus is allocated firm capacity on the basis of maximising the possible firm wind energy penetration in the transmission system as a whole, while preserving traditional network security standards. Index Terms computational geometry, linear programming redundancy, power transmission, wind energy. I. NOMENCLATURE AND UNITS A. Indices d - linear constraint coefficient position index h - time series hourly position index i - network bus position index j - network branch index k - potential wind farm network location index r - DC load flow reference bus position index s - power flow contingency scenario index v - the maximal-vector redundancy stop-criterion set index 13

29 B. Constants α ijs - DC load flow power transfer distribution factors for line j with respect to bus i under contingency scenario s D AVG - average system power demand level (MW) L j - thermal capacity for branch j (MW) λ k - capacity factor of wind farm k λ AVG - capacity factor of system averaged wind power time series G - the number of generation sites in the power system N - the number of network branches in the system S - the total number of contingency scenarios considered T - the length of the wind power time series (years) V - the maximal vector redundancy analysis stop-criterion Δδ - the discrete wind energy penetration target increment C. Time Series t kh - nominal 1MW wind power time series k in hour h (MW) t AVG h - average value of all the nominal 1MW wind power time series in hour h (MW) t APR h - the geographically smoothed total system wind power production value in hour h, (MW) γ jhs - the partial load flow solution of load/conventional plant in branch j, hour h under contingency scenario s (MW) D. Variables C k - the k wind capacity optimisation variables (MW) C APR - the approximated system total wind capacity (MW) f j - the power flow in branch j (MW) M - the minimal-cardinality maximal-vector subset size σ - the wind energy penetration target proportion ω d - the linear constraint coefficient at position d II. INTRODUCTION Due to its environmentally friendly and cost-predictable nature, wind power is widely recognised as a promising alternative electric power generation source at a time of uncertain fossil fuel costs and concern over the harmful effects of climate change [1]. Many countries are considering ambitious future wind energy penetration scenarios [2]. Significant lead-time and investment are associated with the transmission system expansion required to facilitate wind connection at sites distant from traditional load and generation centres however [3]. Furthermore, public acceptance of large scale new-build infrastructure is currently low [4]. In the short-term, maximum use of existing transmission system resources could be made by an optimal wind capacity allocation strategy. 14

30 In some power systems (e.g. Great Britain [5]), wind developers may request a firm connection offer for investment certainty reasons. A firm connection offer is generally given if a generator does not cause network overload subsequent to plausible system security contingencies. Firm connection feasibility study is usually considered as a planning timeframe problem. System re-dispatch of generation using operational timeframe techniques to manage congestion [6] is not considered in this context as this would impinge upon the firm status of other generators connected to the network, and the true bulk system reliability could be limited by transmission constraints. Firm connections are therefore suitable for transmission networks where system operators prefer to have little or no grid congestion. Large standalone conventional plant firm connection was traditionally assessed using heuristic techniques at deterministic basecase scenarios (e.g. incremental-transfer-capability studies at the winter-day-peak and/or the summer-night-valley [7]). Given that the worst-case-scenarios for distributed wind power related flows could conceivably occur at off-peak conditions, clearly a more extensive analysis is required. The statistical interdependence of geographically distinct wind sites, their fluctuating power output, and the large volume of connection applications mean that a collective analysis of all potential wind plant sites over an extended timeframe must be at the kernel of any optimisation approach. The advantage of transmission analysis carried out over a wider range of operating points is that the power flow worst-case-scenarios may be identified without any simplistic base-case assumptions. Some analytical power flow techniques have been reported in [8], [9] modelling a greater variation in generation power injections and customer demand patterns. As discussed in [10] however, a multivariate statistical sampling approach is more accurate for wind-related power-flow studies given the elaborate nature of power system multidimensional statistical dependency distributions with high wind penetration. However a purely random Monte-Carlo sampling technique such as in [11] cannot account for any inter-temporal power system dependencies. The capability of a year-long multivariate sequential time-series method to model the influence of increased wind power inter-temporal variability on system unit commitment (applying conventional generation parameters such as start-up times, minimum up- and downtimes etc.) was highlighted in [12]. Maintaining an integer unit-commitment solution, while including the large number of security constraints that accompany a year-long system load flow analysis, presents a significant dimensionality challenge to the optimal wind capacity placement problem. A simple approach to represent multi-period wind variations was implemented in [12], using a multidimensional binning technique to group (and thereby reduce) the number of relevant power flow scenarios this did not incorporate network security contingencies however. Significant dimensionality 15

31 reduction can be carried out by investigating the structure of the firm capacity optimisation problem if linearised load flow methods are used, then many of the power flow security constraints over the extended time series are redundant and can be removed by an efficient preprocessing scheme [13]. The approach of [13], with a defined optimisation cost function to minimise a wind turbine infrastructure economic cost criterion (by choosing the best wind resource sites), assumed a specific wind energy penetration target was initially feasible using firm wind capacity connection. This would be unrealistic in most power systems with significant targets and present transmission limitations - instead this paper attempts to maximise the firm wind energy potential of the existing system in the short-term prior to transmission expansion in the long-term. To this end the approach of [13] is improved and extended, with incremental firm wind energy penetration targets applied from a lower initial level until a limit is reached when the optimisation model becomes infeasible. Section III of this paper outlines the methodology in more detail. III. METHODOLOGY A. Optimisation Methodology Overview A flowchart of the overall methodology is given in steps in Fig. 1. It is assumed that the transmission network is initially uncongested (i.e. existing conventional plant also has firm connection status), and that the addition of new firm wind capacity should preserve this situation i.e. generation curtailment due to network constraints is inapplicable. Year-long nodal load and nominal 1 MW multivariate wind power hourly production time series are available as inputs. A low firm wind energy penetration target is initially selected for investigation if this level of firm wind energy integration is feasible (through optimised allocation of individual wind capacities), the target can be increased in small discrete steps until additional firm wind energy connection is not possible without system congestion. For each incremental energy target, the overall firm wind capacity allocation problem (incorporating linear wind capacity investment variables, hourly integer unit-commitment/dispatch variables and hourly security-constrained network limitations) is separated into more tractable sub-problems. An approximated total system wind time series facilitates the separation of the existing firm conventional plant mixed-integer scheduling task from a linear programming firm wind capacity allocation analysis. This problem separation into sub-tasks reduces the dimensionality and complexity of the overall firm capacity optimisation problem, and allows an improved solution within practical computing capabilities. Given the separation of the individual sub-tasks, absolute global optimality cannot be guaranteed however. 16

32 Start With Low Firm Wind Energy Penetration Target 1 Approximate System Wind Power Time Series 2 (MIP) Unit Commitment and Economic Dispatch 3 - Formulate the Load Flow Inequality Constraints 4 - Remove Redundant Linear Constraints 5 (LP) Firm Wind Energy Feasibility Test Y Feasible? N 6 - Update Total System Wind Power Time Series N Converged? Y 7 Increment the Firm Wind Energy Penetration 8 Stop Algorithm and Output Results Fig. 1. The firm wind energy maximisation methodology flowchart. The required approximate system-total wind power hourly production time series, (consistent with each incremental wind energy penetration target, and representing the system wind inter-hour variability pattern), can be generated using the multivariate nominal 1 MW wind power time series of the individual locations (step 1, Section III-B-1). Using this initial wind power time series a sequential mixed-integer program (MIP) unit commitment and dispatch study is carried out to determine the MW outputs of existing firm conventional plant to serve the net load time series (step 2, Section III-B-2) this integer optimisation does not require a network model as the system is initially assumed uncongested. A DC load-flow network model subsequently uses this existing firm conventional plant hourly MW dispatch time series and the individual wind farm candidate locations nominal 1 MW time series to define linear network security constraints from a set of critical line and generator contingency scenarios (step 3, Section III-B-3). These linear constraints are formulated so that the power flows related to the optimised firm wind power capacities are fitted-around the power flows resulting from the firm conventional plant dispatch (step 2), while maintaining network security and power balance in each hour. The linear inequality constraints are subsequently preprocessed using a specifically tailored algorithm to remove any constraint redundancy over the extended timeframe of operation (step 4, Section III-B-4), and reduce the model size significantly with no compromise in accuracy [13]. The remaining non-redundant power flow linear constraints are integrated within a simple linear programming (LP) model to test the feasibility of this wind energy penetration target 17

33 level with respect to the network capabilities (step 5, Section III-B-5). The optimisation variables are the individual wind capacity allocations to each candidate node. The resulting total system wind power output time series (as defined by the LP optimised wind capacities and their individual nominal 1 MW hourly time series) may not be precisely consistent with the systemwide approximate hourly input time series used to carry out the initial unit-commitment/dispatch of the firm conventional plant (step 6, Section III-B-6). Steps 2, 3, 4 and 5 are therefore reiterated with this updated resultant system-total hourly time series as the input until power balance converges from one iteration to the next. If a converged solution is feasible, then the wind energy target can be incremented (step 7, Section III-B-7) and the process repeated. On the other hand, if the LP model is deemed infeasible at any stage, the algorithm ends with the previous known feasible firm wind energy capacity allocation solution (step 8). B. Optimisation Methodology Sub-Tasks 1) Approximating System Wind Power Output Time Series A total system-wide wind power production time series is necessary to carry out the scheduling and dispatch of conventional plant. Simultaneous year-long nodal load and nominal 1 MW capacity wind power time series are available as inputs for each transmission network location, representing any geographic statistical and temporal multivariate dependencies independently of the wind plant capacity to be determined later. Each potential wind location (be it a single farm or a collection of local sites) will have its own capacity factor and individual variations in its nominal time series of power production. Therefore the total system wind power output time series required for the MIP conventional generation scheduling sub-task (step 2) cannot be precisely defined prior to completing the individual wind plant capacity LP optimisation sub-task (step 5). However it can be initially assumed that the total system-wide wind power time series converges to a geographically-smoothened or approximate time series regardless of where the output wind power capacities are subsequently allocated. This is acceptable as within a power system of reasonable size and geographical wind capacity spread, the total wind power output is more influenced by the significant degree of interdependency contained in the collection of wind farm time series and the overall weather pattern in the wider geographical region, as opposed to any localised effects. Principal component analysis [14] or independent component analysis [15] can be carried out to study the interdependency present in the multiple wind power time series and substantiate this assumption. In this paper, a simple hourly average of the multivariate nominal 1 MW wind power time series t AVG h was used to generate the initial system-wide hourly wind power output time series for the unit commitment stage of the first wind energy penetration target level. An approximate 18

34 turbine MW capacity C APR required to serve a δ proportion of total yearly load energy demand by wind can be determined in (1) given the average system MW load of D AVG and the average capacity factor of all the potential wind sites λ AVG are known. The initial approximate hourly time series t APR h is determined by scaling the hourly average of the recorded nominal wind farm power time series t AVG h by C APR as in (2). This wind power time series satisfies the annual wind energy penetration target δ while respecting system-wide inter-hourly wind variations, and is thus suitable for the initial unit-commitment and dispatch step. AVG APR D C = δ (1) λ AVG t (2) APR h APR AVG = C t h 2) (MIP) Unit Commitment and Economic Dispatch Connecting new wind generation will not only impact network flows by virtue of its own power injections, but also due to the consequential displacement of existing firm conventional plant in the overall power system operational context. Thus the multivariate interdependence of customer load, existing/new wind generation and existing conventional plant must be determined in conjunction with the task of finding the individual optimal firm wind capacity allocations. Increased wind penetration will lead to greater variability in the net power system load to which conventional plant must respond [16]. Investigating significantly increased wind capacity connection to transmission systems may therefore require a MIP scheduling model [12], with an accurate representation of the conventional plant inter-hourly constraints such as starting times, start costs, minimum up- and minimum down-times, ramp rate limits etc. This may be of significant importance in power systems with inflexible conventional generation [17]. The geographically-smoothed time series of total wind power production t APR h from Section III-B-1 can be subtracted from the total system load, and the resultant net system load time series input to a unit-commitment and dispatch model. This scheduling model can be applied in single-day or multi-day segments as required, with hourly integer resolution. Net system load variability effects over a typical year of operation are thus explicitly accounted for in this algorithm. Provision of primary reserve is included to respond to any single generation contingencies modelled in Section III-B-3. The unit commitment and dispatch stage of this algorithm does not apply a network-constrained re-dispatch as it is assumed the firm connection status of generation already present in the system should be respected - i.e. it is assumed that the system is initially uncongested, and should remain so subsequent to the firm wind capacity addition. The output variables of this step are the least-cost MW power and reserve hourly dispatch of each firm conventional generator in the system for the given wind penetration level δ. Stochastic unit-commitment of conventional plant accounting for short-term wind 19

35 forecast uncertainty information is not applied in this paper (i.e. perfect forecasting of the wind power time series is assumed), though could be included if desired unit commitment methods that account for such forecast uncertainty by producing flexible conventional plant schedules have been reported in [18], [19]. Stochastic scheduling tools are very computationally expensive however. At lower levels of wind penetration, operational timeframe wind forecast uncertainty may have little effect on long-term power transmission planning problems. Longer-term uncertainty relating to input parameters such as fuel or carbon price, peak customer load etc. can be reflected in alternative scenarios. 3) Formulating the Load Flow Inequality Constraints DC load-flow [6] uses the transmission line reactance values to determine a set of linear coefficients α ij (or power transfer distribution factors (PTDFs)) that along with a designated power flow reference bus r, define the power flow solution f j in each branch j as a linear combination of the power injections P i at every other bus i, as in (3). An important advantage of DC load-flow is that linear constraints can be formulated to represent network power flow security criteria at the LP optimisation stage - as will be seen in Section III-B-4, this advantage is critical to reducing the LP wind capacity optimisation sub-problem dimensionality to a manageable size. f α P (3) j = i r ij DC load flow is often used in planning studies as a very good approximation to the transmission network s active power transport requirement. Other important network operational characteristics such as steady-state and dynamic voltage behaviour, short-circuit levels as well as transient stability are more suited to detailed in-depth analysis [20], may not be fully amenable to a manageable optimisation process across such a range of power flow conditions, and are typically assessed once the grid is known to be thermally secure. The algorithm outlined in this paper can be viewed as an optimistic total wind capacity allocation from which to carry out such analyses, with due consideration of advanced technology solutions [21]. If desired, spare line capacity can be set aside for related reactive power flows or known system stability constraints. Fine-tuning of assumed thermal line capacity limits can be carried out in each hour with updated information from subsequent system dynamic and full AC load flow studies this may be necessary as wind generators, specifically squirrel-cage induction machines, can have a high reactive power demand. On the other hand, studies have shown that the addition of wind capacity to a network can improve the voltage profile in some areas if controlled reactive power output is applied using wind turbine power electronics [10],[22]. i 20

36 Assessing power system capability with respect to traditional security criteria across the entire time-length of the time series study implies a very large number of power flow scenarios. In the limit this value is defined by (4), for T years of hourly data analysis, considering each contingency at each hour of the year with respect to each other line s capacity (any single generator (G) and/or simultaneous single network branch (N) contingency state is modelled in this paper). The localised impact of any possible branch contingency would suggest that not all such contingency scenarios realistically need to be modelled. Comparing the probability density functions of the initial system s yearly load flows in each line under intact-network and each other branch-outage condition can be used as a heuristic contingency screening method load flow probability density functions with wide spreads and the most extreme maximum values correspond to the most onerous contingency cases for that line. An initial visual check of these load flow spreads can be carried out to select the network configurations of relevance for subsequent study. Alternatively, network contingency selection could be carried out in a more automated manner on the basis of the network s line-outage distribution factors [6]. For each generation contingency, an appropriate reserve response is allocated from the reserve dispatch of step 2 (Section III-B-2) in each hour so that load balance is maintained. In all, a combined total of S network power injection (i.e. each intact and post-generation-contingency MW power dispatch) and network branch (i.e. each set of PTDFs for the intact network and relevant line outage states) contingencies are applied to each hour of the time series. T 8760 N ( N 1) G (4) As the principle of superposition applies to linear DC load flow, the power flow contribution from both load and conventional generation can be evaluated as numerical partial load flow solution value γ jhs for each hourly security scenario using the relevant MW dispatch information (step 2) and the DC load flow coefficients. The collective net contribution of the k wind farms to the power flows as superimposed values in each of S hourly intact/contingency system configurations is as of yet unknown. The wind capacity allocation optimisation variables C k when scaled by their respective nominal 1 MW hourly time series values t kh and the relevant DC load flow coefficients will determine this. The inequality constraints of (5) are thus included for each of S system configurations in each hour h to ensure that the wind capacity allocations do not overload any of the network thermal line capacities L j, in either the forward or backward flow directions. The optimised firm wind capacity related power flows are essentially fitted-around the existing firm conventional plant production power flows from Section III-B-2, ensuring both network security and power system balance under each critical contingency. The double-sided inequalities of (5) can be represented as single-sided inequalities by the algebraic manipulation of 21

37 (6),(7). Each wind plant s capacity is modelled as a continuous optimisation variable, as individual turbine size is relatively small in comparison. The turbine capacity optimisation variables will of course be non-negative, as specified by (8). Wind plant contingencies can be modelled by assuming their wind power time series t kh is zero in the hours of interest, but the related primary reserve bus power injection required to replace their power production generation contingency can only be modelled for subsequent re-iterations (Section III-B-6) as the individual wind plant capacities are initially unknown. [ L α t C + γ L ] (5) h, j, s j kjs kh k jhs j k [ h, j, s α kj t ( ) 0] S k C h k L j γ (6) j hs k [ h, j, s α kj t ( + ) 0] S k C h k L j γ (7) j hs k C 0 (8) k k 4) Removing Redundant Linear Constraints Optimisation algorithm computational requirements are sensitive to both the number of variables and constraints in the mathematical model applied [23]. A refined version of the algorithm introduced in [13] is applied here to reduce the LP model constraint dimensionality. An approximate method of contingency selection before formulation of the hourly line flow constraints was described in Section III-B-3. Additional model efficiency can be achieved if redundancy in the inter-hourly behaviour of these selected network cases as described by (6) and (7) is also considered. Constraint redundancy will occur when inequality constraints as represented by lines L 2 and L 4 in Fig. 2 below do not intersect the optimisation feasible space, as defined by the inner convex polygon of inequality constraints L 1, L 3 and the variable axes (note the capacity variables cannot be negative as specified in (8)). They will thus not influence the solution of any optimisation model. Dulá in [24] gives an extensive mathematical treatment of linear programming redundancy, outlining the frame or extreme-point subset of the convex-hull of multidimensional data points representing the dual problem constraint matrix formulation as the source of the nonredundancy in the primal representation. As the convex hull of any general type of multidimensional point dataset can define every point within its multidimensional convex volume, it is the minimal complete representation of the multidimensional dataset [25], and all points interior to the convex hull may be considered redundant. Dulá et al. in [26] also describe an 22

38 Fig. 2. Linear program constraint redundancy no feasible space intersection. efficient computational method to determine the frame of the convex hull of a point dataset by testing the feasibility of successively larger linear programs. The presence of (8) in the LP optimisation constraint set allows an intuitively simpler interpretation of the linear constraint redundancy issue for this paper. Consider two arbitrary linear constraints as in (9.a) and (9.b) below. As X, Y are 0 in this problem, then if all the coefficients and the constant of (9.a) (i.e. a 1, a 2,.. const a ) are at least greater than or equal to their equivalents in (9.b), then (9.b), (corresponding to line type L 4 in Fig. 2) will intuitively be a less extreme constraint and is thus made redundant by (9.a). a X + a Y const 0 (9.a) 1 2 a b X + b Y const 0 (9.b) 1 2 b The geometric dual of a line (and thus a linear programming optimisation inequality constraint) is a point [25], and similarly each constraint such as (9.a) and (9.b) can be uniquely represented with the geometric points p a and p b for example pa a1 const a 2, const const, const a ( a1 a 2 a3 a a a ) ( ω, ω, ω ) (10.a) pb b1 const b 2, const const, const ) ( ω, ω, ω ) b ( b1 b2 b3 b b b (10.b) As outlined by the authors in [13], the complete set of inequality constraints in (6) and (7) can therefore be equally defined in this simple manner as a very large cardinality multidimensional point dataset. The problem of quickly removing many redundant linear inequality 23

39 constraints in this LP optimisation model is analogous to finding points that do not form the maximal vector subset of the geometric point dataset see the floating currency analogy to (9.a),(9.b) in [27]. A data-point is a maximal vector of a point dataset if it is not dominated by any other data-point this concept was first introduced in [27] and [28]. p a dominates p b if ω b1 < ω a1, and ω b2 < ω a2, and ω b3 < ω a3 etc. Other computational geometry problems such as the Pareto set problem and the skyline problem [29] have direct correspondence to the maximal vector problem. The maximal vector subset of an arbitrary two-dimensional point dataset is graphically illustrated in Fig. 3 below. The shaded area in Fig. 3 corresponds to the area containing points dominated by maximal vector point O. Fig. 3. Maximal vector subset of a 2-D dataset. Maximal vectors with individual point co-ordinate values that are all greater than average for that dimension will tend to dominate many other points. Instead of a purely naïve sequential search and comparison, a much more efficient redundancy assessment should therefore incorporate an intelligent pre-processing scheme. Godfrey et al. in [30] discussed a data vector coefficient entropy sum to order an arbitrary dataset of points prior to the maximal-vector search and comparison process. If data-points that correspond to very dominant maximal vectors are cycled through the dataset first, they will remove proportionally a much greater than average number of redundant data points, and an efficient redundancy removal process will result. The M data-points still present at the end of the comparison scheme correspond to the dominant maximal vector subset. The discarded or dominated points correspond back to redundant constraints in the LP optimisation model (e.g. line L 4 in Fig. 2) [13]. However, within the set of M remaining maximal vector points there may remain a small number of constraints of the type L 2 in Fig. 2 i.e. those that do not define the feasible space yet are not fully dominated by any other single constraint. This is not critical, as the objective of the LP pre-processing stage is to remove a 24

