On the Behavior of Responsive Loads in the Presence of DFACTS Devices

Transcription

1 On the Behavior of Responsive Loads in the Presence of DFACTS Devices Javad Mohammadi, Student Member, IEEE, Gabriela Hug, Member, IEEE, Soummya Kar, Member, IEEE. Abstract The paper studies a constrained economic dispatch problem in electricity networks with price-dependent demand, and in which the transmission lines are potentially instrumented with DFACTS devices with a view to mitigating system congestion. Specifically, the network objective concerns the joint adjustment of consumption profiles in response to the electricity price and the design of variable reactance (DFACTS settings) that lead to a form of welfare optimization. To this end, the paper introduces an iterative minimization approach that seeks to optimize the constrained dispatch objective through successive (iterative) adjustments of consumption profiles at the responsive loads based on the difference between the price estimated by the loads and the price resulting from the economic dispatch. Simulations demonstrating the efficiency of the proposed approach in the presence of two different types of DFACTS devices are presented on the IEEE 4-bus system. In particular, the numerical findings provide a quantitative understanding of how the presence of DFACTS lead to reduced locational marginal prices and increased demand accommodation capacity in networks of responsive loads. A. Parameters and Constants NOMENCLATURE D 0 Customer s initial load (MW) at each hour ρ 0 Initial (predicted) hourly electricity price ($/MW) E Hourly self-elasticity of demand side a i, b i, c i Generators cost function parameters ($/MW) X ij Reactance of line (i, j) F ij Line flow limit of line (i, j) (MW) P i Maximum generation of generator at bus i P i Minimum generation of generator at bus i X ij Minimum reactance of DFACTS in line (i, j) X ij Maximum reactance of DFACTS in line (i, j) N B Number of buses in the system Number of generators in the system N G B. Variables B(D) D Customer s benefit as a function of consumption ($) Customer s load after responding to price signal (MW) The authors are with the Department of Electrical and Computer Engineering, Carnegie Mellon University, Pittsburgh, PA, 523 USA. jmohammadi@cmu.edu, ghug@ece.cmu.edu, soummyak@andrew.cmu.edu. P i Power generated by generator at bus i ($) θ i Phase angle of bus i (rad) X ij Reactance of DFACTS in line (i, j) X ij,tot Reactance of line (i, j) with DFACTS ρ Hourly electricity price ($/MW) I. INTRODUCTION Restructuring and load growth has caused the transmission grid to be operated closer to its capacity limits and therefore has significantly increased the stress on today s transmission grid []. The resulting bottlenecks in the grid limit cheaper generation from being fully dispatched and forces dispatching of more expensive generators in the system. As the power grid is highly interconnected, even a single overload may prevent many transactions from taking place [2]. However, extending the transmission system is challenging, especially in a deregulated environment. An overview over proposed approaches for congestion management in competitive power markets is provided in [3]. Demand response, i.e. the use of price signals as motivation for electricity market s participants to reduce their consumption and therefore participate in congestion management, is one possible approach [4]. Another possibility to resolve congestions is power flow optimization in which specific parameters in the power system are controlled to maximize the usage of the existing transmission capacity [2], [5] [7]. Power flow control defined as steering the electric power to a desired path by adjusting line impedances, bus voltage magnitudes or phase angle differences is a means for power flow optimization. It can potentially increase the transfer capacity of the transmission system by diverting the power from congested lines to the lines with spare capacity [7]. Flexible AC Transmission Systems (FACTS) are devices capable of altering transmission system parameters as discussed above, and therefore can be used for power flow optimization. However, the adoption of FACTS devices for the purpose of power flow control has been slow due to their high costs and the fact that such devices are mostly custom designed with long build-times [7]. Also, the performance of FACTS devices depends on their location in the system with space requirements limiting the number of potential locations [5]. With the objective to serve the same purpose with smaller and more reliable controllers, a distributed version of FACTS, namely Distributed FACTS (DFACTS), has recently been proposed [8]. These single-phased devices can be clamped onto existing conductors allowing for easy installation and on-site repair. In addition, by design, the impact on system

