Bringing Wind Energy to Market

Page 1 of 8 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 IEEE PES Transactions on Power Systems Bringing Wind Energy to Market E.Y. Bitar, R. Rajagopal, P.P. Khargonekar, K. Poolla, P. Varaiya Abstract—Wind energy is a rapidly growing source of renewable energy. However, the current extra-market approach to assimilating wind energy into the electric grid will not scale at deep penetration levels. In our work, we investigate how an independent wind power producer might bid optimally into competitive electricity markets for energy. Starting with a simple model of the uncertainty in the production of power from a wind turbine farm and a model for the electric energy market, we derive analytical expressions for the optimal bid amount and corresponding expected optimal profit. Moreover, as wind is an inherently variable source of energy, we explore the sensitivity of optimal expected profit to uncertainty in the underlying wind process. We also explore the uncertainty in generation. We obtain analytical expressions for marginal profits from investing in improved forecasting and local auxiliary generation. Index Terms—Wind Energy, Smart Grid, Electricity Markets I. I NTRODUCTION Global warming, widely regarded as one of the most critical problems we face, has led to great emphasis on renewable energy sources such as solar, wind, and geothermal. Many nations have set ambitious goals for renewable energy penetration. Wind energy is expected to be a major contributor to the realization these goals. At deep penetration levels, the significant uncertainty and inherent variability in wind power pose major challenges in integrating it into the electricity grid. In this paper, we focus on the scenario in which wind power producers (WPP) must sell their energy using contract mechanisms in conventional electricity markets. Our goal is to formulate and solve problems of optimal contract sizing, value of sensor information, value of local auxiliary generation, value of storage, and cost of increased reserves needed to accommodate the uncertainty in wind power production. We start with a simple stochastic model for wind power production and a model for the conventional electricity market for energy. With these models, we derive explicit formulae for optimal contract size and the optimal expected profit. Our results cleanly capture the trade-off between penalty for contract shortfall and the need to spill some of the wind energy to increase the probability of meeting the contract. We show that extra information from meteorological models and data increases the expected optimal profit. We also make explicit the relationship between penalty for contract shortfall and the marginal impact of wind uncertainty on optimal expected Supported in part by OOF991-KAUST US LIMITED under award number 025478, the UC Discovery Grant ele07-10283 under the IMPACT program, and NSF under Grant EECS-0925337, and the Florida Energy Systems Consortium. Corresponding author: E. Bitar is with the Department of Mechanical Engineering, U.C. Berkeley ebitar@berkeley.edu P. Khargonekar is with the Department of Electrical and Computer Engineering, University of Florida ppk@ece.ufl.edu R. Rajagopal, K. Poolla and P. Varaiya are with the Department of Electrical Engineering and Computer Science, U.C. Berkeley ramr, poolla, varaiya [@eecs.berkeley.edu] profit. We consider the scenario in which the WPP schedules a capacity reservation to “hedge” against potential shortfalls corresponding to offered contracts and derive a formula for optimal contract size. We also explore the role of local generation and energy storage in managing the operational and financial risk driven by the uncertainty in generation. We obtain analytical expressions for marginal profits from investing in local generation and energy storage. Due to space constraints, the storage analysis is not included, but can be found in an extended version [12]. The remainder of this paper is organized as follows: In Section 2, we provide more detailed background on wind energy and electricity markets. Our problem formulation is described in Section 3 and our main results are contained in Sections 4 through 6. We conduct an empirical study of our strategies on wind power data obtained from Bonneville Power Authority in Section 7. Concluding comments and discussion of current and future research are contained in Section 8. II. BACKGROUND A. Wind Energy Inherent variability of the power output is the most significant difference between wind and traditional power generators. This variability occurs at various time scales: hourly, daily, monthly, and annually. Since loads are also uncertain and variable, the issue of wind power variability does not cause major problems at low wind energy penetration levels, but at deep penetration levels it presents major engineering, economic and societal challenges. Recently, National Renewable Energy Laboratory, has released two major reports [9], [14] on integration of large amounts of wind power (20-30%) into the Eastern and Western electric grid interconnections in North America. These studies show that limitations on the transmission system, increased need for reserves, impact of unpredicted large ramps, limited accuracy in wind forecasting, coordination among and conflicting objectives of independent power producers, system operators, and regulatory agencies, are some of the major issues in achieving increased penetration of wind and solar energy. Integration of wind power into the power system has been the subject of many academic and industry studies. Morales et al [23] formulate and solve a short term optimal trading strategy problem for a wind power producer. They show how their problem reduces to a linear programming problem. In their work, wind uncertainty is dealt with via creation of scenarios trees. For some earlier work along this approach, see [2], [21], [24]. Cavallo [7], [8] has studied compressed air energy storage for utility scale wind farms, and Greenblatt et al [17] compared gas turbines and compressed air energy storage (CAES) in the context of wind as part of base-load electricity generation. Economic viability of CAES in wind energy systems in Denmark has recently been studied in [20]. IEEE PES Transactions on Power Systems 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 Prediction of wind power generation [18], [26] is vitally important in wind power integration. Prediction errors decrease with shortening of the prediction horizon and expansion of the geographic area over which averaging is done. Large fast ramps in power output cause serious difficulties in the operation of the power system. Prediction of these ramp events is challenging and is subject of our current research. B. Electrical Energy Markets The deregulation of the electric power industry led to the development of open markets for electrical energy. The two dominant modes of trading are bilateral trading and competitive electricity pools. The former involves two parties (a buyer and a seller) negotiating a price, quantity, and auxiliary conditions for the physical transfer of energy from the buyer to the seller at some future time. In contrast to bilateral trading, a competitive electricity pool consists of numerous buyers and sellers participating in a single market cleared by a third party – commonly the ISO. Essentially, each supplier (consumer) submits an offer (bid) for energy at a desired price to the ISO. The ISO then combines these bids and offers to construct aggregate supply and demand curves, respectively. The aggregate supply curve is constructed by stacking energy offers in order of increasing offer price. The aggregate demand curve is similarly constructed by stacking energy quantities in order of decreasing bid price. The intersection of these curves determines the market clearing price (MCP). All suppliers that submitted bids at prices below the MCP are scheduled. Likewise, all consumers that submitted offers at prices above the MCP get scheduled. All scheduled parties pay or are paid at the MCP. For a comparison between bilateral and pool trading, see [31]. For concreteness in our studies, we assume that the wind power producer (WPP) is participating in a competitive electricity pool, although much of our analysis is portable to the bilateral trading framework. A common pool trading structure ([22], [23], [19]) consists of two successive ex-ante markets: a day-ahead (DA) forward market and a real-time (RT) spot market. The DA market permits suppliers to bid and schedule energy transactions for the following day. Depending on the region, the DA market closes for bids and schedules by 10 AM and clears by 1 PM on the day prior to the operating day. The schedules cleared in the DA market are financially binding and are subject to deviation penalties. As the schedules submitted to the DA market are cleared well in advance of the operating day, a RT spot market is employed to ensure the balance of supply and demand in real-time. This is done by allowing market participants to adjust their DA schedules based on current (and more accurate) wind and load forecasts. The RT market is cleared five to 15 minutes before the operating interval, which is on the order of five minutes. For those market participants who deviate from their scheduled transactions agreed upon in the ex-ante markets, the ISO normally employs an ex-post deterministic settlement mechanism to compute asymmetric imbalance prices. This asymmetric pricing scheme for penalizing energy deviations reflects the energy imbalance of the control area as a whole and the ex-ante clearing prices. For example, if the overall Page 2 of 8 system imbalance is negative, those power producers with a positive imbalance with respect to their particular schedules will receive a more favorable price than those producers who have negatively deviated from their schedules, and vice-versa. For a more detailed analysis of electricity market systems in different regions, we refer the reader to [5], [6], [30], [31]. III. M ODELS