40 lot of redundant constraints quickly, rather than to fully identify the minimum set of constraints defining the LP feasible space refer to [24],[26] for such an algorithm. As in [13] this paper uses the product of the rank of each point coordinate ω d (i.e. its ordered hierarchical position with respect to the other constraint coefficients for that optimisation variable, a term unrelated to linear algebra matrix rank) to determine the ordered position of the data-point p at the search and comparison stage as defined in (11). This is a particularly efficient pre-processing scheme as coefficient ranks are naturally a good measure of each data-point s dominance in the overall dataset. Redundancy removal progress will slow down as data-points with lower overall coefficient rank product are tested for maximal vector dominance. However the total time to carry out the LP optimisation stage is the sum of the preprocessing and optimisation solver times, so it is wise to implement a stopping criterion, i.e. when the remaining number of constraints is less than a defined number V, which is well within the capabilities of the optimisation solver in use. k + 1 1, ω2,... ωk+ 1)] rank[ ωd ] d= 1 order[ p( ω (11) 5) Linear Programming Firm Wind Energy Feasibility Test A linear programming model can be applied to the wind capacity optimisation model feasibility test for each incremental wind energy penetration target δ, as all of the constraints are linear in nature and the simple linear cost function of (12) was applied here. The outputs of this singlestage LP model are the collectively optimised firm wind capacities that ensure the feasibility of this wind energy penetration level (i.e. so that the wind energy target is achieved and that no network branches are overloaded pre- or post-contingency during the extended time-period of investigation), while minimising the chosen cost function. In theory, the feasibility of each Δδ firm wind energy connection target is determined by the linear constraints alone and should not be sensitive to the cost function applied. The cost function choice may affect the convergence of the overall methodology solution and individual allocations of wind capacity however. The energy contribution of a wind farm is defined by its capacity factor value. The LP wind capacity allocation must satisfy the total system wind energy penetration δ assumed in the scheduling stage of Section III-B-2. This is ensured with the inclusion of (13), where λ k is each wind farm s capacity factor, and the respective wind capacity allocations C k are the optimisation variables. For the given firm wind energy connection target, the constraint of (8), as well as the V remaining network power flow inequality constraints set of (14) and (15) (originally a sub-set of (6) and (7)) will also apply. The complete linear programming model is summarised by (8), (12), (13),(14) and (15). 25

41 v V Cost = Min( ) (12) k C k k C k AVG λ = D δ (13) k [ α t C ( L γ ) 0] (14) k kjv kv k j jv [ α t C ( L + γ ) 0] (15) v V k kjv kv k j jv 6) Updating the System-Total Wind Power Time Series The formulation of the MIP unit-commitment/dispatch problem in Section III-B-2 used an approximate geographically-smoothened total wind power output time series. This may not necessarily correspond precisely to the total wind power time series resulting from the LP output wind turbine capacities when scaled by their individual nominal 1 MW wind power time series. In order to preserve power system balance at each operational hour, steps 2, 3, 4 and 5 of the methodology can be reiterated (within each wind penetration target level δ) using the new total wind power time series resulting from the LP model of the last iteration. The LP output wind turbine capacities may change slightly from one re-iteration to the next; however after a few iterations the total wind power output time series resulting from the LP model converges to that used as input to the unit-commitment/dispatch sub-task, and the turbine capacities from the LP model will subsequently remain unchanged. This re-iteration ensures consistency between the separated MIP unit-commitment/dispatch and the LP wind capacity allocation sub-tasks. 7) Incrementing the Firm Wind Energy Penetration Target Estimating the maximum possible wind energy integration using firm wind capacity can be approached as a series of simpler LP model feasibility tests - by increasing the wind energy target δ in small discrete increments of Δδ, steps 2, 3, 4, 5, and 6 of the methodology are repeated until a feasible solution to the LP model of Section III-B-5 using firm wind capacity allocation to distinct transmission buses no longer exists. The methodology outlined in this paper is broken into smaller optimisation sub-problem steps, as illustrated in Fig. 1. This ensures the ability to separate the MIP system scheduling and large-scale LP wind capacity placement tasks and ensure a practical computational implementation (by exploiting the linear constraint redundancy as detailed in Section III-B-5). While this overall methodology will improve firm wind energy connection feasibility, given the inclusion of MIP variables and the practical separation of the individual subtasks of the optimisation problem, it is unlikely that the absolute global optimal solution can be found however, nor any measure defined with regard to how far the proposed solution is from global optimality. 26

42 IV. METHODOLOGY APPLICATION TO A TEST-SYSTEM The test network used for the illustration of this firm wind energy connection maximisation methodology was a simple 35-bus, 54-line model as depicted in Fig. 4 (with network parameters based on a subset of the Irish All-Island 220/275/400 kv high voltage transmission system). Historical synchronously recorded wind power output data of one year s length, load time series data, and a subset of the existing conventional plant from the Irish power system were combined with the test network of Fig. 4 for study. The total conventional generation capacity was MW. The assumed peak load for the test system was MW. The average load was MW. The network branch capacity values chosen were commensurate with the zero wind integration power flow scenarios, while providing some extra capacity for subsequent wind connection. It was assumed that seven potential wind generation sites were available, at buses 12, 14, 15, 25, 27, 29, and 30. The synchronously recorded wind time series from the existing wind farms were normalised to 1 MW capacity for use in this analysis. High wind capacity factor time series were arbitrarily located on bus 15, 14, 27, and 12, while lower capacity factor wind farms are available at bus 30, 29 and 25. The unit commitment/dispatch step was carried out using the deterministic scheduling (perfect wind forecasting assumed) software tool PLEXOS [31] in 24-hour segments - conventional plant were dispatched on the basis of energy cost. Further test system details are contained in the Appendix I section of this thesis. Fig. 4. The test system transmission network schematic. 27

43 V. RESULTS A. Maximising the Firm Wind Energy Penetration Level An initial firm wind energy penetration target δ of 5% was found to be feasible using the methodology of Sections III-B-2,3,4,5,6. The energy penetration target was updated in small increments until a 10% target was found to be infeasible. The 9.5% target (i.e. δ = 0.095) was therefore estimated by this decomposed optimisation methodology as the maximised firm wind energy penetration level. A table of the collectively optimised firm wind capacity allocations at each wind energy penetration increment is given in Table I. TABLE I OPTIMISED WIND ENERGY PENETRATION CAPACITY ALLOCATIONS (MW) Energy % Wind Farm Bus Number Penetration <infeasible> The firm capacity allocation to the wind farm at bus 15 in successive steps is particularly interesting. As the system wind energy integration target is increased the firm capacity allocation to this node (which has the highest capacity factor) initially increases but then decreases. This illustrates the trade-off between individual wind farm capacity allocation and each farm s contribution to facilitating the maximum possible firm wind energy penetration in the system as a whole. It also underlines the benefit of collectively considering all potential wind farms with the formal optimisation model as outlined in Section III. B. Convergence of the System-Total Wind Power Time Series As described in Section III-B-6, the methodology of Sections III-B-2,3,4,5 was successively repeated using the updated LP model wind capacity allocation results, to remove any load/generation mismatches. Some changes in the integer unit-commitment and linear dispatch variables (e.g. for conventional generators at the margin in each hour) do occur in step 2 from one re-iteration to the next until the solution converges between iterations. A table of the LP optimal capacity allocations at each of the methodology re-iterations for the largest feasible wind energy 28

44 penetration target (9.5%) is given in Table II below. Histograms of the load/generation mismatch error at each hour between successive re-iterations for this firm wind energy penetration level are also illustrated in Fig. 5. Reiteration of the sub-tasks from the first iteration results gradually converges to a solution preserving consistency between the assumed input time series for MIP unit commitment/dispatch and the time series determined by the LP wind capacity allocation model output solution (though as discussed in Section III-B-7, Fig. 5 does not prove convergence to the global optimum). A similar effect was observed for each of the other wind energy penetration levels. TABLE II 9.5% WIND ENERGY RE-ITERATED WIND CAPACITY ALLOCATIONS (MW) Iteration Number Wind Farm Bus Number ITERATION 1 ITERATION 2 ITERATION 3 number of occurrences error, (MW) Fig. 5. Histogram of power imbalance between methodology re-iterations. C. Investigating the Solution Load Flow Results At the end of the firm wind capacity allocation methodology, the optimised wind farm capacities can be used to carry out the resultant time series load flow to ensure that system security is indeed maintained at each of the individual hours. For completeness, this should include all possible line and generation contingencies from (4), irrespective of both the initial contingency 29

45 screening of Section III-B-3 and the mathematically rigorous constraint redundancy elimination of Section III-B-4. The histogram of the year-long power flows in the network branch from bus 6 to bus 12 under all possible single generation contingency events and a single outage contingency in the branch from bus 11 to bus 17 is illustrated in Fig. 6. No cases result in power flow exceeding the thermal line capacity of 100 MW hence the advantage of the year-long extended timeframe analysis. As the very edge of the line flow distribution corresponds to the branch thermal capacity 100 MW, it is clear that this line is a binding constraint on the firm wind energy connection process. x 10 4 Number of Occurences Power Flow (MW) Fig. 6. Load flow distribution for branch 6-12 under branch failure. D. Computational Requirements The year-long conventional generation scheduling and dispatch task, using a rounded-relaxation solution method in PLEXOS, took approximately 20 minutes on average for each of the iterations on the test system s generation portfolio, implemented on a 3.6GHz Pentium dual-core driven, 4 GB RAM enabled Dell Optiplex GX620 desktop PC. Considering the initial LP problem dimensionality of (4) for the simple test system in Section IV would have resulted in ~ 400 x10 6 line power flow security cases for the entire year of study. The contingency screening technique based on the initial zero-wind time series load flow investigation of Section III-B-3 reduced this to ~ 49.34x10 6 constraints using a simple visual test this contingency set was identified and applied to all of the subsequent optimisation row iterations in Table I. The MATLAB [32] software 30

46 environment was used to implement the efficient constraint redundancy pre-processing stage of Section III-B-4, trimming the original ~ 49.34x10 6 possible line flow constraints to a practical userdefined stopping criterion V of 2.5x10 5 in approximately 14 minutes (on average) for each of the iterations. Scaling of the constraint coefficients by the magnitude of the constant terms (as in (10a) and (10b)) was found to refine the computational time efficiency of this paper s investigation by ~ 40% with respect to the original approach in [13]. The benefit of a preprocessing stopping criterion was also investigated. At the 3 rd iteration of the final feasible 9.5% wind energy penetration, when the constraint redundancy analysis was allowed to generate the absolute minimal cardinality set of M non-dominated maximal vectors, only 6.823x10 3 constraints remained, though the computational time trade-off of minutes was substantial. The LP optimisation solver employed was the MATLAB LINPROG medium-scale interior-point algorithm. This determined an optimised solution to the reduced LP problem of Section III-B-5 (with 2.5x10 5 constraints) in approximately 3.14 seconds (the minimal set of 6.823x10 3 constraints was solved in 0.47 seconds in comparison). Post-optimal processing indicated that only 4 of the line flow constraints were binding on the output solution for the 9.5% firm wind energy connection level i.e. four of the lines maximum flow values reached their defined capacity levels at some stage over the year of analysis. A typical graph of the increasing order of constraint rank-product values from (11) can be seen in Fig. 7. As Fig. 7 illustrates, the sparsity of extreme constraints in the firm wind capacity power system planning problem investigated here suggests that relatively few constraints will dominate most of the others, underlining the benefit of applying the ordered maximal-vector redundancy pre-processing step of Section III-B rank product value ordered sample number x 10 5 Fig. 7. A typical graph of ordered rank-product values for pre-processing. 31

47 VI. DISCUSSION A methodology to estimate the maximum firm wind energy that can be integrated to a given power transmission network is outlined in this paper. Wind fluctuations and statistical dependence have been explicitly accounted for through the use of multivariate year-long historical recorded time series. This paper applied fixed contingency security criteria for each of the hours of the yearly duration time series. This generated a very large dimensionality system power flow model. A decomposition of the firm capacity allocation problem into separate MIP unit-commitment/dispatch and LP wind capacity placement steps using a geographical-smoothing approximation of system-total wind power output allowed the design of a pre-processing algorithm to exploit the overwhelming redundancy in the structure of the associated LP model sub-task. Decomposing the overall firm capacity optimisation problem into separate sub-tasks reduces the problem complexity and allows a practical implementation with standard computing resources to achieve improved feasible firm wind energy integration solutions. An associated difficulty however is the inability to guarantee global optimality or determine the solution optimality gap. Using initial contingency screening combined with an efficient and rigorous maximalvector based redundancy algorithm, a reduction of the test-system s original constraint dimensionality by > 99.9% (to 2.5x10 5 ) was possible. Similar computation efficiency should be achievable in larger power systems, with the possible use of parallelised pre-processing. Implementation of the pre-processing algorithm in a faster execution environment than MATLAB could also produce greater efficiency. Commercial optimisation solvers such as CPLEX also use some model pre-processing approaches and report that they can handle millions of constraints and variables [33]. The linear inequality constraint dimensionality in even a medium-sized power system could stretch to the order of billions however, as suggested by (4), so one would expect some user pre-processing may need to be carried out before attempting a solution. The algorithm of Section III-B-4 was tailored specifically for the LP model in this paper, though other linear constraint redundancy removal algorithms have also been proposed previously [26], [34]. While the maximal-vector method in this paper illustrates the overwhelming redundancy present in the LP constraint structure of the applied problem, future work could benchmark its performance against alternative algorithms to evaluate its efficiency. The maximal-vector method is by no means restricted to LP optimisation problems in the power systems domain - as long as the conditions of Section III-B-4 are satisfied then it should be applicable to other problems. While the overall goal of the paper is the integration of wind energy, the methodology uses firm wind capacity only to achieve this (i.e. no transmission congestion allowed). While the maximal-vector subset cardinality M of 6.823x10 3 constraints was less than % of the 32

48 total constraint set, a post-optimal analysis of the linear programming steps furthermore indicated that only a handful of these M constraints were actually binding on the solutions obtained. This would suggest that wind plant behaviour in a few hours of the year determined the firm wind capacity allocation in the entire system. Given the particularly low capacity credit of wind power, and acknowledging the rarity of such worst-case scenarios with a low capacity factor source of generation such as wind (see the relative spread of the extremities of Fig. 6 and the ordered shape of Fig. 7), the results of this paper suggest it would be more prudent in future study to have the connection of non-firm wind capacity (that can economically accept a defined level of energy curtailment over a fixed time period [35]), as the central optimisation variables of interest. However that optimisation problem complexity is certainly of a non-trivial nature it would imply a very large scale hourly security-constrained-opf problem with linear dispatch variables, solved within a daily mixed integer unit-commitment problem, both of which are furthermore coupled to the overall wind capacity investment variables spanning the entire length of the applied time series. The results of the algorithm application to the test system network in this paper are based on a representative set of multivariate historical wind power time series of 1 year s length (though obviously multiple years of data could be used if available). Wind power output profiles may change from year to year using a number of years wind data will improve accuracy and solution robustness. Wind data growing techniques such as those proposed in [36], could also be applied. Any successful wind data growing technique must preserve both the individual wind plant cross-correlative behaviour (for accurate load-flow) but also the auto-correlative trends of total system wind power output (for system unit commitment). Diurnal or seasonal influences in wind production patterns should also be accounted for. The nominal 1 MW wind time series required in Section III-B-1 are assumed to be independent of the optimised wind capacity solution outputs determined by Section III-B-5. This is acceptable in reality as the optimised wind capacity to be allocated to each transmission system bus would likely be the sum of capacities of dispersed (possibly distribution-connected) wind farms in the nearby area, and as the hourly nominal wind time series data described in Section IV was scaled from existing large multi-mw wind farms installed on the Irish power system. A connection queue is typically used in modern deregulated power systems for generation project connection sequencing. The algorithm outlined in this paper attempts to estimate the maximum possible wind energy connection with firm capacity as new transmission projects are implemented. A larger subset of the original firm connection queue projects can be satisfied, though not necessarily in accordance with their actual position in the queue. It may therefore seem more related to the composite system planning techniques of the traditional 33

49 vertically integrated utility. Until significant new transmission capacity is built however, optimal use of networks could be made in the interest of accelerating wind power connection in the short to medium term. Recognising the fundamentally different characteristics of wind energy, a return to some level of integrated planning may be required [37]. VII. CONCLUSION This paper presents a methodology with the aim to maximise the firm wind energy penetration to any given power transmission network. Applying security criteria over the extended sequential timeframe required to model wind power variations generates a high-dimensionality optimisation model separation of the MIP unit-commitment/dispatch and LP firm wind capacity placement problems allows a maximal-vector based pre-processing technique to filter out a substantial proportion of constraint redundancy present in the LP model. While solution global optimality is not guaranteed with this approach, a significant advantage is the reduction in model complexity. The optimised capacity results indicate a balance between each wind site s capacity factor (or position in a connection queue), and the overall goal of maximising the total firm wind capacity connection potential of the power system as a whole, is prudent. Post-optimal analysis underlines the importance of considering wind as a non-firm energy source rather than a firm capacity source in future methods i.e. for currently congested systems where much more wind might be integrated if limited curtailment is acceptable. VIII. REFERENCES [1] Delivering Energy and Climate Solutions EWEA 2007 Annual Report", European Wind Energy Association (EWEA), March Available online - [2] 20% Wind Energy by 2030 Increasing Wind Energy s Contribution to U.S. Electricity Supply, U.S. Department of Energy, May [3] All Island Grid Study, Workstream 4 Analysis of Impacts and Benefits, Irish Government Department of Communications, Energy and Natural Resources/United Kingdom Department of Enterprise, Trade and Investment, Jan Available online - [4] A Study on the Comparative Merits of Overhead Electricity Transmission Lines Versus Underground Cables, Ecofys, available [5] R. Moreno, C.V. Konstantinidis, D.Pudjianto and G.Strbac The New Transmission Arrangements in the UK, presented at the IEEE PES General Meeting, Calgary, July [6] A. J. Wood and B. F. Wollenberg, Power Generation, Operation and Control, 2 nd Ed. New York: Wiley, [7] Transmission Forecast Statement , Eirgrid, Aug Available online

50 [8] P. Zhang and S. T. Lee "Probabilistic Load Flow Computation Using the Method of Combined Cumulants and Gram- Charlier Expansion," IEEE Trans. Power Systems, vol. 19, No.1, Feb [9] C.L. Su Probabilistic Load-Flow Computation Using Point Estimate Method, IEEE Trans. Power Systems, vol. 20, No.4, Nov [10] G. Papaefthymiou, Integration of Stochastic Generation in Power Systems, PhD Thesis, TU Delft, The Netherlands, [11] G. Papaefthymiou, D. Kurowicka Using Copulas for Modeling Stochastic Dependence in Power System Uncertainty Analysis, IEEE Trans. Power Systems, vol. 24, No.4, Feb [12] D. Burke and M.J. O Malley Optimal Wind Power Location on Transmission Systems A Probabilistic Approach, presented at the IEEE PMAPS Conference, Puerto Rico, May [13] D. Burke and M.J. O Malley Optimal Firm Wind Capacity Allocation To Power Systems With Security Constraints, presented at the IEEE PSCE Conference, Seattle, March [14] J.E. Jackson, A Users Guide to Principal Component Analysis, Wiley New York [15] A. Hyvarinen J. Karhunen and Erkki Oja, Independent Component Analysis, Wiley New York [16] T. Ackermann (Editor), Wind Power in Power Systems, Wiley [17] M. Hu, J. Kehler and D. McCrank, Integration of Wind Power into Alberta s Electric System and Market Operation, presented at the IEEE PES General Meeting, Pittsburgh, July [18] C. Weber, P. Meibom, R. Barth, H. Brand, WILMAR: A Stochastic Programming Tool to Analyze the Large-Scale Integration of Wind Energy Chapter-19, pp , Optimization in the Energy Industry, Springer Berlin Heidelberg [19] A.Tuohy, P.Meibom, E. Denny and M.J. O Malley Unit Commitment for Systems with Significant Wind Penetration, IEEE Trans. Power Systems Vol. 24, No. 2, May [20] C.W. Taylor, Power System Voltage Stability - International Edition, McGraw-Hill Singapore, [21] D. Sullivan, M. Takeda, A. Johnson, J. Paserba, S. Yasuda, R. Tucker, et al., Dynamic Voltage Support with the Rector SVC in California s San Joaquin Valley, presented at the IEEE T&D Conference and Exposition, Chicago, April [22] E. Vittal, A. Keane and M.J. O Malley, Varying Penetration Ratios of Wind Turbine Technologies for Voltage and Frequency Stability, presented at the IEEE PES General Meeting, Pittsburgh, July [23] P. Gill, W. Murray, M. Wright, Practical Optimization, Academic Press, London [24] J.H. Dula Geometry of Optimal Value functions with Applications to Redundancy in Linear Programming, Journal of Optimization Theory and Applications Vol.81, No.1, April [25] M. DeBerg, M. Van Kreveld, M. Overmars, O. Schwarzkopf Computational Geometry Algorithms and Applications 2nd Edition, Springer-Verlag Berlin [26] J.H. Dula, R.V. Helgason A New Procedure for Identifying the Frame of the Convex Hull of a Finite Collection of Points in Multidimensional Space, European Journal of Operational Research , [27] F.P. Preparata, M.I. Shamos, Computational Geometry An Introduction, Springer-Verlag New York [28] H.T. Kung, F. Luccio, F.P. Preparata On Finding the Maxima of a Set of Vectors, Journal of the ACM Vol. 22 No. 4, October [29] D. Kossman, F. Ramsak and S. Rost Shooting Stars in the Sky An Online Algorithm for Skyline Queries, Proceedings of the 28 th VLDB Conference, Hong Kong, [30] P. Godfrey, R. Shipley, J. Gryz Maximal Vector Computation in Large Data Sets, Proceedings of the 31 st VLDB Conference, Trondheim, [31] Energy Exemplar, online 35

51 [32] The Mathworks, online [33] ILOG CPLEX, online [34] S. Paulrag, C. Chellappan, T.R. Natesan A Heuristic Approach for Identification of Redundant Constraints in Linear Programming Models, International Journal of Computer Mathematics Vol. 83, Nos. 8 9, August September 2006, pp [35] J. Kabouris, C.D. Vournas, Application of Interruptible Contracts to Increase Wind Power Penetration in Congested Areas, IEEE Transaction on Power Systems Vol.19, No.3, August 2004 pp [36] B. Klockl Multivariate Time Series Models Applied to the Assessment of Energy Storage in Power Systems, presented at the IEEE PMAPS Conference, Puerto Rico, May [37] S.T. Lee, For the Good of the Whole, IEEE Power and Energy Magazine Vol. 5, No.5, September/October

