Optimization and Equilibrium in Energy Economics

Transcription

1 Optimization and Equilibrium in Energy Economics Michael C. Ferris University of Wisconsin, Madison IPAM Workshop Los Angeles January 11, 2016 Ferris (Univ. Wisconsin) IPAM 2016 Supported by DOE/LBNL 1 / 30

2 Welcome and thanks What: Physical system + markets + stochastics + modeling + computation Who: power system engineers, economists, mathematicians, operations researchers, and computer scientists Why: create more dialogue between researchers in this area with different expertize How? IPAM: Christian Ratsch and Roland McFarland Organizing committee: Benjamin Hobbs, Antonio Conejo, Andrew Philpott, Claudia Sagastizabal All of you for attending and preparing presentations Outcomes: new ideas and models, white paper summary This tutorial aims to provide some context and vocabulary for the meeting Ferris (Univ. Wisconsin) IPAM 2016 Supported by DOE/LBNL 2 / 30

3 Power generation, transmission and distribution Determine generators output to reliably meet the load Gen MW Load MW, at all times. Power flows cannot exceed lines transfer capacity. Ferris (Univ. Wisconsin) IPAM 2016 Supported by DOE/LBNL 3 / 30

4 Single market, single good: equilibrium Spatial extension: Locational Marginal Prices (LMP) at nodes (buses) in the network Walras: 0 s(π) d(π) π 0 Market design and rules to foster competitive behavior/efficiency Supply arises often from a generator offer curve (lumpy) Technologies and physics affect production and distribution Ferris (Univ. Wisconsin) IPAM 2016 Supported by DOE/LBNL 4 / 30

5 The PIES Model (Hogan) - Optimal Power Flow (OPF) min c(x) x cost s.t. Ax q balance Bx = b, x 0 technical constr Ferris (Univ. Wisconsin) IPAM 2016 Supported by DOE/LBNL 5 / 30

6 The PIES Model (Hogan) - Optimal Power Flow (OPF) min c(x) x cost s.t. Ax d(π) balance Bx = b, x 0 technical constr q = d(π): issue is that π is the multiplier on the balance constraint Such multipliers (LMP s) are critical to operation of market Can solve the problem iteratively or by writing down the KKT conditions of this QP, forming an LCP and exposing π to the model Existence, uniqueness, stability from variational analysis EMP does this automatically from the annotations Ferris (Univ. Wisconsin) IPAM 2016 Supported by DOE/LBNL 5 / 30

7 Reformulation details 0 Ax d(π) µ 0 0 = Bx b λ 0 c(x) A T µ B T λ x 0 empinfo: dualvar π balance replaces µ π Ferris (Univ. Wisconsin) IPAM 2016 Supported by DOE/LBNL 6 / 30

8 Reformulation details 0 Ax d(π) π 0 0 = Bx b λ 0 c(x) A T π B T λ x 0 empinfo: dualvar π balance replaces µ π LCP/MCP is then solvable using PATH π z = λ, F (z) = x A T B T A d(π) B z + b c(x) Ferris (Univ. Wisconsin) IPAM 2016 Supported by DOE/LBNL 6 / 30

9 Reformulation details (VI formulation) 0 Ax q π 0 0 = Bx b λ 0 c(x) A T π B T λ x 0 q = d(π) 0 = p(q) + π q Inverse demand p(q): π = p(q) q = d(π) [ ] c(x) (x, q) C, 0 + N p(q) c (x, q) (x, q) solves VI (F, C), F (x, q) = ( c(x), p(q)) T New solvers for VI: PATHVI, decomposition, distributed solution Straightforward to extend to more general production functions and cost functions Ferris (Univ. Wisconsin) IPAM 2016 Supported by DOE/LBNL 6 / 30

10 Extensions: maximizing profit and multiple agents max π T x c(x) x s.t. Ax d(π) Bx = b, x 0 profit balance technical constr Issue is that there are multiple producers i The price is now determined by total production max x i p( j x j ) T x i c i (x i ) profit s.t. B i x i = b i, x i 0 technical constr Ferris (Univ. Wisconsin) IPAM 2016 Supported by DOE/LBNL 7 / 30

11 Special case: perfect competition max p( x j ) x i π T x i c i (x i ) j s.t. B i x i = b i, x i 0 profit technical constr 0 i x i d(π) π 0 When there are many agents, assume none can affect π by themselves Each agent is a price taker Two agents, d(π) = d π, d = 24, c 1 = 3, c 2 = 2 KKT(1) + KKT(2) + Market Clearing gives Complementarity Problem x 1 = 0, x 2 = 22, π = 2 Ferris (Univ. Wisconsin) IPAM 2016 Supported by DOE/LBNL 8 / 30

