Probabilistic Model of Payment Cost Minimization Considering Wind Power and Its Uncertainty

716 IEEE TRANSACTIONS ON SUSTAINABLE ENERGY, VOL. 4, NO. 3, JULY 2013 Probabilistic Model of Payment Cost Minimization Considering Wind Power and Its Uncertainty Yao Xu, Student Member, IEEE, QinranHu, Student Member, IEEE, and Fangxing Li, Senior Member, IEEE Abstract The penetration of wind energy sources to power systems has significantly increased in recent years. With variable and uncertain wind power output, the payment and market-clearing price (MCP) may vary in different cases. In this paper, a methodology to quantitatively model the payment cost minimization (PCM) considering the effects of wind power from a probabilistic viewpoint is presented. The autoregressive moving average (ARMA) method with normal distribution of wind forecast error is used to model a time series of wind speed. Based on the wind turbine power curve, the probability distribution of wind power output can be obtained. Then, Monte Carlo simulation (MCS) is used to produce random samples of wind speed, and the genetic algorithm is applied to solve PCM for each sample. The proposed methodology and its solution are verified with simulation studies of two sample systems. The probabilistic distribution results can give consumers an overview of how much they should pay in a probabilistic sense. Further, the simulation results can serve as a lookup table to provide useful input for more refined unit commitment, and also provide a benchmark for future research works on PCM considering wind power. Index Terms Bid cost minimization (BCM), deregulated electricity market, genetic algorithm, market clearing price (MCP), Monte Carlo simulation (MCS), normal distribution, payment cost minimization (PCM), renewable energy. I. INTRODUCTION I N THE U.S., deregulated wholesale electricity markets are operated by independent system operators (ISOs). Generally, ISOs adopt bid cost minimization (BCM) to select generation offers, demand bids and their respective amounts for energy and ancillary services. A settlement mechanism is then used to calculate the payments based on market clearing prices (MCPs). Therefore, this BCM mechanism may give rise to higher consumer payment. Previous research works [1] [8] investigated the payment cost minimization (PCM) method as an alternative approach to determine generation dispatches. There are some additional previous works that discussed the PCM model. The PCM using surrogate optimization is introduced in [9], which can be viewed as a simplified, combined unit commitment (UC) and economic dispatch (ED) model, since it Manuscript received April 16, 2012; revised December 10, 2012; accepted January 08, 2013. Date of publication March 07, 2013; date of current version June 17, 2013. This work was supported by the National Science Foundation under Grant ECCS-1001999, by the Engineering Research Center Program of the National Science Foundation and the Department of Energy under NSF Award EEC-1041877, and by the CURENT Industry Partnership Program for financial and facility support in part to complete this research. The authors are with the Department of Electrical Engineering and Computer Science, The University of Tennessee (UT), Knoxville, TN 37996 USA (e-mail: fli6@utk.edu). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TSTE.2013.2242908 includes multi-hour model, startup cost, and minimum generation cost. In the problem formulation process, the MCP is used instead of the locational marginal pricing (LMP) and the line constraints are not considered. Improved methods are discussed in [10] and [11] which present the PCM with transmission capacity using surrogate optimization. Since the line limits are considered, LMP is also introduced in [10] and [11] to address locational price difference and congestions. In [12], the PCM with the transmission capacity constraints and the line losses are considered using the objective switching method. As the primary energy source, coal is widely used in power production, which causes serious concerns related to environment and sustainability. Thus, renewable energy sources have been developed significantly, leading to a high penetration of renewables. Unfortunately, renewable energy, especially wind, tends to be variable and uncertain because the prime mover, i.e., the wind, depends on natural and meteorological conditions [13], [14]. Previous PCM studies have not considered effects of wind power. To study the wind power effects on PCM and BCM, this paper presents a customized genetic algorithm (GA) method to solve the PCM model with a high-penetration variable wind power. An autoregressive moving average (ARMA) time series was used to model wind speed, and normal distribution was used to model wind speed forecast error. Based on wind turbine power curve, the wind power probability distribution can be obtained. With the consideration of wind