52 CHAPTER 3 CONSIDERING OPERATIONAL WIND MANAGEMENT STRATEGY IMPACTS ON TRANSMISSION PLANNING Electricity Research Centre working paper, in preparation for journal submission. Mr. Daniel J. Burke, Dr. Aidan Tuohy and Prof. Mark J. O Malley. Abstract While the time series sampling often used to represent wind power in transmissionrelated optimisation models will inevitably lead to higher-dimensionality optimisation problems, their associated complexity will be significantly compounded if advanced unit commitment techniques for operational wind variability and forecast uncertainty management are also included. Stochastic mixed-integer scheduling is applied to a suitable test system in this paper to investigate the power flow modelling impacts of such advanced operational management strategies, and determine whether any additional precision justifies the very-large-scale computational burden associated with merging the traditionally separate operational and planning timeframes. Results indicate that power flow modelling is only significantly influenced in a small subset of the network branches associated with major interconnection points and flexible/inflexible conventional generation locations. Model sensitivity analysis also suggests that even at high wind penetration such power flow differences may still be overshadowed by the impact of general uncertainty in demand profile and fuel price volatility that is systemic to the long-term transmission planning problem in many power systems. Index Terms-- power generation scheduling, power transmission, uncertainty, wind energy. I. INTRODUCTION Increasing wind energy penetration is recognised as a key contributor to reducing carbon emissions and maintaining diversity of primary energy supply [1]. Detailed wind integration studies have been carried out in many power systems [2]-[4], with transmission limitations universally acknowledged as a significant challenge. Prudent allocation of wind capacity [5],[6], or optimal transmission development plans could be determined to accelerate wind connection to transmission networks. A transmission access study for large-scale centralized conventional generation was traditionally carried out in a deterministic manner at onerous snapshot hours such as the winter-day-peak and/or the summer-night-valley of system load profile. In contrast wind 37

53 power is a low capacity-factor, geographically-distributed and statistically interdependent source of power generation clearly transmission planning methods require suitable adaptation over a much broader number of study cases to incorporate such characteristics. Equally importantly however, the recent development of advanced unit commitment scheduling strategies to account for wind variability and forecast uncertainty through stochastic optimisation [7]-[9] now necessitates a consideration of how the traditional separation of operational and planning timeframe power flow assumptions may not be as distinct as often assumed in the past. Appropriate choice of model formulation for combined wind-generation/transmission optimisation studies therefore requires detailed consideration, and is the subject matter of this paper. Analytical probability based methods have been proposed in [10],[11] to model a wider range of power flow situations, yet the non-parametric marginal statistical distributions and multivariate statistical dependencies associated with wind power production, least-cost system dispatch and network power flows realistically do not conform to the simplifying assumptions required by such models methods based on sampling may have greatest relevance. A random Monte-Carlo sampling approach to power system statistical dependency modelling using copula theory is outlined in [12], but simple random sampling methods cannot recreate sequential hourto-hour wind variability patterns. Basic auto-regressive moving-average time series synthesis was proposed in [13], and more promising approaches using limited auto-regressive integrated moving-average modelling [14], or statistical transformations [15] have also been reported. In general, wind production profiles based on historical behaviour are widely applied in practice for wind integration study [4]. A historical wind time series data approach is presented for distribution system analysis in [16], and is integral to the optimal firm wind capacity placement methodology of [5]. While a practical drawback is that sometimes there may not be enough data available to give completely statistically-robust conclusions, the benefit of using historical data is that any multivariate statistical and auto-correlative sequential dependencies are implicitly contained in the recorded data set. A major advantage of sequential sampling or historical data profiling approaches is that the effects of the increased operational variability and forecast uncertainty associated with wind penetration can be incorporated to the transmission planning model, if so desired. For example a simplifying geographical wind power time series smoothing assumption was applied to the problem of optimal firm wind capacity allocation to uncongested networks in [5], allowing a separation of the integer unit-commitment and linear wind capacity investment variables and thus accelerating the solution process. Including operational considerations in the more general optimal non-firm wind/transmission planning problem for congested networks will have 38

54 significant computational implications however. If integer variables are retained to model the hourly unit commitment process, then while acknowledging the availability of decomposition techniques for such highly-structured problems [17], the problem complexity will still be greatly increased. The wind generation/transmission optimisation problem must also be formulated with some consideration of long-term model parameter uncertainty while future customer demand growth and conventional plant fuel prices are always difficult to predict accurately, the impacts of electric transportation or smart-meter efficiency applications [18] on the future power system load flow patterns are furthermore greatly uncertain at this present time. Models incorporating short-term operational issues such as stochastic unit commitment, applied over the extended number of samples necessary to represent wind variation characteristics, and under a number of alternative long-term demand-profile/fuel-price uncertainty scenarios will be of almost intractable complexity and dimensionality. The power flow modelling significance of this increased computational burden has not yet been considered in longer-term transmission planning optimisation applications. Varying levels of operational scheduling complexity are applied to a suitable test power system in this paper. A key issue explored is whether the additional computational rigor is justified, or whether a judiciously simplified operational model would be reasonably adequate with pragmatic acknowledgement of the general uncertainty in the long-term power system model itself. Section II outlines the operational complexity issues considered, while the test system is presented in Section III. Results, discussions and conclusions are outlined in Sections IV and V respectively. II. SYSTEM POWER FLOW MODELING A. Impact of operational wind issues in planning timeframes With significant wind capacity installed, more flexible and robust conventional plant commitment and dispatch schedules must be produced so that the system can balance with respect to the wind that actually occurs at the operational time-steps of the near future. This operational wind forecast uncertainty can be represented with a spread of probability-weighted scenarios [19]. Techniques such as rolling-planning and stochastic mixed-integer programming (MIP) using wind forecast scenarios have been reported for generation production-cost analysis in the literature [7],[8]. As the power production, and crucially for this paper, the transmission system load flow patterns, will now deviate somewhat from those modelled by a simple economic dispatch alone (which is generally assumed in most transmission planning models [20]), then the relevance and merits of including the additional model complexity in the long-term transmission planning model should now be assessed. 39

55 From a real-time operational perspective, any one of the wind forecast uncertainty scenarios could potentially occur for each stage of the hour scheduling horizon ahead [21]. Various alternative network congestion management plans would need to be prepared accordingly [2]. From the transmission planning perspective however, only one of the wind power forecast scenarios can ever actually occur at any given operational time-step and therefore result in a specific wind-related power flow contribution which the network design must accommodate. Therefore if a historical wind power time series of reasonable yearly-length is available to clarify the actual wind-power-flow requirements in the transmission planning model, the wind-related operational power flow uncertainty in itself is not relevant in the transmission planning timeframe. This is a key distinction between the implications of wind forecast uncertainty for the transmission network operational and planning problem contexts. However the unit commitment task must be carried out in the hours prior to real-time due to conventional plant inter-temporal constraint limitations (i.e. start-up times, minimum up/down times, ramp limits etc), before the true resultant wind power production is known. Therefore the specific conventional-plant-related power flow contributions will indeed be influenced by the choice of operational wind management strategy, and thus may need to be considered in the transmission planning model. B. Scheduling model power flow investigations Three different system operational wind management strategies are investigated for the same fixed load and wind time series profiles in each case: (1) Merit-order (MO) or simple economic dispatch applied only without any start-up costs or inter-temporal unit commitment constraints linking the separate hours, and no account of wind forecast uncertainty, this option is analogous to studying the power flow outcomes of a random Monte-Carlo simulation such as in [12]. (2) Deterministic unit-commitment applied with the simplistic assumption of perfect wind/load forecasting (DUC-PF) this option includes a full MIP deterministic optimisation and allows a consideration of the effect of system operational variability alone on the network power flow model. (3) Stochastic unit-commitment (SUC) using a wind forecast error scenario tree tool and a stochastic MIP optimisation model, this option allows a complete analysis of both operational variability and forecast uncertainty effects on transmission network power flow modelling. The stochastic programming methodology proposed in [8] for generation production costing studies is used for the SUC analysis of this paper. This is a stochastic mixed-integer hourly- 40

56 resolution model of the operational unit commitment and economic dispatch problem, incorporating load and wind uncertainty scenario trees, conventional generation forced outages, spinning and replacement reserve, fuel, carbon and start-up costs, and detailed conventional plant inter-temporal constraint limitations. Operational wind and load forecast uncertainty is updated with a rolling planning timeframe of 3-hourly periods so the system schedule is effectively re-planned eight times per day. The concise DUC-PF and SUC scheduling model formulations as applied here are thoroughly detailed in [8]. For very high wind penetration, additional constraints were also implemented to ensure a minimum number of large conventional plants remain online for inertial support reasons [22]. The generation power dispatch results can be taken from the three different scheduling investigations and subsequently input to a linear DC network power flow assessment [23]. Using histograms to compare the probability density functions (pdf) of yearly power flows in each transmission branch under the three different operational strategy modelling options allows a direct investigation of the value of additional wind/transmission optimisation model complexity. Transmission network capacity limits are not enforced with an optimal power flow specification in this study as the unconstrained power flow requirement of the network must be observed for planning purposes, and the power flow pdf edges would be truncated at the network branch capacity levels otherwise. As the exact same wind and load time series are applied in the three scheduling approaches described above, then it follows from the reasoning of Section II-A that any differences between the power flow pdfs will therefore be caused only by the differing generating patterns of conventional plant when both variability and forecast uncertainty are accounted for at the operational stage. C. Long-term uncertainty sensitivity analysis Multiple sensitivity analyses were also carried out using the simpler economic dispatch approach to understand the influence of long-term transmission planning model parameter uncertainty on the system power flow model: Case I - Load profiles were linearly scaled across the system to 105% and 95% of their base case patterns to investigate the influence of projected peak load growth uncertainty on network power flow requirements. Case II - Load profile was scaled at each node by a random linear coefficient set, each chosen uniformly and independently from ranges between 92.5% and 102.5% of their original values. This allows a comparative investigation of the impact of spatial demand growth uncertainty on network power flows. 41

57 Case III - Gas, oil and distillate fuel costs were scaled to 75% and 125% of their base case values to illustrate the impact of long-term fuel price volatility on network power flow requirements. The range of load profile and fuel price sensitivities arbitrarily chosen here is consistent with previously observed parameter deviations for example the customer electrical energy demand in the Republic of Ireland dropped by ~ 7% in the year 2008 alone due to unforeseen economic conditions [24], and significant gas price volatility is routinely observed in international commodity markets [25]. Comparing the differences in the estimated network power flow requirements due to long-term model uncertainty with those resulting from different levels of scheduling complexity applied in Section II-B allows a pragmatic consideration of the importance of including detailed operational issues in the long term transmission planning model. III. TEST SYSTEM The test system used in the analyses of this paper is illustrated in Fig. 1. This has a 35-bus, 54-line network, denoted as Area 1 (based on a very simplified model of the Irish All-Island 220/275/400 kv high-voltage transmission system). It contains a mixture of base-load and midmerit fossil-fuel (coal and peat) steam turbine generation, combined-heat-and-power gas plants (CHP), combined-cycle gas turbines (CCGTs), higher-efficiency aero-derivative gas turbines (ADGTs), lower-efficiency open-cycle gas turbines (OCGTs), as well as a few gas/oil-distillate peaking units, amounting to 10.4 GW conventional plant capacity overall. 500 MW of HVDC interconnection capacity to a much larger separate power system denoted as Area 2 (based on an approximate model of the Great Britain generation portfolio) is available at both buses 12 and 34. Conventional plants in Area 2 are grouped approximately into multiple generation capacity blocks of similar plant-type, all connected at a single transmission node. Conventional plant performance data, seasonal natural gas fuel price variations, load profile, load magnitude (accounting for projected load growth to a maximum peak value of 9.61 GW), and the assumed load geographic distribution are consistent with [4]. Additional information on the test network branch reactance parameters, the assumed system geographical load spread, indicative conventional plant unit-commitment inter-temporal constraints, and the conventional generation portfolio network locations as applied in this investigation are presented in the Appendix II section of this thesis. Recorded historical wind power output from the Irish All-Island power system, at hourly resolution over a time period of one year was used for this study this power output data was linearly scaled depending on the total wind capacity level under investigation in Area 1. Wind capacity connection to buses 3, 5, 7, 9, 11, 13, 15, 17, 25 and 33 was investigated. Up to 6 GW of installed wind capacity (corresponding to up to ~34% wind energy penetration) 42

58 was studied, with total capacity equally spread amongst the ten wind plant locations. All model development was carried out in MATLAB [26], GAMS [27], or using the MATLAB/GAMS interface available at [28], and summarised in Appendix 3 of this thesis. Fig. 1 the test power system under investigation. IV. RESULTS A. Scheduling model plant capacity factor impact results As illustrated in Table-I and consistent with previous studies in [8], major system interconnection points and a few of the mid-merit conventional plants such as CCGTs and ADGTs will exhibit different capacity factors under the three different operational scheduling approaches individual plant flexibility/inflexibility may require that it be brought online or kept offline (i.e. out-of-merit) for a particular operational situation. Base-load plants generally operate similarly across the three models. For the test power system in this paper, the large size of the Area 2 system with respect to Area 1 often results in the two HVDC interconnectors being used as sources/sinks for least-cost system variability and uncertainty management in Area 1 hence the significant deviations in their usage. It should be noted that while generation capacity factors will merely influence the transmission network average power flow values, changes in their respective values suggest that the system is being dispatched slightly differently depending on the 43

59 TABLE-I PLANT CAPACITY FACTORS FOR DIFFERENT SCHEDULING APPROACHES 6 GW INSTALLED WIND Unit Type MO DUC-PF SUC Coal Steam Turbine 67.12% 69.02% 70.5% CCGT % 79.06% 77.13% CCGT % 79.75% 79.83% CCGT % 47.04% 43.1% ADGT % 8.25% 8.65% OCGT % 2.48% 2.97% HVDC Interconnector 30.26% 9.6% 29.02% operational wind management strategy applied, and that more general differences in the line power flow pdf extreme values (which are of primary importance for the network planning context of this paper) may be evident also. B. Scheduling model power flow impact results The pdf of yearly line power flows from bus 12 to bus 19 is illustrated in Fig. 2 for 6 GW of wind capacity installed. As implied from Table-I, this transmission line (adjacent to the HVDC interconnection point to Area 2) exhibits a different spread of possible power flows depending on the scheduling model complexity applied. For example the DUC-PF model overestimates the maximum power flow requirement by ~100 MW when compared to the MO or SUC results - power flow model differences have greatest significance if they occur at distribution tails, which most influence system congestion and reliability indices. Similar differences (though less extreme) occur in lines adjacent to other conventional plants that are scheduled differently due to increased operational variability and uncertainty in the system see the pdf of yearly line flows from bus 6 (the location of an ADGT plant) to bus 11 in Fig. 3. However, a reasonable majority of the transmission lines exhibit little or no difference in power flows, as illustrated by Fig. 4, implying there is no additional value obtained from the stochastic MIP scheduling model in their case. Furthermore, it is worth noting that power flow modelling differences in a transmission line will have most significant influence on the solution of a network optimization model only if that line is congested, which in practice may be the case for a limited subset of the network branches only. 44

60 500 MO SUC DUC-PF Number of Occurences Power Flow (MW) Fig. 2 power flow histograms for line adjacent to HVDC interconnection (line 12-19) MO SUC DUC-PF Number of Occurences Power Flow (MW) Fig. 3 power flow histograms for line adjacent to flexible ADGT plant location (line 6-11). 45

61 MO SUC DUC-PF Number of Occurences Power Flow (MW) Fig. 4 sample power flow histogram from elsewhere in the system (line 17-21). C. Long-term planning model uncertainty sensitivities 1) Load profile uncertainty The impact of customer demand profile peak uncertainty (Case I) on transmission power flow modelling is illustrated using the pdf of power flows in the line from bus 1 to bus 3 in Fig. 5. Future peak customer demand projection errors will obviously affect the customer load bus injections themselves, but more importantly they will also significantly impact the usage of specific mid-merit and peaker conventional generators and their resultant network power flow injections. It was observed that the worst-case power flow modelling impact of peak load demand profile uncertainty was generally of lesser absolute magnitude than that of the operational strategy model results in Section IV-B. Notably however, such distributed load profile related uncertainty impacted power flow modelling in a greater number of lines throughout the system, whereas the detailed operational model power flow differences are evident in the few lines associated with flexible/inflexible conventional plant and interconnection locations only. The impact of customer demand profile geographic spatial spread uncertainty (Case II) on the transmission power flow estimation process is also illustrated for the line from bus 10 to 16 in 46

62 'base-case' 105% scaled case 95% scaled case 1200 Number of Occurences Power Flow (MW) Fig. 5 sample pdf of line flows under the influence of peak customer demand uncertainty (Case I, line 1-3) 'base-case' spatial load sensitivity Number of Occurences Power Flow (MW) Fig. 6 sample pdf of line flows under the influence of spatial customer demand uncertainty (Case II, line 10-16). 47

63 Fig. 6. It was generally observed in Case II that the network power flow patterns were not as significantly impacted by the spatial demand uncertainty sensitivity. As the spatial uncertainty modelling outlined in Section II-C assumed independent scaling of distributed loads, the overall system total load profile is more or less similar to that of the base case investigation, and therefore individual conventional plants have similar capacity factors and bus power flow injections. 2) Conventional plant fuel price uncertainty The impact of the conventional plant fuel price volatility (as modelled in Case III) on the relative merit order positions of typical baseload coal and CCGT generators is illustrated in Fig. 7. Natural gas fuel price can exhibit reasonably strong seasonal dependence due to increased space-heating demand in the colder winter months, with coal prices generally more stable throughout the year. As evidenced by Fig. 7 for the base-case fuel price assumptions, the CCGT plant unit average energy costs may become cheaper than those of the coal plants in the summer months, and thus replace them in the merit order. With the respective 75% or 125% average trend shifting sensitivities applied in Case III however, the coal plants are either more expensive or less expensive for the full 12 months of the year, (all other parameters kept fixed). Such merit order position uncertainty will have significant impact on the system dispatch patterns, and thus the unconstrained network power flows. This is evidenced by the considerable deviations in Fig. 8 for the power flow modelling in the network branch from bus 19 to bus 22, and from bus 9 to 15 in Fig. 9. The power flow pdf differences in Fig. 8 and Fig. 9 due to such long-term model uncertainty are clearly of similar or greater magnitude than those resulting from alternative operational wind euro/mwh baseload coal CCGT 100% CCGT 125% CCGT 75% time, (months) Fig. 7 Seasonal baseload coal and CCGT average unit costs (Case III). 48

64 'base-case' 125% gas/oil 75% gas/oil 700 Number of Occurences Power Flow (MW) Fig. 8 sample pdf of line flows under the influence of fossil fuel price uncertainty (Case III, line 19-22) 'base-case' 125% gas/oil 75% gas/oil 800 Number of Occurences Power Flow (MW) Fig. 9 sample pdf of line flows under the influence of fossil fuel price uncertainty (Case III, line 9-13). 49

65 management strategies as evident in Fig. 2 and Fig. 3. Analysis of the pdfs of other line flows in the test system suggests that the impact of fuel price uncertainty is furthermore much more widespread in the network, and thus may overshadow the power flow modelling effects of operational wind management strategy. V. DISCUSSION AND CONCLUSIONS A detailed consideration of the transmission planning model implications of simplified operational timeframe assumptions has been outlined in this paper. Various levels of scheduling complexity were investigated using a stochastic mixed-integer unit commitment model, with reasonably different network power flows evident in specific parts of the test transmission network depending on the operational wind management strategy applied. These power flow differences were contrasted with the impact of long term load-profile/fuel-price parameter uncertainties using sensitivity analysis however. In general, the network flow differences associated with the long term uncertainty were more widespread throughout the test network, and sometimes of much greater magnitude than those associated with the operational model strategy investigation. For the test system used in this paper, it is therefore reasonable to conclude that the inclusion of stochastic mixed-integer unit commitment models within the long term transmission planning context will not give significant added value with respect to the associated computational burden. This is especially relevant given that the long-term load-profile and fuelprice parameters can anyways only be subjectively included in optimization models due to their more general uncertainty e.g. it may difficult to objectively propose any particular probability weighting of the alternative gas-price scenarios in Fig. 7. If an optimal wind/transmission investment planning model formulation with sequentially independent operational samples can be justified on the basis of the analysis outlined in this paper, then the simple block-diagonal constraint matrix structure that results can be very effectively exploited by well-known decomposition schemes [6]. For systems with significant installed energy-limited storage capacity, then of course the individual hourly samples cannot be completely decoupled in the investment problem constraint matrix due to block-staircase structure. Nested decomposition schemes [29] and/or stochastic recombining-tree approximations of wind variability [30] will perhaps be of relevance in such situations, though a pragmatic consideration of the long term model uncertainties must still be retained. For generation expansion considerations alone (e.g. determining the optimal amount of flexible generation in the overall generation portfolio) or for transmission system studies requiring significant precision (such as a reliability analyses or non- 50

66 firm wind-curtailment assessments of specific locations on the present transmission network), additional operational model complexity could perhaps be of greater benefit. The conclusions of the study presented here apply directly to the given test power system, and other similar isolated wind/gas-dominated systems. Given that the operational wind management strategy power flow differences as illustrated in Section IV-B are most appreciable at network points of generation flexibility/inflexibility, it is reasonable to suggest that system generation portfolio flexibility characteristics are an important factor in generalising the results of this analysis to other systems. Combining significant wind capacity with less flexible coal or nuclear dominated system portfolios might lead to different conclusions to those for the test system of this paper. The general performance of wind forecasting will obviously have strong impact also in some systems where wind forecast uncertainty may be significant due to geographical factors, the operational issues may be of greater significance [31]. For geographically larger systems, their generally lower wind variability and uncertainty will be balanced by a larger number of conventional units and thus operational issues may be of lesser importance for transmission planning considerations. Given the diversity in size and composition of the many power systems around the world therefore, no single conclusion on this issue can be proposed for all therefore. Detailed power flow modelling investigations and pragmatic sensitivity analyses such as outlined in this paper will help inform on the most appropriate type of combined wind/transmission planning optimisation models to be formulated in each case. VI. REFERENCES [1] M. Bazilian and F. Roques, (Editors), Analytical Methods for Energy Diversity and Security A Tribute to the Work of Dr. Shimon Awerbuch, Elsevier, [2] Integrating Wind, Developing Europe s Power Market for the Large Scale Integration of Wind Power, TRADEWIND Final Report, available online - [3] Eastern Wind Integration and Transmission Study, Final Report, prepared for the National Renewable Energy Laboratory USA, by Enernex Corporation, available online - [4] All Island Grid Study, Workstream 4 Analysis of Impacts and Benefits, Irish Government Department of Communications, Energy and Natural Resources/United Kingdom Department of Enterprise, Trade and Investment, Jan Available online - [5] D.J. Burke, M.J. O Malley Maximizing Firm Wind Connection to Security Constrained Transmission Networks, IEEE Transactions on Power Systems, Vol.25, No.2, May [6] D.J. Burke, M.J. O Malley A Study of Optimal Non-Firm Wind Capacity Connection to Congested Transmission Systems, IEEE Transactions on Sustainable Energy, (In Review). 51

