AS the penetration of wind power in power systems

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1 1 Robust Bidding Strategy for Wind Power Plants and Energy Storage in Electricity Markets Anupam A. Thatte, Student Member, IEEE, Daniel E. Viassolo, Member, IEEE, andlexie,member, IEEE Abstract This paper explores a robust optimization-based bidding strategy for operating a wind farm in combination with energy storage devices in electricity markets. Through coordination with moderate capacity of energy storage, variable wind resources can be utilized in multi-time-scale electricity market operations, as opposed to being utilized only as real-time nondispatchable energy producers. Given the inherent uncertainties in electricity market prices and available wind generator output, a robust optimization-based approach is formulated to determine the bidding strategy. Case studies on day-ahead and hour-ahead markets show that robust-optimization based bidding strategy provides computationally practical and economically efficient approach to operating wind farms and co-located storage when uncertainties are severe. Index Terms Electricity market, robust optimization, wind power, energy storage, virtual power plants. I. INTRODUCTION AS the penetration of wind power in power systems increases, the variable and uncertain nature of the wind resource poses challenges to electricity grid operations. Despite improvements in forecasting methods it is difficult for system operators to dispatch wind generators as they dispatch conventional generators. Storage technologies can help in firming the output of wind generation and provide benefits to the system over different time scales. As an example, storage can help in exploiting arbitrage opportunities due to temporal variations in electricity prices over a duration of several hours [1]. In addition, fast acting storage technologies can allow wind generators to participate in ancillary services such as frequency regulation [2]. Improved utilization of wind energy can help in improving the operational economic performance of wind generation. Researchers have proposed techniques to coordinate the power output of wind generation with energy storage devices. Reference [3] considers a combined wind and pumped storage facility and determines the optimal operational strategy based on deterministic linear optimizations for scenarios generated using a Monte Carlo simulation approach. The coordination of wind and flywheels for energy balancing and frequency regulation has been proposed in [4]. A dynamic programming algorithm for optimal scheduling of the combination of wind with a generic energy storage device is presented in [5]. Reference [6] formulates the joint optimization of a wind This work was supported in part by Vestas Technology R&D. A. A. Thatte and L. Xie, are with Department of Electrical and Computer Engineering, Texas A&M University, College Station, TX, USA. s: thatte@tamu.edu, lxie@ece.tamu.edu. D. E. Viassolo is with Energy Systems Department, Vestas Technology R&D, Houston, TX, USA. davia@vestas.com farm and a pumped-storage facility as a two-stage stochastic programming problem. In fact there are many examples of the stochastic programming approach applied to power systems to deal with uncertainty [7] [9]. However the stochastic programming approach is computationally challenging due to the large number of scenarios that have to be considered. Over the past few years Robust Optimization (RO) has been receiving increasing attention from researchers in electric power system operations. The main feature of the RO approach is that it uses a non-probabilistic approach to deal with the uncertainty. Uncertainty is addressed by constructing an uncertainty set and the solutions obtained are robust to all realizations of uncertain data within that uncertainty set. This definition of uncertainty leads to a more tractable problem. Furthermore, the uncertainty set can be determined based on historical data or by using a confidence interval. The RO approach has been applied across a variety of domains including portfolio optimization, supply chain management, network flows, circuit design, wireless networks and model parameter estimation [10], [11]. In power systems recently the RO approach has been applied to the unit commitment problem [12]. In this paper, we present a robust optimization-based bidding strategy for the combination of a wind farm and onsite storage in a deregulated electricity market. This bidding strategy takes into account uncertainty in both wind power forecasts and electricity price forecasts. The wind farm is assumed to be a price taker, and leverages the storage for energy arbitrage application. The combination of the wind farm and storage can also be used for other applications such as grid frequency regulation. We present case studies on the energy arbitrage formulation for day-ahead and hour-ahead electricity markets with the objective of maximizing the total profit for a given day. We analyze and compare the performance of the robust optimization approach to the deterministic one, specifically considering the worst case scenario of wind power and price forecast error. The impact of the choice of uncertainty set on the optimality of the result is examined. The rest of this paper is organized as follows. In Section II we discuss the background on the robust optimization approach. In Section III we present the formulations of the bidding strategy for energy arbitrage, using (i) deterministic optimization approach, and (ii) robust optimization approach. In Section IV we present case studies on the energy arbitrage application, compare the robust optimization approach to the deterministic optimization approach and discuss the results. Finally in Section V we present the conclusions and future work /12/$ IEEE