generation uncertainty, MCP can be obtained in a probabilistic model. Then, the expected value of the minimum payment can be calculated, and generation dispatch can be obtained. The Monte Carlo method is used to produce a series of wind speed, and genetic algorithm is applied to solve PCM in this paper. In short, the paper provides a methodology and its solution to the PCM problem considering wind power and demonstrates that PCM can give rise to less consumer payment than BCM in a probabilistic sense with variable wind power. The probabilistic results can serve as a lookup table to provide useful input for further refinement of UC considering stochastic features, and also provide a benchmark for future research works on PCM considering wind power. This paper is organized as follows. Section II illustrates the problem formulation including how to model wind speed, wind power, the BCM model, and the PCM model. Section III presents the solution methodology using GA for PCM with variable wind power. Section IV presents two case studies for the PJM 5-bus system and the IEEE 118-bus system. Section V concludes the paper. II. PROBLEM FORMULATION In this section, with the effect of the wind energy in power systems, the PCM problem is formulated for day-ahead energy 1949-3029/$31.00 2013 IEEE

XU et al.: PROBABILISTIC MODEL OF PCM CONSIDERING WIND POWER AND ITS UNCERTAINTY 717 market. Usually, as shown in the PCM model in [9] [12], a multi-hour optimization model is employed as a combined UCED formulation. Although the actual practice in UC and ED takes more complicate processes such as multiple rounds of optimization, the multi-hour optimization model in the previous PCM and BCM studies in [9] [12] addresses multiple hours scheduling, startup cost, minimum generation cost, and thus, can be viewed as a simplified and integrated UCED problem. This paper takes the basic optimization model in [9] as the foundation for the proposed probabilistic study of the PCM and BCM models under high penetration of wind energy. In addition, spinning reserve is considered in this work. Since both PCM and BCM are based on the same UCED formulation, the comparison is fair and meaningful. A. Wind Speed Model Time Series and Forecast Error Distribution Wind energy tends to be variable and uncertain because of the effects of the natural and meteorological conditions. In addition, wind speed at a specific hour is related to previous hours. To model wind speed, an ARMA [15] [17] time series was used. The general ARMA model is given as follows: Fig. 1. Wind turbine power curve. (1) where is the time-series value at time ; ( ) is the autoregressive average of the model; ( ) is the moving average of the model; and [ ] is a normal white noise where NID means normally independently distributed [15] [17]. Based on previous works [18], [19], a normal distribution is applied to the wind speed forecast error in this paper. The model is written as follows: where = wind speed forecast error at time ; denotes normal distribution; = mean value of ; =varianceof ; = probability density function of ; = cumulative density function of. Then, the hourly simulated wind speed can be obtained based on, and B. Wind Power Output Model Wind power output from a wind turbine is determined by the wind speed. In order to build the wind power model, the relationship between wind speed and wind power must be considered. The classic wind power curve [20] is shown in Fig. 1. (2) (3) (4) Fig. 2. Wind speed and the projected wind power probability distribution around 10 m/s. The wind power curve shows strong nonlinearity between power output and wind speed. Based on the wind forecast and the wind turbine power curve, the wind power probability distribution model can be easily obtained. For example, with the assumption that the wind speed forecast is 10 m/s (i.e., ) and wind speed forecast error follows normal distribution with, then the projected probability distribution of wind power can be obtained from the simulated values of wind speed using the wind turbine power curve. The probability distribution results are shown in Fig. 2. Therefore, the probabilistic model of wind power over a duration like 24 hours can be obtained as follows: 1) Obtain the average wind speed at each time spot (e.g., hourly) based on historical meteorological data. 2) Use the ARMA model in (1) (4) to generate samples which stand for the sequence of hourly wind speed. 3) Combine the results in Step 2 with the power curve to calculate the corresponding sequences of wind power. Thus, a number of random samples of hourly sequential wind power output can be obtained. Note, the random samples of hourly sequential wind power output will serve as the input for Monte Carlo simulation (MCS), which is commonly employed to capture the stochastic distribution of a system s output, which will be the consumer s payment in this paper, as discussed in Sections III and IV.