2 performance in case of a failure of one of these devices is limited. Hence, DFACTS could provide a cost-effective mean for power flow control and congestion management. The impacts of DFACTS utilization by studying the sensitivities of power system quantities such as voltage magnitude, line power flows, etc with respect to line impedance are examined in [2]. Also, DFACTS installation effects on power system performance are investigated in [7] in several contexts such as power system operation, economic, etc. Resolving congestions by means of power flow control allows for a more cost-effective generation dispatch which on the other hand impacts the electricity price. Consequently, employing power flow control in an environment in which demand response (DR) is part of balancing supply and demand, power flow optimization will not only affect the paths of the power flows but also the level of consumption. Generally, there are two categories of DR programs: incentive based DR programs and time based rate programs [9]. Incentive based DR programs are designed to modify the consumers consumption pattern by providing rewards and penalties whereas time based programs use price signals which reflect the actual cost of generation and the state of the power system. Using DR as a resource to handle the transmission system congestion has been surveyed in [4]. The impact of load elasticity in congestion management is investigated and the resulting elasticity effects on consumers and producers is evaluated. In [0], it is suggested to value load reduction offers in electricity market based on each consumers contribution to the congestion management and loss reduction. As many system operators are in the process of implementing programs to benefit from demand side participation [], it is of importance to study the mutual impact of power flow control and demand side response on generation dispatch and the resulting electricity prices. Simultaneous usage of DR and FACTS for congestion management has been proposed in [5], in which the total market cost is analyzed by exerting penalties and rewards on consumers in addition to the usage of FACTS. The present paper studies the behavior of elastic loads and the overall system performance in the presence of DFACTS in the power system using an iterative central optimization approach in which both load response and the optimal settings of the DFACTS are calculated in each iteration. The rest of the paper is structured as follows: DFACTS and demand response modeling approaches are discussed in Section II. Section III presents the problem formulation and the proposed market clearing procedure. Section IV describes the test case system and presents simulation studies. Finally, Section V concludes the paper. II. MODELING This section provides models for power flow control devices and demand response behavior to be used in the sequel. A. DFACTS Modeling Two types of DFACTS are investigated: Distributed Series Reactance (DSR) and Distributed Series Impedance (DSI) [7]. Both can be modeled as variable reactances X ij in series Fig.. X ij DFACTS modules on power line. Xij with the line reactance X ij as shown in Fig.. DSRs can steer power from the line in which they are placed by increasing the reactance of that line, i.e. with X ij,tot = X ij + X ij,dsr () 0 X ij,dsr X ij (2) As a result, the power will be diverted to the less congested areas of the system. DSIs, in addition to increasing the line reactance, are also capable of decreasing its reactance, with X ij,tot = X ij + X ij,dsi (3) X ij X ij,dsi X ij (4) thus being able to draw power to the line in question from more congested areas of the network. B. Demand Response Modeling The DR modeling approach used in this paper is adopted from [9]. Here, the derivation of the demand as a function of its elasticity E and the market price ρ is summarized. The reader is referred to [9] for more details. The authors start by defining consumer elasticity E as E = D ρ ρ0 D 0 (5) and the hourly demand side s profit from electricity consumption as S(D) = B(D) D ρ (6) where B(D) denotes customers benefit as a function of consumption. Hence, the maximization of the flexible load s profit is achieved by S D = B(D) D ρ = 0 (7) Taking into account (5) - (7), it further can be concluded that 2 B D 2 = ρ D = E ρ0 (8) D 0 The authors further discuss that as the flexible load reduces its normal consumption D 0 to D, the benefit function will changes as a consequence of the consumer s elasticity and the market price. This change can be modeled by using Taylor Series expansion of the consumer s benefit function around D 0 : B(D) = B(D 0 ) + (D D 0 ) B(D 0) D B(D 0 ) D 2 (D D 0 ) 2 (9)

