Energy Procurement Strategies in the Presence of Intermittent Sources

Transcription

1 Energy Procurement Strategies the Presence of Intermittent Sources Jayakrishnan Nair Centrum Wiskunde & Informatica Sach Adlakha California Institute of Technology Adam Wierman California Institute of Technology ABSTRACT The creasg penetration of termittent, unpredictable renewable energy sources such as wd energy, poses significant challenges for utility companies tryg to corporate renewable energy their portfolio In this work, we study the problem of conventional energy procurement the presence of termittent renewable resources We model the problem as a variant of the newsvendor problem, which the presence of renewable resources duces supply side uncertaty, and which conventional energy may be procured three stages to balance supply and demand We compute closed form expressions for the optimal energy procurement strategy and study the impact of creasg renewable penetration, and of proposed changes to the structure of electricity markets We explicitly characterize the impact of a growg renewable penetration on the procurement policy by considerg a scalg regime that models the aggregation of unpredictable renewable sources A key sight from our results is that there is a separation between the impact of the stochastic nature of this aggregation, and the impact of market structure and forecast accuracy Additionally, we study the impact on procurement of two proposed changes to the market structure: the addition and the placement of an termediate market We show that addition of an termediate market does not necessarily crease the efficiency of utilization of renewable sources Further, we show that the optimal placement of the termediate market is sensitive to the level of renewable penetration 1 INTRODUCTION Society s satiable appetite for energy and growg environmental concerns have led many states the United States to enact renewable portfolio standards The standards mandate that utility companies must procure a certa percentage of their electricity from renewable sources [39 For example, California has set the goal that 33% of its electricity should come from renewable sources by 2020 Among possible renewable sources, wd energy is expected to play a major role There has been an explosive growth stalled wd capacity over the last few years [15 due to the ease of stallation and low operational costs However, current electricity markets that govern energy procurement were designed for a scenario where there is very little uncertaty More specifically, until now, supply side uncertaty has been low, arisg maly due to generator failures, which are rare Furthermore, accurate demand forecastg ensures that the uncertaty demand is small However, gog forward, the troduction of large volumes of highly termittent and unpredictable renewable generation will crease supply side uncertaty dramatically Thus, corporatg wd energy 1 to the energy portfolio of utilities 1 For the remader of the paper, we will use wd energy is a challengg task that requires rethkg how electricity is procured [34, 12, 9 This paper seeks to provide sights to the impact of creasg supply side uncertaty on the efficiency of procurement Utility companies typically procure electricity via two modes of operation - bilateral long term contracts and competitive electricity markets [22, 35, 16 In the former, utility companies sign long term bilateral contracts with various generators to supply certa amounts of electricity for specified periods Currently, most utility companies purchase the bulk of their generation through long term bilateral contracts This is feasible because the aggregate demand is highly predictable and because most conventional generators have very little uncertaty To account for daily (or hourly fluctuations demand, the utility companies purchase the remader of their electricity competitive electricity markets These markets, which utility companies are buyers, and generators are sellers, are run by a third party called the dependent service operator (ISO There are typically two markets: a day-ahead or forward market, and a real time or spot market A utility company may buy electricity both markets order to ensure that it has enough supply to meet the demand Integration of wd energy to current electricity markets has attracted considerable attention recent years; excellent surveys can be found [10, 34 Broadly speakg, there are two different approaches to tegratg wd to current electricity markets In one approach, wd power producers participate only competitive electricity markets eg, California s participatg termittent resource program (PIRP Such a scenario has been analyzed, fostance, [5 In the second approach, the wd power producers do not participate electricity markets; stead, they sign long term multi-year contracts with utility companies In such contracts, the utility company acquires rights to the energy generated from a certa wd farm stallation return for a predetermed payment For example, Southern California Edison (SCE, a major utility company servg the greater Los Angeles area, has signed various contracts spanng 2 10 years with various wd farms to procure wd energy rangg from 666 MW to 115 MW [30 In this paper, we study the consequences of followg the second approach In particular, we study a settg where utility companies have procured large volumes of termittent, unpredictable renewable energy via long term contracts Because the realized amount of wd energy is variable and unpredictable, the utility company must still procure conventional generation via the electricity markets Our goal is to study the impact of this long term commitment to renewable generation on the procurement strategies for conterchangeably with renewable energy; the models and the sights of our paper apply to any form of termittent resource