718 IEEE TRANSACTIONS ON SUSTAINABLE ENERGY, VOL. 4, NO. 3, JULY 2013 C. Needed Number of Monte Carlo Samples The number of Monte Carlo samples should be sufficient enough to obtain the output distribution, but not more than a necessary number. Otherwise, it is a waste of computing time. The number of samplings can be obtained by setting criterion for the MCS. According to [15] [17], the stopping criterion is shown mathematically as follows: where is the output variable from the MC simulation (i.e., the consumer s payment as discussed in the next section); is the number of the Monte Carlo samples; is the expected value of ; is the standard variation of ;and is a specific convergence threshold. If the specific convergence threshold is determined, the least number of samples for MCS can be calculated. If is small, then is large, which leads to more computation or simulation time to maintain a good enough accuracy; vice versa for a larger value of. Therefore, in real-world practice, a reasonable convergence threshold needs to consider trade-off between accuracy and computing time. In this paper, the approach to obtain a reasonable number of samples is shown as follows. 1) Set an initial guess of, e.g., 100. 2) An MCS run with 100 trials is processed. 3) The statistical results from these 100 MCS, i.e., and, can be used to verify whether (5) is satisfied. If so, then stop and return with all results; otherwise, a new, minimum can be calculated with and from the previous 100 samples based on (5). 4) The whole process in Steps 1 3 will be repeated with additional samples based on the updated until (5) is met. D. Conventional Bid Cost Minimization (BCM) Model The conventional BCM problem can be modeled as the minimization of the total production cost subject to energy balance. Again, transmission constraint is ignored in this work. Considering startup cost, minimum generation cost, and spinning reserve in the 24-hour duration, we have the BCM modeled formulated as follows: (5) where = number of loads; = number of traditional generators; = number of wind plants; = bid price of traditional generator ($/MWh); = cost at minimum generation ($); = bid price of wind plant ($/MWh); = output of traditional generator at time ; = output of wind plant at time ; = maximum output of traditional generator ; = minimum output of traditional generator ; = power of load at time ; = start up cost of traditional generator ; = spinning reserve at hour ; = 1/0: generator is ON/OFF at time ; = 1/0: wind plant is ON/OFF at time. After obtaining the optimal solution of generation dispatch, we can calculate the MCP, which is the maximum bid price of selected generators. is defined as Then, the total payment is given by (7) (8) (9) (10) (11) In (11), and are known based on the solution of (6) (9), and can be obtained with (10). (6) E. Payment Cost Minimization (PCM) In the PCM model, the consumers payment is directly minimized. With the consideration of the startup cost, minimum gen-

XU et al.: PROBABILISTIC MODEL OF PCM CONSIDERING WIND POWER AND ITS UNCERTAINTY 719 eration cost, and spinning reserve in the 24-hour duration, the PCM model can be formulated as (12) Subject to: (7) (9). Note that PCM is also subject to (7) (9), the energy balance equation, and the generation upper/lower bounds. If compared with the BCM model in (6) (9), the problem formulation of PCM is much more complicated, because,,and are unknown in (12). In the BCM model, the generation dispatch can be obtained from (6) (9) first, and then the can be obtained afterwards using (10). This is a sequential process. In contrast, in (12) is not explicitly known since the dispatch is not known. Therefore, the original linear integer programming solution applied to the PCM model in (6) (9) is not suitable. In Section III, the genetic algorithm is presented to solve the PCM model in (12), subject to (7) (9). F. Expected Value of Payment The variable wind output power gives variable MCP for different PCM scenarios. Based on the probabilistic distribution of wind generation output, a Monte Carlo method is applied to generate a sequence of random wind power output of 24 hours based on our wind speed model. By generating different cases, we can obtain different values of payment for both PCM and BCM as,. Note each of these cases contains 24 hours. Then, we can simply use the mean value of cases as the expected value (13) (14) Based on the expected value of the PCM objective function, i.e., payment, the total optimal customer payment with the effects of the renewable energy can be calculated, and the actual multi-hour generation scheduling results can be obtained. III. SOLUTION