12 MOPEC min θ i (x i, x i, p) s.t. g i (x i, x i, p) 0, i x i p solves VI(h(x, ), C) equilibrium min theta(1) x(1) g(1)... min theta(m) x(m) g(m) vi h p cons Trade/Policy Model (MCP) (Generalized) Nash Reformulate optimization problem as first order conditions (complementarity) Use nonsmooth Newton methods to solve Solve overall problem using individual Split model (18,000 vars) via region optimizations? = + Gauss-Seidel, Jacobi, Asynchronous Ferris (Univ. Wisconsin) IPAM 2016 Supported by DOE/LBNL 9 / 30

13 rvoir and one thermal plant, as shown in Figure 1. We Hydro-Thermal System (Philpott/F./Wets) nd de ne Figure 1: Example hydro-thermal system. Competing agents (consumers, or generators in energy market) Each agent maximizes objective independently (profit) Market prices are function of all agents activities Ferris (Univ. Wisconsin) IPAM 2016 Supported by DOE/LBNL 10 / 30

14 Simple electricity system optimization problem SO: max d k,u i,v j,x i 0 s.t. W k (d k ) C j (v j ) + V i (x i ) k K j T i H U i (u i ) + v j d k, i H j T k K x i = xi 0 u i + hi 1, i H u i water release of hydro reservoir i H v j thermal generation of plant j T x i water level in reservoir i H prod fn U i (strictly concave) converts water release to energy C j (v j ) denote the cost of generation by thermal plant V i (x i ) future value of terminating with storage x (assumed separable) W k (d k ) utility of consumption d k Ferris (Univ. Wisconsin) IPAM 2016 Supported by DOE/LBNL 11 / 30

15 SO equivalent to CE (price takers) Consumers k K solve CP(k): max d k 0 Thermal plants j T solve TP(j): max v j 0 Hydro plants i H solve HP(i): max u i,x i 0 W k (d k ) π T d k π T v j C j (v j ) π T U i (u i ) + V i (x i ) s.t. x i = xi 0 u i + hi 1 Perfectly competitive (Walrasian) equilibrium is a MOPEC CE: d k arg max CP(k), k K, v j arg max TP(j), j T, u i, x i arg max HP(i), i H, 0 π U i (u i ) + v j d k. i H j T k K Ferris (Univ. Wisconsin) IPAM 2016 Supported by DOE/LBNL 12 / 30

16 General Equilibrium models (static) (C) : max x k X k U k (x k ) s.t. π T x k i k (y, π) (I ) :i k (y, π) = π T ω k + j α kj π T g j (y j ) (P) : max π T g j (y j ) y j Y j (M) : max π 0 πt k x k k ω k j g j (y j ) s.t. π l = 1 l This is an example of a MOPEC: strategic, top-down, policy analyses Can extend these models in several ways: more goods (not just energy), more agents (refineries, farmers), different behavior patterns: who is driving the bus? Ferris (Univ. Wisconsin) IPAM 2016 Supported by DOE/LBNL 13 / 30

17 Bus or Taxi: two agents (duopoly) max x i p( j x j ) T x i c i (x i ) profit s.t. B i x i = b i, x i 0 technical constr Cournot: assume each can affect p by choice of x i Two agents, same data KKT(1) + KKT(2) gives Complementarity Problem x 1 = 20/3, x 2 = 23/3, π = 29/3 Exercise of market power (some price takers, some Cournot) Ferris (Univ. Wisconsin) IPAM 2016 Supported by DOE/LBNL 14 / 30

18 UBER: Bilevel Program (Stackelberg) Assumes one leader firm, the rest follow Leader firm optimizes subject to expected follower behavior Follower firms act in a competitive Nash manner Bilevel programs: min f (x, y ) x,y s.t. g(x, y ) 0, y solves min v(x, y) s.t. h(x, y) 0 y model bilev /deff,defg,defv,defh/; empinfo: bilevel min v y defv defh EMP tool automatically creates the MPCC min f (x, y ) x,y,λ s.t. g(x, y ) 0, 0 v(x, y ) + λ T h(x, y ) y 0 0 h(x, y ) λ 0 Ferris (Univ. Wisconsin) IPAM 2016 Supported by DOE/LBNL 15 / 30

