The Analysis of Electricity Storage Location Sites in the Electric Transmission Grid

Transcription

1 Proceedings o the 2010 Industrial Engineering Research Conerence A. Johnson and J. Miller, eds. The Analysis o Electricity Storage Location Sites in the Electric Transmission Grid Thomas F. Brady College o Engineering and Technology Purdue University North Central, Westville, IN Andrew L. Liu School o Industrial Engineering Purdue University, West Laayette, IN Abstract Wind power is a growing sector o the United States electricity generation system. The integration o wind energy into the electrical grid presents capacity-related challenges to policy makers. The introduction o large-scale storage capabilities or wind power could lead to cost and reliability beneits i the interace locations are chosen wisely. The location o storage capability in the electrical grid relative to sources o demand and supply becomes a key research question. This paper describes the development o a conceptual ramework using marginal pricing and associated modeling requirements necessary to analyze siting issues o storage capacity into the electricity grid. Keywords Wind, storage, locational marginal price, transmission. 1. Introduction To eectively harness the power o electricity generated rom wind, signiicant inrastructure challenges exist. Individual wind turbines must be sited and constructed as part o a wind arm and then connected to the electrical grid inrastructure, and the generated power should be managed accordingly. Due to its stochastic nature, wind energy cannot be controlled; it must be managed. The integration and management o wind power within the highly complex, interconnected electrical inrastructure o the United States presents numerous policy making and decision analysis scenarios. A signiicant wind power challenge is that wind turbines generate electricity only when the wind speed is above a certain threshold. Practical, cost eective means o storing, or inventorying electricity do not yet exist but are being developed rapidly. The introduction o storage mechanisms into the electricity generation and distribution inrastructure will present both challenges and opportunities to system s reliability and economic operation. There have been relatively extensive studies rom both reliability and cost-beneit point o view on wind plants integration into a transmission grid, such as the reports rom NERC [1] and NREL [6]. Studies on the beneits o electricity storage in a transmission grid with variable-output resources are also available, such as the Sandia report [4] and NREL report [2]. While the studies take a holistic approach to analyze the true beneits o various storage technologies, they do not consider a more detailed yet important aspect o storage its impacts on transmission congestion. A notable work along this line is by Denholm and Sioshansi [3] (and see the reerences therein). However, they only consider highly aggregated transmission networks that do not have loop lows. This paper presents a mathematical programming model that can quantiy the economic beneits resulting rom the alleviation o transmission congestion by properly siting grid-level electricity storage. In addition, a stochastic version o the model is also presented that is more closely resemble to the two-settlement system (day-ahead and real-time) in a Regional Transmission Organization (RTO) market. This work will hence provide one more account o storage s economic beneits and may contribute to urther deployment o various electricity storage technologies on a transmission grid. The rest o the paper is organized as ollows. Section 2 introduces the optimal power low models that are used to derive locational marginal prices with storage and variable-output resources in a transmission network. Section 3 provides an example to show that when sited properly, a storage acility can reduce nodal electricity prices. Section 4 discusses methods to solve the optimal power low models; while Section 5 concludes the paper with suggestions to uture research directions.