67 [7] C. Weber, P. Meibom, R. Barth, H. Brand, WILMAR: A Stochastic Programming Tool to Analyze the Large-Scale Integration of Wind Energy Chapter-19, pp , Optimization in the Energy Industry, Springer Berlin Heidelberg [8] A.Tuohy, P.Meibom, E. Denny and M.J. O Malley Unit Commitment for Systems with Significant Wind Penetration, IEEE Trans. Power Systems Vol. 24, No. 2, May [9] J.M. Morales, A.J. Conejo and J. Perez-Ruiz Economic Valuation of Reserves in Power Systems with High Penetration of Wind Power, IEEE Transactions on Power Systems, Vol.2, No.24, May [10] H. Yu, C.Y. Chung, K.P. Wong and J.H. Zhang A Chance Constrained Transmission Network Expansion Planning Method With Consideration of Load and Wind Farm Uncertainties, IEEE Transactions on Power Systems, Vol.24, No.3, August [11] P. Zhang and S. T. Lee, "Probabilistic Load Flow Computation Using the Method of Combined Cumulants and Gram- Charlier Expansion," IEEE Trans. Power Systems, Vol. 19, No.1, Feb [12] G. Papaefthymiou, D. Kurowicka Using Copulas for Modeling Stochastic Dependence in Power System Uncertainty Analysis, IEEE Trans. Power Systems, vol. 24, No.4, Feb [13] R. Billinton and W. Wangdee, Reliability-Based Transmission Reinforcement Planning Associated With Large-Scale Wind Farms, IEEE Transactions on Power Systems, Vol. 22, No.1, February [14] P. Chen, T. Pedersen, B. Bak-Jensen, Z. Chen ARIMA-Based Time Series Model of Stochastic Wind Power Generation, IEEE Transactions on Power Systems, Vol.25, No.2, May [15] B. Klockl Multivariate Time Series Models Applied to the Assessment of Energy Storage in Power Systems, presented at the IEEE PMAPS Conference, Puerto Rico, May [16] T. Boehme, A. R. Wallace and G. P. Harrison, "Applying Time Series to Power Flow Analysis in Networks With High Wind Penetration," IEEE Trans. Power Systems, Vol. 22, No.3, August [17] J.F. Benders Partitioning Procedures for Solving Mixed-Variables Programming Problems, Numerische Mathematik 4, (1962) [18] R. Sioshansi Evaluating the Impacts of Real-Time Pricing on the Cost and Value of Wind Generation, IEEE Transactions on Power Systems, Vol.25, No.2, May [19] P. Pinson, G. Papaefthymiou, B. Klockl, H.Aa. Nielsen, H. Madsen From Probabilistic Forecasts to Statistical Scenarios of Short-Term Wind Power Production, Wind Energy 12(1) 2009, pp [20] G. Latorre, R Dario Cruz, J.M. Areiza and A. Villegas Classifications of Publications and Models on Transmission Expansion Planning, IEEE Trans. Power Systems, Vol. 18, No.2, May [21] G. Papaefthymiou and P. Pinson, Modeling of Spatial Dependence in Wind Power Forecast Uncertainty, presented at the IEEE PMAPS Conference, Puerto Rico, May [22] R. Doherty, A. Mullane, G. Nolan, D. Burke, A. Bryson, M.J. O Malley An Assessment of the Impact of Wind Generation on System Frequency Control, IEEE Trans. Power Systems, Vol.25, No.1, February [23] A. J. Wood and B. F. Wollenberg, Power Generation, Operation and Control, 2 nd Ed. New York: Wiley, [24] Generation Adequacy Report , Eirgrid. Available online - [25] S.P.A. Brown, M.K.Yucel What Drives Natural Gas Prices?, Energy Journal, 2008, Vol. 29 Issue 2, p [26] MATLAB, available at [27] General Algebraic Modeling System, GAMS available online [28] Matlab and GAMS Interfacing Optimization and Visualization Software, by M.C. Ferris available online at 52

68 [29] M.A.H. Dempster, R.T. Thompson Parallelization and Aggregation of Nested Benders Decomposition, Annals of Operations Research 81 (1998), pp [30] D.J. Swider, C. Weber The Costs of Wind s Intermittency in Germany Application of a Stochastic Electricity Market Model, European Transactions on Electrical Power Vol.17, [31] M. Hu, J. Kehler and D. McCrank, Integration of Wind Power into Alberta s Electric System and Market Operation, presented at the IEEE PES General Meeting, Pittsburgh, July

69 CHAPTER 4 A STUDY OF OPTIMAL NON-FIRM WIND CAPACITY CONNECTION TO CONGESTED TRANSMISSION NETWORKS Journal paper in review with the IEEE Transactions on Sustainable Energy, Mr. Daniel J. Burke, and Prof. Mark J. O Malley. Abstract As wind is a low capacity factor source of power generation, a non-physically-firm connection strategy is key to its cost-effective and timely integration to presently constrained transmission networks. This paper outlines the study of an optimal non-firm wind capacity allocation as applied to a test system network. The structured large scale linear programming problem resulting from the multivariate wind sampling representation is easily exploited by the well-known Benders decomposition scheme. Various wind capacity target levels are considered, and sensitivity analyses performed for multiple load profiles, wind profiles, and fuel price parameter values. The possible value of combining wind connection with advanced postcontingency network remedial action schemes is also highlighted. Index Terms-- power generation planning, power transmission, wind energy. A. Indices i - network bus index in Area 1. k - network bus index in Area 2. I. NOMENCLATURE h j g w f - time series hourly index. - Area 1 network branch index. - Area 1 conventional generation index. - Area 1 wind generation index. - Area 2 conventional generation index. n - HVDC interconnection line index between Areas 1, 2. η ζ - Area 1 network branch contingency index (η=1 for intact-network). - Area 1 generation contingency index. B. Constants ε - wind capacity investment cost, (euro/mw). β σ - generation or HVDC operating cost, (euro/mwh). - wind capacity connection target for Area 1, (MW). 54

70 P MAX - dispatch variable maximum value, (MW). α ijη - DC load flow sensitivity of branch j to power injection at bus i under network contingency scenario η. L - network branch thermal power capacity, (MW). C. Time series ω wh - nominal 1MW wind power time series for potential wind farm location w, (MW). γ ih, γ kh - customer load demand time series, (MW). D. Optimization Decision Variables C w - wind investment capacity at potential wind site location w, (MW). P gh P wh P fh P nh - hourly power generation of conventional plants in Area 1, (MWh). - hourly power generation of wind farms in Area 1, (MWh). - hourly power generation of conventional plants in Area 2, (MWh). - hourly dispatch of HVDC interconnection lines, (MWh). II. INTRODUCTION The generation mix of many power systems is currently experiencing a significant transition, with wind power considered to have a key role to play in the evolution of less carbon intensive and more environmentally sustainable electricity supply systems. Transmission network limitations are an almost universal impediment to the rapid deployment of wind capacity needed to satisfy such ambitious renewable integration policy targets however [1]. Prudent use of existing transmission assets could be made with an optimal wind capacity allocation strategy, ensuring the most efficient and cost effective integration of wind energy in the short- to medium-term while any necessary long-term transmission expansion is simultaneously in development. The low capacity factor and low capacity credit [2] of wind generation have important implications for transmission design criteria. Physically-firm transmission connections [3] imply that no curtailment should occur due to network congestion. Post-optimal analysis from [4] however indicates that the provision of completely physically-firm connections for wind generation projects (to guarantee wind power export at all times) will result in a largely overdesigned network. A more carefully designed non-physically-firm connection could allow export of most of the annual available wind energy with significantly less infrastructure requirement [5]. A tradeoff between two principal factors will therefore influence the optimal non-firm wind capacity allocation solution for a given power system, namely the relative wind capacity factors in each region, and the network congestion impacts of new generation connection at each respective node. A competent optimization model formulation should accurately capture both of these factors in a concise manner. 55

71 A chance-constrained, stochastic programming based genetic algorithm is proposed in [6] for the optimal wind-transmission expansion planning problem. However the analytical formulation of this approach based on convolution (implying the assumption of independence), may not be suited to intricate multivariate statistical dependencies that apply to distributed wind production and load demand in reality. Historical recorded data sampling over extended timeframes is often used to model wind production characteristics in many wind integration studies [1], [4], [7], [8]. A combined wind/conventional-generation optimization placement study was outlined for the British power system in [9], yet limited representation of wind production using a small number of samples and very simplistic modeling of the transmission network are applied. An interesting stochastic-recombining-tree model is proposed to limit wind/conventional generation portfolio optimization problem size while maintaining sequential sample dependence in [10]. The value of including operational timeframe sequential issues such as wind variability and forecast uncertainty management within a long-term transmission planning timeframe was investigated in [11]. Power flow modeling was only significantly influenced in a small subset of network locations if a stochastic mixed-integer unit commitment solution is retained, particularly when compared to the impact of uncertain long-term demand profiles and conventional fuel price volatility. Therefore the additional precision associated with including stochastic unit commitment within the optimal wind capacity deployment problem does not justify the almost intractable associated computational burden for network optimization studies in many power systems. While optimization problems based on extensive wind profile time series sampling and relatively simpler operational models will still be of considerable size, most large scale problems have suitable structure which can be exploited by a decomposition scheme. For example, if operational timeframe issues linking individual hours can be ignored without impacting solution accuracy, then the optimal non-firm wind capacity placement problem constraint matrix is block diagonally separable. Overall wind integration targets are generally defined on a policy basis [12], or perhaps from a generation mix portfolio analysis [13]. Using the well known Benders decomposition scheme [14] this paper considers the subsequent medium-term transmission planning task of where to optimally allocate this fixed quantity of non-firm wind capacity on a given network. Extensive sensitivity analysis of the problem solution is performed for multiple wind capacity connection targets, different fuel price and customer demand levels, inter-yearly wind profile variations, and the possible use of wide area remedial action schemes for post-contingency rather than pre-contingency optimal power flow management. The test system is presented in Section III, followed by the linear programming optimization model formulation and sensitivity analysis 56

72 descriptions in Sections IV and V respectively. Results, discussions and conclusions are given in Sections VI and VII. III. TEST POWER SYSTEM The test system used in the analyses of this paper is illustrated in Fig. 1. This has a 35-bus, 54-line network, denoted as Area 1 (based on a very simplified model of the Irish All-Island 220/275/400 kv high-voltage transmission system). It contains a mixture of base-load and midmerit fossil-fuel (coal and peat) steam turbine generation, combined-heat-and-power gas plants (CHP), combined-cycle gas turbines (CCGTs), higher-efficiency aero-derivative gas turbines (ADGTs), lower-efficiency open-cycle gas turbines (OCGTs), as well as a few gas/oil-distillate peaking units, amounting to 10.4 GW conventional plant capacity overall. 500 MW of HVDC interconnection capacity to a much larger separate power system denoted as Area 2 (based on an approximate model of the Great Britain generation portfolio) is available at both buses 12 and 34. Conventional plants in Area 2 are grouped approximately into multiple generation capacity blocks of similar plant-type, all connected at a single transmission node. Conventional plant performance data, seasonal natural gas fuel price variations, load profile, load magnitude (accounting for projected load growth to a maximum peak value of 9.61 GW), and the assumed load geographic distribution are consistent with [7]. Load profile information for Area 2 was sourced from [15]. Additional information on the test network branch reactance and thermal capacity parameters (chosen so that no congestion occurs at the zero wind integration level), the assumed system geographical load spread, and the conventional generation portfolio network locations as applied in this investigation are given in the Appendix II section of this thesis. Synchronously recorded historical wind power data from 10 geographically distributed existing wind farms on the Irish power system was used for this study. This data was taken at eight-hourly sampling resolution over a time period of four years (from ), giving 4380 samples overall. This multivariate power output data was linearly re-scaled to nominal 1 MW wind capacity profiles ω w for the optimisation formulation of Section IV. These nominal wind time series will inherently represent both the quality of the wind resource at each individual site, as well as any multivariate spatial power output dependencies between geographically distinct regions. Capacity factor details are given in Table-I, Section V. Wind capacity connection to Area 1 buses 3, 5, 7, 9, 11, 13, 15, 17, 25 and 33 was investigated, for a number of different total wind capacity connection targets. Wind capacity installation in Area 2 was assumed to be zero the performance of the optimal non-firm wind capacity allocation model in Area 1 is of primary interest. All model development for this paper was carried out in MATLAB [16] and GAMS [17], using the MATLAB/GAMS interface available at [18]. 57

73 Fig. 1 the test power system under investigation. IV. OPTIMAL NON-FIRM WIND ALLOCATION MODEL A. Linear Programming Model Formulation The optimal non-firm wind capacity allocation problem, fully outlined by equations (1)-(5), consists of both investment and operational timeframe decisions. As the Area 1 total wind capacity target level σ is assumed to be predetermined and fixed with the equality constraint of (2a), therefore the turbine capacity investment cost ε (furthermore assumed equal for each potential wind plant location) will have no impact on the capacity allocation choice and can be defined zero in the optimization cost function (1). The effect of the problem cost function is therefore to determine the optimal wind capacity allocations so that the overall conventional plant fuel cost for Areas 1 and 2 is minimized over the time series of interest. Wind energy dispatch and HVDC interconnection operating marginal costs β w and β n are also assumed to be zero. Conventional plants in Areas 1 and 2 are modelled using single cost energy bids β g, β f. Separate load balance equality constraints (3a) and (3b) apply for each area and each hour of the time series, with the HVDC decision variable flow polarities defined as sources in Area 1 and sinks in Area 2. The HVDC interconnection transfers between Areas 1 and 2 are assumed lossless. Generation dispatch and HVDC interconnection flow transfers are constrained between minimum and maximum nameplate capabilities with inequalities (4a),(4b) and (4c). The product of the wind 58

74 capacity decision variables C w and nominal 1 MW wind time series profiles ω w ensures that wind dispatch in any hour cannot exceed the available resource for each farm (4d). The transmission network branch thermal limitations are enforced with inequalities (5a) and (5b). A lossless linear DC load flow model [19] is applied to the transmission network. A DC network model is sufficient for the sensitivity analyses of this paper, though in reality the impacts of induction-generator wind capacity on steady-state voltage, network stability, fault current issues etc are also of considerable importance [2]. An N-1 security constrained optimal power flow (SCOPF) model is applied with any single generation or network contingency covered. For each transmission branch, normal intact-network operation (η=1) or an outage in any other branch is modelled with the relevant DC load flow coefficient power-transfer distribution-factors α ijη input to (5a). The effects of generation contingency are implied by inequality (5b), where the power flow impact of removing each generator in turn from the intact-network is modelled. The optimal wind capacity allocation to each transmission node was assumed to be distributed amongst 4 separate farms, and thus only 25% of each wind variable s dispatch in any hour is considered as its generation contingency level. All of the optimal non-firm wind capacity allocation problem decision variables and constraints are linear, so linear programming can be tasked with solution. Cost = Min{ ε w. C w + ( β g. Pg h + i i i i β w. P i wih ) + n Pn + i h i, w h, i g w h, i, k ( β β P } (1) n. f. k f k h ) k ik f C w i = σ i, w (2a) 0 C wi w, i Pg h + Pw h + Pn h ) = i i i k i, k g w n i ih (2b) ( γ (3a) h P f h Pn h = k k, f P i, k P i k k γ kh h 0 g i h MAX g i, g, h i 0 P f k h PMAX f k, f, h k PMAX P n i n k i h PMAX k n k, i, n, h ik 0 P ω. C i, w, h w i h w i h w i (3b) (4a) (4b) (4c) (4d) L L j j α ij η i, k, g, w, n α ( P + P + P γ g i h wi h ni k h ih ) L ij ( η = 1) ( Pg h + Pw h + Pn h ih ) i g k w n i i i k,, (, ς ) γ j L j (5a) j,η, h, j, h (, w ) ς g (5b) 59

75 B. Model Decomposition and Solution A reasonably large number of time series samples h is required to accurately recreate the wind capacity factor and multivariate statistical dependency in (1)-(5). Therefore even if only the most relevant SCOPF contingencies are included for each branch over each hour of the time series (acceptable due to the localized network impact of most contingencies), a large number of constraints and variables will still result. For the small test system of Fig. 1, a direct solution attempt of (1)-(5) would imply ~ {98* } optimization variables and ~ {1988 * } constraints. Like most large scale optimization problems however, the optimal non-firm wind capacity placement problem of (1)-(5) has significant structure in its constraint matrix. The relationship between investment and operational variables forms a classic block-diagonal constraint pattern as illustrated in Fig. 2. This is easily exploited by the well-known Benders decomposition scheme [14] - by fixing the investment variables C w, the much smaller hourly time series SCOPF problems can be solved separately, as depicted in Fig. 3. From an initial wind capacity allocation solution guess, linear programming dual information from the sub-problems (which give an upper-bound on the optimal cost function value) is used to generate successive feasibility and optimality cuts for the investment master problem (which gives a lower-bound for the optimal cost function value), improving the solution until the difference between upper and lower bounds reaches a specified tolerance. Tractability is the main computational advantage - instead of attempting to solve one very large problem, lots of smaller problems are each iteratively solved many times. A detailed tutorial on the application of Benders decomposition in power system planning problems is given in [20] see Appendix 4 of this thesis also. The test system of Fig. 1 is already adequate and secure at the zero wind integration level, and thus it follows that the Benders decomposition SCOPF sub-problems are always feasible regardless of the Benders master problem solution estimate. From (1)-(5), both master and sub-problem stages are still linear programming problems. Fig. 2 block diagonal constraint matrix structure. 60

76 Fig. 3 Master/sub-problem Benders decomposition scheme. V. MODEL INVESTIGATION & SENSITIVITY ANALYSES The solution of the optimal firm wind capacity allocation problem as outlined in Section IV is investigated for a number of different wind capacity target levels and a wide selection of model parameter uncertainty sensitivities. In each case, the Benders decomposition restricted master solution initial guess uses an equal allocation of the wind capacity target to each potential wind node as decomposition algorithm progresses this capacity spread is redistributed in an optimal manner. For un-ambiguous solution comparison purposes, a very small Benders decomposition upper-bound/lower-bound tolerance stopping criterion of 1*10 (-7) % is applied in each case, ensuring that each optimization solution almost reaches the exact optimum point. For consistency, the same conventional plant portfolio is also employed for all wind target levels and sensitivity analyses. Further detail on the sensitivities investigated is outlined in the subsections below. A. Increasing Wind Capacity Targets The total wind capacity target for Area 1, σ, is varied from 1 GW to 7 GW in steps of 1 GW, corresponding to a maximum wind energy penetration level of approximately 35-40%. B. Wind Resource Profile Sensitivities The wind resource profile gradient between the respective wind locations is a key factor in the determination of the optimal non-firm capacity allocation. Lack of suitable historical data length can sometimes be a hindrance for wind integration studies wind capacity factor is known to exhibit significant variation from one year to the next as illustrated by the historical data in Fig. 4, with a number of real Irish wind farms studied over an eight year period (from ). Fig. 4 illustrates that there is a reasonable degree of dependence between the individual capacity factor variations over the period of interest (2007 was a particularly poor wind year for the system as a whole), yet for some individual years the comparative capacity factor ranking between separate 61

77 locations is also reversed, falsely making some locations appear more advantageous for development than others. An investigation of how a limited wind profile data-set influences the optimal wind capacity allocation is carried out using wind data from the same 10 Irish wind farm locations for the year 2007 alone (sampled at 2-hourly resolution to still give a consistent 4380 samples overall), instead of the 4 year period as outlined in Section III. A comparison of the variation in the test system s 10 wind farm capacity factors from the 4-year base-case data-set to the 1-year sensitivity model is given in Table-I. Fig. 4 Irish wind farm inter-yearly capacity factor variation (specific y-axis info not given for commercial sensitivity reasons). TABLE-I TEST SYSTEM WIND CAPACITY FACTOR INFORMATION (%) System Node 4-year dataset 1-year dataset C. Load Profile Uncertainty Sensitivities A generation capacity connection feasibility study requires a reasonably long-term consideration of the system load profile trends. While load peak values, daily load curve shape and spatial demand distribution prediction has always been an approximate process at best (for example Irish customer energy demand dropped unexpectedly by 7% in year 2008 alone due to economic conditions [21]), the impacts of smart-metering efficiency policies and electric transportation 62

78 development on transmission system load flow and congestion patterns are furthermore greatly uncertain at the present time. Multiple sensitivity analyses for the influence of Area 1 load profile uncertainty on the optimal wind capacity solution were carried out load profile was either scaled linearly across the entire system, at a small subset of the potential wind plant node locations only, or randomly at each nodal location with uncorrelated samples taken from a uniform distribution of limited range. Specific sensitivity analysis case details are as follows: Case C1 Area 1 load profile linearly scaled across the system to 90% of its original value. Case C2 Area 1 load profile linearly scaled across the system to 95% of its original value. Case C3 Area 1 load profile linearly scaled at bus locations 11, 17 and 33 alone, to 90% of original values. Case C4 Area 1 load profile scaled at each node by random linear coefficients chosen uniformly and independently between 92.5% % of original values. Case C5 Area 1 load profile scaled at each node by an alternative random linear coefficient set also chosen between 92.5% % of original values. D. Fuel Price Uncertainty Sensitivities Fuel price volatilities can modify the system s economic dispatch merit-order or locational marginal costs, and thus impact upon congestion management in SCOPF models. Conventional plant fuel price volatility has been observed regularly in the past long term fuel price trends are thus generally quite uncertain, and assuming fixed deterministic values for their parameters in the non-firm capacity allocation problem might be a fallacy. To this end the following fuel price sensitivity analyses were performed on the optimal wind capacity solution: Case D1 Area 1 and Area 2 gas, oil and distillate fuel prices linearly-scaled to 125% of base case values. Case D2 Area 1 and Area 2 carbon dioxide (CO 2 ) price scaled to 125% of its base case value of 30 euro/tonne. Case D3 Area 1 and Area 2 gas, oil and distillate fuel prices linearly scaled to 75% of base case values. E. Contingency Management Policy Sensitivity The optimal non-firm wind capacity allocation formulation of Section IV applies a DC SCOPF model to ensure that the test system network thermal power transfer design criteria are not exceeded. 63