METHODOLOGY Essentially, the proposed optimization problem is to find the optimal dispatch of generators to minimize PCM value under certain scenarios of wind power output. This problem is an NP problem with exponential time. In order to solve this problem, the MCS method is used to produce a series wind speed, and then the genetic algorithm (GA) is applied to solve PCM in each MCS sample. GA, an intelligent search algorithm using stochastic operations, is customized in this work to solve the model given by (12), (7) (9). Because the PCM optimization model is a complicated nonlinear, noncontinuous problem with in the objective function depending on other unknown variables and in the objective function, it is reasonable to apply the heuristic search algorithm GA here for solving PCM. Starting with randomly generated solutions, GA searches the improved solutions through a number of iterations called generations. The performance of each solution is evaluated by a fitness function. In each generation, relatively good solutions will be selected for reproduction of offspring by crossover and mutation. Crossover combines materials from parents to produce their children. Crossover provides pressure for improvement or exploitation while mutation makes small local changes to feasible solutions in order to provide the variability of the population. The reproduction cycle goes on until there is no further improvement in consecutive generations or the maximum number of iterations is met [21]. This heuristic searching approach has been applied to solve optimization problems in power systems [22] [25]. As discussed in Section II, (7) (9) and (12) are integrated to the functions for each random MCS sample as follows: (15) (16) With the application of GA, we can find out which generator to dispatch and the amount of the generator output in time. Therefore, the known parameters here are,,,,,and ; and the unknown parameters are,,and in the proposed formulation. Next, the customized GA for PCM is discussed in details. A. Initial Solution Generation In the first step of this algorithm, an initial set is randomly generated to form an initial population. Since in some cases the generated sets are obviously infeasible under practical consideration, the bad sets should be detected and deleted. The criterionusedhereinthisparticular GA application to ensure a feasible solution is given by (17)

720 IEEE TRANSACTIONS ON SUSTAINABLE ENERGY, VOL. 4, NO. 3, JULY 2013 Fig. 3. Structure of one individual. Fig. 4. Structure of whole population. Once infeasible cases are identified, they will be removed immediately and new ones will be regenerated to replace them. This can be repeated until all the population are feasible. B. Population Structure The structure of an individual in a population is shown in Fig. 3. Each individual records all generators statuses in 24 hours, which are noted as in our problem formulation. Thus, 24 bits can represent one individual, where is the number of GA generations. Since we have already defined the structure of an individual, the structure of the whole population in each iteration is shown in Fig. 4. Here, the population size is. C. Fitness Function In order to achieve the objective function in (15), the fitness function that stands for the probability of being chosen can be formulated as follows: Fig. 5. Flowchart of GA for PCM. (18) where is the value of fitness for the th solution in population and is the population size. With an initial solution, the solution structure, and the fitness function, the optimal solution for generation scheduling can be obtained based on crossover and mutation. Since this paper uses the standard process for crossover and mutation, which has been well documented in the literature, the detailed description on these two steps is ignored here. D. Flow Chart of GA for PCM The flowchart of GA for PCM is shown in Fig. 5. Note is the iteration count in the figure. Also note that this flowchart is for one random sample of a wind speed series. Then, MCS, which is the computational algorithm based on repeated random sampling to model stochastic events, is used here to calculate the PCM values with randomly generated wind speed samples, as discussed in Sections II-A and II-B. Also, the needed number of MC samples has been discussed in the Section II-C. IV. NUMERICAL STUDY In this section, two studies based on the PJM 5-bus and the IEEE 118-bus systems are presented. The solutions are implemented with MATLAB. A. PJM 5-Bus System The parameters of the PJM 5-bus system with slight modification are shown in Table I. The parameters of GA are set as below: Generation ; Population Size ; Crossover Rate ;and Mutation Rate. In this system, the demand data for 24 hours are shown in Fig. 6. The spinning reserve is set as.bid1isassumed to be from a wind power plant. Mean value and standard deviation of wind speed at the wind plant of Bid 1 is calculated based on the Regina wind speed data from [17] and [26] during 2007 2012 for demonstration here and is shown in Fig. 7. Based on [17], the value of the MCS stopping criterion given in (5)