2 2. Problem Formulation To study the siting o storage acilities in a transmission constrained network with intermittent resources, such as wind arms, we consider an optimal power low (OPF) problem solved by the system operator o a transmission grid 1. In such a problem, the system operator s the costs o meeting electricity demand through dispatching to ensure that the demand is served by the lowest-cost resources irst and that the grid is used optimally. An underlying assumption is that the system operator knows the true production costs o each generation unit, which is equivalent to the no-market-power assumption in a deregulated electricity market. To start with a manageable model, we consider a lossless, DC-load low network. The Kirchho Laws (Kirchho Circuit Law and Kirchho Voltage Law), which uniquely determine the load low in a transmission network with loops, can be represented by the power transmission and distribution actors (see, or example, Metzler [7]). Also to reduce the problem dimension, we adopt the same hub-spoke type o network models as in Hobbs [5]. In such a network model, an arbitrary node is designated as a hub, and a low rom node i to node j is considered as rom i to the hub irst and then rom the hub to j. Beore presenting the mathematical ormulation o the OPF problem, we irst list out the notation to be used. Sets, Indices and Dimensions i, j N Set o nodes in a transmission grid (can be a production or a demand node, or both); N = N G Set o all electricity generators; G = F v V Set o generators with variable outputs; V = V ; G/V being the set o non-intermittent generators. l,k A Set o (directional) transmission links in the network N ; i.e., link i- j is dierent than link j-i; A = 2J n( ) = i n( ) is a mapping, with n( ) = i meaning that the electric power producer is at node i Variables q q ω+ q ω y ω i s i s ω+ i, s ω i Electrical energy that is scheduled to be produced by generator ; [MWh] The extra amount o energy generated above q by generator in real time; [MWh] The amount o energy that is short o the scheduled generation q by generator in real time; [MWh] MWs transmitted in real time (i.e., ater random events occurred) rom the hub to node i; [MW] Energy scheduled to be stored or discharged at node i (with positive meaning charging and negative meaning discharging); [MWh] Extra energy to be stored (+) or discharged (-) at node i in real time; [MWh] Parameters d i (ω) Electricity demand at node i, which can be a random variable deined on a probability space o (Ω,F,P) to relect load variations; [MWh] c (q ) The cost unction o generator with respect to producing q units o energy; [$] K (ω) The generation capacity o generator, which can be a random variable to relect the variable availability o an intermittent resource, such as a wind arm; [MW]. S i The storage capacity at node i ( S i represents the discharge capacity at node i); [MW] PTDF li The (l,i)-th element o the matrix consisting o power transmission-distribution actors T l The transmission capacity bound on line l; [MW] r The payments to unit that is orced to reduce its generation due to large amount o energy rom variable-output generating resources in real-time; [$/MWh]. 2.1 Deterministic OPF with Storage To present the concept o locational marginal prices on a transmission grid with respect to the shadow prices o the market clearing conditions, we irst consider a deterministic version o the OPF problem. 1 Here we consider a snapshot o the OPF problem, say, in an hour. Multiperiod models will be discussed in Section 5 as uture research.

3 q, y, s C(q) := c (q ) G subject to y i = G:n( )=i q d i s i, i N, (µ i ) y i = 0, (σ) PT DF li y i T l, l A, (λ l ) 0 q K, G, S i s i S i, i N, (γ, γ, ζ i, ζ i ). (1) The variables in the parentheses in (1) are the Lagrangian multipliers o the corresponding constraints. The irst two sets o constraints together guarantee market clearing; that is, total system demand can be met by total system supply. The Lagrangian multipliers (µ i s) associated with the irst set o constraints are the locational marginal prices (LMPs), which are deined as the marginal cost to serve one-more unit o load at the corresponding location. The storage variable s i can be either positive or negative as indicated in the notation section, with positive indicating storage charging and negative meaning discharging. Note that we do not have a irm (or a generator) 2 index o the storage variables. The underlying assumption is that we can aggregate all storage acilities at each node and consider them as one acility, and