2 2 II. BACKGROUND A. Robust Optimization In the stochastic programming approach knowledge of the probability distribution of the uncertainty is needed. For problems in power systems it may be difficult to accurately estimate the probability distribution of the uncertain variable. Further a large number of scenarios have to be considered in order to get a reasonable guarantee on the solution of the optimization, which leads to large problem size. The robust optimization approach offers a simpler way to deal with the uncertain variable, through the use of an uncertainty set. The objective function is calculated for the worst case of uncertainty as defined by the uncertainty set. Therefore feasibility of the solution can be guaranteed for all realizations of the uncertain data within the uncertainty set. The second advantage is tractability, since for many classes of optimization problems the RO formulation is tractable [10]. Robust optimization solves the following generic min-max problem min max f(x, u) x X u U s.t. g(x, u) 0 (1) where x is a vector of decision variables which belong to set X, f and g are the objective function and constraints respectively, u are the uncertain parameters which take values in the uncertainty set U. However this approach leads to a result that may be too conservative. A method for selecting the uncertainty set based on the decision maker s risk preference is proposed in [13]. This leads to polyhedral uncertainty set for which the robust counterpart of the uncertain linear programming problem is equivalent to a linear optimization problem. The general uncertain linear programming problem is given as min x X ct x s.t. ã T x b (2) where without loss of generality the uncertainty is assumed to affect only the constraint coefficients ã. Every element of the vector ã is assumed to be subject to uncertainty and belongs to a symmetrical interval [â Δa, â+δa], where â and Δa represent the nominal values and deviations respectively. If information about the variance of the uncertain coefficients is available then that information can be used to construct polyhedral uncertainty sets [14]. The polyhedral uncertainty set can be defined as U = { ã i n i=1 ã i â i σ i } Γ where ã i â i = Δa i = α â i, α being a scalar constant in the set [0, 1] which gives the relation of the deviation Δa i to the nominal value â i. σ i is the standard deviation of coefficient a i and Γ is referred to as the budget of uncertainty which is used to adjust the level of conservatism of the solution (3) [15]. When Γ=0the problem reduces to the deterministic case which solves the problem using the nominal values i.e., expected values of the uncertain coefficients. Γ can be adjusted according to the trade-off between decision maker s risk preference and the conservatism of the solution. B. Choice of uncertainty set for price Equation (3) assumes that the standard deviations σ i of the coefficients a i are known. For the case of electricity price, if the covariance matrix Σ of the uncertain coefficients is available then it can be included in the choice of uncertainty set [14]. Reference [16] presents a method to estimate the covariance matrix of day d if the historical data on true prices and estimates is available for d 1 days. Σ est = 1 D D i=1 (Λ true i Λ est i )(Λ true i Λ est i ) T (4) where D is a convenient number of days for which data is available. The vectors Λ true i and Λ est i are the historical true values and estimates of electricity price, respectively. Accordingly the definition of the uncertainty set (3) is updated as follows U = { } λ Σ 1/2 α pˆλ 1 Γ Thus based on the decision maker s choice of Γ a corresponding value of α p is obtained, which leads to the choice of uncertainty set for price as [(1 α p )ˆλ, (1 + α p )ˆλ]. III. FORMULATIONS The bidding strategy for the combined wind and storage unit is formulated as a linear optimization problem with the goal of maximizing the total profit in the deregulated electricity market. The decision on charging or discharging of the energy storage, and output to grid is determined based on forecast of electricity prices, forecast of wind farm power output and technical constraints. The abbreviation DKK is used for Danish Kroner. The nomenclature used is given in Table I. A. Deterministic optimization-based bidding strategy The aim of the energy arbitrage strategy is to leverage the storage to take advantage of temporal variations in the electricity price. This is done by storing energy from the wind farm in the storage device when the price is low and then returning this energy to the grid when the price increases. The objective function and constraints are expressed as follows. min P w(k),p d s (k),p c s (k),(k) (5) N [ ˆλ(k)(P w (k)+ps d (k) Ps c (k)) k=1 + C w P w (k)+c s (P d s (k)+p c s (k)) + C e E s (k)] (6)