FOR W IND P OWER AND M ARKETS A. Wind Power Model Wind power w(t) is modeled as a scalar-valued stochastic process. We normalize w(t) by the nameplate capacity of the wind power plant, so w(t) ∈ [0, 1]. For a fixed t ∈ R, w(t) is random variable (RV) whose cumulative distribution function (CDF) is assumed known and defined as F (w, t) = P(w(t) ≤ w). The corresponding density function is denoted by f (w, t). In this paper, we will work with marginal distributions defined on the time interval [t0 , tf ] of width T = tf − t0 . Of particular importance are the time-averaged density and distribution defined as Z 1 tf f (w, t)dt (1) f (w) = T t0 Z w Z 1 tf F (w) = F (w, t)dt = f (x)dx (2) T t0 0 Also, define F −1 : [0, 1] → [0, 1] as the quantile function corresponding to the CDF F . More precisely, for β ∈ [0, 1], the β-quantile of F is given by F −1 (β) = inf {x ∈ [0, 1] : β ≤ F (x)} (3) The quantile function corresponding to the time-averaged CDF will play a central role in our results. B. Market Model The energy market system considered in this paper consists of a single ex-ante DA forward market with an ex-post imbalance penalty for scheduled contract deviations. Contracts offered in the DA market are structured as power levels that are piecewise constant over contract intervals [typically hour long]. In the absence of energy storage capabilities for possible price arbitrage, the decision of how much constant power to offer over any individual hour-long time interval is independent of the decision for every other time interval. Hence, the problems decouple with respect to contract intervals and our analysis focuses on the problem of optimizing a constant power contract C scheduled to be delivered continuously over a single time interval [t0 , tf ]. We define p ($/MW-hour) as the clearing price in the forward market and q ($/MW-hour) as the imbalance penalty price for [uninstructed] contract shortfalls. The wind power producer (WPP) is assumed to be a price taker in the forward market, because the WPP is considered small relative to the whole market. Moreover, the WPP is assumed to have a zero marginal cost of production. Remark 3.1: In this formulation p and q are assumed to be fixed and known. However, this assumption can be relaxed to p and q random and time varying without affecting the tractability of the results as long as they are assumed to be independent of the wind process w(t). Also, most of the results Page 3 of 8 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 IEEE PES Transactions on Power Systems to follow can be generalized to the case in which p and q belong to a class of functions concave in C and convex in the deviation C − w(t), respectively.  The profit acquired, the energy shortfall, and the amount of energy spilled by the WPP over the time interval [t0 , tf ] are defined respectively as Z tf Π(C, w) Σ− (C, w) Σ+ (C, w) + = = = t Z 0tf pC − q [C − w(t)] dt (4) + (5) + (6) [C − w(t)] dt t Z 0tf [w(t) − C] dt t0 where x+ := max{x, 0}. As wind power w(t) is modeled as a random process, we will be concerned with the expected profit J(C), the expected energy shortfall S− (C) and the expected amount of spilled (curtailed) wind energy, S+ (C): J(C) = E Π(C, w) (7) S− (C) = E Σ− (C, w) S+ (C) = E Σ+ (C, w) (8) (9) Here, the expectation is taken with respect to the random wind power process w = {w(t) | t0 ≤ t ≤ tf }. IV. O PTIMAL C ONTRACTS We begin by defining a profit maximizing contract C ∗ as C ∗ = arg max J(C). C≥0 (10) Theorem 4.1: Define the time-averaged distribution F (w) as in (2). (a) An optimal contract C ∗ is given by p C ∗ = F −1 (γ) where γ = . (11) q (b) The optimal expected profit, the shortfall, and the spillage are given by Z γ ∗ ∗ J (C ) = J = qT F −1 (w)dw (12) 0 Z γ  ∗  ∗ C − F −1 (w) dw (13) S− (C ∗ ) = S− = T 0 Z 1   −1 ∗ F (w) − C ∗ dw. (14) S+ (C ∗ ) = S+ = T γ Proof: Consider item (a). It is straight forward to prove concavity of J(C) in C on [0, 1] by using the properties: (1) x+ is sub-additive and (2) x+ is homogeneous of degree one. Now, notice that J(C) can be rewritten as Z 1 Z tf [C − w]+ f (w, t)dt dw J(C) = pCT − q 0 t0 Z 1 = pCT − qT C = pCT − qT Z 0 0 [C − w]+ f (w)dw (C − w)f (w)dw. Clearly J(C) is also continuous in C on [0, 1] for any probability density function f (w). It follows that the set of all maxima, denoted by C ⊆ [0, 1], is convex. Hence, any C0 ∈ C is a global maximum. Moreover, C0 is a global maximum if and only if zero is contained in the subdifferential set of J at C0 . For technical simplicity in the proof, we assume that f (w) is continuous on [0, 1]. It follows that J(C) is differentiable in C on [0, 1]. Hence, the set of all global maxima is given by C = {C ∈ [0, 1] : dJ/dC = 0}. Application of the Leibniz integral rule yields Z C dJ = pT − qT f (w)dw = pT − qT F (C). dC 0 Setting dJ/dC = 0 yields the desired optimality condition γ = F (C). Now consider item (b). The result is easily proven by considering the change of variables θ = F (w). We prove only the result for J ∗ as expressions for the remaining quantities, ∗ ∗ S− and S+ are established analogously. Using a change of variables, we have Z C∗ ∗ ∗ (C ∗ − w)f (w)dw J = pC T − qT 0 Z γ = pC ∗ T − qT (C ∗ − F −1 (θ))dθ 0 Z γ ∗ F −1 (θ)dθ, = C T (p − γq) +qT | {z } 0 =0 which gives us the desired result.  