XU et al.: PROBABILISTIC MODEL OF PCM CONSIDERING WIND POWER AND ITS UNCERTAINTY 721 TABLE I SUPPLY OFFER PARAMETERS FOR PJM 5-BUS SYSTEM Fig. 8. Monte Carlo hourly wind speed. TABLE II PJM-5 BUS SYSTEM PAYMENT IN BCM: MEDIAN SAMPLE HOURLY VALUES Fig. 6. PJM 5-bus system demand. Fig. 7. Mean value and standard deviation of wind speed at Bid 1. is set to 0.05. After an initial 100 samples, the mean and deviation of the results from these initial samples suggest about 1000 samples for MCS to converge. Note, the MCS method is used to produce 1000 random 24 hours wind speed samples in the simulation. These 1000 cases of hourly wind speed are shown in Fig. 8. For each case, based on the GA algorithm, the optimal generation dispatch and the lowest consumer payment can be obtained. The generation dispatches in 24 hours in the BCM and PCM formulations for a particular sample, the median sample basedonsortedpcmvalues,areshownintablesiiandiii,respectively, for illustrative purpose. Hourly mean values of all samples in BCM and PCM are shown in Tables IV and V, respectively. From Tables II and III, the results represent the generation scheduling under the proposed combined UCED model in PCM and BCM. Moreover, all the results can be used to build a lookup table as the input for a more refined unit commitment study, if we want to apply this to a more practical, detailed UCED model. Based on the results of all random samples of 24 hours wind power output, the discretized probability mass function of and are shown in Fig. 9. The expected values of the payment inpcmandbcmare and, respectively. From Fig. 9, the overall comparison of PCM and BCM with wind power considered matches the general conclusion from the literature that the PCM model gives lower consumer payment than BCM. The methodology presented in this paper is a comprehensive and systematic approach to develop the impact to these two models under probabilistic consideration. The probability distribution results can give the consumer an overview of how much they should pay with an associated probability. The results also verify that PCM is a more economically efficient

722 IEEE TRANSACTIONS ON SUSTAINABLE ENERGY, VOL. 4, NO. 3, JULY 2013 TABLE III PJM-5 BUS SYSTEM PAYMENT IN PCM: MEDIAN SAMPLE HOURLY VALUES TABLE V PJM-5 BUS SYSTEM PAYMENT IN PCM: MEAN HOURLY VALUES TABLE IV PJM-5 BUS SYSTEM PAYMENT IN BCM: MEAN HOURLY VALUES Fig. 9. Probability distribution of PCM and BCM for PJM 5-bus system. mechanism for consumers under a high-penetration of variable wind generation. B. IEEE 118-Bus System The study results on the IEEE 118-bus system are brieflypresented in this subsection to further demonstrate the applicability of the proposed methodology, while similar conclusions to those for the PJM 5-bus system can be drawn. The IEEE 118-bus system has 54 generators. In the original IEEE 118-bus system, there is no generator bidding data, which

XU et al.: PROBABILISTIC MODEL OF PCM CONSIDERING WIND POWER AND ITS UNCERTAINTY 723 TABLE VI GENERATOR STARTUP COST FOR IEEE 118-BUS SYSTEM Fig. 11. Probability distribution of PCM and BCM for IEEE 118-Bus Study. than the BCM model. The presented PCM model and the solution methodology are able to be applied in a larger system and are robust models for the optimal generator dispatches. Again, while the result itself represents the generation scheduling output under the combined UCED optimization framework, it can serve as a lookup table to provide useful input for a more refined UC considering the stochastic nature of wind power when applied to actual practice. Fig. 10. IEEE 118-bus system demand. are indispensable to perform the economic study here. Therefore, the generator bidding data per MWh are assumed as follows for illustrative purpose [27]: 20 low-cost generators with bidding from $10.0 to $19.5 with $0.5 increments; 20 expensive generators with bidding from $30 to $49 with $1 increments; and 14 most expensive generators with bidding from $70 to $83 with $1 increments. The startup cost is shown in Table VI. In the simulation,. The generators at 10, 26, and 40 are set as wind units with no startup cost. Note that other units with a low to zero startup cost may represent a gas or