2 ventional generation Given the complexity of electricity procurement, we must consider a simplified model to be able to obta analytical results To that end, we consider a settg that ignores many complexities of generation and transmission (eg, ramp constrats and le capacities, but models explicitly the multitimescale nature of electricity procurement Moreover, we assume that the utility company is a price-take the markets for conventional generation, ie, it cannot fluence the prices these markets through its actions These assumptions are standard the literature and, though arguable, they enable us to derive a closed form characterization of the optimal energy procurement strategy of the utility company, which leads to several useful, counter-tuitive sights Contributions of this paper The ma contributions of this work fall, briefly, to two categories: (i we characterize the optimal procurement strategy the presence of long-term contracts fotermittent, unpredictable generation; and (ii we study the impact of creasg renewable penetration and proposed changes to market structure on the optimal procurement strategy We describe these each more detail the followg The first contribution of this work is to characterize the optimal procurement strategy for a utility company that has a long term contract with an unpredictable generation source, eg, wd energy (see Section 3 More specifically, we derive closed-form formulas for the optimal procurement that a utility company needs to make both long term and day ahead markets This result is a generalization of solutions for the classical newsvendor problem [1, 33, 20 A key feature of our result is that the optimal procurement quantities can be viewed terms of reserves, where these reserve quantities are the additional purchases that the utility needs to make to balance the current uncertaty of the supply and the higher cost of procurg energy future markets The second contribution of this work is to study how the optimal procurement strategy changes as the penetration of renewable energy grows (see Section 4 In particular, we consider a scenario where the quantity of renewable generation contracted for grows, and ask how the procurement changes The scalg focreasg penetration that we consider allows for a wide variety of models for how the unpredictability of renewable generation changes with creased penetration For example, it cludes scalg via additional sources with eithedependent or highly correlated generation Our ma result from this section yields a simple, formative equation summarizg the impact of renewable generation on the procurement of conventional generation Specifically, Theorem 2 states that the average total procurement of conventional generation the presence of a long term contract for wd is d αγ δγ θ where d is the demand, α is the average generation of a sgle wd farm, γ is the number of wd farms (ie, the renewable penetration, θ is a constant capturg the dependence between the generations of different wd farms, and δ is a constant that depends on the details of the market structure The way to terpret this equation is as follows d αγ represents the mimum average procurement, sce this is the amount of demand that is not met by the wd Thus, the extra generation required because of the uncertaty of the wd is δγ θ The key pot about this term is that γ θ is purely dependent on the degree of renewable penetration and the correlation between renewable sources; thus the impact of market structure is limited to δ The third contribution of this work is to study the impact of proposed changes to market structure [34, 9 on the optimal procurement strategy In particular, there are two types of changes to the market structure that are most commonly considered: changg the placement of the day ahead market, eg, by movg it closer to real time; and addg markets, eg, addg a new market between the day ahead and real time markets [28 The results Sections 5 and 6 address the impact each of these possibilities First, Section 5 studies the impact of the placement of markets on procurement This is a particularly salient issue because one might expect that as the penetration of renewable energy creases, it is beneficial to shift markets closer to real time, order to take advantage of the improved prediction accuracy of the renewable generation Our results highlight that this tuition may not be true Specifically, we prove that, under very general assumptions, the placement of the day ahead market that mimizes the average total cost of procurement is dependent of the penetration of renewables (Theorem 3 Next, Section 6 studies the impact of additional markets on procurement The addition of markets is often suggested as a way to help corporate renewable generation by providg new markets closer to real time where predictions about renewable availability are more accurate Our results highlight that one needs to be careful when considerg such a change Specifically, we contrast procurement a two level market with procurement a three level market order to understand the role of addg an termediate market Of course, the cost of procurement always decreases as additional markets are troduced However, with environmental concerns md, the key question is not about cost but about the amount of conventional generation procured Perhaps surprisgly, additional markets do not always reduce the amount of conventional generation procured Specifically, if we consider the addition of an termediate market, then the average amount of conventional generation may drop or grow dependg on the quality of the estimates for renewable generation, ie, δ Equation (1 may decrease ocrease Informally, if the estimation error is, a sense, light-tailed (eg, Gaussian, then the addition of an termediate market reduces procurement of conventional generation (Theorem 5; but if the error has a heavy-tail (eg, power-law, heavy-tailed Weibull, then the addition of an termediate market can have the opposite effect (Theorem 4 Interestgly, it is typical to assume analytical work that forecast errors are Gaussian [18, 38, 28, whereas empirical work on wd power generation suggests that a Weibull distribution may be a more accurate description [19, 6 2 MODEL Our goal this paper is to understand how the presence of long term contracts fotermittent, unpredictable renewable generation impacts the procurement of conventional generation Such long term contracts are a common, effective way of corporatg renewable energy to a utility s portfolio [30; however they create challenges for a utility company s procurement of conventional generation Thus, at the core of the paper, is a model of the electricity markets for conventional generation utility companies participate, which we describe this section The key features we seek to capture are (i the multi-timescale nature of electricity markets, (ii price volatility, and (iii the uncertaty of renewable generation Specifically, the procurement of conventional generation typically happens through participation a multi-tiered set of electricity markets cludg a long-term market (typically bilateral contracts, which could take place years or months ahead of time; an termediate market, aka forward market, which could take place a day or several hours ahead of time; and a real time market, aka spot mar-

