MOPEC: Multiple Optimization Problems with Equilibrium Constraints

Transcription

1 MOPEC: Multiple Optimization Problems with Equilibrium Constraints Michael C. Ferris Joint work with Roger Wets University of Wisconsin Scientific and Statistical Computing Colloquium, University of Chicago: Tuesday March 26, 2013 Ferris (Wisconsin) MOPEC SSC 1 / 24

2 The PIES Model (Hogan) min x s.t. c T x Ax = d(p) Bx = b x 0 cost balance technical constr Issue is that p is the multiplier on the balance constraint of LP Extended Mathematical Programming (EMP) facilitates annotations of models to describe additional structure Can solve the problem by writing down the KKT conditions of this LP, forming an LCP and exposing p to the model EMP does this automatically from the annotations Ferris (Wisconsin) MOPEC SSC 2 / 24

3 Reformulation details empinfo: dualvar p balance replaces µ p 0 = Ax d(p) µ 0 = Bx b λ 0 A T µ B T λ + c x 0 LCP/MCP is then solvable using PATH p z = λ, F (z) = x A T B T A p d(p) B λ + b x c Ferris (Wisconsin) MOPEC SSC 3 / 24

4 Transmission Line Expansion Model (F./Tang) 1 min x X π ω ω i N d ω i p ω i (x) Nonlinear system to describe power flows over (large) network Multiple time scales Dynamics (bidding, failures, ramping, etc) Uncertainty (demand, weather, expansion, etc) p ω i (x): Price (LMP) at i in scenario ω as a function of x Use other models to construct approximation of p ω i (x) Ferris (Wisconsin) MOPEC SSC 4 / 24

5 Solution approach Use derivative free method for the upper level problem (1) Requires p ω i (x) Construct these as multipliers on demand equation (per scenario) in an Economic Dispatch (market clearing) model But transmission line capacity expansion typically leads to generator expansion, which interacts directly with market clearing Interface blue and black models using Nash Equilibria (as EMP): empinfo: equilibrium forall f: min expcost(f) y(f) budget(f) forall ω: min scencost(ω) q(ω)... Ferris (Wisconsin) MOPEC SSC 5 / 24

6 Generator Expansion (2): f F : min π ω C j (y j, qj ω ) r(h f y j ) y f ω j G f j G f s.t. j G f y j h f, y f 0 G f : Generators of firm f F y j : Investment in generator j qj ω : Power generated at bus j in scenario ω C j : Cost function for generator j r: Interest rate Market Clearing Model (3): ω : min C j (y j, q z,θ,q ω j ω ) s.t. f j G f z ij = dj ω j N( pj ω ) q ω j i I (j) z ij = Ω ij (θ i θ j ) b ij (x) z ij b ij (x) u j (y j ) q ω j u j (y j ) (i, j) A (i, j) A z ij : qj ω : θ i : Ω ij : b ij (x): u j (y), u j (y): Real power flowing along line ij Real power generated at bus j in scenario ω Voltage phase angle at bus i Susceptance of line ij Line capacity as a function of x Generator j limits as a function of y Ferris (Wisconsin) MOPEC SSC 6 / 24

7 MOPEC min θ i (x i, x i, y) s.t. g i (x i, x i, y) 0, i x i equilibrium min theta(1) x(1) g(1)... min theta(m) x(m) g(m) is solved in a Nash manner Allows multipliers from one problem to be used in another problems dualvar p g(1) Ferris (Wisconsin) MOPEC SSC 7 / 24

8 Feasibility KKT of min π ω C j (y j, qj ω ) r(h f y j ) f F (2) y f Y ω j G f j G f KKT of min C j (y j, q (z,θ,q ω j ω ) ω (3) ) Z(x,y) f j G f Models (2) and (3) form a complementarity problem (CP via EMP) Solve (3) as NLP using global solver (actual C j (y j, qj ω ) are not convex), per scenario (SNLP) this provides starting point for CP Solve (KKT(2) + KKT(3)) using EMP and PATH, then repeat Identifies CP solution whose components solve the scenario NLP s (3) to global optimality Ferris (Wisconsin) MOPEC SSC 8 / 24

9 Scenario ω 1 ω 2 Probability Demand Multiplier SNLP (1): Scenario q 1 q 2 q 3 q 6 q 8 ω ω Ferris (Wisconsin) MOPEC SSC 9 / 24