Remark 4.2: (Newsboy) The structure of our expected profit criterion and optimal policy are closely related to the classical Newsboy inventory problem in economics [25].  Remark 4.3: (Non-uniqueness of C ∗ ) Clearly, any contract C that solves γ = F (C) is profit maximizing with respect to problem (10). Because the CDF F is only guaranteed to be monotone non-decreasing on it’s domain [0, 1], it may have intervals in its domain on which it is constant, which allows for non-uniqueness of the optimizer C ∗ . Hence, it is straight forward to see that C ∗ is unique if and only if the set Γ(F, γ) := {x ∈ [0, 1] : γ = F (x)} is a singleton. As stated in Theorem 4.1-(a), a particular choice for an optimal contract is C ∗ = F −1 (γ) – the γ th - quantile of F , as specified in equation (3). Although the optimal expected profit J ∗ is independent of the choice of C ∗ ∈ Γ(F, γ), it is straightforward to see that C ∗ = F −1 (γ) is the minimizer of ∗ the expected optimal shortfall S− among all contracts C ∈ ∗ Γ(F, γ). The opposite is true for S+ . The effect of alternative ∗ ∗ and S− is quantified as choices of C ∗ from Γ(F, γ) on S+

−1 ∗ −1 ∗ (15) S− (C ) = S− F (γ) + γ C − F (γ)

∗ −1 ∗ −1 S+ (C ) = S+ F (γ) − (1 − γ) C − F (γ) (16) for C ∗ ∈ Γ(F, γ).  Remark 4.4: (Graphical Interpretation) Theorem 4.1-(b) provides explicit characterizations of the optimal expected ∗ ∗ profit J ∗ , energy shortfall S− , and energy spilled S+ . These three quantities can be graphically represented as areas bounded by the time-averaged CDF F (w) as illustrated in Figure 1 for γ = 0.5. J ∗ = qT A1 , ∗ S− = T A2 , ∗ S+ = T A3 IEEE PES Transactions on Power Systems From Figure (1), it is apparent that a reduction of “statistical dispersion” in the time-averaged distribution F (w) will result in an increase in optimal expected profit (A1 ) and a decrease in the optimal expected energy shortfall and spillage (A2 , A3 ) – all of which are favorable consequences.  1 A3 0.8 F (w) 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 0.6 γ = 0.5 0.4 A1 0.2 0 0 A2 C∗ 0.2 0.4 0.6 0.8 w (MW generation/capacity) 1 ∗, Fig. 1. Graphical interpretation of (A1 ) optimal profit J ∗ , (A2 ) deficit S− ∗ where γ = 0.5. and (A3 ) spillage S+ Remark 4.5: (Price Elasticity of Supply) Under certain assumptions, the quantile rule (11) in Theorem 4.1 can be interpreted as the supply curve for the WPP. Of primary importance is the assumption that the WPP is a price taker in the DA forward market, ensuring that it weild no influence over the market price. This is reasonable given the relatively low penetration of wind energy in existing markets. For a fixed deviation penalty price q, one can interpret the optimal quantile rule (11) as indicating the amount of energy that the WPP is willing to supply at a price p. Specifically, the supply curve   is given by p −1 C(p) = F q The total energy that the WPP is willing to supply at price p over the time interval [t0 , tf ] is then T C. With this explicit characterization of the WPP’s supply curve, the price elasticity of supply, EC , can be readily derived as EC := dF −1 (γ) γ γ d ln C(p) = −1 = . d ln p F (γ) dγ Cf (C)  Remark 4.6: (Role of γ, VaR) It follows from the γ-quantile structure of the optimal solution that the optimal contract C ∗ is chosen to be the largest contract C such that the probability of a shortfall occurrence – with respect to the time averaged distribution F (w) – is less than or equal to γ. In the finance literature, C ∗ is referred to as the γ-Value-at-Risk (VaR), where 1 − γ is interpreted as a confidence level. Clearly then, the price-penalty ratio γ = p/q plays a critical role in implicitly controlling the probability of shortfall with respect to optimal offered contracts C ∗ = F −1 (γ). Consider the scenario in which the ISO has direct control over the shortfall deviation penalty price q. As the penalty price q becomes more harsh, (i.e., larger), the price ratio γ decreases – resulting in smaller offered contracts C ∗ . This follows from the fact that the quantile function F −1 (γ) is non-decreasing in γ (non-increasing in q). Consequently, the probability of shortfall F (C ∗ , t) with respect to the optimal contract C ∗ , is non-increasing in q.  Page 4 of 8 ∗ Remark 4.7: (Spillage) The expected optimal shortfall S− can be further interpreted as the expected amount of energy supplied by the ISO to balance the shortfalls in the WPP’s contractual obligation. A straightforward corollary of Theorem ∗ ∗ 4.1 is that the expected optimal shortfall S− and spillage S+ are monotonically non-decreasing and non-increasing in γ, respectively. This makes explicit the claim that some wind energy must be spilled in order to reduce the amount of operational reserve capacity needed to hedge against uncertainty in the wind power.  