3 ket This multi-tiered structure means that when the utility company purchases conventional generation the long term otermediate markets it does not know how much renewable generation will be realized, nor does it know what the price will be the spot market So, its decisions must be made usg only forecasts of these quantities, and, given the volatility of both renewable generation and prices, this creates a challengg procurement problem for the utility 21 Model overview To keep the model simple enough to allow analytic study, we ignore issues such as generator rampg constrats and transmission network capacity constrats our model and further assume that the utility company has no access to large scale energy storage capacity These assumptions allow us to focus on a sgle stant of time, which we denote by t = 0, and to consider only aggregate supply and demand We denote the electricity demand the utility company under consideration faces by d, and assume it is fixed and known ahead of time This assumption is not restrictive our settg, sce demand uncertaty can be corporated to the uncertaty of renewable generation To meet the demand d, the utility company combes long term contracts for renewable generation (for simplicity we will often refer to this simply as wd with participation a typical threetier set of electricity markets for conventional generation: (i A long term market, which the purchase commitment is made at time T lt The price this market is denoted p lt, and when makg its purchase commitment, the utility has a forecast ŵ lt of the wd generation that will be realized at t = 0 (ii An termediate market (forward market, which the purchase commitment is made at time T The price this market is denoted p, and when makg its purchase commitment, the utility has a forecast ŵ of the wd generation that will be realized at t = 0 (iii A real time market (spot market, which the purchase commitment is made at time t = 0 The price this market is denoted p rt, and utility knows the actual realization of the wd generation w at this time Of course, T lt < T < 0 The key feature of this model is the evolution of prices (p lt, p, p rt and renewable forecasts (ŵ lt, ŵ, w across markets We describe our stochastic models for these evolutions the followg 22 Evolution of prices Prices electricity markets are typically uncerta and volatile Thus, for example, when decidg the procurement strategy the long term market, a utility company does not know what the prices will be the termediate or real time markets However, general, conventional energy tends to be more expensive markets closer to real time The reason for this is that the margal costs of production tend to be highe spot markets than forward or long term markets because any conventional energy that is demanded closer to real time is provided by generators that have low start up time and these generators typically are more expensive than generators that require several hours to start up Our model for the evolution of prices across the markets focuses on the two features described above price volatility and creasg costs closer to real time Specifically, we assume that p lt is the known fixed price the long term markets Sce long term purchase commitments are made via bilateral contract where the utility company knows the price, this assumption is very mild and typically true reality Next, let p be the random price the termediate market We make the assumption that E[p > p lt This assumption reflects the fact that the generators used to supply electricity the termediate markets tend to have a higher margal cost of production, and hence on average, the price the termediate market is higher that the price long term market [31 Note that the utility company knows the exact realization of the termediate price at time of procurement the termediate market (ie, at time T However, when the utility company is makg a purchase commitment the long term market, it is uncerta about the price the termediate market Similarly, let p rt be the random price the real time (or spot market We make the assumption that E[p rt p > p This assumption states that given any realization of the termediate price, the real time prices are higher on average This reflects the fact that the electricity generated real time market comes from fast ramp up generators which have a higher margal cost of production Note that at t = 0, the utility company knows the exact realization of the real time price, but its value is uncerta at the time of the long term and termediate purchase commitments Importantly, the assumptions on the price evolution imply that E[p rt > E[p > p lt However, any particular realization of the prices may have the price real time market less than the termediate price, or the price the termediate market less that the long term price Additionally, we make the followg mild regularity assumptions Let [p, p denote the support of the random variable p, where p > 0 We assume that p is associated with a density function f p ( which is contuous over (p, p, and f p (p > 0 for p (p, p Also, E[p rt p = p is contuous with respect to p over (p, p Fally, we assume that E[p, E[p rt < Our model is a generalization of the well-known martgale model of forecast evolution [17, 14, 18, which assumes additionally that the random variables p lt p and p p rt are dependent and normally distributed Fally, it is important to note that our model for price evolution assumes that the utility company cannot impact the price This corresponds to assumg that the utility company under consideration is a small participant the market, and hence is a price-taker This is a common assumption literature [5, 21 and is typically true if there is enough competition from the demand side electricity markets Of course it would also be terestg to consider a more