combined cycle unit, while a high startup cost unit may represent a steam turbine unit. In this system, the demand data for 24 hours are shown in Fig. 10. The spinning reserve is set as. Similar to the PJM 5-bus system study, the mean value and the standard deviation of wind speed at Bid 10, 26, 40 are obtained from actual operational data set as Regina, Saskatoon, and Swift Current wind speed in [17] and [26]. Basedonthedistributionofwindgenerationindifferent cases, the discretized probability mass functions of and are shown in Fig. 11. The expected values of the payment are with BCM and with PCM, respectively. It is apparent that the expected value of the payment from the PCM model in this case gives much lower consumer payment V. CONCLUSION With the development of the wind power in recent years, the effects of the variable wind power on the power market operation becomes an increasingly hot topic. Meanwhile, a number of papers addressed the BCM and PCM models in deregulated power markets. However, few have discussed the PCM model with the effects of the wind energy. This challenge has been addressed in this paper. The contribution of this paper can be summarized as follows: In this paper, a methodology to quantitatively model PCM considering the effects of the wind power from the probabilistic viewpoint has been discussed. The ARMA method with normal distribution of wind forecast error was used to model a time series of wind speed. Based on wind turbine power curve, the wind power probability distribution can be obtained. The MCS method was used to produce randomly generated series of wind speed data, and the genetic algorithm was applied to solve PCM for each sample. The probabilistic distribution results, obtained with the proposed methodology, can give consumers an overview of how much they probably should pay in the probabilistic sense. While the result itself represents the generation scheduling output under the simplified, combined UCED model, it can serve as a lookup table to provide useful input for a more refined UC to consider stochastic features when applied to actual practice of UC with wind power. Since both PCM and BCM take the same UCED model, the comparison is fair and meaningful. Future work may include transmission constraints considering network model as well as the probabilistic characteristics

724 IEEE TRANSACTIONS ON SUSTAINABLE ENERGY, VOL. 4, NO. 3, JULY 2013 of load variation, especially under the smart grid initiatives for demand response. ACKNOWLEDGMENT The authors are grateful to Prof. P. B. Luh from the University of Connecticut and Dr. J. H. Yan from Southern California Edison for their useful discussions on PCM. REFERENCES [1] C. Vazquez, M. Rivier, and I. J. Perez-Arriaga, Production cost minimization versus consumer payment minimization in electricity pools, IEEE Trans. Power Syst., vol. 17, no. 1, pp. 119 127, Feb. 2002. [2] S. Hao, G. A. Angelidis, H. Singh, and A. D. Papalexopoulos, Consumer payment minimization in power pool auctions, IEEE Trans Power Syst., vol. 13, no. 3, pp. 986 991, Aug. 1998. [3] F.Zhao,P.B.Luh,Y.Zhao,J.H.Yan,G.A.Stern,andS.Chang, Bid cost minimization vs. payment cost minimization: A game theoretic study of electricity markets, in Proc. Power Eng. Soc. General Meeting, Tampa, FL, USA, Jun. 2007. [4] J. Alonso, A. Trias, V. Gaitan, and J. J. Alba, Thermal plant bids andmarket clearing in an electricity pool: Minimization of costs vs. Minimization of consumer payments, IEEE Trans. Power Syst., vol. 14, no. 4, pp. 1327 1334, Nov. 1999. [5]J.H.YanandG.A.Stern, Simultaneousoptimalauction and unit commitment for deregulated electricity markets, Electricity J., vol. 15, no. 9, pp. 72 80, Nov. 2002. [6]G.A.Stern,J.H.Yan,P.B.Luh,andW.Blankson, What objective function should be used for simultaneous optimal auctions in the ISO/RTO electricity market, in Proc. IEEE Power Eng. Soc. General Meeting, Montreal, Canada, 2006. [7] J. H. Yan, Bid cost minimization versus payment cost minimization in the ISO/RTO markets, in Proc. IEEE PES General Meeting, Tampa, FL, USA, Jun. 2007. [8] S. Hao and F. Zhuang, New models for integrated short-term forward electricity markets, IEEE Trans. Power Syst., vol. 18, no. 2, pp. 478 485, May 2003. [9] P.B.Luh,W.E.Blankson,Y.Chen,J.H.Yan,G.A. Stern, S.-C. Chang, and F. Zhao, Payment cost minimization auction for deregulated electricity markets using surrogate optimization, IEEE Trans. Power Syst., vol. 21, no. 2, pp. 568 578, May 2006. [10] F. Zhao, P. B. Luh, J. H. Yan, G. A. Stern, and S.-C. Chang, Payment cost minimization auction for deregulated electricity markets considering transmission capacity constraints, in Proc. IEEE ISAP, 2005, pp. 470 475. [11] F.Zhao,P.B.Luh,J.H.Yan,G.A.Stern,and S.