3 3 TABLE I NOMENCLATURE ˆλ(k) Forecast price of electricity in time period k (DKK/MWh) P w (k) Power injected by wind farm directly into grid (MW) ˆP w (k) Forecast output of wind farm based on wind speed (MW) Ps d (k) Discharge power of storage device (MW) Ps c (k) Charge power of storage device (MW) E s (k) Energy level of storage device (MWh) max Upper limit on energy level of storage device (MWh) min Lower limit on energy level of storage device (MWh) C w Marginal cost of wind (DKK/MW) C s Charging discharging (degradation) cost (DKK/MW) C e Energy storage cost (DKK/MWh) Pw r Ramping constraint of wind farm (MW/h) N Number of time periods (for one day N = 24) s.t. Pw r P w (k) P w (k 1) Pw r (7) 0 P w (k) ˆP w (k) (8) E min s E s (k) max (9) 0 Ps c (k) Ps max (10) 0 Ps d (k) Ps max (11) E s (k) E s (k 1) (Ps d (k) Ps c (k)) Ps loss (k) ( ˆP w (k) P w (k)) (12) The objective function (6) consists of (i) revenue from the sale of power from both the wind farm and storage, and (ii) various costs including the marginal cost of wind, degradation costs associated with charging and discharging, and energy storage costs. By minimizing the negative of the total profit, (6) effectively maximizes the total profit. The change in power output of the wind farm between consecutive time periods is subject to ramping up/down constraints (7). The amount of wind power directly injected into the grid can t exceed the forecast maximum wind power available (8). The amount of energy that can be stored in the storage device as well as its charging and discharging rate have certain upper and lower limits (9-11). The amount of energy in the storage device in any time period, depends on the charge/discharge history of the storage device, i.e. the storage dynamics, which are given by (12). The model assumes that energy storage device can only be charged using wind power and not by the grid. This assumption is reflected in the equation for the storage dynamics (12). The term ˆP w (k) P w (k) in (12) is the firming power provided by storage to compensate for wind power forecast errors. The result of the optimization is the total power to be sold in the Day Ahead Market (DAM) for each hour of the operating day. Bid(k) =P w (k)+p d s (k) P c s (k), fork =1, 2,..., 24. (13) B. Robust optimization-based bidding strategy Using the min-max method described in Section II the robust optimization based strategy can be formulated as an extension to the deterministic optimization based strategy. Uncertainty exists in the amount of wind power and electricity price due to inaccuracy of forecasts. In such case the robust optimization problem can be stated as follows. min max P w(k),ps d(k),p s c(k),(k) λ U, P w V s.t. N [ λ(k)( P w (k)+ps d (k) Ps c (k)) k=1 + C w P w (k)+c s (P d s (k)+p c s (k)) + C e E s (k)] (14) Pw r P w (k) P w (k 1) Pw r (15) 0 P w (k) P w (k) (16) min E s (k) max (17) 0 Ps c (k) Ps max (18) 0 Ps d (k) Ps max (19) E s (k) E s (k 1) (Ps d (k) Ps c (k)) Ps loss (k) ( P w (k) P w (k)) (20) U =[(1 α p )ˆλ, (1 + α p )ˆλ] (21) V =[(1 α w ) ˆP w, (1 + α w ) ˆP w ] (22) where λ is the uncertain electricity price variable, Pw is the uncertain available wind power, U is the uncertainty set for electricity price, V is the uncertainty set for available wind power, and α p and α w are the scalar parameters that define the respective uncertainty sets. IV. CASE STUDIES In this section, we present case studies to compare the performance of the robust optimization approach with the deterministic optimization approach. These approaches are applied for determining the optimal bidding strategy of the wind farm and storage device combination, for the energy arbitrage application presented in Section III. The characteristics of the wind farm and a generic energy storage device are presented in Table II. A. Day Ahead Market - One Day (Deterministic vs. Robust) Electricity price data from Nordpool for West Denmark is used for the simulations. The hourly bids for sale and purchase of energy in the day ahead market for the entire operating day have to be submitted on the previous day, 12 hours before the beginning of the operating day (Fig. 1). It is assumed that any excess wind generation is sold in the Hour-Ahead market whereas any deficit has to be purchased from the Hour-Ahead