3 Using (7) and (8), the following simplified benefit function is derived: B(D) = B(D 0 ) + (D D 0 ) ρ ρ 0 D 0 E (D D 0) 2 (0) By taking the derivative of (0) and using (7), it follows: ( B(D) D = ρ = ρ 0 + D D ) 0 () E D 0 and finally ( D = D 0 + E (ρ ρ ) 0) ρ 0 (2) which corresponds to a behavior model of the flexible load in response to the price signal. III. PROBLEM FORMULATION An economic dispatch taking into account line constraints using a DC power flow (DCED/DCOPF) is used to determine the consumption levels, the electricity price and the settings of the DFACTS devices. Hence, the problem formulation is given by s.t. P k D k (ρ) = P kn N G min (a i Pi 2 + b i P i + c i ) (3) i= n Ω k X kn,tot (θ k θ n ), k =,..., N B (4) X kn,tot (θ k θ n ) P kn, (k, n) Ω L (5) X kn X kn X kn, (k, n) Ω L (6) where X kn is the variable reactance introduced by DFACTS, D k (ρ) is the demand side s consumption function as given in (2), Ω k are the buses to which bus k is connected and Ω L are the lines in the system. If no generation and/or no load is connected to bus k, P k and/or D k for that bus are equal to zero. It should be noted that, in our formulation the flexible demand side is assumed to respond to the electricity price based on its elasticity. As DFACTS affect the market price, the demand side s level of consumption will be modified in response to price signals. On the other hand, the resulting elastic load s response may modify the line loading and further induce a new set of DFACTS settings, hence, we use the iterative approach depicted in Fig. 2 to solve the optimization problem given by (3)-(6). In the iterative procedure, the demand initially predicts the resulting electricity price ρ 0 and sets its consumption level D 0. The system operator will then clear the market using the DCED/DCOPF formulation (3)-(6) for this fixed load. After the first iteration of implementing the DCED/DCOPF, if the calculated electricity price ρ differs from the demand side s predicted price, the demand will alter its level of consumption using (2). The optimization problem will be solved again Fig. 2. YES End Estimate LMP Set Demand 0 D0 Set Iteration q = Solve DCED/DCOPF => DFACTS settings Xkn and LMPs q q q- Market Clearing Procedure. ε? NO Update load D=D( q) Set Iteration q=q+ with the updated consumption. This process continues until the difference between demand side s estimated prices, which the consumption is based on, and calculated price becomes less than a certain threshold ǫ. An iterative process also has been employed in [2] and [5] to integrate DR into the generation scheduling problem. However, in [2] the responsive load s behavior is modeled simply using the definition of self-elasticity. The authors of [5] use an iterative procedure to handle congestion by exerting penalties and setting incentives for demand side participation in addition to using FACTS but the incentives and penalties are assumed as predetermined values. In the other words, although the discussed incentives and penalties play an important role in power flow control, they are not considered as optimization variables. A. Test System IV. SIMULATION RESULTS The IEEE 4-bus test system shown in Fig. 3 is used to investigate the mutual impacts of demand side response and power flow control devices. The cost parameters chosen for the generators are given in Table I. The simulations are implemented in MATLAB and solved using the KNITRO solver of Tomlab. The KNITRO solver is an interior point method package that uses trust regions method to improve convergence [3]. Two cases with regards to the initial estimates of the prices are considered: The first case assumes that the consumer has over-predicted the price initially, whereas, the second set of simulations concerns the case of under-prediction of the initial price. In each case the following scenarios are considered: Scenario I: no responsive load, no DFACTS Scenario II: no responsive load, DSRs