the system operator has ull control in determining the real-time usage o these storage acilities. More detailed modeling o dierent storage technologies are discussed in Section 5. The last three sets o constraints in (1) provide bounds on transmission, generation, and storage, respectively. To derive an explicit expression o the LMPs, we deine the Lagrangian unction as ollows. L(q,y,s) = C(q) + µ j (y j + G G:n( )= j [γ (q K ) γ q ] q + d j + s j ) σ y j + + [ζ j (s j S j ) ζ j s j ]. k A λ k (T k PTDF k j y j ) (2) Assume that the cost unction o each irm at each node is convex and dierentiable; then so is the aggregated unction C(q). With the linearity o the constraints, the irst-order optimality condition (a.k.a the KKT condition) is both necessary and suicient or an optimal solution to the OPF problem (1). Hence, or any optimal solution (q,y,s ), the ollowing equation must hold. yi L(q,y,s ) = µ i σ PTDF ki λ k = 0, i N. (3) k A which gives the explicit expression o LMPs (or the ease o arguments, the asterisk is dropped, with the understanding that the equation below corresponds to the optimal solutions to the OPF problem (1)): µ i = σ + PTDF ki λ k, i N. (4) k A A general LMP ormulation consists three components, energy, transmission congestion, and energy losses. Since we consider a lossless network, ormula (4) only consists o the irst two parts, with the irst being energy price at the hub node and the second being the economic rents created by transmission congestions. With the problem deined in (1) and the ormula or calculating LMPs given in (4), we can study the eects on LMPs with storage at dierent locations within a constrained transmission grid. An example will be given in Section 3 to illustrate such eects. 2.2 Stochastic OPF with Storage In real world situations, uncertainty is an inherit eature in transmission grid operation. Electricity demand constantly changes over time, and generation or transmission capacities may vary due to orced outages. Such randomness is aggravated when there is large amount o variable-output generation capacities on the grid, such as wind generators. As a result, it is important to establish an OPF model that explicitly considers various random events that could 2 A irm may own several generation units. The distinction between irms and units is not essential under the perect competition assumption.

4 happen over a short period o time. A two-stage stochastic optimization model o the OPF problem is presented in this section to account or the randomness in real-time grid operation. In the irst stage, the system operator dispatches available units and storage acilities beore the realization o various random actors, such as the real-time load and outputs rom wind power plants. The second stage represents the system operator s decisions ater all uncertainties are realized. Such a two-stage stochastic problem closely resembles the two-settlement system employed by most RTOs in the US, (such as PJM, ISONE, NYISO and MISO), in which a day-ahead settlement determines the generation schedules or the next day and adjustments to such schedules are made in real time. The problem s ormulation is provided below. q, s Θ(q,s) := E ξ(ω) C(q, q+, q ) := c (q + q ω+ q ω+, q ω, y ω, s ω+, s ω q ω ) + r q ω G G/V subject to y ω i = G/V n( )=i q + G/V n( )=i q ω+ G/V n( )=i y ω i = 0, (σ ω ) PT DF li y ω i T l, l A, (λ ω l ) q ω + v V n(v)=i q q ω 0, q + q ω+ K, G/V, (γ ω, γω ) s i s ω i S i, s i + s ω+ i S i, i N, (ζ ω i, ζω i ) q ω+, q ω, s ω+ i, s ω i 0, F, i N. K v (ω) d i (ω) s i s ω+ i + s ω i, i N, (µ ω i ) In the stochastic OPF model (5), ξ(ω) is a random vector in R N+V and ξ(ω) = (d 1 (ω),...,d N (ω),k 1 (ω),...,k V (ω)) T. With each realization o a random event ω Ω, ξ is reerred to as a scenario. The superscript o ω over y, q +, and s +, is used to emphasize that they are not random variables. Instead, they are the actions corresponding to each realization o a random event. In the objective unction, in addition to production costs, there are also payments to the units that are orced to reduce their generation rom their schedules due to excessive energy rom the variable-output resources. Here we assume that the payments r are exogenous or each G. This is similar to the rules in several RTO markets, such as in MISO, where such payments are reerred to as Revenue Suiciency Guarantee (RSG) payments. Those who are required to produce more are compensated by the real-time locational marginal prices at the corresponding nodes, and hence do not appear in the cost-minimizing objective unction. The