general model where the prices arrive endogenously from market behavior However, corporatg multiple forward markets to multi-stage models is notoriously difficult and, as a result, the assumption of perfect foresight (no prediction error is typically needed to obta analytic results these cases, eg, [26, 7 23 Evolution of wd forecasts A fundamental challenge of corporatg long-term contracts for renewable energy is the uncertaty about how much renewable energy will be realized real time and the fact that the utility company must guarantee that it procures enough generation to meet the demand despite this uncertaty In particular, at the time of each market, the utility company needs to forecast how much wd generation will be available at time t = 0 Note that if the available wd energy was known with certaty, the utility company could purchase its entire remag demand exactly the long term market; however, because the amount of available wd energy at time t = 0 is highly uncerta at the time of the long term market, the utility company needs to balance its lowest cost purchase long term with better estimates of wd energy closer to real time To capture this, we consider a model where the forecast accuracy of the wd generation improves upon movg closer to real time, ie, t = 0

4 p h(r p lt p F p E1 (r (pf p (p dp E[p rt p = pp (E 1 E 2 > r; E 1 < r (p f p (p dp = 0 (6 p=p p=p Specifically, we assume that the utility company comes to know the value of wd realized at time t = 0, denoted by w, only when purchasg generation the real time market Durg the long term and termediate markets the utility only has estimates of w, denoted by ŵ lt and ŵ respectively Clearly, general, uncertaty about the wd generation w decreases as one moves closer to real time Formally, we capture this by assumg that ŵ = ŵ lt E 1, and w = ŵ E 2, (1 where E 1 and E 2 are zero mean dependent random variables, dependent of the prices p and p rt In other words, we assume that the forecast of w evolves with dependent crements where the random variables E 1 and E 2 capture these crements Note that ŵ lt is a coarser estimate of w than ŵ, sce the long term forecast error ŵ lt w = E 1 E 2 is more variable than the termediate forecast error ŵ w = E 2 Our model is a generalization of the well-known martgale model of forecast evolution [17, 14, 18, which makes the additional assumption that E 1 and E 2 follow a Gaussian distribution Let [L 1, R 1 and [L 2, R 2 denote respectively the supports of the random variables E 1 and E 2, where L 1, L 2 { } R and R 1, R 2 R { } We make the followg regularity assumptions on the distributions of E 1 and E 2: they are associated with contuously differentiable density functions, denoted by f E1 ( and f E2 ( respectively, with f Ei (x > 0 for x (L i, R i This concludes our discussion of the model of this paper The followg notation is used heavily the remader of this paper We use P (E to denote the probability of an event E, and F X to denote the complementary cumulative distribution function associated with the random variable X, ie, F X(x = P (X > x Fally, [x := max{x, 0} 3 OPTIMAL PROCUREMENT In this section, we formalize the utility s procurement problem, and then characterize the optimal procurement strategy This strategy is the basis for the explorations of the impact of creased penetration of renewables and changes to market structure subsequent sections 31 The procurement problem To beg, note that the procurement decision of the utility each market can depend only on the formation available to the utility company at that time Specifically, each market, we consider procurement strategies that depend only on (i the wd estimate available at the time of purchase, (ii the price of conventional generation the current market, and (iii the total conventional generation that has already been procured 2 Accordgly, let q lt (ŵ lt, p lt denote the quantity of conventional generation procured the long term market, given the long term wd estimate ŵ lt and long term price p lt Similarly, let q (ŵ, q lt, p denote the quantity of conventional generation procured the termediate market, given the correspondg wd estimate ŵ, the realized price p, and q lt (the quantity procured already Fally, let q rt(w, q lt q, p rt denote the the quantity of conventional generation procured the real time market, 2 Given our stochastic model for prices and wd forecast evolution, it is easy to show that there is no loss of optimality restrictg policies to this class which depends on the realized wd w, the actual price the real time market p rt, and the quantity procured already, ie, q lt q For notational convenience, we often drop the arguments from these functions and simply write q lt, q, q rt We make the assumption that the utility cannot sell power any market 3, and thus the quantities procured all three markets must be non negative When determg these procurement quantities, we assume that the utility company is seekg to mimize its expected total cost while ensurg that the total quantity purchased these three markets satisfies the residual demand (ie, the demand mus the available wd real time Thus, we can express the optimal procurement problem as follows: m E [p lt q lt p q lt,q,q q p rtq rt (P rt subject to q lt 0, q 0, q rt 0 q lt q q rt w d Recall that q lt, q and q rt are functions that depend on the correspondg wd estimate, price of conventional generation, and the total procurement so far Here, the expectation is with respect to the randomness associated with the wd forecast evolution and price volatility The optimal procurement problem posed above is mathematically equivalent to a variant of the classical newsvendor problem [20, 27 In Section 7, we discuss the relationship between our work and the literature on the newsvendor problem A fal comment about the the optimal procurement problem above is that we have considered a three tiered market structure that models the common current practice In general, the optimal procurement problem is simply a Markov decision process, and can be studied for arbitrary numbers of tiers, eg, see [28 We limit ourselves to a three tiered structure this paper to keep the analysis simple and the resultg formulas terpretable 32 The optimal procurement strategy The followg theorem is the foundational result for the remader of the paper It characterizes the optimal procurement strategy for a utility company a three tiered market with price volatility Theorem 1 The optimal procurement strategy for the utility company the three tiered market scenario of Problem (P is: where q lt = [d ŵ lt r lt (2 q = [d ŵ q lt (p (3 q rt = [d w q lt q, (4 (p = ( 1 F E 2 and r lt is the unique solution of (6 p E[p rt p, (5 The proof of Theorem 1 is given at the end of this section A key feature of this theorem is that the structure of 3 Our model can be extended to relax this assumption However, we do not consider this generalization because it adds complexity without providg additional sight