79 The SCOPF concept generally implies an N-1 constrained pre-contingency dispatch, enforcing that system security must always be guaranteed in the immediate aftermath of a contingency, even if the probability of such contingencies (for network branches at least) is quite small. Alternative operational studies [22] have proposed the application of a carefully designed postcontingency re-dispatch policy (assuming this is technically feasible to implement) to reduce the operational cost while still maintaining acceptable system security. A sensitivity analysis investigating the value of such advanced wide area remedial action schemes for the optimal non-firm wind capacity placement problem was also carried out, with a simple OPF model used to replace SCOPF equations (5a), (5b) in Section IV. Therefore only the relevant intact-network values (η=1) for α ij are used in (5a), and no generation contingencies at all are included. This simple model therefore implicitly assumes that a feasible re-dispatch is always possible if a contingency ever did occur (though specific constraints could be formulated to ensure this with the methods in [23]), and that the recourse cost of contingencies that do occur has relatively little overall cost-function impacts in (1) as they are relatively improbable. VI. OPTIMAL NON-FIRM CAPACITY MODEL RESULTS A. Increasing Wind Capacity Targets The optimal non-firm wind capacity allocations for increasing Area 1 wind capacity target levels σ are presented in Table-II clearly the wind capacity factors in Table-I have strong influence on the optimal solution. As the overall target is increased, capacity is first allocated to the high capacity factor nodes, until an economic congestion level is reached additional wind capacity is then distributed amongst the less advantageous wind resource areas. Wind resource quality is not the only factor of influence however, as evidenced by the optimal capacity allocation to node 9. As this is the location of the three most economical baseload coal generators in Area 1, wind capacity addition to this node would constrain off the system s most cost-effective plant and thus add to the overall energy fuel cost. Table-II also indicates that the optimal non-firm wind capacity solution for each individual location is not always monotonically increasing with increasing overall system target level σ for example the optimal capacity additions to nodes 15 and 17 fluctuate somewhat. The corresponding wind energy curtailment percentages are given in Fig. 5. A common trend from Fig. 5 is that greater wind energy curtailment generally occurs at the higher wind capacity factor sites, as they will produce more energy when there is no system congestion to compensate for this. The cost saving through optimal non-firm wind capacity placement will of course be conditional upon the initial Benders restricted master problem solution guess at the zero-th 64

80 TABLE-II OPTIMAL NON-FIRM WIND CAPACITY ALLOCATION, (MW) System Node σ (GW) wind energy curtailment,(%) farm 3 farm 5 farm 7 farm 9 farm 11 farm 13 farm 15 farm 17 farm 25 farm x10 3 wind capacity,(mw) Fig. 5 Wind energy curtailments for increasing total wind capacity targets. iteration. Therefore the full envelope (i.e. the Benders upper bound progression) of the cost difference between each progressive iteration of the optimal 6 GW Benders algorithm and the final optimal solution cost value is illustrated in Fig. 5 a subset of the corresponding restricted master problem solutions at different algorithm iteration stages are also referenced in Table-III. Total combined Area 1 and 2 cost savings in Fig. 6 are reported as percentages of the Area 1 cost alone, as this is where the wind capacity allocations are decided interestingly though, the net cost of Area 1 alone was found to increase from the initial restricted master problem solution guess value at some of the wind target levels. 65

81 TABLE-III 6 GW WIND CAPACITY ALLOCATION SOLUTION PROGRESSION, (MW) System Node Iterate (end) Fig. 6 Value of optimal non-firm wind capacity allocation, as % of Area 1 total cost, at each progressive Benders iteration. B. Wind Resource Profile Sensitivities The effect of having an incomplete model of the power system s relative wind resource quality for the optimal 6 GW wind capacity placement problem can be seen in Table-IV. Using one-year of wind power output data alone for the model of Section IV clearly gives a significantly different estimate of the true optimal capacity allocation solution from that obtained by the four year wind data set. Clearly the more historical data available for study, the more stable is the implicit wind profile model that results, and the more consistent is the optimal wind capacity solution obtained. 66

82 The sub-optimality implications of the results in Table-IV on the overall system operational cost are mild however. Applying the third-row of Table-IV as the wind capacity allocation for a SCOPF study using the more representative original 4-year wind dataset lead to a total (i.e. Area 1 + Area 2) increased operational cost of just 0.21% of the Area 1 cost, when compared to that given by the second-row of Table-IV when applied to the 4-year wind dataset. TABLE-IV EFFECT OF LIMITED WIND PROFILE MODEL- 6 GW CAPACITY SOLUTION (MW) System Node 4-year dataset 1-year dataset C. Load Profile Uncertainty Sensitivities The results of the optimal non-firm wind capacity allocations with respect to the load profile sensitivity analyses outlined in Section V-C are presented in Table-V. It is quite interesting to note that aside from some deviations in the wind capacities allocated to nodes 3, 7 and 17, the solution deviations are generally quite small with respect to the original model. TABLE-V EFFECT OF LOAD PROFILE UNCERTAINTY- 6 GW CAPACITY SOLUTION (MW) System Node Base-Case Case C Case C Case C Case C Case C D. Fuel Price Uncertainty Sensitivities The results of the optimal non-firm wind capacity allocations with respect to the fuel price sensitivity analyses as outlined in Section V-D are presented in Table-VI. The gas/oil/distillate price sensitivities in Cases D1 and D3 are observed to have an influence on the optimal wind turbine capacity allocations to nodes 3, 13 and 17. The carbon price sensitivity has a somewhat lesser impact. Optimal placement of wind capacity at the best wind resource sites will result in proportionally more wind energy generation when no network congestion is present, and less 67

83 wind curtailment when it is if more wind energy can be produced from a given installed wind capacity then less conventional plant fuel will be consumed. Therefore the economic benefit of optimal wind capacity placement, as illustrated in Fig. 6, will also be somewhat conditional upon conventional plant fuel prices. TABLE-VI EFFECT OF FUEL PRICE UNCERTAINTY- 6 GW CAPACITY SOLUTION (MW) System Node Base-Case Case D Case D Case D E. Contingency Management Policy Sensitivity The results of the optimal non-firm wind capacity allocations with respect to the optimal power flow contingency management sensitivity analysis as outlined in Section V-E are presented in Table-VII. Operating the power system in post-contingency re-dispatch mode, as opposed to enforcing a preventive security-constrained dispatch strategy, will allow a significant relaxation of transmission network congestion limitations (provided a fast-acting and reliable wide area control mechanism can be implemented). Much more wind turbine capacity can be installed at the better wind capacity factor resource locations, as illustrated in Table-VII for nodes 15, 17 and 33. The overall operational cost saving is 3.31% of the Area 1 cost. TABLE-VII EFFECT OF CONTINGENCY MANAGEMENT- 6 GW CAPACITY SOLUTION (MW) System Node Base-Case SCOPF OPF Method VII. DISCUSSION AND CONCLUSIONS This paper outlines a detailed study of the optimal non-firm wind capacity allocation problem for a given transmission network. With real historical wind data sampling used to model multivariate spatial wind dependency, the well known Benders decomposition method was applied to exploit the resultant block diagonal constraint matrix structure. Extensive wind profile, load demand 68

84 profile, fuel price and contingency management sensitivities were examined, and their effect on the optimal wind capacity allocation solution clearly shown. In congested large scale transmission networks, reasonable economic benefits and timely policy advancement could result from combining new renewable generation projects with advanced communication and control mechanisms for real-time network contingency management, provided such schemes are demonstrated to be sufficiently reliable [22]. Block-diagonal separability with linear programming decomposition is a key simplifying assumption in this paper whether this is allowable in every power system requires careful consideration of operational timeframe wind management issues such as is outlined for the test system of this paper in [11]. Any attempt to include stochastic integer unit commitment in the long term wind-transmission planning problem will have massive computational complexity implications however, even if a decomposition scheme could still probably be applied to exploit any constraint matrix structure present. Most large scale optimization problems have regularized structure of some kind for wind power expansion planning problems of non-standard block constraint layout, more generic block-permutation methods might be applicable [24]. If desired, the impact of load and fuel price model parameter uncertainties could be included directly within the optimization model as alternative or probability-weighted scenarios, with the simple effect of multiplying the constraint matrix diagonal length the number of decomposition sub-problems would increase significantly though, with associated computational consequences. There is no specific inclusion of power system reliability issues in the analyses of this paper. The same conventional portfolio is maintained throughout to allow comparisons such as given in Table-II, though the higher wind capacity integration cases will of course be slightly more reliable as a result. In a real system, the conventional plant portfolio would be tailored on the basis of the wind capacity credit at that particular wind integration level [13]. Wind capacity geographic spatial spread is known to have an influence on its reliability contribution [25], and should also be considered in the optimal wind capacity allocation solution of a real power system this is an important future research topic. The operational cost-minimization paradigm as adhered to in this paper (as opposed to a possible welfare-maximization or other such formulations [26]) may require adaptation for deregulated power system studies. The cost-minimization study approach is often used as the first-step in many wind integration studies however [7], in order to examine the combined wind/transmission physical capabilities of a given power system - market considerations can follow subsequently. Non-firm wind capacity connections will generally result in wind curtailment for congestion management reasons, without financial compensation. While the non-physically-firm 69

85 wind connection strategy clearly has significant system-wide development cost reduction benefits, in a deregulated market the individual wind farms themselves may be exposed to nonnegligible additional financial risk as a result. Given that wind capacity investment constitutes a relatively large up-front capital outlay, particularly with respect to future network parameter uncertainties, the challenge of quantifying and managing such long-term curtailment risk within the market environment will be a significant enabling factor for cost-effective renewable energy integration. VIII. REFERENCES [1] Eastern Wind Integration and Transmission Study, Final Report, prepared for the National Renewable Energy Laboratory USA, by Enernex Corporation, available online - [2] T. Ackermann, (Editor) Wind Power in Power Systems, Wiley, [3] R. Moreno, C.V. Konstantinidis, D.Pudjianto and G.Strbac The New Transmission Arrangements in the UK, presented at the IEEE PES General Meeting, Calgary, July [4] D.J. Burke, M.J. O Malley Maximizing Firm Wind Connection to Security Constrained Transmission Networks, IEEE Transactions on Power Systems, Vol.25, No.2, May [5] J.Kabouris, C.D.Vournas, Application of Interruptible Contracts to Increase Wind Power Penetration in Congested Areas, IEEE Transaction on Power Systems Vol.19, No.3 August [6] H. Yu, C.Y. Chung, K.P. Wong and J.H. Zhang A Chance Constrained Transmission Network Expansion Planning Method With Consideration of Load and Wind Farm Uncertainties, IEEE Transactions on Power Systems, Vol.24, No.3, August [7] All Island Grid Study, Workstream 4 Analysis of Impacts and Benefits, Irish Government Department of Communications, Energy and Natural Resources/United Kingdom Department of Enterprise, Trade and Investment, Jan Available online - [8] T. Boehme, A. R. Wallace and G. P. Harrison, "Applying Time Series to Power Flow Analysis in Networks With High Wind Penetration," IEEE Trans. Power Systems, Vol. 22, No.3, August [9] K. Neuhoff et al Space and Time Wind in an Investment Planning Model, Cambridge Energy Policy Research Group (Working Paper) - [10] D.J. Swider, C. Weber The Costs of Wind s Intermittency in Germany Application of a Stochastic Electricity Market Model, European Transactions on Electrical Power Vol.17, [11] D.J. Burke, A. Tuohy and M.J. O Malley Considering Operational Wind Management Strategy Impacts on Transmission Planning, (Electricity Research Centre Working Paper, Chapter 3). [12] Delivering Energy and Climate Solutions EWEA 2007 Annual Report", European Wind Energy Association (EWEA), March Available online - [13] R. Doherty, H. Outhred, M.J. O Malley Establishing the Role That Wind Generation May Have in Future Generation Portfolios, IEEE Transactions on Power Systems, Vol. 21, No.3, August [14] J.F. Benders Partitioning Procedures for Solving Mixed-Variables Programming Problems, Numerische Mathematik 4, (1962) [15] [16] MATLAB, available at [17] General Algebraic Modeling System, GAMS available online [18] Matlab and GAMS Interfacing Optimization and Visualization Software, by M.C. Ferris available online at [19] A. J. Wood and B. F. Wollenberg, Power Generation, Operation and Control, 2 nd Ed. New York: Wiley,

86 [20] M. Shahidepour, Y.Fu Benders Decomposition in Restructured Power Systems, (Tutorial) - [21] Generation Adequacy Report , Eirgrid. Available online - [22] J. Wen, P. Arons, W.H. Edwin Liu The Role of Remedial Action Schemes in Renewable Generation Integrations, presented at the IEEE PES Innovative Smart Grid Technologies Conference, Maryland, January [23] F. Capitanescu, L. Wehenkel Improving the Statement of the Corrective Security Constrained Optimal Power Flow Problem, IEEE Transactions on Power Systems, Vol. 22, No.2, May [24] J. Gondzio, R. Sakrissan Parallel Interior Point Solver for Structured Linear Programs, Mathematical Programming Vol.96, No.3, June 2003 [25] B. Hasche, A. Keane, M.J. O Malley Capacity Value of Wind Power, Calculation and Data Requirements: the Irish Power System Case, IEEE Transactions on Power Systems, (Accepted, In Press). [26] M.O. Buygi, G. Balzer, H.M. Shanechi, M. Shahidepour Market-Based Transmission Expansion Planning, IEEE Transactions on Power Systems, Vol.19 No.4, November

87 CHAPTER 5 FACTORS INFLUENCING WIND ENERGY CURTAILMENT Journal paper in review with the IEEE Transactions on Sustainable Energy, Mr. Daniel J. Burke, and Prof. Mark J. O Malley. Abstract Non-physically-firm wind generation connections may be necessary for significant wind integration to congested transmission networks. A study of factors influencing associated wind energy curtailment is therefore of timely importance. In this paper, the wind curtailment estimation effects of natural inter-yearly wind profile variability, system demand-profile/fuelprice parameter uncertainty, and minimum system inertial constraints are studied in detail. Results indicate that curtailment estimation error can be reduced by appropriate wind data yearlength and sampling-rate choice, though a pragmatic consideration of system parameter uncertainty should be maintained. Congestion-related wind energy curtailment risk due to such parameter uncertainty exhibits appreciable inter-locational dependency, suggesting there may be scope for effective curtailment risk management. The coincidence of wind energy curtailment estimated due to network thermal congestion and system-wide inertial-stability issues also has commercial significance for systems with very high wind energy penetration targets, suggesting there may be appreciable interaction between different sources of curtailment in reality. Index Terms-- power transmission, power generation dispatch, power system economics, uncertainty, wind energy. I. INTRODUCTION The low capacity factor of wind energy as an alternative form of electric power generation has significant implications for wind farm transmission access and transmission network design criteria [1]. Wind is most appropriately considered as a variable energy source in long-term network integration studies as it rarely reaches nameplate capacity production in many locations. If optimal transmission system design implies an accommodation of distributed wind energy production for most but not all of the time (i.e. it is uneconomic to design transmission networks for all of the available wind energy [2]), then some level of wind curtailment will be an obvious consequence. Both the expected value, and equally importantly the risk or uncertainty of wind curtailment estimates will have considerable relevance for non-firm wind capacity investment in 72

88 deregulated power systems. A detailed consideration of the various and somewhat interdependent factors influencing curtailment is therefore necessary. As wind energy is a fluctuating and partially dispatchable generation source, curtailment investigation must be considered within a probabilistic rather than deterministic study context. While advanced wind power time series simulation methods have been reported in the literature [3],[4], wind production data based on historical behaviour is often the basis for many wind transmission integration studies applied in practice [5],[6]. Synchronously recorded historical power output data is useful in that it will inherently represent any multivariate spatial dependencies, though often there is only a limited amount of data available for study. Wind profiles may exhibit both significant inter-yearly variation as well as appreciable short term intra/inter-hourly variability in some areas - important questions arise such as how many years of historical data and what data sampling frequency are required to accurately estimate respective wind energy curtailment indices. Historical wind power data-timeframe considerations of this nature have been shown to strongly impact wind capacity credit estimation in [7] for example. While such wind profile timeframe modeling issues will no doubt influence wind curtailment estimation, long-term uncertainty associated with other power system parameters will also be of importance. For example the power flow implications of future customer demand shaping with smart-metering and electric transportation, combined with fossil fuel/carbon price volatility, are relatively unknown at present, and may even fluctuate dynamically as the future system evolves in time. Such model parameter uncertainty contributes to wind curtailment estimate variation, i.e. curtailment risk. Excessive wind curtailment risk, even for network locations where the expected curtailment level in itself is quite low, will be problematic from an investment security perspective as wind capacity is a relatively capital-intensive investment option. Given that wind development is usually distributed at multiple locations in the power system network however, then a study of the co-dependency of wind curtailment estimate variations between distinct locations allows an investigation of how such long term curtailment uncertainties might possibly be overcome from a risk management perspective. Anti-correlated curtailment risks will be particularly advantageous in this respect. At low to medium wind energy penetration levels, network congestion will be the principal factor influencing wind curtailment values. At very high penetration levels however, sometimes the total wind generation available may approach or even exceed the total customer load demand in small regional or island power system areas. Some wind power may also have to be curtailed for load balancing, system inertial stability or network dynamic stability reasons therefore [8],[9]. It is presently unclear what the coincidence of such inertial-stability concerns with network thermal-congestion problems will be, as sufficiently detailed studies of these issues 73

89 are often completed separately [10]. For example if there is already wind curtailment required due to local network congestion, then the load balancing/inertial-stability excess in total wind power availability may not occur in the first place. Whether the overall net level of wind curtailment will be equal to or less than the algebraic addition of these separate results is furthermore an issue of considerable economic significance for wind farm owners in reality. This paper presents a detailed study of the effect and possible coincidence of these factors influencing overall wind energy curtailment patterns in congested transmission networks. Extensive multi-year historical recorded wind power data is initially investigated to quantify the impacts of natural inter-yearly wind profile variation and data-sampling rate on curtailment estimation. With a suitably stable and compact time-frame representation of the wind data chosen to negate such inherent wind profile variability effects, the influence of power system parameter uncertainty and inertial-stability constraints on wind curtailment risk is subsequently investigated. Section II describes the characteristics of the paper s test system and Section III gives the practical detail of the respective case study implementations. The results in Section IV outline the historical wind data timeframe, power system parameter uncertainty, inter-locational risk dependency, and coincident system inertial constraint effects on wind energy curtailment patterns. Relevant discussions and conclusions are given in Sections V and VI respectively. II. TEST POWER SYSTEM DETAILS The test system used in the analyses of this paper is illustrated in Fig. 1. This has a 35-bus, 54-line network, denoted as Area 1 (based on a very simplified model of the Irish All-Island 220/275/400 kv high-voltage transmission system). It contains a mixture of base-load and midmerit fossil-fuel (coal and peat) steam turbine generation, combined-heat-and-power gas plants (CHP), combined-cycle gas turbines (CCGTs), higher-efficiency aero-derivative gas turbines (ADGTs), lower-efficiency open-cycle gas turbines (OCGTs), as well as a few gas/oil-distillate peaking units, amounting to 10.4 GW conventional plant capacity overall. 500 MW of HVDC interconnection capacity to a much larger separate power system denoted as Area 2 (based on an approximate model of the Great Britain generation portfolio) is available at both buses 12 and 34. Conventional plants in Area 2 are grouped approximately into multiple generation capacity blocks of similar plant-type, all connected at a single transmission node. Conventional plant performance data, seasonal natural gas fuel price variations, load profile, load magnitude (accounting for projected load growth to a maximum peak value of 9.61 GW), and the assumed load geographic distribution are consistent with [6]. Load profile information for Area 2 was sourced from [11]. Additional information on the test network branch reactance and thermal 74

90 Fig. 1 the test power system under investigation. capacity parameters (chosen so that no congestion occurs at the zero wind penetration level), the assumed system geographical load spread, and the conventional generation portfolio network locations as applied in this investigation, are given in the Appendix II section of this thesis. Synchronously recorded historical multivariate wind power data from multiple geographically distributed existing wind farms on the Irish power system, recorded over varying numbers of years and at 15-minute sampling resolution, was used as the database for the wind energy curtailment studies of this paper. This multivariate power output data was linearly rescaled to model different installed wind capacity levels positioned at various locations on the test system network, as appropriate for each study further information is detailed as necessary in Section III below. Coincident 15-min resolution load time series data was taken from the Irish power system for use with the test power system of Fig. 1, with inter-year normalization by peakload applied to remove any demand-growth patterns present. III. WIND CURTAILMENT STUDY IMPLEMENTATION A lossless linear DC security-constrained optimal-power-flow (SCOPF) dispatch model was sufficient for the curtailment sensitivity-analysis context of the three studies outlined below. This model applied any single network or generation N-1 contingency of the test system as the operational security criteria to be satisfied by the generation dispatch solution at each time-step. Unit commitment for variability and forecast uncertainty modelling is not included [12], and therefore wind curtailment due to conventional generation flexibility or ramping constraints is not 75

91 considered here. All model development was carried out in MATLAB [13] and GAMS [14], using the MATLAB/GAMS interface available at [15]. A. Historical Data Timeframe Modeling Case Study Eight consecutive years of recorded historical wind power output data was available at 15-minute sampling frequency from 4 separate existing wind farms located on the Irish power system (wind data from the other sites was available with lesser timeframe length). This wind data was linearly re-scaled to an arbitrary 750 MW capacity wind farm connected to buses 9, 11, 13 and 17 on the test power system in Fig. 1. In total therefore, ~ 280,320 individual SCOPF analyses were carried out to model the wind energy curtailment at each respective wind farm over the entire historical time series dataset. The SCOPF results were subsequently filtered at 15-min, 30-min, 1-hourly, 2- hourly, 4-hourly, 8-hourly, 12-hourly and 24-hourly sequential time-segment resolution to investigate data sampling frequency impacts on the wind energy curtailment estimation. To preserve any diurnal characteristics in the wind data, the low frequency sampling rates were carried out randomly in each respective sequential data segment. The SCOPF results were also filtered for various year-length timeframes from 1 year of data alone to the full 8 year dataset for example there are 28 ( 8 C 2 ) possible ways to select any two years of data from the original 8- year set. This timeframe-filtering of the SCOPF results allows an investigation of the wind energy curtailment estimation error associated with a limited historical data timeframe at low sample resolution, when compared to the original 8-year 15-min dataset. Two separate SCOPF sensitivity analyses were also carried out with respect to conventional generation gas fuel price and the customer load demand profile for the 8-year, 15-min historical database. Gas price was arbitrarily increased by 25% from the base case scenario and the total system demand profile was reduced to 95% of its base case pattern. Observing the curtailment uncertainty effects of these limited sensitivity analyses puts the historical data inter-yearly/sample-rate curtailment estimation variations in context of typical power system parameter uncertainty effects, allowing a prudent choice of the number of historical data samples to retain for subsequent investigations. The presentation and discussion of results for this case-study is outlined in Section IV-A. B. Inter-locational Curtailment Risk Dependency Case Study In this case study, to investigate wind curtailment risk dependency across a suitably large number of network locations, 10 distinct wind farm installations were assumed connected at buses 3, 5, 7, 9, 11, 13, 15, 17, 25 and 33. On the justification of the historical data timeframe study as outlined in Section III-A, wind power output profiles were modeled using an appropriate year-length and 76