Remark 4.8: We are able to derive additional insight into the structure of the optimal solution by considering an equivalent reformulation of the optimal expected profit J ∗ (12) in terms of conditional expectation. Straightforward manipulations yield J ∗ = pT C̃, where C̃ = E [ W | W ≤ C ∗ ] and expectation is taken with respect to the time averaged distribution F (w) conditional on the shortfall event {W ≤ C ∗ }. In the finance literature, the quantity C̃ is referred to as the conditional Value-at-Risk (CVaR) [27] – the mean of the γ probability tail. This reformulation reveals that optimal expected profit J ∗ is equal to the revenue associated with C̃. ∗ Moreover, it follows that the expected optimal shortfall S− ∗ −1 is proportional to the gap between the VaR, C = F (γ), and the CVaR, C̃.   ∗ = γT C ∗ − C̃ S−  V. ROLE OF I NFORMATION Intuitively, an increase in uncertainty in future wind power output will increase contract sensitivity to the price-penalty ratio γ = p/q. Hence, it is of vital importance to understand the effect of information [such as available implicitly through forecasts] on expected optimal profit. Consider a simple simple scenario in which the WPP observes a random variable Y that is correlated to the wind process w(t). The random variable Y can be interpreted as an observation of a meteorological variable relevant to the wind. Using the results in theorem 4.1, it is natural to define the optimal expected profit J ∗ (y) conditional on information Y = y as Z γ J ∗ (y) = qT F −1 (w|y)dw. 0 1 T R tf F (w, t|y)dt and F (w, t|y) is the where F (w|y) := t0 CDF of w(t) conditioned on the realization Y = y. Theorem 5.1: E [ J ∗ (Y ) ] ≥ J ∗ ∗ Proof: Define C (y) as a profit maximizing contract conditional on the observation Y = y. More precisely, C ∗ (y) = arg max E[Π(C, w) | Y = y], C∈[0,1] where expectation is taken with respect to the time-averaged conditional distribution. The following inequality holds for all C0 ∈ [0, 1] by optimality of C ∗ (y). Z 1 J ∗ (y) = pC ∗ (y) − q [C ∗ (y) − w]+ f (w|y)dw 0 Z 1 ≥ pC0 − q [C0 − w]+ f (w|y)dw 0 Page 5 of 8 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 IEEE PES Transactions on Power Systems Taking expectation with respect to y of both sides of the inequality yields Z Z 1 E[J ∗ (Y )] ≥ pC0 − q [C0 − w]+ f (w|y)dwf (y)dy 0 y = pC0 − q Z 1 0 [C0 − w]+ f (w)dw. The equality follows from a straightforward application of Bayes rule. Finally, maximizing the right hand side with respect to C0 yields the desired result.  Remark 5.2: It follows from Theorem 5.1 that information helps in the metric of expected profit. Moreover, figure 1 offers some intuition as to how a reduction in “statistical dispersion” of the CDF F results in increased expected optimal profit.  A. Quantifying the effect of uncertainty It is of interest to quantify the marginal improvement of expected optimal profit with respect to information increase in various metrics of dispersion. In practice, there are numerous deviation measures of dispersion of probability distributions (e.g. standard deviation, mean absolute deviation). In [27], the authors take an axiomatic approach to construct a class of deviation measures for which there is a one-to-one correspondence with a well known class of functionals known as expectationbounded risk measures. We refer the reader to [1], [27] for a detailed exposition on topic. For our purposes, it suffices to Z realize that 1 γ −1 Dγ (X) = E[X] − F (x)dx (17) γ 0 is a valid deviation measure [27] for all square-integrable random variables X with CDF F (x) and γ ∈ (0, 1). It is sometimes referred to as the conditional value-at-risk (CVaR) deviation measure. This particular choice of the CVaR deviation measure Dγ is special in that it permits the analytical computation of the marginal improvement of optimal expect profit J ∗ with respect to the wind variability, as measured by Dγ . Simple algebraic manipulation of the formula for optimal expected profit (12) reveals J ∗ to be an affine function in Dγ (W ), where W is distributed according to the time averaged distribution F (w). J ∗ = pT ( E[W ] − Dγ (W ) ) (18) This result quantifies the increase in expected profit that results from a reduction in Dγ (W ) using sensors and forecasts. Further, it makes explicit the joint sensitivity of optimal expected profit J ∗ to uncertainty and prices. Remark 5.3: (Role of γ) As we discovered earlier, the pricepenalty ratio, γ = p/q, plays a role in controlling the probability of shortfall with respect to optimal bids C ∗ = F −1 (γ). In a related capacity, the price-penalty ratio γ also acts to discount the impact of uncertainty in the underlying wind process, w(t), on optimal expected profit J ∗ . This assertion is made rigorous by the fact that Dγ (W ) is monotone non-increasing in γ for a fixed W . Its limiting values are given by lim Dγ (W ) = E[W ] − inf W = E[W ] lim Dγ (W ) = 0 γ→0 γ→1  Remark 5.4: (Effect of Uncertainty) This interpretation of optimal expected profit agrees with intuition. As the “uncertainty” in the random wind process decreases, we have that Dγ (W ) decreases monotonically to zero resulting in J ∗ approaching pT E[W ]. In the limit where the uncertainty goes to zero (i.e., Dγ (W ) → 0) we have that Z tf w(t)dt lim J ∗ = p Dγ (W )→0 t0 where {w(t) | t ∈ [t0 , tf ]} is known for Dγ (W ) = 0.  