5 the the optimal procurement strategy gives a natural terpretation to r lt and (p as reserve levels Specifically, at the time of purchase the long term market, d ŵ lt can be terpreted as an estimate of the conventional procurement that is required to meet the demand Then r lt is the additional reserve purchased by the utility to balance the current wd uncertaty and the higher cost of conventional energy subsequent markets The reserve (p has a similaterpretation We note that the termediate reserve level is a function of the price of conventional generation the termediate market For notational simplicity, we often drop the argument of this function, and simply write We henceforth refer to r lt and as the (optimal long term and termediate reserves respectively It is important to pot out that the reserves r lt and may be positive or negative A negative reserve implies that it is optimal for the utility to mata a net procurement level that is less than the currently anticipated residual demand, and purchase any shortfall subsequent markets The values of the optimal reserves depend on the price volatility (via the distribution of termediate and real time prices as well as the accuracy of the wd forecasts ŵ lt and ŵ (via the distributions of the random variables E 1 and E 2 Note that the optimal reserves do not depend on the demand d, or on procurements made prior markets As is evident from Equation (5, the optimal reserve the termediate market is structurally similar to the critical fractile solution of the classical newsvendor model [1 This is because, at the time of the termediate market, the utility company conjectures the average real time price to be E[p rt p, and havg already purchased the quantity q lt, it faces a problem similar to the classical newsvendor problem Fally, Theorem 1 highlights the need for the followg additional assumption order to avoid trivial solutions Assumption 1 We assume that the demand is large enough that the utility company procures a positive quantity the long term market That is, d ŵ lt r lt > 0 This assumption ensures that the optimal procurement problem the three market has a non-trivial solution the followg sense: if d ŵ lt r lt 0, then the utility company procures q lt = 0 the long term In this case, the procurement problem effectively reduces to a two market scenario Intuitively, Assumption 1 will hold as long as the demand d exceeds the peak capacity of the renewable stallations, which is of course true most current practical scenarios Assumption 1 allows us to rewrite the optimal procurement quantities three markets as follows: q lt = d ŵ lt r lt, q = [E 1 r lt (p, (7 q rt = [E 2 r lt m{e 1, r lt (p }, where we abuse notation to let qlt, q and qrt represent the (random quantities that are purchased the long term, the termediate, and the real time markets respectively We conclude this section with the proof of Theorem 1 Proof of Theorem 1 Sce the utility is required to procure enough generation to satisfy its demand, it it easy to see that the optimal strategy real time is to procure just enough conventional energy to exactly meet the demand, ie, qrt = [d w q lt q This ensures that the last constrat the optimization (P is always satisfied Havg decided the optimal strategy for real time, the optimization problem (P can now be re-written as m E [ p lt q lt p q lt,q q p rt [d w q lt q subject to q lt 0, q 0 This problem is a 3-stage Markov decision process [4, with the stages correspondg to the long term, termediate, and real time markets The state each market is the tuple consistg of (i the current wd estimate, (ii the price of conventional generation the current market, and (iii the total conventional generation procured so far The action each stage corresponds to the procurement decision that market, and the stage cost is the cost of that procurement We summarize the Markov decision process associated with the optimal procurement problem the table below stage state action stage cost 1 S 1 = (ŵ lt, p lt, 0 q lt p lt q lt 2 S 2 = (ŵ, p, q lt q p q 3 S 3 = (w, p rt, q lt q p rt [d w q lt q We can now compute the optimal procurement strategy for the termediate and the long term markets usg the dynamic programg algorithm [4 In the termediate market, the optimal procurement is the mimizer of the expected cost to go; it is therefore the solution of the followg optimization m q 0 E [ p q p rt [d w q lt q S 2 (8 Usg Equation (1 to write w = ŵ E 2 and makg the substitution q = d q lt ŵ r, we can write the objective function the above mimization problem as ξ(r = p (d q lt ŵ r E[p rt p E [E 2 r Here, we can thk of r as the additional reserve required the termediate market By direct differentiation, we get that ξ (r = p E[p rt p FE2 (r Sce ξ (r is nondecreasg, ξ( is convex Moreover, (p (L 2, R 2 is the unique mimizer of ξ( over R It is now easy to see that mimization (8 is convex, and that the optimal procurement the termediate market is given by (3 The derivation of the optimal procurement the long term market is more cumbersome Due to space constrats, we omit some algebraically tensive calculations here, and only outle the ma steps of the derivation Denotg the optimal cost to go from the termediate stage onwards by J2 (S 2, the optimal procurement the long term market is the solution to the followg optimization m q lt 0 E [p lt q lt J 2 (S 2 S 1 (9 In the above objective, we make the substitution q lt = d ŵ lt r, with the terpretation that r is the additional reserve the long term Denotg the objective now by ϕ(r, it can be shown that ϕ (r = h(r, where h(r is defed Equation (6 Differentiatg aga, we obta ϕ (r = h (r = E[p rt p = p f E1 (r z f E2 (z f p (p dz dp p Here, the tegral with respect to the variable p is over the terval [p, p, ie, the support of the random termediate price p Also, we terpret a density function to be zero outside the support of the correspondg random variable, if the support is fite Sce ϕ ( is nonnegative, ϕ( is convex Let := f p (p, p (p Sce f Ei (x > 0 over x (L i, R i, it follows that ϕ (r > 0 for r (L 1, R 1 R 2, and that ϕ (r = 0 for all r < L 1 and r > R 1 R 2 This implies that ϕ (r is