-C. Chang, Payment cost minimization auction for deregulated electricity markets with transmission capacity constraints, IEEE Trans. Power Syst., vol.23, no. 2, pp. 532 544, May 2008. [12] X. Han, P. B. Luh, J. H. Yan, G. A. Stern, and S.-C. Chang, Payment cost minimization with transmission capacity constraints and losses using the objective switching method, in Proc. IEEE PSCE, 2011, pp. 532 544. [13] S. Mathew, Wind Energy, Fundamentals, Resource Analysis and Economics. Berlin, Germany: Springer-Verlag, 2006. [14] S. Heier and R. Waddington,GridIntegration of Wind Energy Conversion Systems, 2nd ed. Hoboken, NJ, USA: Wiley, 1998. [15] R. Billinton, H. Chen, and R. Ghajar, A sequential simulation technique for adequacy evaluation of generating systems including wind energy, IEEE Trans. Energy Convers., vol. 11, no. 4, pp. 728 734, Dec. 1996. [16] R. Billinton and G. Bai, Generating capacity adequacy associated with wind energy, IEEE Trans. Energy Convers., vol. 19, no. 3, pp. 641 646, Sep. 2004. [17] W. Wangdee and R. Billinton, Considering load carrying capability and wind speed correlation of WECS in generation adequacy assessment, IEEE Trans. Energy Convers., vol. 21, no. 3, pp. 734 741, Sep. 2006. [18] H. Mori and E. Kurata, Application of Gaussian process to wind speed forecasting for wind power generation, in Proc. IEEE Int. Conf. Sustainable Energy Technologies, 2008 (ICSET 2008), Nov. 2008, pp. 956 959. [19] M. Lange, On the Uncertainty of Wind Power Prediction, analysis of the forecast accuracy and statistical distribution of errors, J. Solar Energy Eng., vol. 127, no. 2, pp. 177 184, May 2005. [20] General Specification Vestas, V90-3.0MW, May 2011 [Online]. Available: http://www.gov.pe.ca/photos/sites/envengfor/file/ 950010R1_V90-GeneralSpecification.pdf [21] M. C. Chan, C. C. Wong, B. K. S. Cheung, and G. Y. N. Tang, Genetic algorithms in multi stage portfolio optimization systems, in Proc. 8th Int. Conf. Society for Computational Economics, Computing in Economics and Finance, Aix-en-Provence, France, 2002. [22] S. O. Orero and M. R. Irving, Economic dispatch of generators with prohibited operating zones: A genetic algorithm approach, IEE Proc. Gen., Transm., Distribut., vol. 143, no. 6, pp. 529 534, 1996. [23] F. Li, J. D. Pilgrim, C. Dabeedin, A. Chebbo, and R. K. Aggarwal, Genetic algorithms for optimal reactive power compensation on the national grid system, IEEE Trans. Power Syst., vol. 20, no. 1, pp. 493 500, Feb. 2005. [24] C.-P. Cheng, C.-W. Liu, and C.-C. Liu, Unit commitment by Lagrangian relaxation and genetic algorithms, IEEE Trans. Power Syst., vol. 15, no. 2, pp. 707 714, May 2000. [25] E. Zio, P. Baraldi, and N. Pedroni, Optimal power system generation scheduling by multi-objective genetic algorithms with preferences, Reliab. Eng. Syst. Saf., vol. 94, pp. 432 444, Feb. 2009. [26] National Climate Data and Information Archive, [Online]. Available: http://www.climate.weatheroffice.gc.ca/climatedata/canada_e.html last accessed Jul. 19, 2012 [27] R. Bo and F. Li, Probabilistic LMP forecasting considering load uncertainty, IEEE Trans. Power Syst., vol. 24, no. 3, pp. 1279 1289, Aug. 2009. Yao Xu (S 11) received the B.S. and M.S. degrees in electrical engineering from Changsha University of Science and Technology, China,in2006and2009,respectively. She is working toward the Ph.D. degree at The University of Tennessee, Knoxville, TN, USA. Her research interests include utility application of power electronics, power system control, and power markets. She received the Min Kao Graduate Fellowship in Electrical Engineering. Qinran Hu (S 11) received the B.S. degree in electrical engineering from Southeast University, Nanjing, China, in 2010. He is working toward the Ph.D. degree at The University of Tennessee, Knoxville, TN, USA. His research interests include power system optimization, smart grid, and renewable energy integration. Fangxing Li (M 01 SM 05), also known as Fran Li, received the BSEE and MSEE degrees from Southeast University, Nanjing, China, in 1994 and 1997, respectively, and then the Ph.D. degree from Virginia Tech in 2001. He is an Associate Professor at The University of Tennessee, Knoxville, TN, USA. He had been a senior and then a principal engineer at ABB Electrical System Consulting (ESC) in Raleigh, NC, USA, from 2001 to 2005, prior to joining UTK in August 2005. His current interests include renewable energy integration, power markets, distributed energy resources, and smart grid. Dr. Li is a registered Professional Engineer (P.E.) in the state of North Carolina, and a Fellow of IET (formerly IEE).