Example 5.5: (Uniform Distribution) It is informative to consider the case in which F (w) is taken be a uniform distribution having support on a subset of [0, 1]. Under this assumption, it is straightforward to compute the optimal expected profit as   √ J ∗ = pT E[W ] − σ 3(1 − γ) , where σ is the standard deviation of W – the most commonly used measure of statistical dispersion. The marginal expected profit with respect to wind uncertainty, as measured by σ, is √ dJ ∗ = −pT 3(1 − γ). dσ A direct consequence is that the expected profit’s sensitivity to uncertainty, σ, increases as the penalty price q becomes more harsh – or equivalently, as γ → 0. VI. ROLE OF R ESERVE M ARGINS G ENERATION AND L OCAL A. Reserve Margins In order to maintain reliable operation of the electric grid, the ISO is responsible for procuring ancillary services (AS) to balance potential deviations between generation and load. The various underlying phenomena responsible for these deviations result in system imbalances with varying degrees of uncertainty on differing time scales. In order to absorb this variability on the different time scales, multiple ancillary services must be procured. Broadly, these services consist of regulation, load-following, reserve (spinning and non-spinning), voltage control, and reactive power compensation. Based on the scheduled energy, the ISO first determines the total reserve requirement for the entire control area needed to satisfy pre-specified reliability criteria. The ISO then assigns to each participating load serving entity (LSE) a share of the total reserve requirement based roughly on its scheduled demand, because of the uncertainty in load [16]. Each LSE has the option to procure all or a portion of its reserve requirement through bilateral contracts or forward markets. The remaining portion of the reserve requirement not provided by the LSE is procured by the ISO through ancillary service markets. A detailed exposition on ancillary services can be found in [31]. Wind power is inherently difficult to forecast. Moreover, it exhibits variability on multiple time scales ranging from single-minute to hourly. It follows then that regulation, loadfollowing, and reserve services will be necessary to compensate imbalances resulting from fluctuations in wind[13]. To simplify our analysis, we will lump all of these ancillary services into a single service that we refer to as “reserve IEEE PES Transactions on Power Systems 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 margin”. Under the current low capacity penetration levels of wind power (∼ 1%), the added variability of wind is largely absorbed by existing reserve margins used to cover fluctuations in the load. As the capacity penetration of wind increases, its affect on operating reserve margins will become more pronounced [13], [14]. Moreover, it will become economically infeasible to continue the socialization of the added reserve costs, stemming from wind variability, among participating LSEs. Hence, it is likely that the wind power producer (WPP) will have to bear the added cost of reserve margins [10], [34]. Consider now a departure from the current deterministic practice of scheduling reserve margins. Assume that the WPP is responsible for procuring enough reserve power such that the loss of load probability (LOLP), with respect to an offered contract C, is less than or equal to ǫ. More precisely, given an offered contract C on interval [t0 , tf ], the WPP procures reserve capacity R(C, ǫ) satisfying
R(C, ǫ) = min R s.t. P R ≤ [C − w(t)]+ ≤ ǫ R≥0 for all t ∈ [t0 , tf ]. It is straightforward to show that the reserve capacity R(C, ǫ) is given by + R(C, ǫ) = [C − δ(ǫ)] , (19) where δ(ǫ) := mint F −1 (ǫ, t). Remark 6.1: (Interpretation of δ(ǫ)) The probabilistic quantity δ(ǫ) can be interpreted as the highest power level that the WPP can supply constantly over the time interval [t0 , tf ] with probability greater than or equal to 1 − ǫ. Hence, in order to satisfy the 1 − ǫ reliability criteria with respect to an offered contract C, the WPP will procure enough reserve power capacity R(C, ǫ) to cover the additional risk introduced by contracts larger than δ(ǫ). For contracts C less than δ(ǫ), the ISO schedules no reserve power, as indicated by (19).  Ex-ante, the WPP must make a capacity payment for the operating reserve at the capacity price qc . Ex-post, if the WPP under produces with respect to the offered contract C, it must make an energy payment. For shortfalls less than R(C, ǫ), the WPP pays at the energy price qe . For shortfalls larger than R(C, ǫ), the WPP pays at the imbalance energy penalty price q. All of the prices are in units of ($/MW-hour). This augmented penalty mechanism is captured by the following penalty function φ : R × R+ → R+ .   