19 Representative decision-making timescales in electric power systems Transmission Siting & Construction Power Plant Siting & Construction Closed-loop Control and Relay Setpoint Selection Maintenance Scheduling Long-term Forward Markets Load Forecasting Closed-loop Control and Relay Action Day ahead market w/ unit commitment Hour ahead market Five minute market 15 years 10 years 5 years 1 year 1 month 1 week 1 day 5 minute seconds ure Many 1: Representative interacting levels/hierarchies, decision-making with timescales different in time electric scaledpower decisions systems at each level - collections of models needed. presents. As an example of coupling of decisions across time scales, consider e siting Ferris of(univ. major Wisconsin) interstate transmission IPAM 2016 lines. One Supported of thebygoals DOE/LBNL in the 16 / exp 30

20 Complications and myriad of acronyms Size/integrity: AC/DC models, reactive power, new devices Day ahead, regulation, FTR s, co-optimization Semidefinite programming (global), EPEC s Discrete: Unit commitment (DAUC, RUC, RT) Topology optimization (e.g. Transmission line switching, siting) Design of flexible computing systems How do we price discrete decisions? (make-good, fix ints,...) Dynamic: Design/operation Multi-period, unit commitment, minimum up and down time Demand response, load shedding, demand bidding Stochastic: Security constraints - failures/reserves (SCED/SCUC) Stochastic demand Renewables/storage/feed-in prices Ferris (Univ. Wisconsin) IPAM 2016 Supported by DOE/LBNL 17 / 30

21 Discrete: Totally Unimodular Congestion Games 2,3,5 2,3,6 s 1,2,8 t 4,6,7 1,5,6 N = 3, m = 5. Each player chooses a path from s to t. Costs for one, two or three players using arc are given by triplets. The goal is to find a pure Nash equilibrium, i.e. a state x = (x 1,..., x N ) X such that, for each player i and x i X i : c i (x 1,..., x i,..., x N ) c i (x 1,..., x i,..., x N ). Theorem (DelPia/F./Micini) There is a strongly polynomial-time algorithm for finding a pure Nash equilibrium in symmetric TU congestion games. Ferris (Univ. Wisconsin) IPAM 2016 Supported by DOE/LBNL 18 / 30

22 Dynamic: PJM buy/sell storage Storage transfers energy over time (horizon = T ). PJM: given price path p t, determine charge q + t and discharge q t : max h t,q + t,q t T p t (qt q t + ) t=0 s.t. h t = eq t + qt 0 h t S 0 q t + Q 0 qt Q h 0, h T fixed Uses: price shaving, load shifting, transmission line deferral What about real-time storage, or different storage technologies? Ferris (Univ. Wisconsin) IPAM 2016 Supported by DOE/LBNL 19 / 30

23 Stochastic: Agents have recourse? Agents face uncertainties in reservoir inflows Two stage stochastic programming, x 1 is here-and-now decision, recourse decisions x 2 depend on realization of a random variable ρ is a risk measure (e.g. expectation, CVaR) SP: min c T x 1 + ρ[q T x 2 ] s.t. Ax 1 = b, x 1 0, T (ω)x 1 + W (ω)x 2 (ω) d(ω), x 2 (ω) 0, ω Ω. A T T W igure Constraints matrix structure of 15) problem by suitable subgradient methods in an outer loop. In the inner Ferris (Univ. Wisconsin) IPAM 2016 Supported by DOE/LBNL 20 / 30

24 Risk Measures VaR, CVaR, CVaR + and CVaR - Modern approach to modeling risk aversion uses concept of risk measures CVaR α : mean of upper tail beyond α-quantile (e.g. α = 0.95) Frequency VaR Probability 1 α CVaR Maximum loss Loss mean-risk, mean deviations from quantiles, VaR, CVaR Much more in mathematical economics and finance literature Optimization approaches still valid, different objectives, varying convex/non-convex difficulty Ferris (Univ. Wisconsin) IPAM 2016 Supported by DOE/LBNL 21 / 30

25 Stochastic price paths (day ahead market) min x,h,q +,q c1 (x) + E ω [ T ] p ωt (q ωt + qωt) + c 2 (q ωt + + qωt) t=0 s.t. h ωt = eq + ωt q ωt 0 h ωt Sx 0 q + ωt, q ωt Qx h ω0, h ωt fixed First stage decision x: amount of storage to deploy. Second stage decision: charging strategy in face of uncertainty Ferris (Univ. Wisconsin) IPAM 2016 Supported by DOE/LBNL 22 / 30