index set G/V represents an assumption that only non-variable-output generators have the lexibility to increase or reduce production in real time, such as combustion turbine units. The irst set o constraints represents another assumption that the system operator does not dispatch variable-output generators; that is, the operator will always use whatever amount o energy produced rom them in real time to meet the demand, and such generators always produce at ull capacity. This is relected by the term v V :n(v)=i K v (ω). For simplicity, the generation capacities rom non-variable-output resources or transmission capacities are not treated as random variables. Modeling-wise it is trivial to incorporate such randomness (given that the change o transmission capacities does not change the network topology). Methods or solving the stochastic OPF model will be discussed in Section An Illustrative Example In this section we consider a simple example in which the proper siting o a storage acility lowers the LMPs. Consider a network consisting o three nodes. The network topology is shown in Figure 1. Assume that the resistance on the three lines are the same. Choose Node 3 as the hub and the counterclockwise orientation as positive direction. Then the power transmission and distribution actors can be calculated and are provided in Table 1. Suppose that there are two generators, one sitting at Node 1 with a marginal cost o 10 $/MWh, and one at Node 3 with a marginal cost o 50 $/MWh. Assume that neither o the generator is capacity constrained. The load is at Node 3 o 100 MWh. The line between Node 2 and Node 3 are capacity constrained at 30 MW. The let picture in Figure 1 depicts the situation without storage options. The cheaper generator at Node 1 cannot meet the demand at Node 3, due to Kirchho Laws (5)

5 1 3 $50/MWh 1 3 $50/MWh G1 G2 G1 G2 LMP = 30 MW LMP = $50/MWh 100 MWh LMP = 30 MW LMP! $20/MWh 100 MWh 2 2 storage Figure 1: Comparison o LMPs no storage vs. storage Table 1: PTDFs o the 3-node network (Node 3 is the hub) node \ link (1, 2) (2, 3) (3, 1) (2, 1) (3, 2) (1, 3) combined with the transmission constraint on link 2-3. Hence, the more expensive generator has to be used to serve the load. The resulting LMPs at Node 1 and Node 3 are then 10 $/MWh and 50 $/MWh, respectively. With storage at Node 2, as shown in the right picture in Figure 1, by Kirchho Laws, the lows rom Node 1 to Node 3 and that rom Node 1 to Node 2 are o the opposite direction when passing the link 2-3, hence alleviating the congestion on link 2-3. Consequently, the demand at Node 3 can be supplied solely by the cheaper generator at Node 1. By solving the OPF problem (1) on the example (which is a deterministic linear program according to the data provided), with a 10 MW storage capacity at Node 2, the resulted LMPs at Node 1 and 3 are 10 $/MWh and 20 $/MWh, respectively. 4. Solution Methods The deterministic OPF problem (1) is a convex problem under the convexity assumption o all cost unctions. Furthermore, the constraints are linear under the lossless, DC-load low assumptions. Such problems can be eiciently solved by many existing convex optimization solvers such as MINOS, SNOPT, FILTER, KNITRO, IPOPT etc. The stochastic OPF problem is much more challenging computationally. With a inite sample space Ω, the stochastic problem (5) can be solved as a deterministic problem by writing out all possible scenarios. However, suppose that the sample space is o cardinality m. The total number o scenarios associated with ξ(ω) is then m (N+V ), which can quickly become too large to be handled by any computers. An alternative approach is the sampling average approximation (SAA) method, which uses Monte Carlo simulation techniques to generate scenarios. Suppose that ξ 1,ξ 2,...ξ Φ are the scenarios generated by a Monte Carlo technique. Deine a vector z as z = (q +, q, y, s +, s ) T R 2F+2J+2N and let g h (q,s,z;ξ), h H represent a generic constraint in the stochastic OPF problem (5), with H being the index set o all the constraints in (5). Let Θ Φ (q,s) denote the optimal value unction o the ollowing optimization problem. q, s, z subject to 1 Φ Φ φ=1 C(q, z φ ) g h (q, s, z φ ; ξ φ ) 0, h H, φ = 1,...,Φ. (6) Then instead o solving the stochastic problem (5), one can solve the ollowing deterministic approximation problem q, s Θ Φ (q, s). (7) Under the assumptions o convex cost unctions and a lossless, DC network, the resulting deterministic optimization problem is again a convex problem with linear constraints, which can be solved eiciently with medium to relativelylarge scale problems. By the Strong Law o Large Numbers, or a given (q,s), the sample average Θ Φ (q,s) converges to the corresponding expected value Θ(q, s) almost surely, which justiies the usage o the SAA method. The convergence, however, may be very slow ([8]), meaning that to close the gap o the objective values between Problem (5)