qx − (q − qe )R x ∈ (R, ∞) φ(x, R) = (20) q x x ∈ [0, R]  e 0 x ∈ (−∞, 0) It follows that the fiscal cost and benefit of reserve capacity to the WPP can be explicitly accounted for in the following expected profit criterion Z tf pC − qc R(C, ǫ) − φ (C − w(t), R(C, ǫ)) dt JR (C) = E t0 (21) Fiscal benefit is derived from the assumption that qc ≤ p. Moreover, it is assumed that qe ≤ q, guaranteeing convexity ∗ as of (20) in x. Define a profit maximizing contract CR Page 6 of 8 ∗ CR = arg max JR (C) (22) C≥0 Theorem 6.2: Define the time-averaged distribution F (w) as in (2). ∗ (a) An optimal contract CR is given p − qc ∗ . (23) CR = F −1 (γR ) where γ = qe (b) The optimal expected profit is given by ∗ JR (CR ) = qc T δ(ǫ) [ 1 + F (δ(ǫ)) ] Z Z γR −1 + qe T F (w)dw − qT F (δ(ǫ)) δ(ǫ) F (w)dw (24) 0 Proof: The proof technique parallels that of Thm. 4.1.  Remark 6.3: It is interesting to note that the optimal con∗ tract size CR offered by the WPP does not depend on the LOLP ǫ and the imbalance penalty price q. However, the expected profit certainly does depend on these parameters.  B. Local Generation As ancillary service markets have been known to exhibit price volatility [29], it may be advantageous for the WPP to circumvent the AS markets and procure its reserve margin from a small fast-acting generator co-located with its wind farm. Consider the scenario in which the WPP has at its disposal a co-located generator of power capacity L and operational cost qL ($/MW-hour). Hence, the local plant can be used to mitigate financial risk by covering contract shortfalls up to a limit L at a reduced energy cost qL < q. It is straightforward to capture this scenario by reinterpreting the penalty function (20) and expected profit criterion (21) through an alternative choice of parameters. qc = 0, qe = qL , R(C, ǫ) = L. Because of the significant capital costs associated with the investment in local generation, it’s important to quantify the marginal improvement in profit resulting from the investment in a generator with small power capacity L. The following theorem distills this notion. Theorem 6.4: Define the time-averaged distribution F (w) as in (2). (a) An optimal contract CL∗ is given by any solution C of p = qL F (C) + (q − qL )F (C − L). (b) The marginal expected optimal profit with respect to power capacity L is given by   qL dJ ∗ pT = 1− dL L=0 q Proof: The proof for part (a) follows from direct application of the proof technique for theorem 4.1-(a). Part (b) is proven as follows. It is straightforward to show that the expected profit is given by Z C−L J(C, L) = pCT − T [q(C − w) − (q − qL )L] f (w)dw 0 Z C − T qL (C − w)f (w)dw C−L Page 7 of 8 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 IEEE PES Transactions on Power Systems for any choice of C and L. Taking the derivative with respect to L yields dJ(C, L) = (q − qL )F (C − L)T. dL Taking the limit as L goes to zero and substituting for the first order optimality condition (a) yields the desired result.  Remark 6.5: For operational cost qL ≤ p, the optimal bid is given by CL∗ = F −1 (γ) + L. For qL ≥ q, we have CL∗ = F −1 (γ). Moreover, for L small, we have CL∗ ≈ F −1 (γ) – recovering the optimal policy in Theorem 4.1.  VII. E MPIRICAL S TUDIES Using a wind power time series data set provided by the Bonneville Power Administration (BPA), we are in a position to illustrate the utility and impact of the theory developed in this paper. A. Data Description The data set consists of a time series of measured wind power aggregated over the 14 wind power generation sites in the BPA control area [3]. The wind power is sampled every 5 minutes and covers the 2008 and 2009 calendar years. Accompanying the measured wind power is a time series of rolling one hour-ahead forecasts sampled at the same frequency. To account for additional wind power capacity coming online at various points in time over the 2-year horizon, all of the data are normalized by the aggregate nameplate power capacity of the wind farms. B. Empirical Probability Model As stated earlier, wind power is modeled as a continuous time stochastic process whose marginal cumulative distribution is denoted by F (w, t). While the identification of stochastic models that accurately capture the statistical variability in wind power is of critical importance, this is not the focus of our paper. We will make some simplifying assumptions on the underlying physical wind process to facilitate our analysis. A1: The wind process w(t) is assumed to be first-order cyclostationary in the strict sense with period T0 = 24 hours – i.e F (w, t) = F (w, t + T0 ) for all t [32], [11]. Thus, we are ignoring the effect of seasonal variability. A2: For a fixed time τ , the discrete time stochastic process {w(τ + nT0 ) | n ∈ N} is independent in time (n). Fix a time τ ∈ [0, T0 ] and consider a finite length sample realization of the discrete time process zτ (n) := w(τ + nT0 ) for n = 1, · · · , N . Using this data set, we take the empirical