92 TABLE-I OPTIMAL NON-FIRM WIND CAPACITY ALLOCATIONS, (MW) System Node Total Wind Target (GW) data sampling rate choice to negate the influence of natural wind profile variability effects. Instead of an arbitrary wind capacity allocation assumed connected to each location as applied in Section III-A, this particular study proceeds from the basis of an optimal non-firm wind capacity investment solution determined by the methodology of [16]. This methodology uses the basecase load-profile/fuel-price parameter values, determining a least-cost distributed wind capacity placement for a given total-system wind capacity connection target. The optimal wind capacity placement results therefore implicitly specify the least-cost wind curtailment basis to which sensitivity analysis perturbation is applied in this case-study. A selection of optimal wind capacity allocation solutions are given in Table-I for this test-system, for different total wind capacity target levels. The wind energy curtailment risk of the optimal 6 GW total wind capacity solution was investigated in this curtailment risk case-study, corresponding to a reasonably high ~ 29.7% total wind energy penetration. Distributed system load profile, coal/gas/peat conventional plant fuel-price and carbon-price were the uncertain system parameters allowed to vary in the curtailment risk analysis. 100 random samples were taken from the system parameter uncertainty model to set-up 100 distinct background power system scenarios, to each of which a separate SCOPF multi-year time-series wind curtailment investigation was then applied. The choice of how to model fuel-price/load-profile uncertainty is generally subjective to some extent (i.e. it may be difficult to objectively justify any particular fuel price probability model for example), so therefore the curtailment risk impacts of two distinct system parameter uncertainty models were investigated: Case I Fuel and carbon prices were allowed to vary independently of each other with uniform distributions chosen to be centred around the original base-case deterministic values as follows gas (75-125% of base-case value), coal (90-110% of base-case value), peat (90-110% of base-case value) and carbon (80-120% of base-case value). The individual system bus load growth uncertainties were assumed to vary with uniform distributions, 77

93 independently of each other and also independent of the fuel/carbon prices, with a linearscaling parameter spread around % of their original base-case values. Case II In the second parameter uncertainty model, the fuel and carbon price statistical representation was kept the same as Case I, but the individual network bus load growth uncertainties were instead assumed to have a correlated Gaussian statistical dependency. The bus loads were assumed to have a mean uncertainty value of 96.25% of their base case values, a standard-deviation of 3.125% of their base case values, and an inter-locational correlation coefficient of 0.7. The presentation and discussion of results for this case-study is outlined in Section IV-B. C. Inertial/Congestion Curtailment Dependency Case Study The 7, 8 and 9 GW optimal non-firm wind capacity solutions in Table-I (corresponding to ~ 35-40% total wind energy penetration levels) were used as the system configuration basis for this particular case-study. With this approach applied (as in Section III-B) the initial network congestion related curtailment levels have a minimum-cost justification [16]. To model power system inertial issues within the SCOPF analyses, a simple inertial constraint approximation was implemented using a rounded-relaxation procedure that ensured the equivalent of more than 5 large-scale synchronous conventional units are maintained online at or above their minimum generation levels at all times. For example, four large CCGT generators and two smaller peat generators, or three large coal generators and two CCGTs would be sufficient. Any wind generation causing the net-load to drop below this critical minimum conventional generation level would have to be curtailed. Using the same historical data year-length and sampling rate choice as justified by the wind profile variability analysis of Section III-A, three separate case study investigations were implemented for each of the 7, 8, 9 GW wind capacity levels: Case A In this case, the minimum inertial constraint was applied without including SCOPF network constraints this models curtailment from detailed dynamic studies without network limits included [8]. Case B In this case, the SCOPF network constraints were included but no inertial constraint was applied this models the results from network analyses that do not consider practical inertial problems with high wind. Case C In this case, both the inertial and SCOPF network constraints were included together, modelling the least-cost operational patterns and overall wind curtailment likely to occur in reality. The presentation and discussion of results for this case-study is outlined in Section IV-C. 78

94 IV. WIND CURTAILMENT RESULTS A. Historical Data Timeframe Considerations A sample illustration of the effect of limited data timeframe length on the estimation of wind energy curtailment at Farm-9 is given in Fig. 2, with the vertical columns representing all the various possible individual data-year combinations (each applied with 15-min data sample resolution). Depending on the year in question, if only 1 year of study data was available for example, the estimated wind energy curtailment could vary anything from 1.4% to 2.4% of total available energy, compared to the full 8-year dataset value of 1.86%. Analogous to the wind capacity credit studies in [7], more years of data available progressively reduces the variance of the curtailment estimation error. The corresponding effect of limited data year-length on the estimated Farm-11 wind energy curtailment is illustrated in Fig. 3. Similar effects are evident for the wind farms at buses 13 and 17. Fig. 2 variations in wind energy curtailment at Farm-9 with respect to number of years of data (15-minute sample resolution). Fig. 3 variations in wind energy curtailment at Farm-11 with respect to number of years of data (15-minute sample resolution). 79

95 The mean absolute value of the wind energy curtailment percentage error, averaged over the four wind farms in the system, is summarized in Fig. 4 for all such possible historical data yearlength and sampling frequency combinations. For example, 4-years of wind data sampled at 8- hourly resolution give a system-averaged expected curtailment error of approx 8%. For additional years of data greater than 4 years, and for data sampling rates more frequent than once every 8 hours, the wind energy curtailment mean absolute error tends to reduce linearly. With much less data available however, the mean absolute error reduces quite non-linearly for any additional data available at all. For more frequent than 12-hour sampling rate, the slope of the surface in Fig. 4 generally suggests that additional years of historical data may improve the accuracy more effectively than increased sampling rate. Wind data timeframe modeling issues will have an effect on estimated wind capacity factor also. The corresponding mean absolute value of the wind energy capacity factor error, again averaged over the four wind farms, is given in Fig. 5. Interestingly the capacity factor error reduces linearly with respect to timeframe yearly length across all parts of the surface, and sampling resolution has much less of an influence when compared to the wind farm curtailment error in Fig. 4. Wind power output rarely reaches maximum rated capacity over extended time periods of study, and thus wind curtailment estimation accuracy will effectively be based on much fewer occurrences compared to wind farm capacity factor estimation. Fig. 4 system-averaged mean absolute wind energy curtailment error with respect to number of years of data and data sampling resolution. 80

96 Fig. 5 system-averaged mean absolute wind capacity factor error with respect to number of years of data and data sampling resolution. Fig. 6 A comparison of capacity factor and wind energy curtailment variations over the 8 year period for Farm 9 (15- min sampling). Fig. 7 A comparison of capacity factor and wind energy curtailment variations over the 8 year period for Farm 11 (15- min sampling). 81

97 TABLE-II WIND ENERGY CURTAILMENT %- EFFECT OF SENSITIVITY ANALYSIS Farm 9 Farm 11 Farm 13 Farm 17 Base Case % Load Profile % Fuel Price A comparison of the co-variations in capacity factor and wind energy curtailments for wind farms at bus 9 and bus 11 are given in Fig. 6 and Fig. 7 respectively (note that specific capacity factor information is not given for commercial sensitivity reasons). It is important to consider this relationship, as if data time-frame modelling limitations over/underestimate the true wind energy curtailment values coincidently with over/underestimation of the true wind capacity factor values, then a similar net wind energy export capability would be concluded regardless. Fig. 6 and Fig. 7 illustrate that while there are some weak trend similarities, there are many notable exceptions when the yearly wind energy curtailment increased/decreased when the capacity factor exhibited the opposite fluctuation. The variation in the wind energy curtailments for the different power system sensitivity analyses is given in Table-II. The wind energy curtailment estimate variation for these wind farms due to power system parameter uncertainty is of the order of 5-10% of the base case values. Comparing this parameter uncertainty effect with the natural inter-yearly wind profile and sampling frequency variations illustrated in Fig. 4 allows a pragmatic consideration of the value of additional sample data in wind energy curtailment estimation studies. For this test system example, 4-years of wind data sampled at an 8-hourly frequency gives curtailment accuracy (on average, though outliers will exist) comparable to that associated with typical uncertainty in the test power system model itself therefore the value of additional wind timeframe sampling inclusion in excess of a suitable level must be considered with regard to the additional computational burden. This is especially important in network optimization applications where repeated multi-year wind time series SCOPF routines are often sub-problem steps of iterative decomposition schemes [16] even if many years of high-frequency data was available for study it may not be computationally sensible or even necessary to use all of it to get suitably good model solutions for such problems. On the justification of these historical data timeframe study results, wind power output profiles were modeled using 4 years of multivariate wind power data sampled at an 8-hourly period for the analyses outlined in Sections III-B and III-C. 82

98 B. Inter-locational Curtailment Risk Dependency Case Study The impact of future power system model parameter uncertainty on the network congestion related wind energy curtailment indices was illustrated with Table-II, for two simple sensitivity analyses. This type of wind curtailment model uncertainty constitutes a direct risk to wind farm investment. However, the columns of Table-II illustrate that the impacts of load profile reduction and gas price increase had opposite impacts on the individual curtailments of wind farms at buses 11, 13 and 17. Interestingly, the rows of Table-II also illustrate that the wind energy curtailment at buses 13 and 17 increased in the high gas price scenario with respect to the base case, while the curtailment at bus 11 simultaneously decreased. Table-II therefore underlines the possible variations of wind energy curtailment estimation at each bus for alternative parameter uncertainty scenarios, and indeed curtailment variation inter-dependencies for wind plants installed at different network locations this curtailment risk diversity characteristic is worthy of significant investigation. The mean wind energy curtailment percentages for the different wind farms, with respect to the two alternative system parameter uncertainty model sample sets described in Section III-B, are presented in Table-III. No curtailment occurred for the farms at buses 3 and 9. The scatter plots of wind energy curtailment risk dependency between Farms 5 and 11, and Farms 13 and 15 are illustrated in Fig. 8 and Fig. 9 respectively for the Case I parameter uncertainty model. The spread of curtailment risk in each wind farm due to model parameter uncertainty again puts the inherent wind profile variability related curtailment error characteristics of Fig. 4 in perspective. Fig. 8 and Fig. 9 also indicate that the curtailment risk is clearly locational in nature Farms 5 and 11 have a slightly correlated curtailment risk (that is they both tend to be over/under curtailed together), while the curtailment risks at Farms 13 and 15 are anti-correlated (when either is curtailed more than expected, the other is curtailed less). Wind curtailment risks that are independent or as anti-correlated as possible may be useful from a collective risk sharing perspective for example the total wind curtailment risk across both Farms 13 and 15 is much lower than that across Farms 5 and 11 considered together, as Farms 13 and 15 will generally compensate one another. TABLE-III MEAN WIND ENERGY CURTAILMENTS, (%) Wind Farm Case I Case II

99 Fig. 8 Wind curtailment risk dependency for Farms 5 and 11 (Case I). Fig. 9 Wind curtailment risk dependency for Farms 13 and 15 (Case II). TABLE-IV DISTRIBUTED WIND ENERGY CURTAILMENT RISK CORRELATIONS CASE I WIND FARM SYSTEM TOTAL

100 TABLE-V DISTRIBUTED WIND ENERGY CURTAILMENT RISK CORRELATIONS CASE II WIND FARM SYSTEM TOTAL The overall curtailment risk dependencies are summarized with linear correlation metrics in Table-IV and Table-V respectively for the Case I and Case II system parameter uncertainty assumptions. The right-hand column gives the curtailment risk correlation of each individual wind farm with variation in the total curtailed wind energy in the system as a whole. For the Case I uncertainty model in Table-IV there are quite a number of anti-correlated inter-locational risk dependencies, due to adjacent network locations or proximity to conventional plants of particular fuel-types. Wind energy curtailment risk at Farm 33 in particular is anti-correlated to some extent with almost every other wind farm location. The risk dependency of each individual site with the system-total wind energy curtailed is also quite low on average, indicating that if the Case I uncertainty model were accurate (which assumes all parameter uncertainties are independent) then both effective inter-locational and system-wide curtailment risk sharing mechanisms might be conceptually feasible through an intelligent wind plant portfolio location choice. Table-V illustrates the strong impact of the uncertainty modeling assumptions on the overall risk dependency estimation process however. The Case II correlated Gaussian load uncertainty case causes much greater positive dependency in the curtailment risk estimates. For example curtailment risks at buses 5, 7 and 11 are much more dependent than in Case I, though Farms 25 and 33 are still somewhat independent of the general system-wide wind energy curtailment pattern. The standard deviation of the system total wind energy curtailment risk in Case II is also double that of Case I, as the variance of a sum of strongly correlated risks will always be greater than the variance of a sum of independent/anti-correlated risks. Effective system-wide risk sharing will thus be more difficult if Case II is an accurate model of the power system 85

101 parameter uncertainties, though for each wind farm there is still at least one other location that has low or even negative curtailment risk dependence, as evident in Table-V. C. Inertial/Congestion Curtailment Dependency Case Study The system-total wind energy curtailment results for Cases A, B and C at the 7, 8, and 9 GW installed wind capacity levels are given in Table VI as percentages of the respective total available wind energy. At the high levels of installed wind capacity under investigation, Case A illustrates that some level of inertial-constraint related wind curtailment is necessary at very high instantaneous wind power output. However the negligible differences between the system-total wind curtailment results for Cases B and C (for all three wind capacity installation levels) indicate that the inertial constraint wind curtailment instances identified by Case A are almost entirely contained as a subset of the network congestion related wind curtailment instances in Case B. A typical scatter-plot of the Case A and Case B curtailment instances is given in Fig. 10 for the 8 GW installed wind capacity level, with the corresponding time series plot given in Fig. 11 (where the x- axis corresponds to sample position in the chosen 4-year/8-hourly data-set). These illustrations further underline the coincidence of the curtailments identified by the two separate analyses. V. DISCUSSION Non-physically-firm wind farm connections will allow the harvesting of much more wind energy from a given transmission network investment. Wind farm development is very capital-intensive, with revenue pay-back over a long timeframe. Effective curtailment risk management schemes in deregulated power systems will be a key enabling factor in supporting non-firm wind investment therefore. This paper has identified the physical existence of inter-locational and system-wide curtailment risk diversity, though how such characteristics are exploited with respect to financial or regulatory mechanisms is equally important. Curtailment is not the only risk to wind development of course if wind farm operators compete freely as price-makers in the market [17] (as opposed to depending purely on renewable support schemes [18]) then the effect of fuel or demand uncertainties on the basic energy price revenues may overshadow any energy volume curtailment risks. The significant differences in remuneration and support schemes for renewable energy in many power systems preclude a universal conclusion on such issues in this paper only curtailment volume risk due to network congestion and/or inertial stability has been considered in this analysis. Previous studies have identified the avoidance of curtailment due to excess system-wide wind power availability and minimum system inertial constraints as a key factor to improving the cost-effectiveness of very-large-scale energy storage investment [19]. However, the results of this 86

102 TABLE-VI WIND ENERGY NETWORK/INERTIAL CURTAILMENT VALUES, (%) Wind Level 7 GW 8 GW 9 GW Case Study A B C Fig. 10 Scatter plot of inertial/network-congestion curtailment 8 GW wind. Fig. 11 Time series of inertial/network-congestion curtailments 8 GW wind. 87

103 paper indicate that in a transmission system with a non-physically-firm wind connection strategy, a study of the economics of such centralized storage services may be much more complex than determined by such a generation production-costing study alone. Wind is typically distributed in nature, so therefore the excess instantaneous wind energy, that appears to be available for storage and usage later, may not be transferable to large centralized storage units if most economic dimensioning of transmission infrastructure for wind energy sources is applied. Further study is required to investigate this issue in greater detail. Other factors of influence not included in this paper s analysis, such as ramp-rate unit commitment limits and voltage stability, may also affect the overall curtailment estimates. Many of the issues raised in this paper will become most apparent at medium to high wind penetration levels. With large-scale wind investment, transmission expansion will alleviate wind energy curtailment due to network congestion, and greater interconnection may reduce excess wind availability above the load-balancing requirement the tradeoff between the factors discussed in Section IV-C will be dependent on such investment decisions. Active network management with remedial action schemes managing congestion may also reduce wind curtailment in the short term until long-term investment projects materialize [20]. VI. CONCLUSIONS This paper has illustrated the influence of wind power data historical timeframe modeling, power system parameter uncertainty, and minimum system inertial constraints on the curtailment indices of distributed wind energy. There can be appreciable inter-yearly variation in estimated wind energy curtailment due to natural wind profile variations, and very low data recording frequency will also lead to equally significant sampling error. Additional data availability will reduce the estimation error appropriately, but curtailment study dimensionality selection should always be framed within the context of inherent power system load-profile and fuel-price uncertainties, among other variable parameters. Their influence on curtailment estimate risk may be more pronounced. There may be appreciable network congestion related curtailment risk dependency between different power system locations, potentially giving scope for effective risk management strategies. Precise evaluation of inter-locational curtailment risk dependency is heavily influenced by the power system uncertainty modeling strategy though. Interaction between different sources of wind curtailment will be important to study for example wind curtailment estimates due to inertial constraints may be a subset of curtailments already caused by network congestion, and thus the net effect on wind farm investment profitability may not be as extreme as if they were totally independent. 88

104 VII. REFERENCES [1] J.Kabouris, C.D.Vournas, Application of Interruptible Contracts to Increase Wind Power Penetration in Congested Areas, IEEE Transaction on Power Systems Vol.19, No.3 [2] D.J. Burke, M.J. O Malley Maximizing Firm Wind Connection to Security Constrained Transmission Networks, IEEE Transactions on Power Systems, Vol.25, No.2, May [3] A. Lojowska, D. Kurowicka, G. Papaefthymiou and L. Van Der Sluis Advantages of ARMA-GARCH Wind Speed Time Series Modeling, presented at the IEEE PMAPS Conference, Singapore, June [4] P. Chen, T. Pedersen, B. Bak-Jensen, Z. Chen ARIMA-Based Time Series Model of Stochastic Wind Power Generation, IEEE Transactions on Power Systems, Vol.25, No.2, May [5] European Wind Integration Study Final Report, available at [6] All Island Grid Study, Workstream 4 Analysis of Impacts and Benefits, Irish Government Department of Communications, Energy and Natural Resources/United Kingdom Department of Enterprise, Trade and Investment, Jan Available online - [7] B. Hasche, A. Keane, M.J. O Malley Capacity Value of Wind Power, Calculation and Data Requirements: the Irish Power System Case, IEEE Transactions on Power Systems, (Accepted, In Press). [8] R. Doherty, A. Mullane, G. Nolan, D. Burke, A. Bryson, M.J. O Malley An Assessment of the Impact of Wind Generation on System Frequency Control, IEEE Trans. Power Systems, Vol.25, No.1, February [9] D. Gautam, V. Vittal and T. Harbour Impact of Increased Penetrations of DFIG Based Wind Turbine Generators on Transient and Small Signal Stability, IEEE Transactions on Power Systems, August [10] All-Island TSO Facilitation of Renewables WP3 Final Report, prepared for Eirgrid by Ecofys/DIgSILENT, June Available at - [11] [12] D.J. Burke, A. Tuohy and M.J. O Malley Considering Operational Wind Management Strategy Impacts on Transmission Planning, (Electricity Research Centre Working Paper, Chapter 3). [13] MATLAB, available at [14] General Algebraic Modeling System, GAMS available online [15] Matlab and GAMS Interfacing Optimization and Visualization Software, by M.C. Ferris available online at [16] D.J. Burke, M.J. O Malley A Study of Optimal Non-Firm Wind Capacity Connection to Congested Transmission Systems, IEEE Transactions on Sustainable Energy, (In Review). [17] J.M. Morales, A.J. Conejo and J Perez-Ruiz Short-Term Trading for a Wind Power Producer, IEEE Transactions on Power Systems, Vol.25, No.1, February [18] C. Hiroux, M. Saguan Large-Scale Wind Power in European Electricity Markets Time for Revisiting Support Schemes and Market Designs?, Energy Policy, 2009, doi: /j.enpol [19] A. Tuohy, M.J. O Malley Impact of Pumped Storage on Power Systems with Increasing Wind Penetration, presented at the IEEE PES-GM, Calgary, Canada, July [20] J. Wen, P. Arons, W.H. Edwin Liu The Role of Remedial Action Schemes in Renewable Generation Integrations, presented at the IEEE PES Innovative Smart Grid Technologies Conference, Maryland, January

105 CHAPTER 6 A STUDY OF MULTIVARIATE COMPONENT ANALYSIS APPLIED TO SPATIALLY DISTRIBUTED WIND POWER Journal paper in review with the IEEE Transactions on Power Systems, Mr. Daniel J. Burke, and Prof. Mark J. O Malley. Abstract Multivariate dimension reduction schemes could be very useful in limiting the number of random statistical variables needed to represent distributed wind power variations in transmission integration studies. In this paper, principal component analysis (PCA) is applied to the covariance matrix of distributed wind power data from existing Irish wind farms, with the eigenvector/eigenvalue analysis generating a lower number of uncorrelated alternative variables. It is shown that though uncorrelated, these wind components may not necessarily be statistically independent. Independent component analysis (ICA) is therefore applied to investigate if a further optimal linear rotation can generate wind power components with the much richer property of statistical independence. The results of the PCA/ICA studies indicate that while significant dimension reduction can be achieved, the property of statistical independence in retained components may yet be more elusive if a linear mixing matrix must be assumed. Index Terms-- power transmission, statistics, time series, wind energy. I. NOMENCLATURE i t n X C x L E σ Q X X REC R A - multivariate component index - number of multivariate historical wind power data observations - number of original wind power random variables - matrix of multivariate historical wind power data observations - covariance matrix of X - PCA eigenvalue matrix with diagonal values l i - PCA eigenvector matrix with columns e i - number of retained components - principal component time series matrix - vector of original wind variable capacity factor values - reconstructed original variables for σ<n - reconstructed original variable residual errors - ICA linear mixing matrix 90

106 S U Z W Y y GAUSS H(y) p y J(y) M(Y) - independent component time series matrix - pseudo-inverse of A - zero-mean unit-variance scaled PCA results - orthogonal ICA mixing matrix for Z - estimated independent random variable vectors y i - zero mean unit-variance Gaussian random variable - entropy of random variable y - probability density function of random variable y - negentropy of random variable y - mutual information of random variable vector Y II. INTRODUCTION Due to common weather patterns, wind power will exhibit reasonably strong statistical dependency within a small regional area. For more geographically distinct locations, this dependency reduces with increasing separation by distance [1]. While there will be many instances when wind power production at such distinct sites will be similar, it could be a relatively common occurrence that one wind farm might be at maximum power output with simultaneously little or no wind production at the others. Thus while generation expansion studies can justifiably use uni-variate statistical models for total system wind power production [2], in contrast the spatially distributed context of the transmission network planning problem requires an effective representation of multivariate wind statistical interdependency. Maintaining a distinct statistical representation for each potential wind farm location will result in a very large number of random variables. If a more approximate model can capture the salient features of each location s univariate marginal distribution as well as the spatial multivariate interdependency, then considerable computational efficiency or model specification simplicity will result. A multivariate discretisation procedure was studied in [3], whereby individual multivariate wind power historical time series hours were grouped into uniformly sized multi-dimensional histogram bins, thus allowing the collection of hours in each bin to be represented by a single probability-weighted power flow case. The multidimensional wind volume is generally quite full with samples however, as even medium-scale geographical separation of wind locations results in little grouping around the main multidimensional diagonal. Thus while uni-variate probability discretisation was previously used for customer demand variation modeling in transmission planning applications [4], the efficiency of such basic discretisation approaches is generally quite low for the multivariate wind power problem (e.g. ~ 20% reduction in model size for the seven wind farm case study in [3]). Such discretisation approaches may have greater promise for 91