4 that, as the DSI increases the transmission system s capacity more than DSR, the difference between the predicted price and the calculated price increases with usage of DSI. This, in turn implies that the system is able to support more load in the presence of DSI System s Load (MW) Fig. 3. Modified IEEE 4-bus test system. Fig. 4. Test System s Total Load-Over Predicted Price. Scenario III: no responsive load, DSIs Scenario IV: responsive load, no DFACTS Scenario V: responsive load, DSRs Scenario VI: responsive load, DSIs TABLE I COST PARAMETERS FOR GENERATORS Bus a i b i c i G G LMP($/MWh) Bus Bus2 Bus3 Bus4 Bus5 LMPs of the Selected Buses Scenario I serves as the reference case. The system load is equal to the initial guess. In particular, the paper does not consider the DFACTS placement, and DFACTS are assumed to be installed on all the lines. Moreover, line impedance are allowed to change by 0% from their actual values. B. Over-Predicted Electricity Price Since in the case of over-predicted price, the final Local Marginal Price (LMP) is less than the initial predicted price, the level of consumption will increase over the iterations. The impacts of DFACTS on the system operation differ based on their capability to change the line reactances. DSI can improve the transfer capacity of the transmission system more than DSR, due to the fact that it can decrease the reactance of uncongested lines while increasing the reactance of congested lines. The total load and LMPs for all the scenarios in the case of initially over-predicted prices are shown in Figs. 4 and 5, respectively. As expected, system load will increase in response to the difference between the predicted price and the calculated LMP. As the transfer capacity of the system increases with the usage of DFACTS, the price difference will increase. Thus, the system is eventually able to support more load as compared to the case with the absence of DFACTS (Scenario IV). Comparing the final system load in Scenario VI with respect to that in the Scenario V, it may be concluded Fig. 5. Generation Cost ($) Fig LMPs of Buses-Over Predicted Price Generation Cost-Over Predicted Price. The total generation cost for various scenarios are shown in Fig. 6. Comparing the system operation for scenarios I, II and III, it can be concluded that the generation cost is decreasing with the usage of DFACTS in the system, due to the fact that these devices enhance the transfer capacity of the system. So, the cheaper generation unit may be committed more. However, as the system load increases with the usage of DFACTS in presence of flexible load, generation units output and total generation cost increases (compare Scenarios IV, V and VI). Table II presents the social welfare improvement for the studied scenarios with respect to the reference case. Social

5 TABLE II SOCIAL WELFARE IMPROVEMENT-OVER PREDICTED PRICE Percentage of Social welfare improvement Scenario II 7.77% Scenario III 8.35% Scenario IV 0% Scenario V 7.85% Scenario VI 8.48% welfare is defined as the difference between demand side s benefit and generation cost. Thus, the social welfare improvement can be calculated by comparing the generation cost and demand side s benefit variation as given in (0) with respect to the reference scenario. Based on this table, the use of DFACTS increases the social welfare. In other words, although generation cost is increasing for these cases, demand side is consuming more power at a lower price. C. Under-Predicted Electricity Price In this case, the final LMPs are higher than the initially predicted prices of the consumer. Therefore, the level of consumption will decrease over the iterations. Fig. 7 shows the resulting prices. Increasing the transfer capacity of the transmission system by using DFACTS, the system LMPs will decrease. Again, the impacts of DFACTS on the system operation differ based on their capability to change the line reactance. By using DFACTS more power can be bought from the cheapest generator in the system, and LMPs of the load buses decrease. Hence, the difference between the calculated and the predicted LMP reduces which leads to an increased consumption compared to the case where there is no DFACTS installed in the system as shown in Fig. 8. Figure 9 shows the generation cost for the various scenarios. The generation cost increases with the usage of DFACTS in the presence of flexible load (Scenarios IV, V and VI) as a result of the load growth shown in Fig. 8. Table III