distribution F̂N (w, τ ) as an approximation of the underlying distribution F (w, τ ): N 1 X F̂N (w, τ ) = 1 {zτ (n) ≤ w} N i=n (25) Invoking the strong law of large numbers under the working assumptions, it can be shown that the F̂N (w, τ ) is consistent with respect to F (w, τ ) [4]. Figure 2 (a) depicts nine representative marginal empirical distributions identified from the BPA data set described earlier. Note that the times corresponding to the nine distributions are equally spaced throughout the day to provide a representative sample. Figure 2 (b) depicts the trajectory of the empirical median F̂N−1 (0.5, t) and its corresponding interquartile range [F̂N−1 (0.25, t), F̂N−1 (0.75, t)]. C. Optimal Contracts in Conventional Markets Using empirical wind power distributions identified from the BPA wind power data set, we are now in a position to compute and appraise optimal day-ahead (DA) contracts offered by a representative Oregon wind power producer (WPP) participating in the idealized market system described in Section IV. We are also able to examine the effect of γ on ∗ ∗ J ∗ , S− , and S+ using this particular characterization of wind uncertainty. The following empirical studies assume a contract 24 structure {[ti−1 , ti ), Ci }i=1 , where [ti−1 , ti ) is of length one hour for all i. Remark 7.1: (Optimal DA Contracts) Figure 2 (c) depicts ∗ ) for various price ratios γ = optimal contracts (C1∗ , · · · C24 0.3, 0.4, · · · , 0.9. As expected, as the price-penalty ratio γ = p/q decreases, the optimal contract C ∗ decreases. From Figure 2 (c), it is evident that WPPs will tend to offer larger contracts during morning/night periods when wind speed is typically higher than during mid-day (as indicated by Figure 2 (b)).  Remark 7.2: (Profit, Shortfall, and Spillage) Figures 3 (a) and (b) demonstrate the effect of the price-penalty ratio γ on the optimal expected profit, energy shortfall, and energy ∗ ∗ spillage. The units of S− and S+ are (MW-hour)/(nameplate capacity), while the units of J ∗ are in $/(q · nameplate capacity). When γ = 1, the WPP sells all of its energy production at price p = q. In this situation, the expected profit per hour (see Figure 3 at γ = 1) of approximately 6.4 24 equals the ratio of average production to nameplate capacity. This number is consistent with typical values of the wind ∗ ∗ production capacity factor. The spillage S+ and shortfall S− are relatively insensitive to variations in γ [for γ ∈ [0, 0.1]] because the marginal empirical distributions are steep here.  D. Local Generation We now consider the optimal contract sizing formulation in section VI. Figure 3 (c) depicts the marginal expected optimal profit with respect to local generation power capacity L. As q → qL , the marginal value of local generation diminishes. VIII. C ONCLUSION In this paper we have formulated and solved a variety of problems on optimal contract sizing for a wind power producer operating in conventional electricity markets. Our results have the merit of providing key insights into the tradeoffs between a variety of factors such as penalty for shortfall, cost of reserves, value of storage and local generation, etc. In our current and future work, we will investigate a number of intimately connected research directions: improved forecasting of wind power, optimization of reserve margins, making wind power dispatchable, network aspects of renewable energy grid integration, and new market structures for facilitating integration of renewable sources. We are also studying the important case of markets with recourse where the producer has opportunities to adjust bids in successive stages. We are also developing large scale computational simulations which can be used to test the behavior of of simplified analytically tractable models and suggest new avenues for research applicable to real-world grid-scale problems. 1 0.8 0.8 0.6 0.6 0.4 0.4 0.2 0.2 Page 8 of 8 F̂N−1 (.25, t) 0.8 γ = 0.9 0.7 F̂N−1 (.50, t) 0.6 F̂N−1 (.75, t) 0.5 C ∗ (t) 1 0.4 0.3 0.2 0 0 0.2 0.4 0.6 0.8 w (MW generation/capacity) 1 0.1 0 0 5 10 15 t (hours) 0 γ = 0.3 0 5 20 10 15 t (hours) 20 −1 Fig. 2. (a) Empirical CDFs F̂N (w, τ ) for nine equally spaced times throughout the day, (b) Trajectory of the empirical median F̂N (.5, t) and its −1 −1 corresponding interquartile range [F̂N (.25, t), F̂N (.75, t)], (c) Optimal contracts offered in the DA market for various values of γ = 0.3, 0.4, · · · , 0.9. 0.5 8 0.8 ∗ S+ J∗ 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 F̂N (w, t) IEEE PES Transactions on Power Systems 7 0.7 6 0.6 5 0.5 4 0.4 3 0.3 2 0.2 1 0.1 0.4 ∗ S− 0.3 qL = p 0.2 0.1 0 0 0.2 0.4 γ 0.6 0.8 1 0 0 0.2 0.4 γ 0.6 0.8 1 qL = q 0 0.2 0.4 0.6 0.8 L Fig. 3. (a) Optimal expected profit J ∗ as a function of γ, (b) Optimal expected energy shortfall and spillage for the 12th hour interval, as a function of γ, (c) Marginal expected optimal profit with respect to power capacity L of a local generation plant for various operational costs qL ∈ [p, q]. 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