6 strictly creasg over the range r (L 1, R 1 R 2 An elementary application of the domated convergence theorem yields lim r ϕ (r = p lt E[p < 0, lim r ϕ (r = p lt > 0 It therefore follows that the equation ϕ (r = 0 has a unique solution r lt (L 1, R 1 R 2 Moreover, r lt is the unique mimizer of ϕ( over R Fally, it follows as before that the optimization (9 is convex, and that the optimal procurement the long term market is given by (2 4 THE IMPACT OF INCREASING RENEWABLE PENETRATION The penetration of renewable energy, particular of wd energy, is poised for major growth over the comg years A consequence of this is that utility companies will face an ever creasg supply side uncertaty In this section, we explore the impact of this growth on procurement Specifically, we ask: how will the optimal procurement policy change as the volume of termittent wd resources creases? To answer this question, we troduce a scalg regime for wd penetration, which models the effect of aggregatg the output of several wd generators A key feature of our scalg model is that it allows for varyg levels of stochastic dependence between the termittent energy sources beg aggregated For example, our model lets us study the aggregation of dependent sources, as well as perfectly correlated sources Based on this scalg model, we study how the optimal reserves, the amount of conventional generation procured, as well as the cost of procurement scale with creasg wd penetration Our analysis yields clean and easy to terpret scalg laws for these quantities Remarkably, the scalg laws reveal a decouplg between the impact of the level of stochastic dependence between different wd sources and the impact of market structure and wd forecast accuracy 41 A scalg regime We beg by describg our scalg model for wd penetration We start with a basele scenario Let us denote by α the average output of a sgle termittent generator For concreteness, we refer to this as the basele wd farm the followg We let Ẽ1 and Ẽ2 be the error random variables that relate the long term wd estimate ŵ lt to the estimate of the wd the termediate market ŵ and the actual wd realization w (see Equation (1 We assume that the long term market there is no better estimate of the wd than the long term average, ie, ŵ lt = α Note that the optimal procurement the basele scenario (before scalg can be computed usg Theorem 1 For the remader of this section, we use r lt and to denote respectively the optimal long term and the termediate reserve the basele scenario We emphasize that is a function of the realization of the termediate price p, although we suppress this dependence for notational convenience this section To scale the wd penetration, we troduce a scale parameter γ that is proportional to the capacity of wd generation The scale parameter γ may be terpreted as the number of (homogeneous wd farms whose aggregate output is available to the utility company Thus, the average wd energy available to the utility company is given by ŵ lt (γ = γα Clearly, as the capacity of wd generation scales, the error the wd forecast available at both the long term and the termediate markets will also scale, and this scalg will depend on the correlation between the generation at each of the wd farms We use the followg simple, but general scalg model ŵ lt (γ = γα, E 1(γ = γ θ Ẽ 1, (10 E 2(γ = γ θ Ẽ 2, where we let θ [1/2, 1 It is important to pot out that our scalg regime leaves the prices of conventional generation unchanged; we scale only the volume of termittent generation To terpret this scalg, consider first the case θ = 1 In this case, E 1 = γẽ1 and E2 = γẽ2, implyg that ŵ(γ and w(γ scale proportionately with γ This scalg corresponds to a scenario where the aggregate wd output with γ wd farms is simply γ times the output of the basele wd farm One would expect such a scalg to occur if the wd farms are co-located The case of θ = 1/2 (with Ẽ1 and Ẽ2 beg normally distributed corresponds to a central limit theorem scalg, and seeks to capture the scenario where the output of each wd farm is dependent Equation (10 captures this scenario exactly if the forecast evolution distributions for each wd farm follow a Gaussian distribution If not, Equation (10 can be terpreted as an approximation for large enough γ based on the central limit theorem Intuitively, one would expect such a scalg if the different wd farms are geographically far apart Fally, the case θ ( 1, 1 seeks to capture correlations 2 that are termediate between dependence and perfect correlation (see, fostance [2, 36 Note that the forecast error distributions grow slowest when the outputs of different wd farms are dependent, and fastest when the outputs are perfectly correlated 42 Scalg results Given the scalg regime described above, we can characterize the impact of creasg penetration of termittent resources on the procurement of conventional generation First, we analyze how the optimal reserve levels scale with creasg wd penetration Next, we use these results to obta scalg laws for the procurement quantities the three markets, and also the total procurement and the total cost of procurement The scalg of reserve levels The followg lemma characterizes the scalg of the optimal reserves under our wd scalg model It shows the optimal reserve levels the long term and termediate markets follow the same scalg as imposed on the distributions of the forecast errors Lemma 1 Under the scalg regime defed Equation (10, the optimal long term and termediate reserves scale as: r lt (γ = γ θ r lt, (γ = γ θ (11 We emphasize that and (γ are both functions of the termediate price Equation (11 states that the function (γ scales proportionately to γ θ, ie, ((γ(p = γ θ (p Proof From Theorem 1, given any scale parameter γ, the optimal reserve the termediate market (γ is the unique solution of the equation P (E 2(γ > (γ = E[p (12 rt p Now, notg that E 2(γ = γ θ Ẽ 2, and P (Ẽ2 > p = p E[p rt p (by Theorem 1 applied to the basele scenario, we conclude that (γ = γ θ satisfies Equation (12