6 and (7), many samples may be needed rom the Monte Carlo simulation. One condition that may improve the convergence rate is the relative completeness o the recourse problem ([9]). That is, or any (q, s), the easible region in the stochastic problem (5) is always easible. This is equivalent to say that whatever the demand is or the amount o energy is produced by variable-output resources in real time, the system-wide supply and demand can always be balanced. Such an assumption is an issue o reliability, and is an important question that deserves its own treatment. We will leave the detailed implementation o the SAA approach to uture research. 5. Conclusion and Future Research In this paper we presented a modeling and solution ramework to study the (short-term) impacts o storage capacities on locational marginal prices in a transmission-constrained network. The 3-node example discussed in Section 3 clearly indicates that when sited properly, storage acilities may improve the usage o transmission grids by alleviating congestions and lower the LMPs to consumers. Both a deterministic and a stochastic version o the optimal power low model are established. While the deterministic OPF model can be solved eiciently under modest assumptions on cost unctions and the networks, the stochastic OPF model posts a series o computational challenges ranging rom sampling methods to large-scale computation. A natural step ollowing this work is to solve the stochastic version o the OPF problem with more realistic network models and data. Note that an implicit assumption in either version o the OPF model is that the variable-output resources are already connected to the grid. In reality, the siting o variable-output resources, especially wind arms, is a very challenging issue. Considering the siting o wind arms and storage acilities together may bring more beneits to grid operation and to consumers than consider their locations separately. A nontrivial and important extension is to build models to study the co-siting problem. Another direction o uture work is to build a multi-stage model (say, a multi-hour model with the time horizon o a week, which is a common short-term planning period or system operators). With such a model, the engineering details o dierent storage technologies (such as charging and discharging time) and o various generation technologies (such as ramping constraints on combined-cycle units) can be represented more realistically. In addition, such a multi-period model can be naturally ormulated as a unit commitment model, which includes 0-1 integer variables to indicate whether a generator should be committed in the next planning period. A stochastic, multi-period, integer unit commitment problem would certainly require extending the current rontier o both theory and computation methods on stochastic programming and would be a challenging area or uture research. Reerences [1] North American Electric Reliability Corporation. Accommodating high levels o variable generation. http: // [2] P. Denholm, E. Ela, B. Kirby, and M. Milligan. The role o energy storage with renewable electricity generationy. Technical report, National Renewable Energy Laboratory, [3] P. Denholm and R. Sioshansi. The value o compressed air energy storage with wind in transmission-constrained electric power systems. Energy Policy, 37(8): , [4] J. Eyer and G. Corey. Energy storage or the electricity grid: Beneits and market potential assessment guide. Sandia Report SAND , Sandia National Labrotories, [5] B. F. Hobbs. Linear complementarity models o Nash-Cournot competition in bilateral and Poolco power markets. IEEE Transactions on Power System, 16(2): , [6] National Renewable Energy Laboratory. Eastern wind integration and transmission study. gov/wind/systemsintegration/pds/2010/ewits_inal_report.pd, [7] C. B. Metzler. Complementarity models o competitive oligopolistic electric power generation markets. PhD thesis, Department o Mathematical Sciences, The Johns Hopkins University, Baltimore, MD, May [8] A. Shapiro. Monte Carlo simulation approach to stochastic programming. In Proceedings o the 2001 Winter Simulation Conerence, [9] A. Shapiro and A. Philpott. A tutorial on stochastic programming, 2007.