107 distribution system applications however, as later outlined in [5], where wind generators are much more closely located than in the transmission system case. Probability discretisation procedures attempt to reduce the size of the overall problem by grouping historical data observations for a given number of random variables. An alternative approach might instead try to reduce the number of variables with an intelligent dimension reduction scheme. Wind stochastic clusters were proposed in [6] for example, whereby arbitrary groups of close-by wind generators were represented by a single random variable, with multivariate dependency information retained only for the relationship between each distinct cluster. Principal component analysis (PCA) is a useful multivariate dimension reduction technique commonly applied to large statistical datasets. PCA was used to investigate dependency in transmission network flows in [7] for example. By performing eigenvector/eigenvalue analysis of the multivariate wind power covariance matrix, an arbitrarily lower number of alternative statistical variables (the principal components ) could be determined [8]. Usefully, the resultant variables are uncorrelated, and the sum of the eigenvalues corresponding to the retained principal components relates to how much of the original variables variance is explained by the lower dimension transformation. PCA is most suitable to multivariate Gaussian distributions, which are uniquely defined by a covariance matrix. For more general multivariate distributions however, the covariance which is determined by second-order statistical moment information is but a measure of linear statistical dependency only, and zero correlation does not necessarily imply independence [9]. Higher-order statistical moments will be required to describe the nonparametric marginal statistical and multivariate dependency distributions that apply to the distributed wind power production model. More general statistical dependency based dimension reduction techniques such as independent component analysis (ICA) [10], where the objective is to produce a lower number of alternative variables with the much richer property of statistical independence (the independent components ), could potentially be of additional use. In this paper, the application of both principal and independent component linear separation analyses to the distributed wind power multivariate dimension reduction problem is analyzed in detail. Section III outlines the relevant PCA and ICA investigations for the distributed wind power case. Section IV describes the real historical wind power production data from multiple locations on the Irish power system that is used along with a simple test power system for the investigation. Section V gives the results of the PCA and ICA approaches, with discussions and conclusions outlined in Section VI. 92

108 III. WIND POWER COMPONENT ANALYSIS STUDY A. Principal Component Analysis of Wind Power Data For a given t*n matrix X of t observations from n distributed wind power random variables, the wind power covariance matrix C x is a symmetric n*n matrix. Any symmetric nonsingular matrix such as C x can be transformed to a diagonal matrix L through pre-multiplication and postmultiplication by a given orthonormal matrix E as in (1). l 1, l 2,.. l i.. l n, the diagonal values of L, are the eigenvalues of the C x matrix, and can be determined by solution of the classic characteristic equation (2), where I is the identity matrix of size n. The eigenvectors e 1, e 2,.. e i.. e n comprising the columns of E are given by solutions of equations (3a) and (3b). For very large values of n, the eigenvalues and eigenvectors are usually determined by iterative numerical techniques [8]. The n*t principal wind power components matrix Q is determined by the linear orthonormal transform (4) the original variable set X is first centered by subtracting the column mean vector of capacity factor values X, and its transpose is then pre-multiplied by E T. This essentially corresponds to a rotation of the original co-ordinate axes, with each new axis chosen to explain as much of the variance in the original wind power dataset as possible. The rows of Q correspond to the resultant uncorrelated principal components. E T. C. E = L (1) x det( C x li ) = 0 (2) [ C l. I ] k = 0 (3a) x i e i k = i (3b) T k i. k i Q ) T T = E.( X X (4) It can be shown that the trace of matrix L, Tr(L) is equal to Tr(C x ). Each eigenvalue, corresponding to the variance of one principal wind component, therefore relates to how much of the variance in the original multivariate wind power data which that component explains. For highly correlated datasets, the first few principal components will explain the majority of the original variance. The eigenvalues can be plotted in order of decreasing size this approach is sometimes used as an approximate visual technique to determine how many principal components should be retained. The approximate scree or broken-stick tests propose the decreasing eigenvalue plot cornerpoint discontinuities as the natural number of components to retain, though more sophisticated statistical tests have also been proposed [8]. Some PCA implementations advise the use of the correlation matrix instead of the covariance matrix for C x, giving different resultant principal 93

109 components [8]. The correlation matrix is most useful in situations where the original variables have different units of measurement, or when there are other significant differences in their variances. For the wind PCA study of this paper however, the consistent scaling of the nominal 1 MW wind power time series in Section IV justifies the use of the covariance matrix. B. Wind Power Data Reconstruction Residual Effects For any given number of retained principal components σ a reconstructed estimation of the original wind power time series X REC can be determined in (5) by inverting the PCA transform of (4), using only the relevant truncated E and Q matrix rows and columns. If less than the overall n principal components are used, this will result in a non-zero residual error matrix R with respect to each original variable of X in (6). X T 1 REC = ( E ) trunc. Q trunc + X (5) R = X X (6) REC C. Independent Component Analysis of Wind Power Data PCA, which is based on the principle of de-correlation, will be of greatest relevance for multivariate Gaussian datasets, as the linear PCA transformation will then still give Gaussian principal components, and uncorrelated multivariate Gaussian random variables also have the desirable property of statistical independence. As will be illustrated in Section V however, this property does not necessarily extend to other more generally dependent multivariate distributions such as in the distributed wind power case. A different model is therefore required to find independent wind components. A statistical transformation using the marginal cumulative distributions to uniform variables, followed by the inverse Gaussian cumulative distribution, could allow generation of more statistically independent Gaussian components [11]. Such combined transforms would be very non-linear in nature for expression in analytical models though, and the multivariate Gaussianity of such transformed variables must also be carefully tested if the vector ARMA models as proposed in [11] are valid. In addition to PCA dimension reduction, this paper therefore investigates if a more useful linear ICA mixing matrix model can be found instead. The basic noise-free ICA model (7) assumes that the random wind power observations in X are the result of a linear mixing process A performed on an equal or lesser number of statistically independent components S [10]. Given X, ICA algorithms estimate the linear de-mixing matrix U (i.e. the pseudo-inverse of A) together with the independent components themselves (or in practice, components that are as statistically independent as possible). It can be shown [10] that if prior to the ICA estimation process the observations in X are first de-correlated to Q using 94

110 PCA and then scaled to have unit variance as alternative random variables Z, then the required de-mixing matrix W in (8) will be an orthogonal transformation. An orthogonal de-mixing matrix has fewer degrees of freedom and is thus easier to determine. X = A. S (7) S = W.Z (8) From the central limit theorem of statistics, any linear mixture of independent random variables will be more Gaussian than the original components themselves. An intuitive ICA scheme might therefore estimate W and S by maximizing the non-gaussianity of a linear transformation of Z. An appropriate measure of non-gaussianity is therefore required. Some ICA algorithms are based on maximizing the kurtosis of the independent component estimates, given that the kurtosis of the Gaussian distribution is zero [10]. For an observation-data driven technique such as ICA, the higher-order statistical moments based kurtosis measures can be disproportionately sensitive to data-outliers, and may thus perform poorly from a numerical perspective. The entropy H(y) of a single random variable y in (9), where p y is the probability density function of y, can be used as a measure of its randomness or lack of structure. It can be shown [10] that the Gaussian distribution is that which has the maximum entropy H(y GAUSS ) of all possible zero-mean unit-variance distributions. The negentropy J(y) as defined in (10) is therefore an intuitive measure of non-gaussianity, and its maximization can be used to search for the independent components. Usefully it is always non-negative and zero only for Gaussian distributions. H ( y ) = p y ( ξ ). log( p y ( ξ )). d ξ (9) J y ) = H(y ) - H(y) (10) ( GAUSS An efficient ICA algorithm, Fast-ICA, based on negentropy maximization is given in [12]. The algorithm of [12] uses a fixed-point gradient-descent optimization approach, with a nonlinear function approximation of negentropy. As the Fast-ICA algorithm uses gradient descent of non-linear functions, local optimality can be guaranteed only. Multiple iterations of Fast-ICA from different initialization points has been proposed in [13], with post-optimal clustering and visualization procedures for the resultant component estimates. MATLAB implementations of [12] and [13] are available at [14]. Little prior knowledge of the most appropriate number of independent components to search for is known in most applications. Dimension reduction, if 95

111 any, is generally performed at the PCA stage of the overall ICA process, and thus the same residual errors in (6) will be retained by the Fast-ICA process results. The independent wind power components can be estimated separately (termed deflationary ICA) or together ( symmetric ICA). For the deflationary ICA approach, all subsequent component estimations to the first must constrain the appropriate row of W to be orthogonal to the previous rows. In contrast to the PCA technique, the ICA wind power component results cannot be attributed any specific order of importance. D. Measuring Independence Using Mutual Information The classical independent component analysis model, assuming statistically independent hidden latent or explanatory variables formed from a linear de-mixing process, may not necessarily apply directly to the multivariate wind power data case the PCA/ICA algorithms will generally give linearly separated wind components that are as statistically independent as possible. While the techniques outlined in [13] mainly test the convergence robustness of the ICA results, a postoptimal measure of optimality would also be useful. Mutual information M(Y) as defined for a vector of random variables Y in (11), is an information-theoretic based non-parametric measure of statistical independence that is applied in this paper. Mutual information is always non-negative, and is zero only for independent statistical variables. M(Y) corresponds to the Kullback-Leibler divergence probability distance metric between the component joint-dependency density function p Y, and the density function resulting from the product of all the individual component univariate marginal density functions. If the PCA/ICA results are indeed statistically independent, these two density functions would be identical. Analysis of the reduction in mutual information from the PCA component estimates to the ICA component estimates will outline the value of applying the Fast-ICA algorithm to the multivariate wind power case. σ M ( Y ) = H ( y ) H ( Y ) (11) i = 1 i IV. MULTIVARIATE WIND DATA AND TEST POWER SYSTEM A. Irish Wind Power Time Series Data Fig. 1 illustrates the locations of the Irish wind farms used as the database for this study. Geographically adjacent wind farms were arbitrarily grouped into nine wind clusters Zone A, B, C, D, E, F, G, H, and I, with each zone cluster based on real data from 2 to 5 close-by individual existing wind farms. Each wind zone was modeled by summing the respective wind farm power time series, and then rescaling by the zone s total capacity to give consistent 1 MW nominal wind power time series models for each region. Synchronously recorded historical power output data 96

112 from the year 2008 was used (inherently representing the relevant marginal statistics and multivariate dependency), taken at 2-hourly intervals from the original 15-min recordings thus giving 4392 multivariate samples overall. Fig. 1 - Irish wind power zones used in this study (each with 2-5 wind farms). B. Test Power System Information The test power system used for these economic dispatch and power flow studies is illustrated in Fig. 2. This has a 35-bus, 54-line network, denoted as Area 1 (based on a very simplified model of the Irish All-Island 220/275/400 kv high-voltage transmission system). It contains a mixture of base-load and mid-merit fossil-fuel (coal and peat) steam turbine generation, combined-heat-andpower gas plants (CHP), combined-cycle gas turbines (CCGTs), higher-efficiency aero-derivative gas turbines (ADGTs), lower-efficiency open-cycle gas turbines (OCGTs), as well as a few gas/oildistillate peaking units, amounting to 10.4 GW conventional plant capacity overall. 500 MW of HVDC interconnection capacity to a much larger separate power system denoted as Area 2 (based on an approximate model of the Great Britain generation portfolio) is available at both buses 12 and 34. Conventional plants in Area 2 are grouped approximately into multiple generation capacity blocks of similar plant-type, all connected at a single transmission node. Conventional plant performance data, seasonal natural gas fuel price variations, load profile, load magnitude (accounting for projected load growth to an Area 1 maximum peak value of 9.61 GW), and the assumed load geographic distribution are consistent with [15]. Load profile information for Area 2 was sourced from [16]. Additional information on the test network branch reactance and thermal capacity parameters, the assumed system geographical load spread, and the conventional generation portfolio network locations as applied in this investigation are given in 97

113 the Appendix II section of this thesis. The wind power collective Zones A, B, C, D, E, F, G, H, and I are modeled as connected to network buses 3, 4, 9, 17, 12, 25, 15, 28, and 30 respectively. Wind capacity installation in Area 2 was assumed zero the performance of statistical component analysis for wind power output in Area 1 is of primary interest. All model development for this paper was carried out in MATLAB [17] and GAMS [18], using the MATLAB/GAMS interface available at [19]. Fig. 2. Test power system network schematic and wind zone locations. TABLE-I ZONAL WIND CAPACITY ALLOCATION USED FOR SCOPF ANALYSIS, (MW) Zone A B C D E F G H I Capacity C. Residual Error System Power Flow Case Studies The residual error (6) impact of discarding increasing numbers of statistical wind components on transmission network power flow modeling accuracy was also investigated using a linear DC power flow model. Two case studies were implemented with the arbitrary zonal wind capacity allocation in Table-I: Case-I A network unconstrained economic dispatch model illustrating the total power flow requirements. Case-II An N-1 security constrained optimal power flow model (SCOPF) analysis of network congestion indices such as wind farm energy curtailment. 98

114 V. RESULTS A. Multivariate Wind Data The marginal probability density functions of power output from two typical Irish regions, Zone X and Y (X and Y are not linked to Fig. 1 for commercial sensitivity reasons), are illustrated in Fig. 3. Clearly the individual wind power output patterns over an extended timeframe correspond to non-parametric statistical distributions (resulting from passing the Weibull wind speed distribution through the non-linear turbine power curves). Such distributions cannot be uniquely described by first- and second-order statistical moment derived mean and variance information alone. The scatter plot of their joint power production, as in Fig. 4, furthermore outlines their non-parametric bi-variate statistical dependency. Fig. 4 also emphasizes the unsuitability of basic discretisation approaches applied to a raw multivariate wind power dataset as in [3]. If each of the n wind zones were modeled by d discrete binning density, then the spread of the scatter plot in Fig. 4 (each of the 10 2 boxes contains at least 1 data sample) would suggest that d n, the total number of multidimensional probability weighted discrete cases, would be intolerably large if more than 3 or 4 wind zones are studied i.e. the curse of dimensionality. Fig. 3 - Marginal probability density function histograms for Zones X, Y. Fig. 4 Bi-variate dependency scatter plot for Zones X, Y. 99

115 B. PCA Results The correlation matrix for the nine wind zones is given in Table-II. Clearly the inter-zonal correlation reduces from Zone A to Zone I, as might be suggested by the geographical separation in Fig. 1. As Fig. 4 suggests though, these single linear-dependency metrics typically explain only part of the overall dependency structure. The covariances of the PCA results are given in Table-III. As the covariance matrix is diagonal, clearly the resultant principal components have been decorrelated. A plot of the eigenvalues given in decreasing order, as per the diagonal of Table-III, is illustrated in Fig. 5. Clearly there is a rapid reduction in the amount of the original wind variance explained by retention of more than 3 or 4 principal components. Also the first principal component and its associated eigenvalue is significantly more important than the others, due to the relatively high linear correlations in Table-II. On the basis of the arbitrary scree test alone, TABLE-II ZONAL CORRELATION COEFFICIENTS BEFORE PCA TRANSFORMATION ZONE A B C D E F G H I A B C D E F G H I TABLE-III PRINCIPAL COMPONENT COVARIANCE AFTER PCA TRANSFORMATION PC

116 0.7 eigenvalue order of components 0.6 associated eigenvalue resultant component number Fig. 5 Ordered eigenvalue plot associated with each principal component. retaining principal components 6-9 would seem to add very little value. The first principal component corresponds to the common power production patterns in the original multivariate set. The other components do not have such a tangible explanation. A time series plot of the nine principal components is given in Fig. 6. The residual error analysis of (6) for wind Zone A is illustrated in Fig. 7 with different numbers of retained principal components. Retaining a few components alone leads to significant residual error in this statistical variable. Retaining more principal components results in a better estimation of the original Zone A variable, but as Fig. 5 might suggest, the incremental benefit reduces somewhat for the higher-number principal components. The reduction in the root-mean-square (rms) of the residual error in (6) for a selection of the wind zones is given in Fig. 8. Clearly, retaining additional principal components gives a much greater reduction in the rms error for some variables more than others see the disproportionate reduction in the rms error for Zone E with the retention of principal component 3. This illustrates a commonly observed effect in PCA, where the discarded principal components in Fig. 5 can sometimes correspond to specific individual variables in the original multivariate set, rather than being shared amongst all. Thus careful residual error analysis must always be performed for X REC when using PCA to ensure that no single wind zone is overwhelmingly impacted by the dimension reduction process. The impact of residual errors on the network power flow modeling of the test system in Fig. 2 is illustrated in Fig. 9 and Fig. 10. The histograms of the Case-I economic dispatch model power flows in network branch over the extended reconstructed multivariate time series are given in Fig. 9. Wind capacity at zone F is connected to bus 25. Clearly there are some nonnegligible power flow model differences associated with the multivariate reconstruction residual 101

117 error (6) for different numbers of retained components. Interestingly however, viewing the results in Fig. 10 for the SCOPF analysis applied in Case-II, the selected wind zone energy curtailments due to network congestion were not as significantly influenced as might be initially expected from the magnitude of the residual errors observed. One possible reason is that the type of residual errors in Fig. 7 are at least somewhat symmetric, and thus there may be as many hours when the curtailment is overestimated incorrectly as underestimated incorrectly, giving approximately the same net overall yearly percentage value. Note the relationship between Fig. 8 and Fig. 10 for Zone I the curtailment step increments correspond to specific residual error decrements with the inclusion of components 2 and 6. Fig. 6 The resulting principal component time series. 102

118 Fig. 7 Zone A residual error for 1, 3, 5, 7, 9 retained principal components. RMS reconstruction error Zone A Zone C Zone E Zone G Zone I number of retained principle components Fig. 8 Reduction in zonal rms reconstruction error for additional components 103

119 Number of Occurences PC 3 PCs 5 PCs 9 PCs Power Flow (MW) Fig. 9 Case-I : Transmission network power flow modeling residual error (line 15-25). 12 wind curtailment % Zone B Zone D Zone I number of retained principle components Fig. 10 Case-II : Wind energy curtailment % with respect to number of retained components. A scatter plot illustrating the dependency between the first two principal components resulting from the wind data PCA study is given in Fig. 11. While Table-III proves these components to be de-correlated, one can clearly see that there is still some general dependency present. For example choosing samples of principal component 1 at both the lower and upper values of its domain will limit the range of values that can be given by principal component 2 relative to samples chosen in the middle of the domain of principal component 1. This emphasizes the drawbacks of wind component analysis based on second order moment statistics information alone. 104

120 1 PCA scatter plot PC PC 1 Fig. 11 scatter plot of retained principal components 1 and 2. C. ICA Results The ICA results are illustrated in Fig. 12 for 5 retained wind components. Independent component 3 bears a strong similarity to principal component 1, except for a negative scaling factor. A scatter plot of the bi-variate dependency between independent components 2 and 3 is given in Fig. 13. Clearly a non-negligible amount of the multivariate dependency from the PCA outputs is still retained in the linearly separated ICA results. The mutual information measures in Table-IV furthermore indicate the small but unspectacular dependency reduction achieved by the basic linear ICA model. As Fig. 14 outlines, the % reduction in mutual information achieved by the Fast- ICA algorithm is at most 11% for a choice of 5 retained components (note the mutual information values will be higher regardless with increasing number of components retained). Both Fig. 8 and Fig. 14 will together outline the tradeoffs associated with choosing the appropriate number of components to retain. VI. DISCUSSION AND CONCLUSION This paper presents a multivariate dimension reduction study as applied to distributed wind power historical data. Using principal and independent component analysis, it is shown that the strong dependency in the distributed wind power data allows a reasonably significant dimension reduction to retain a large proportion of the original statistical behaviour. A lower number of statistical variables will reduce model dimensionality for computationally efficient power system study applications note the real Irish power system has more than 100 wind farms at present, that on the basis of the results in this paper, might be reasonably accurately represented by many 105

121 Fig. 12 The ICA estimates for 5 retained components. Fig. 13 scatter plot of independent components 2 and 3. TABLE-IV MUTUAL INFORMATION OF PCA AND ICA ESTIMATES, (BITS) # Retained Components PCA ICA

122 Fig. 14 % reduction in mutual information from PCA to ICA results. fewer variables. If a linear ICA model is assumed, the retained components most likely cannot be guaranteed to have the desirable statistical property of pure independence that is aimed for at the outset linearly separated components are more accurately described as being as independent as possible. Careful residual analysis should always be performed for the discarded components, as they sometimes correspond directly to specific wind zones only. The data used in this study was sourced from wind farms on the Irish power system, though generality to other wind regimes should apply. A very desirable characteristic of the PCA/ICA techniques are that they are performed on zero-mean centered data (4), and therefore the precise wind zone capacity factor values are preserved for the original variable reconstructions in (5). Combined wind/transmission optimization problems are quite sensitive to wind resource quality differences in the power system geographical area [20], and preservation of the capacity factor characteristics, in addition to the reasonably stable wind energy curtailment results in Fig. 10, indicates that optimization solution quality with a few retained components describing the wind variations may still be reasonably accurate. The results of Section V-C outline the rather limited impact of the classic linear ICA model on the dependence of the retained principal components. A transformation using the cumulative distribution to multivariate Gaussian random variables could of course be useful [11], either before or after the PCA step, though the significant non-linearity of such transforms may have consequences for optimisation problem models. Both the PCA/ICA input variables and the component results are sequentially dependent time series, though no account of this is taken in the Fast-ICA algorithm. Use of the time series autocorrelation in the data might give the extra information that in addition to searching for a non-linear mixing matrix, could allow generation of 107