4 4 ˆP max P max max TABLE II WIND FARM AND STORAGE DEVICE PARAMETERS w Rated capacity of wind farm (MW) 30 s Rated capacity of storage device (MW) 3 Maximum energy level of storage device 3.75 (MWh) η Round trip efficiency of storage 90% C w Marginal cost of wind (DKK/MW) 5 C s Charging discharging (degradation) cost 1.5 (DKK/MW) C e Energy storage cost (DKK/MWh) 1 lower, part of the wind energy is used to charge the storage device. In hour 11 when the forecast price reaches its peak the stored energy is injected into the grid. Thus the storage device can be used to take advantage of arbitrage opportunities that result from temporal variations in the electricity price. Fig. 4 shows the bidding decision using the robust optimization approach. In this scenario total profit from the wind farm and storage combination for the deterministic approach is DKK 25, whereas the total profit for the robust approach is DKK 25, Thus the economic performance of the robust approach is higher than deterministic by 0.398% for this particular day. market at given price. The decision for the wind farm and storage device hourly power profile is made based on forecasts of the day-ahead electricity price and wind power. The profit is calculated based on the actual values of price and wind. This settlement is done after the end of the operating day. The robust optimization problem is solved using MATLAB along with the YALMIP toolbox [17]. Fig. 2. Electricity prices for Scenario A Fig. 1. Day-Ahead Market (DAM) Timeline In order to analyze the performance of the optimization based bidding strategy Monte Carlo simulation method is used. The Cauchy distribution is considered as the model for the distribution of wind power forecast errors [18]. For electricity price forecast error again Cauchy distribution has been shown to be a reasonable model [19]. Thus for both the wind farm power output and the electricity price an error is generated at random for each hour of the day by sampling a Cauchy distribution within bounds defined by 90% confidence interval. For each hour of the day the realization of the actual value of the input quantity (i.e., wind farm power output and electricity price) is obtained by subtracting the error from the forecast value. Thus M=100 scenarios of actual wind farm power output and electricity price are generated using random sampling. Two particular scenarios are shown for comparing the performance of robust optimization to deterministic optimization. In Scenario A the forecast error is high whereas in Scenario B the forecast error is low. 1) Scenario A: Fig. 2 shows the hourly electricity prices in the day-ahead market for Scenario A. The error between forecast and actual price is high, particularly for hour 17 and 21. Fig. 3 shows the bidding decision for wind and storage for the given day. In hours 2-5 when the forecast price is Fig. 3. Results of Deterministic Optimization Scenario A 2) Scenario B: Fig. 5 shows the hourly electricity prices in the day-ahead market for Scenario B. Compared to Scenario A the actual electricity prices are closer to the forecast. Fig. 6 and Fig. 7 show the bidding decisions for wind and storage using the deterministic and the robust optimization approach respectively. In this scenario since the electricity price forecast error is smaller than Scenario A, particularly for the key time intervals, hours 17 and 21 when the storage charges and discharges, the robust optimization gives a more conservative

5 5 Fig. 4. Results of Robust Optimization for Scenario A Fig. 6. Results of Deterministic Optimization for Scenario B result than the deterministic optimization. In this scenario total profit from the wind farm and storage combination for the deterministic approach is DKK 25, whereas the total profit for the robust approach is DKK 24, Thus the economic performance of the robust approach is lower than deterministic approach by 0.422% for this particular day. Fig. 7. Results of Robust Optimization for Scenario B Fig. 5. Electricity prices for Scenario B 3) Impact of Uncertainty Set: The performance of the robust optimization approach as a function of the budget of uncertainty (Γ) is also analyzed. The variance of the electricity prices is estimated using the method outlined in Section II-B [14]. Historical true values and forecasts of electricity price for past seven days are used to estimate the variance of the hourly prices for the given day. For each value of Γ using (3) the corresponding value of α p is obtained. Then based on available electricity price forecast an uncertainty set can be obtained as [(1 α p )ˆλ, (1+α p )ˆλ]. The mean total profit of the day using the deterministic and robust optimization approach for the 100 scenarios is calculated. The worst case realization profit is also calculated for each case. Table III illustrates how the result is affected as the budget of uncertainty for price increases. The uncertainty set for wind power is fixed and is based on α w =0.01 It is observed that as the budget of uncertainty increases the performance of robust optimization becomes more and more conservative in terms of the mean daily total profit. Also, increasing Γ above 25 has only a small impact on the optimality of result. The result of robust optimization for each case is better than the worst case realization. In what follows, we study the economic performance of robust optimization-based bidding strategy using many sample days. B. Day Ahead Market - Many Days (Deterministic vs. Robust) In this case the robust optimization algorithm is used for determining the bidding strategy for 90 consecutive days. Fig. 8 shows the electricity price forecast and actual data for 10 days. Fig. 9 shows the wind power output forecast and actual values for the same time period. The mean daily total profit of the robust optimization approach is calculated for different choices of price uncertainty bounds and wind uncertainty bounds. Table IV shows the relationship of the mean daily total profit over 90 days to the parameter α w which determines