26 Contingency: a single line failure A network with N lines can have up to N contingencies Each contingency case: Corresponds to a different network topology Requires a different set of equations g and h E.g., equations gk and h k for the k-th contingency. Ferris (Univ. Wisconsin) IPAM 2016 Supported by DOE/LBNL 23 / 30

27 Security-constrained Economic Dispatch Base-case network topology g 0 and line flow x 0. If the k-th line fails, line flow jumps to x k in new topology g k. Ensure that x k is within limit, for all k. SCED model: min c T u Total cost u,x 0,...,x k s.t. 0 u ū GEN capacity const. g 0 (x 0, u) = 0 x x 0 x Base-case network eqn. Base-case flow limit g k (x k, u) = 0, k = 1,..., K Ctgcy network eqn. x x k x, k = 1,..., K Ctgcy flow limit Ferris (Univ. Wisconsin) IPAM 2016 Supported by DOE/LBNL 24 / 30

28 Model structure Base Case Contingency 1, time 0 Contingency 1, time 1 Costminimizing direction ED optimal point SCED optimal point Contingency Contingency 1, time 2 Base Case SCED Feasible Region Rows Contingency Figure : On the u 0 plane, the feasible region of a SCED is the intersection of K+1 polyhedra Columns Figure : Sparsity structure of the Jacobian matrix of a 6-bus case, considering 3 contingencies and 3 post-contingency checkpoints. Decomposition approaches allow solution of realistic sized models with many contingencies in minutes Ferris (Univ. Wisconsin) IPAM 2016 Supported by DOE/LBNL 25 / 30

29 Contracts in MOPEC (Philpott/F./Wets) Can we modify (complete) system to have a social optimum by trading risk? How do we design these instruments? How many are needed? What is cost of deficiency? Facilitated by allowing contracts bought now, for goods delivered later (e.g. Arrow-Debreu Securities) Conceptually allows to transfer goods from one period to another (provides wealth retention or pricing of ancilliary services in energy market) Can investigate new instruments to mitigate risk, or move to system optimal solutions from equilibrium (or market) solutions Ferris (Univ. Wisconsin) IPAM 2016 Supported by DOE/LBNL 26 / 30

30 Example as MOPEC: agents solve a Stochastic Program Buy y i contracts in period 1, to deliver D(ω)y i in period 2, scenario ω Each agent i: min C(x 1 i ) + ρ i ( C(x 2 i (ω)) ) s.t. p 1 x 1 i + vy i p 1 e 1 i (budget time 1) p 2 (ω)x 2 i (ω) p 2 (ω)(d(ω)y i + e 2 i (ω)) (budget time 2) 0 v i y i 0 (contract) 0 p 1 i ( e 1 i xi 1 ) 0 (walras 1) 0 p 2 (ω) i ( D(ω)yi + e 2 i (ω) x 2 i (ω) ) 0 (walras 2) Ferris (Univ. Wisconsin) IPAM 2016 Supported by DOE/LBNL 27 / 30

31 Theory and Observations agent problems are multistage stochastic optimization models perfectly competitive partial equilibrium still corresponds to a social optimum when all agents are risk neutral and share common knowledge of the probability distribution governing future inflows situation complicated when agents are risk averse utilize stochastic process over scenario tree under mild conditions a social optimum corresponds to a competitive market equilibrium if agents have time-consistent dynamic coherent risk measures and there are enough traded market instruments (over tree) to hedge inflow uncertainty Otherwise, must solve the stochastic equilibrium problem Research challenge: develop reliable algorithms for large scale decomposition approaches to MOPEC Ferris (Univ. Wisconsin) IPAM 2016 Supported by DOE/LBNL 28 / 30

32 Conclusions Optimization critical for understanding of power system markets Different behaviors are present in practice and modeled here Modern optimization within applications requires multiple model formats, computational tools and sophisticated solvers Policy implications addressable using MOPEC Stochastic MOPEC enables modeling dynamic decision processes under uncertainty Modeling, optimization, statistics and computation embedded within the application domain is critical Many new settings available for deployment; need for more theoretic and algorithmic enhancements Ferris (Univ. Wisconsin) IPAM 2016 Supported by DOE/LBNL 29 / 30

33 What is EMP? Annotates existing equations/variables/models for modeler to provide/define additional structure equilibrium vi (agents can solve min/max/vi) bilevel (reformulate as MPEC, or as SOCP) disjunction (or other constraint logic primitives) randvar dualvar (use multipliers from one agent as variables for another) extended nonlinear programs (library of plq functions) Currently available within GAMS Ferris (Univ. Wisconsin) IPAM 2016 Supported by DOE/LBNL 30 / 30