7 The optimal long term reserve r lt (γ is the unique solution of Equation (6, with E 1, E 2, and substituted by E 1(γ, E 2(γ, and (γ, respectively As before, usg the characterization of r lt obtaed from Equation (6 applied to the basele scenario, it is easy to verify that r lt (γ = γ θ r lt The scalg of procurement quantities Usg Lemma 1, we can now characterize the optimal procurement amounts the three markets As Section 3, to avoid the trivial solution with zero long term procurement, we restrict the range of scale parameter γ to satisfy d > γα γ θ r lt, (13 This ensures that the optimal long term procurement, given by qlt(γ = [d ŵ lt (γ r lt (γ is strictly positive 4 For the remader of this section, we focus on the range of scale parameter γ that satisfies Condition (13 Intuitively speakg, Condition (13 will hold as long as the demand is larger than the peak wd capacity Under this assumption, usg Equation (7 and Lemma 1, we get that the optimal procurement quantities the long term, termediate, and real time markets is given by qlt(γ = d γα γ θ r [ lt, q(γ = γ θ Ẽ 1 r lt, (14 [ qrt(γ = γ θ Ẽ 2 r lt m{ẽ1, r lt } The above equations reveal that as we crease the penetration of wd, the optimal procurement the termediate and the real time markets creases proportionately to γ θ This creasg procurement markets closer to real time is a consequence of the creasg uncertaty the renewable forecasts Indeed, note that the procurements the termediate and real time markets scale exactly the same manner as the forecast error distributions Therefore, these procurements scale slowest when outputs of different wd farms are dependent, and fastest when the outputs are perfectly correlated From the standpot of the system operator, Equation (14 describes how the stalled capacity of fast ramp conventional generators that can supply energy the termediate and real time markets needs to scale as the capacity of wd generation scales The scalg of the total procurement and the total cost of procurement As we scale the wd capacity, we expect that the amount of total conventional energy procured, as well as the cost of procurement must decrease The followg theorem characterizes the scalg laws for these quantities Let T P (γ and T C(γ denote respectively the total procurement of conventional energy and the total cost of procurement, correspondg to scale parameter γ Theorem 2 For the range of scale parameter γ satisfyg Equation (13, the expected total procurement and the expected total cost are given by E[T P (γ = d αγ δγ θ, (15 E[T C(γ = p lt (d αγ δ γ θ, (16 4 One sufficient θ-dependent condition that ensures that Condition (13 holds is γ < d α r lt This imposes an upper bound on the degree of penetration that we consider where, δ 0 and δ 0 are defed as δ r lt E [Ẽ1 ( r lt E [Ẽ2 r lt m{ẽ1, r lt }, { } δ p lt r lt E p [Ẽ1 ( r lt { } E p rt [Ẽ2 r lt m{ẽ1, r lt } The scalg law (15 has the followg terpretation If there were no uncertaty the wd generation, ie, w(γ = ŵ lt (γ = αγ, then it easy to see that the utility would procure the exact residual demand, ie, d αγ the long term market (sce the prices of conventional generation crease on average as we move closer to real time From (15, we see that on average, the utility has to make an additional procurement equal to δγ θ as a result of the uncertaty the wd generation The fact that this additional procurement grows proportionately to γ θ highlights the benefit of aggregatg dependent wd sources Specifically, if θ = 1, ie, the wd farms are co-located, then the additional procurement is of the same order as the wd capacity On the other hand, if θ = 1/2, ie, the wd farms are geographically far apart, then the additional procurement grows much slower than the wd capacity The scalg law (16 has a similaterpretation As before, note that if there were no uncertaty the wd generation, then p lt (d γα equals the (optimal average cost of conventional procurement From (16, we see that on average, the utility curs an additional cost equal to δ γ θ as a result of the uncertaty the wd generation Once aga, we see the benefit of aggregatg dependent wd sources Thus, tuitively speakg, a utility company would benefit from signg long term contracts with wd generators that are as geographically spread out as possible 5 The structure of the scalg laws (15 and (16 also reveals an terestg separation between the impact of the level of stochastic dependence between different wd sources and the impact of market structure and wd forecast accuracy To see this, consider the additional procurement δγ θ Equation (15 and the additional cost δ γ θ Equation (16 The factor γ θ is purely dependent upon the volume of wd capacity and the stochastic dependence between the outputs of the different wd farms On the other hand, the factors δ and δ depend on only the prices of conventional energy the three markets (and their volatility, and the basele forecast error distributions In other words, δ and δ are variant with respect to the aggregation of wd sources As we discuss the next section, this separation has terestg consequences for market design We now present the proof of Theorem 2, which follows from Lemma 1 and Theorem 1 Proof of Theorem 2 The expected total procurement and the expected total cost follow trivially from Equation (14 To prove that δ 0, we note that the optimization problem P requires that the total procurement exceed d w(γ for every realization of wd w Thus, the expected total procurement must satisfy E[T P (γ d γα, which implies, usg Equation (15, that δ 0 To prove that δ 0, we consider a hypothetical procurement formulation with no wd uncertaty Specifically, consider a scenario which the values of E 1(γ and E 2(γ are revealed to the utility a priori This implies that 5 This above discussion of course assumes implicitly that network capacity is not a bottleneck the utilization of the available wd energy