123 more independent components. Analysis of multivariate dependency in distributed wind speed data as opposed to wind power data may also be worthy of investigation, though retention of the piecewise non-linear wind turbine speed/power curves in power system study formulations could be a significant practical drawback. Investigating more effective and/or more analytically simpler alternative approaches to the distributed multivariate wind data dimension/dependence reduction problem is an interesting future research topic. VII. REFERENCES [1] T. Ackermann, (Editor) Wind Power in Power Systems, Wiley, [2] R. Doherty, H. Outhred, M.J. O Malley Establishing the Role That Wind Generation May Have in Future Generation Portfolios, IEEE Transactions on Power Systems, Vol. 21, No.3, August [3] D. Burke and M.J. O Malley Optimal Wind Power Location on Transmission Systems A Probabilistic Approach, presented at the IEEE PMAPS Conference, Puerto Rico, May [4] G. Latorre, R Dario Cruz, J.M. Areiza and A. Villegas Classifications of Publications and Models on Transmission Expansion Planning, IEEE Trans. Power Systems, Vol. 18, No.2, May [5] L.F. Ochoa, C.J. Dent, G.P. Harrison Distribution Network Capacity Assessment: Variable DG and Active Networks, IEEE Trans. Power Systems, Vol. 25, No.1, February [6] G. Papaefthymiou, A. Tsanakas, D. Kurowicka, P. H. Schavemaker, and L van der Sluis, "Probabilistic Power Flow Methodology for the Modeling of Horizontally-Operated Power Systems," International Conference on Future Power Systems, Amsterdam, Nov [7] S. Deladreue, F. Brouaye, P. Bastard, L. Peligry Using Two Multivariate Methods for Line Congestion Study in Power Systems Under Uncertainty, IEEE Trans. Power Systems, Vol. 18, No.1, February [8] J.E. Jackson, A Users Guide to Principal Component Analysis, Wiley New York [9] G. Papaefthymiou, D. Kurowicka Using Copulas for Modeling Stochastic Dependence in Power System Uncertainty Analysis, IEEE Trans. Power Systems, Vol. 24, No.4, Feb [10] A. Hyvarinen J. Karhunen and Erkki Oja, Independent Component Analysis, Wiley New York [11] B. Klockl Multivariate Time Series Models Applied to the Assessment of Energy Storage in Power Systems, presented at the IEEE PMAPS Conference, Puerto Rico, May [12] A. Hyvarinen Fast and Robust Fixed-Point Algorithms for Independent Component Analysis, IEEE Trans. Neural Networks, Vol. 10, No.3, May [13] J. Himberg, A.Hyvarinen, F. Esposito Validating the Independent Components of Neuro-imaging Time Series via Clustering and Visualisation, Elsevier NeuroImage, Vol. 22, No.3, July [14] [15] All Island Grid Study, Workstream 4 Analysis of Impacts and Benefits, Irish Government Department of Communications, Energy and Natural Resources/United Kingdom Department of Enterprise, Trade and Investment, Jan Available online - [16] [17] MATLAB, available at [18] General Algebraic Modeling System, GAMS available online [19] Matlab and GAMS Interfacing Optimization and Visualization Software, by M.C. Ferris available online at [20] D.J. Burke, M.J. O Malley Maximizing Firm Wind Connection to Security Constrained Transmission Networks, IEEE Transactions on Power Systems, Vol.25, No.2, May

124 CHAPTER 7 SCENARIO REDUCTION APPLIED TO DISTRIBUTED WIND POWER TRANSMISSION STUDIES Electricity Research Centre working paper, in preparation for journal submission. Mr. Daniel J. Burke and Prof. Mark J. O Malley. Abstract Representing spatially distributed multi-period wind power output in transmission system studies can lead to high-dimensionality and computationally demanding problem formulations. This paper investigates the computational efficiencies and accuracy tradeoffs of sequentially independent and probability-weighted discrete-scenario based multivariate wind power representations. The simultaneous-backward scenario reduction scheme is applied to generate the discrete scenarios from historical wind and load demand data. The optimal non-firm wind capacity placement and the wind energy curtailment estimation problems for a constrained transmission network are used as methodology evaluation test-cases. Results indicate that reasonably low numbers of retained scenarios are sufficient to provide solutions within appropriate cost and accuracy tolerances for these respective problems. Scenario reductions schemes are therefore promising with regard to computationally tractable study of wind integration in large-scale transmission networks. Index Terms-- power transmission, statistics, power generation planning, wind energy. I. INTRODUCTION Wind power is a variable and distributed electrical energy resource. Accommodating its distinct characteristics within traditionally deterministic snapshot transmission planning case-studies is not possible, so the study of system operation under a more diverse combination of power flow conditions is necessary [1],[2]. Historical time series data [3]-[5], time series synthesis [6], [7] and random statistical sampling [8], are some of the methods proposed to account for this case-study diversity. However these methods typically require a very large number of individual samples in order to recreate the correct marginal and multivariate wind statistical dependency properties. The computational burden associated with performing typical system studies such as load flow or dynamic stability assessments for such a large number of diverse system operating conditions is a significant challenge. Techniques to reduce the minimum required case-study dimensionality while still preserving the essential characteristics of wind power statistical distributions will be very useful for a wide range of transmission study tasks. 109

125 A study of the most appropriate number of historical data samples to retain for distributed wind energy curtailment assessment was outlined in [9] for example, where the influence of both the number of historical data years and the applied data sampling frequency were investigated. A multivariate discretisation approach was proposed in [10] to reduce the time series dimensionality by grouping individual hourly samples in regularized multidimensional histogram bins of a given probability weighting. Though such discretisation approaches may have significant benefits in distribution system analysis [11], their efficiency is generally quite poor at transmission level for more than 2 or 3 geographically distinct wind zones i.e. the curse of dimensionality [12]. A more intelligent sample grouping approach might use a distance metric to combine similar hours, in contrast to the regularized binning procedure of [10]. Scenario reduction methods [13],[14] aim to reduce stochastic sample dimensionality by combining similar scenarios through addition of their probabilities, and retaining the scenarios that are more diverse. An interesting stochastic recombining-tree approach is outlined in [15] to model wind power variations in power system planning models. The optimal solution accuracy degradation for different recombining-tree branch/node density modeling choices is not reported in [15] though. Furthermore it is not clear how the method of [15] can efficiently account for geographical spatial wind power diversity in transmission planning problems its modeling of multivariate wind production with fixed discrete levels over discrete time steps will surely lead to the same dimensionality challenges experienced by [10]. Even though sequential scenario approaches may be useful for total-system wind power auto-correlative variability modeling, for many power systems without large-scale energy storage the inclusion of operational-timeframe wind variability/forecast-uncertainty in long-term transmission network planning problems may be somewhat excessive from a complexity/accuracy tradeoff viewpoint [16]. Temporal independence of hourly sample scenarios may thus be justifiable in situations where there is little or no installed storage capacity therefore. In most stochastic problem applications, there is typically a given minimum scenario density that adequately represents underlying stochastic variables and their interdependency while keeping the computational burden within reasonable limits. Identifying the cardinality of this ideal scenario density is a key task in any stochastic problem representation, and is the subject matter of this paper for two typical wind/transmission system study applications. Section II outlines the detail of these wind-power/transmission-system case studies investigated and describes the scenario reduction approach used. The test power system information is given in Section III, while the results and discussions of the system analyses are given in Sections IV. 110

126 II. TRANSMISSION STUDIES AND SCENARIO REDUCTION A. Transmission Planning Problems Investigated Efficiently accounting for multi-period and multivariate wind behaviour has many system study applications [5], [17], [18]. Two typical transmission system study tasks are investigated in this paper to analyze the efficiency and accuracy implications of the scenario reduction technique applied to distributed wind power data. These are namely the optimal allocation of non-firm wind capacity to constrained transmission networks [19], and the estimation of distributed wind energy curtailment indices from multi-period optimal power flow studies [9]. Further detail on the specific composition of these individual studies is reported in the subsections below. 1) Optimal Non-Firm Wind Capacity Connection Present transmission limitations are hindering the timely connection of wind capacity in many transmission system networks. Optimal combined generation and transmission expansion planning for wind power could help with reaching ambitious renewable energy penetration targets in a cost-effective manner. The mixed-integer and very large scale nature of the optimal transmission expansion planning problem for wind power is very computationally demanding. There are currently few if any techniques available in the literature to solve this type of large scale complex problem, with the accurate representation of multivariate wind power characteristics obviously a key challenge. To allow a more tractable study of the benefits and drawbacks of probability-weighted scenario based wind power representation, the optimal non-firm wind generation allocation linear programming problem is studied here for a fixed network configuration instead. Any conclusions with regard to the appropriate number of scenarios to retain etc for this simpler problem should also hold for the more difficult network topology optimization problem to be considered later. A time-series based investigation of this optimal non-firm wind capacity allocation linear programming problem was outlined in [19], with the block-diagonal structured multi-period security-constrained optimal power flow constraint matrix exploited by the Benders decomposition technique. For a given overall system wind capacity connection target, the objective function in [19] minimizes the system operational fuel cost by effective placement of the wind capacity to distinct candidate transmission nodes. Each node has its own respective wind resource capacity factor and network congestion characteristics. Multi-year and multi-period data was used to model load and wind power production diversity, thereby giving large numbers of individual sub-problem routines for each separate iteration of the overall decomposition scheme. The number of decomposition iterations multiplied by the number of time series sub-problems gives an overall computational demand that is quite intensive even for the relatively small test 111

127 system investigated in [19]. For realistic-sized power systems, scenario reduction techniques could effectively reduce the number of sub-problems by using an appropriately lower number of probability-weighted samples, and thus make this category of optimization problem more tractable. Outlining the appropriate number of scenarios to estimate a good problem solution within a reasonable computational bound is the important task studied in this paper. 2) Distributed Wind Curtailment Analysis Determining wind energy curtailment indices in congested transmission networks is of considerable economic importance for wind farm investors. A study of historical wind power data timeframe (i.e. number of years of data and data sampling frequency) and power system parameter uncertainty impacts on wind energy curtailment level estimation was outlined in [9]. Uncertain power system input parameters such as load profile and fuel/carbon price will directly influence the risk associated with expected wind energy curtailment levels evaluating their impact required many multi-period optimal power flow investigations, each associated with a wide number of alternative parameter uncertainty values. For power systems of realistic size, the representation of multivariate wind characteristics using a much lower number of probabilityweighted scenarios will have significant computational benefits for wind energy curtailment risk modelling therefore, with evaluation of the wind curtailment estimation error, as outlined in this paper for each number of retained probability scenarios, a necessary prior exercise. B. Application of Scenario Reduction For the typical power transmission case studies to be analyzed in Section II-A, historical recorded load and wind power time series data is used as the initial sample set to which the scenario reduction method is then applied. These multivariate wind power time series will inherently represent both the quality of the wind resource at each individual site, any multivariate spatial power output dependencies between geographically distinct regions, as well as any weaker dependency between wind power output and system load demand patterns. The historical load and wind time series data is contained by matrix X, with the q columns corresponding to distributed wind power and system-wide load demand variables, and h rows corresponding to sequential observations. Each single data observation is considered as one separate scenario in this paper s analysis. Scenario reduction algorithms are usually applied to scenario sets each consisting of multiple sequentially dependent observations in the situation when it can be assumed the operational variability and uncertainty of wind power has much lesser relevance in the long-term transmission planning timeframe [16], and if the system has limited large-scale 112

128 storage capacity, the individual time series hours can simply be considered as single-observation equally-weighted scenarios themselves. The optimal subset of scenarios to discard is that subset which ensures a probabilitydistance metric between the original multivariate probability distribution and that given by a retained-scenario based probability distribution estimate is a minimum. For a given number of scenarios to retain, finding the set of optimal scenarios to discard is a difficult combinatorial setcovering problem. Thus except for the trivial case of discarding only one scenario or retaining only one scenario, it is difficult in most scenario reduction problems to find the exact optimum subset solution. However, intuitive practical approaches have been proposed in the literature for example if the scenarios are iteratively discarded one at a time then the exact optimal scenario to discard can be found that minimizes the probability distance metric for each separate iteration s scenario set with respect to that of the last [13]. Repeated over a number of iterations until the desired retained-scenario cardinality is reached, this backward reduction approach will generally lead to reasonably good overall solutions. A more refined version of this methodology is given with the simultaneous-backward scenario reduction algorithm [13],[14]. The method of [14] ensures that the set of previously discarded scenarios is considered together with each candidate scenario to discard at every stage, so that the next discarded scenario chosen is the one that effectively ensures the collective distance is evaluated with respect to the original dataset, rather than the retained scenario set of the last iteration. Once the required number of scenarios has been reached, the probability associated with the discarded scenarios is re-distributed among the closest retained scenarios, with each retained scenario allocated a non-zero probability. It is important to note that scenario reduction is a non-parametric technique that can be applied to any observational data such as wind and load data. The scenario reduction stage of any stochastic optimization problem will take some time in itself, but the set of retained scenarios is then kept fixed for all the subsequent optimization problem investigations and sensitivity analyses etc. The simultaneous-backward algorithm in [14] is applied to the multivariate wind/load dataset in this paper. An inter-scenario difference measure is integral to most scenario reduction algorithms. For the analysis of this paper, the simple Euclidean distance d h1h2 is used as in (1) for the distance between X row scenario indices h 1 and h 2, with each column of X first normalized by its maximum value so that no variable dominates any others. ( ( ( 1h = 2 X h h q 2 1/ 2 1, q) X ( h2, q)) ) d (1) III. TEST POWER SYSTEM The test system used in the analyses of this paper is illustrated in Fig. 1. This has a 35-bus, 54-line 113

129 network, denoted as Area 1 (based on a very simplified model of the Irish All-Island 220/275/400 kv high-voltage transmission system). It contains a mixture of base-load and midmerit fossil-fuel (coal and peat) steam turbine generation, combined-heat-and-power gas plants (CHP), combined-cycle gas turbines (CCGTs), higher-efficiency aero-derivative gas turbines (ADGTs), lower-efficiency open-cycle gas turbines (OCGTs), as well as a few gas/oil-distillate peaking units, amounting to 10.4 GW conventional plant capacity overall. 500 MW of HVDC interconnection capacity to a much larger separate power system denoted as Area 2 (based on an approximate model of the Great Britain generation portfolio) is available at both buses 12 and 34. Conventional plants in Area 2 are grouped approximately into multiple generation capacity blocks of similar plant-type, all connected at a single transmission node. Conventional plant performance data, seasonal natural gas fuel price variations, load profile, load magnitude (accounting for projected Area-1 load growth to a maximum peak value of 9.61 GW), and the assumed load geographic distribution are consistent with [20]. Load profile information for Area 2 was sourced from [21]. Additional information on the test network branch reactance and thermal capacity parameters (chosen so that no congestion occurs at the zero wind penetration level), the assumed system geographical load spread, and the conventional generation portfolio network locations as applied in this investigation are given in the Appendix II section of this thesis. Synchronously recorded historical wind power data from 10 geographically distributed existing wind farms on the Irish power system was used for this study. As with [9], [19], this data was taken at eight-hourly sampling resolution over a time period of four years (from ), initially giving 4380 samples overall. Original capacity factor details are given in Table-I. The wind and load data were input to the scenario reduction process as outlined in Section II-B, with the 4380 original observations reduced to 10, 20, 30, 40, 50, 60, 70, 80, 90, 100, 150, 200, 250, 300, 400, 500, 600, 700, 800, 900, 1000, 1200, 1400, 1600, 1800, 2000, 2250, 2500, 2750, 3000, 3250, 3500, 3750, and 4000 retained probability weighted scenarios for the transmission system studies of Section II-A. While 4380 samples were initially chosen here, in reality any number of multi-year samples could be used as the initial scenario set, depending on the availability or synthesis of suitable data. Optimal wind capacity connection to Area 1 buses 3, 5, 7, 9, 11, 13, 15, 17, 25 and 33 was investigated (as outlined in Section-II-A-1), for a total wind capacity connection level of 6 GW. For this optimal wind capacity allocation, a security-constrained optimal power flow wind energy curtailment precision investigation is subsequently performed (as outlined in Section-II-A- 2) for the various numbers of retained probability-weighted scenarios. Wind capacity installation in Area 2 was assumed zero the performance of the scenario reduction process applied to the multivariate wind and load data observations for Area 1 is of primary interest. All model 114

130 development for this paper was carried out in MATLAB [22] and GAMS [23], using the MATLAB/GAMS interface available at [24]. Fig. 1 the test power system under investigation. TABLE-I TEST SYSTEM WIND FARM CAPACITY FACTORS (%) System Node Capacity Factor IV. RESULTS A. Retained Multivariate Wind Samples A typically representative scatter plot of 500 randomly-chosen historical data samples from the two-dimensional bi-variate statistical dependency of wind power outputs at Farm 3 and Farm 5 is illustrated in Fig. 2. A majority of these scenarios are located in the bottom-left corner on the main diagonal line of the plot, yet a significant number of outliers are also contained in the high power output regions of one or both of the wind farms. The scenario reduction scheme should aim to preserve these outlier samples as they may be of greater relevance for wind power related transmission system congestion, while effectively combining the redundant scenario positions on the main diagonal that are likely of lesser significance. The effect of the simultaneous-backward 115

131 scenario reduction scheme [14] is illustrated in Fig. 3 and Fig. 4 for an arbitrary reduction in retained scenario cardinality from the original 500 samples to 250 and 100 samples respectively. Clearly the scenario reduction scheme does indeed retain the most interesting cases and generally combines those sample positions that are most similar to one another. Of course, not all the retained scenarios have equal probability weight those retained in the bottom left hand corner of the plot will have acquired the associated probability of the many discarded samples in that region. B. Optimal Non-Firm Wind Capacity Solution Variation The results of the optimal non-firm wind power capacity allocation problem as described in Section II-A-1 are given in Table-II for the various retained `scenario group cardinalities. Fig. 2 Bi-variate scatter plot of 500 hourly wind power samples. Fig. 3 Effect of scenario reduction 250 samples retained. 116

132 Fig. 4 Effect of scenario reduction 100 samples retained. The corresponding number of Benders decomposition master/sub-problem iterations for each number of retained scenarios is given in Fig. 5 for example with more than two hundred scenarios retained the number of iterations required remains reasonably stable at approximately 150. For additional scenarios above this level therefore, the overall computational burden (i.e. the total number of security-constrained optimal power flow sub-problems) increases linearly. A plot of the wind capacity investment solution convergence in Table-II is also illustrated in Fig. 6 (note the x-axis scales are not linear, with each point corresponding to incremental rows of Table-II). As from Table-II and Fig. 6, the optimal wind capacity investment solution is more or less stabilized for many of the wind farm locations with only a few hundred scenarios retained. The lower capacity factor farms 3 and 7 are perhaps the only noteworthy exceptions. A graph of the estimated optimal cost function variation for different numbers of retained scenarios is given in Fig. 7. Clearly there is still non-negligible variation in the estimated optimal cost function value even for the higher numbers of retained scenarios. It is important to stress however, that these are simply estimates of the cost function value taken directly from the scenario reduction and decomposition scheme results, and not the problem cost function values that would apply in reality. To determine the true cost degradation of the approximate problem representation, the optimal capacity allocation estimates from each row of Table-II were input to a securityconstrained optimal power flow analysis applied to the more representative 4380-sample original wind and load data-set. A graph of the true-cost function value deviations is illustrated in Fig. 8, given as proportions of Table-II s 4380-sample optimal solution dispatch cost for Area-1 (as this is where the decision variables are located). While including more scenarios does reduce the real 117

133 TABLE-II OPTIMAL NON-FIRM WIND CAPACITY ALLOCATION SOLUTION VECTORS MW) System Node Retained Scenarios

134 Fig. 5 Corresponding number of Benders decomposition iterations required. Fig. 6 Optimal wind capacity allocations solution variations. cost function value, even at the very low cardinalities of 40 or 50 retained scenarios the cost function error penalty is just % of the true Area-1 yearly cost. The significance of this result for optimal wind/transmission integration studies relates to the associated computational effort. The solution of the 50-retained-samples decomposition problem required only about 0.75% of the computational time effort associated with the retained-samples case, for example. It is clear then, that reasonably stable solutions can be found 119

135 Fig. 7 Estimated cost function values for different numbers of retained scenarios. Fig. 8 True optimal cost function values for different numbers of retained scenarios expressed as percentage deviations of Area-1 total cost. with somewhat approximate representations of the stochastic inputs to the problem, and hence the value of applying the scenario reduction algorithm to the multivariate wind power data. The 120

136 computational effort will still be reasonably onerous though in very large transmission systems 50 security-constrained optimal power flow decomposition sub-problems each solved 100 times relates to 5,000 individual large-scale linear programming tasks. A major advantage of decomposition schemes of course is that parallel computations can be applied to each subproblem - this might indeed be required for large-scale network analyses. It should also be noted that though there is a very small cost differential between the 50 and 4380 scenario problem representations, their precise solutions may be reasonably different as indicated by comparing the respective rows in Table-II. The optimal wind capacity investment problem should give due consideration to longterm power system demand and fuel-price volatilities. If the original multivariate time series is used directly in the optimization problem, then for computational limitation reasons, this uncertainty consideration would likely have the form of sensitivity analysis only, as applied in [19]. Given that this paper suggests that a much lower number of probability-weighted scenarios can approximate the optimal solution quite effectively for a given customer-demand/fuel-price background scenario, then it may now be possible to integrate the longer-term planning uncertainties directly in the optimization model formulation. Using alternative long-term demand/fuel-price uncertainty background scenarios, to each of which the wind power/load demand probability scenarios are superimposed, the simple effect is a multiplication of the length of the constraint matrix main diagonal. This is a key advantage of the scenario reduction method as suggested by the results of this paper. C. Wind Energy Curtailment Study Variations The variation in estimated wind energy curtailment percentages for different numbers of retained scenarios is illustrated in Fig. 9. For more than ~ 400 probability-weighted scenarios retained, the estimated wind energy curtailment values stabilize for most of the wind farm locations - the main exception is Farm 17 which still exhibits some variation with respect to its true curtailment value for even the highest numbers of retained scenarios. However such curtailment modelling error should still be considered in the context of load/fuel-price parameter uncertainty influences that nevertheless apply to any future power system study [9]. The curtailment uncertainty patterns as highlighted in [9] require multiple curtailment studies to be performed with respect to many alternative parameter uncertainty values. The scenario reduction approach will therefore be valuable in reducing the computational effort required to evaluate curtailment risk dependency in realistic large-scale transmission systems. 121

137 Fig. 9 Estimated wind energy curtailment percentage variations. V. CONCLUSION This paper investigates the applicability of scenario reduction schemes to the representation of distributed wind power variations in transmission network study problems. The precision of both the optimal non-firm wind capacity allocation problem and the wind energy curtailment estimation task was investigated for different numbers of retained scenarios. For the optimal nonfirm wind investment problem, it was shown that a very low number of probability-weighted scenarios gave a very effective representation of the wind power stochastic variables with little overall economic cost degradation. This concise method of wind power representation will be essential for mixed-integer optimal transmission expansion planning investigations. A relatively small number of retained scenarios also gave reasonably good accuracy for the distributed wind curtailment investigation task, suggesting that scenario reduction schemes will also be quite useful for computationally efficient inter-locational curtailment risk dependency estimation in large power systems. 122