presents the social welfare improvement for the studied scenarios with respect to the reference case. According to this table, the use of DFACTS increases the social welfare. LMP($/MWh) Fig Bus Bus2 Bus3 Bus4 Bus5 LMPs of the Selected Buses LMPs of Buses-Under Predicted Price. V. CONCLUSIONS In this paper, we have investigated the behavior of flexible loads in the presence of two types of DFACTS devices in System s Load (MW) Fig. 8. Generation Cost ($) Fig Test System s Total Load-Under Predicted Price. Generation Cost-Under Predicted Price. TABLE III SOCIAL WELFARE IMPROVEMENT - UNDER PREDICTED PRICE Percentage of Social welfare improvement Scenario II 7.77% Scenario III 8.35% Scenario IV 0.7% Scenario V 8.07% Scenario VI 8.4% the electric power system. We have introduced an iterative procedure for minimizing an economic dispatch objective that incorporates line constraints and further takes into account the price dependent behavior of loads. Specifically, to cope with the nonlinear dependence of the flexible demand on the price, our method involves successive (iterative) adjustments of consumption profiles at the responsive loads based on the difference between the price estimated by the loads and the price resulting from the economic dispatch. For numerical studies, we have considered both the scenarios in which the consumer initially over-predicts the price and the case in which the initial price is under-predicted. For each of these we have simulated six different scenarios corresponding to whether the demand is responsive or not, and the nature of the DFACTS being deployed, i.e., DSRs, DSIs, or no DFACTS at all. We have observed that in the case of responsive loads the presence of DFACTS lead to reduced locational marginal prices. This finding is consistent with and intuitively a consequence of the fact that DFACTS can potentially increase the network transmission capacity by resolving congestions. Further, we have noted that the above effect of locational marginal price reduction (and increased demand accommo-

6 dation capacity) is more pronounced in the presence of DSIs with respect to DSRs - this may be explained by the ability of the former to introduce both positive and negative reactances into the lines as opposed to the latter which may only induce positive reactance. ACKNOWLEDGMENTS This paper is the result of a term project in the course Smart Grids and Future Electric Energy Systems. We would like to acknowledge the inputs of the course instructor Prof. Marija D. Ilic to this term project. REFERENCES [] R. D. Christie, B. F. Wollenberg, and I. Wangensteen, Transmission management in the deregulated environment, Proc. IEEE, Vol.88, pp.70-95, Feb, [2] K. M. Rogers, Power system control with distributed flexible ac transmission system devices, M.S. thesis, Dept. Electr. Eng., University of Illinois at Urbana-Champaign, IL, [3] A. Kumar, SC. Srivastava, and S. N. Singh, Congestion management in competitive power market: a bibliographical survey, Electr. Power Syst. Res, Vol.76, pp , Sep, [4] E. Bompard, E. Carpaneto, G. Chicco, and G. Gross, The role of load demand elasticity in congestion management and pricing Power Engineering Society Summer Meeting, pp , [5] A. Yousefia, T. T. Nguyena, H. Zareipourb, and O. P. Malik, Congestion management using demand response and FACTS devices, INT J ELEC POWER, Vol.37, pp , May, 202. [6] G. Hug-Glanzmann, and G. Andersson, Coordinated control of FACTS devices in power systems for security enhancement, irep Symposium, pp.-0, Aug, [7] H. Johal, Distributed series reactance: a new approach to realize grid power flow control, PhD dissertation, Dept. Elect. Eng., Georgia Institute of Technology, GA, [8] D. Divan, and H. Johal, Distributed FACTS-a new concept for realizing grid power flow control, IEEE Trans. Power Electron, pp , June, [9] H. Aalami, G. R. Yousefi, and M.P. Moghadam, Demand response model considering edrp and tou programs, IEEE/PES. T&D conference, pp. - 6, [0] J. Mohammadi, H. Ghasemi, J. Saebi, and M. K. Sheikh-el-Eslami, Using responsive loads as a tool for congestion management and system loss reduction, EnergyCon, pp , 200. [] F. Rahimi, and A. Ipakechi, Demand response as a market resource under the smart grid paradigm, IEEE Trans. Smart Grid, Vol., pp , June, 200. [2] D. S. Kirschen, G. Strbac, P. Cumperayot, and D. de Paiva Mendes, Factoring the elasticity of demand in electricity prices, IEEE Trans. Power Syst, Vol. 5, pp , May, [3] TOMLAB/KNITRO User s Guide. Available