8 the utility knows the exact realization of the wd generation, and therefore the exact residual demand, before makg its procurement decisions for conventional generation This conventional generation can be procured the same threetiered market structure as our origal formulation, with the same price volatility The problem of optimal procurement this hypothetical settg, seekg to mimize the average cost of procurement (and subject to satisfyg the demand d, can be formulated as before Sce the prices are ordered on average, it is easy to show that the optimal policy this case is to buy the entire residual demand (equal to d w(γ the long term market However, yet another feasible strategy is to make procurements accordg to Equation (14 Clearly, the average cost for this latter strategy must be at-least the average cost of the optimal policy, ie, ( E p,p rt p lt (d γα γ θ r [ lt p γ θ Ẽ 1 r lt [ p rtγ θ Ẽ 2 r lt m{ẽ1, r lt } p lt (d w(γ Note that the expectation above is only taken with respect to the prices Now, takg expectations with respect to E 1(γ and E 2(γ, we conclude that E[T C(γ p lt (d αγ, which implies, usg Equation (16, that δ 0 This completes the proof 5 THE OPTIMAL PLACEMENT OF THE INTERMEDIATE MARKET In current electricity markets, the termediate market takes place about 14 hours before t = 0 (this market is called the day ahead or forward market [8 Because the demand can be predicted reasonably accurately by this time, this allows the utility to procure most of its generation much before the time of use However, accurate prediction of wd the current day ahead markets is not feasible Thus, it is commonly suggested that as the penetration of wd creases, the system operator may decide to move the termediate market closer to real time to allow for better prediction of the wd In this section, we consider the optimal placement of the termediate market and study how this optimal placement changes as we crease the penetration of wd energy Recall that, our model, the termediate market takes place T time units prior to the time of use (ie t = 0 If the system operator were to move this termediate market closer to real time, this would imply better estimates of available wd at that time However, movg the market closer to real time, the procured conventional generation must come from generators that have faster ramp up times These generators are typically more expensive and hence we would expect that the price of the conventional generation the termediate market would crease (on average as we move the market closer to real time We defe the optimal placement of the termediate market as the one that mimizes the average total procurement cost Note that this optimal placement balances an improvg forecast of the wd generation movg closer to real time with an creasg price of conventional energy The optimal placement is clearly a function of how the forecast errors Ẽ1 and Ẽ2 change as the function of the termediate market time T It also depends on how the jot distribution of the prices of conventional generation the termediate and the real time market changes as T changes Let us denote by T the optimal placement of the termediate market that mimizes the expected total cost of procurg conventional generation Importantly, Theorems 1 and 2 allow the computation of T numerically However, there is little structural sight for this placement that can be provided But, an important question that we can provide analytic sight about is the followg: how does the optimal placement of the termediate market change as we crease the penetration of wd energy? The answer to this question has significant implications for future market design Remarkably, based on the scalg regime developed the previous section, the followg theorem tells us that the optimal placement does not depend upon the scale parameter γ or the aggregation parameter θ Theorem 3 For the wd scalg satisfyg Equation (13, over the range of γ satisfyg Condition (13 for all considered placements of the termediate market, the optimal placement of the termediate market is dependent of the scale parameter γ and the correlation parameter θ Proof From Theorem 2, we note that the expected total cost of procurement is given by E[T C(γ = p lt (d αγ δ γ θ Furthermore, the effect of placement is only on the parameter δ (via the distributions of Ẽ1, Ẽ2, p, and the prices p and p rt which is dependent of γ and θ Thus, the optimal placement is dependent of scale parameter γ and the aggregation parameter θ Theorem 3 has important implications for market design It says that the optimal termediate market placement can be decided dependently of how many wd farms are there the system, and how correlated their outputs are Thus, the system operator can keep the placement of this market fixed as more wd energy is corporated to the system 6 THE VALUE OF ADDITIONAL FORWARD MARKETS One of the key objectives of this paper is to study the impact of creased renewable penetration on the structure of electricity markets As seen the previous section, the optimal placement of the termediate market is dependent of the amount of wd present the system In this section, we look at another important market design question: can we facilitate the penetration of renewable energy by providg additional forward markets? It is commonly suggested that havg additional markets would be beneficial, sce this allows the utility to better exploit the evolution of the wd forecast Indeed, the tuition is true terms of procurement cost havg additional forward markets benefits the utility company (and thus the end consumer by lowerg the average cost of conventional energy procurement However, mimizg cost is not the only goal Another important question is if additional forward markets also lower the total amount of conventional energy that a utility company needs to procure From the environmental viewpot, this is an extremely relevant question; reducg conventional energy use is one of the key drivg factors for the renewable portfolio standards Indeed, one would desire that the policy decision of addg a forward market for conventional energy creases the efficiency of the available renewable sources by decreasg our consumption of conventional generation In order to address the impact of additional markets on total procurement, we study the effect of the addition of a sgle termediate (forward market on the (average total conventional procurement Specifically, we compare the average total procurement under the three market scenario with the average total procurement under a scenario where there is no termediate market Let us denote the average total procurement under the three market scenario by