6 6 TABLE III RESULTS OF MONTE CARLO RUNS (DO: DETERMINISTIC OPTIMIZATION, RO: ROBUST OPTIMIZATION, RESULT =MEAN DAILY TOTAL PROFIT, CHANGE =%CHANGE RELATIVE TO DO CASE, WC=WORST CASE REALIZATION PROFIT) Γ α p Result Change WC (DKK) % (DKK) DO ,750 26,413 RO , , , , , , , , , , , , , , , , , , , , , ,304 TABLE IV IMPACT OF CHOICE OF UNCERTAINTY SET OF WIND POWER α p =0.01 α w Mean Daily Total Profit (DKK) % Change 0 (Det) 70, , , , , , profit over 90 days to the parameter α p which determines the uncertainty set for price. As we observe, the mean daily total profit is more sensitive to the choice of wind uncertainty set than price uncertainty in this particular case. More generalized study of the impact of uncertainty set on wind farms profits will be part of our future work. TABLE V IMPACT OF CHOICE OF UNCERTAINTY SET OF PRICE α w =0.01 α p Mean Daily Total Profit (DKK) % Change 0 (Det) 70, , , , , , Fig. 8. Fig. 9. Forecast and Actual Electricity Price for 10 days Forecast and Actual Wind Farm Power Output for 10 days the uncertainty set for wind. Table V shows the relationship of the mean daily total V. CONCLUSIONS This paper exploits a method for the determining the bidding strategy of a wind farm with co-located energy storage in a deregulated electricity market based on robust optimization. The combination of wind and storage leads to better utilization of the uncertain wind resource and increased economic performance through participation in energy arbitrage. In the worst case scenario of significant wind generation and electricity price forecast error, robust optimization-based strategy gives a better economic performance than the deterministic approach. When forecast error is low the robust optimization gives a more conservative result. The uncertainty set for the robust optimization approach can be determined based on historical data of forecast error of the uncertain variable as well the decision maker s risk preference. Applying concepts of robust optimization in wind farms decision making process opens several research directions. One possibility is the quantification of wind uncertainty and the resulting choice of uncertainty set. Another important direction is to integrate robust optimization-based bidding strategies into look-ahead market dispatch framework. Last but not least, one could also investigate other applications of the wind and energy storage combination, including ancillary services such as grid frequency regulation.

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Milligan, Wind power forecasting error distributions over multiple timescales, in IEEE Power and Energy Society General Meeting, Jul. 2011, pp [19] W. K. Gatterbauer, Interdependencies of electricity market characteristics and bidding strategies of power producers, Master s thesis, Massachusetts Institute of Technology, Anupam A. Thatte (S 02) received the B.E. degree in electrical engineering in 2004 from Pune University, India, and the M.S. degree in electrical and computer engineering in 2005 from Carnegie Mellon University, Pittsburgh, PA. He is currently a Ph.D. student at Texas A&M University, College Station, TX. His industry experience includes an internship with ABB at their Corporate Research Center in Raleigh, NC. His research interests include modeling and control of power systems, renewable energy and smart grids. Daniel E. Viassolo is a Principal Engineer with Vestas Wind Systems, in Houston, Texas. His expertise lies in the areas of Control Systems & Optimization, with a vast experience in applications to Energy Systems such as power generation, wind turbines, combined-cycle power plants, gas turbines. Daniel holds 10 US patents, and he has published 25+ papers in conference proceedings and journals of IEEE, ASME, AIAA, etc. Before joining Vestas in January 2010, he was with General Electric Co. in upstate New York. Daniel obtained his PhD from Purdue University in Le Xie (S 05-M 10) is an Assistant Professor in the Department of Electrical and Computer Engineering at Texas A&M University, where he is affiliated with the Electric Power and Power Electronic Group. He received his B.E. in Electrical Engineering from Tsinghua University, Beijing, China in He received S.M. in Engineering Sciences from Harvard University in June He obtained his PhD from the Department of Electrical and Computer Engineering at Carnegie Mellon University in His industry experience includes an internship in 2006 at ISO-New England and an internship at Edison Mission Energy Marketing and Trading in His research interests include modeling and control of large-scale complex systems, smart grid applications in support of renewable energy integration, and electricity markets.