10 Scenario ω 1 ω 2 Probability Demand Multiplier SNLP (1): Scenario q 1 q 2 q 3 q 6 q 8 ω ω EMP (1): Scenario q 1 q 2 q 3 q 6 q 8 ω ω Firm y 1 y 2 y 3 y 6 y 8 f f Ferris (Wisconsin) MOPEC SSC 9 / 24

11 Scenario ω 1 ω 2 Probability Demand Multiplier SNLP (2): Scenario q 1 q 2 q 3 q 6 q 8 ω ω Ferris (Wisconsin) MOPEC SSC 10 / 24

12 Scenario ω 1 ω 2 Probability Demand Multiplier SNLP (2): Scenario q 1 q 2 q 3 q 6 q 8 ω ω EMP (2): Scenario q 1 q 2 q 3 q 6 q 8 ω ω Firm y 1 y 2 y 3 y 6 y 8 f f Ferris (Wisconsin) MOPEC SSC 10 / 24

13 Observations But this is simply one function evaluation for the outer transmission capacity expansion problem Number of critical arcs typically very small But in this case, pj ω volatile are very Outer problem is small scale, objectives are open to debate, possibly ill conditioned Economic dispatch should use AC power flow model Structure of market open to debate Comparing the different types of objective functions LMP LMP and Generator Cost LMP with interest rate Types of generator expansion also subject to debate Suite of tools is very effective in such situations Ferris (Wisconsin) MOPEC SSC 11 / 24

14 EMP: variational inequalities Allows (GAMS) models to be manipulated to form other problems of interest via a simple EMP info file: VI(f, C): vi f x cons 0 f (x) + N C (x) generates a variational inequality where C defined by cons Either generates the equivalent complementarity (KKT) problem, or provides problem structure for algorithmic exploitation Extension of (square) nonlinear systems and mixed complementarity problems QVI can be specified in the same manner Ferris (Wisconsin) MOPEC SSC 12 / 24

15 MOPEC and 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 is solved in a Nash manner Ferris (Wisconsin) MOPEC SSC 13 / 24

16 MOPEC and min θ i (x i, x i, p) s.t. g i (x i, x i, p) 0, i x i h(x, p) = 0 equilibrium min theta(1) x(1) g(1)... min theta(m) x(m) g(m) vi h p cons is solved in a Nash manner Ferris (Wisconsin) MOPEC SSC 13 / 24

17 Water rights pricing (Britz/F./Kuhn) Ferris (Wisconsin) MOPEC SSC 14 / 24

18 The model AO max q i,x i,wo i 0 s.t. q i p x i,f w f i f {int,lab} q i (x i,f + e i,f ) ɛ i,f f x i,land e i,land wo i 1 = x i,wat + wo i 0 0 i i q i d(p) p 0 e i,lab x i,lab w lab 0 i Ferris (Wisconsin) MOPEC SSC 15 / 24

19 The model IO max q i,x i,wo i 0 s.t. ( q i p ) x i,f w f f q i (x i,f + e i,f ) ɛ i,f f x i,land e i,land wo i 1 = x i,wat + wo i 0 0 i i q i d(p) p 0 e i,lab x i,lab w lab 0 i Ferris (Wisconsin) MOPEC SSC 15 / 24

20 The model IO max q i,x i,wo i,wri b,wri s 0 s.t. ( q i p x i,f w f wri b f q i (x i,f + e i,f ) ɛ i,f f (w wr + τ) + wr s i w wr ) x i,land e i,land wo i 1 = x i,wat + wo i wr i + wri b x i,wat + wri s 0 i q i d(p) p 0 0 e i,lab x i,lab w lab 0 i i 0 wri s wri b w wr 0 i i Ferris (Wisconsin) MOPEC SSC 15 / 24

21 Figure 4 Click here to download high resolution image Different Management Strategies Ferris (Wisconsin) MOPEC SSC 16 / 24

22 is Hydro-Thermal paper we will System illustrate (Philpott/F./Wets) the concepts using the voir and one thermal plant, as shown in Figure 1. Figure 1: Example hydro-thermal system. Ferris (Wisconsin) MOPEC SSC 17 / 24

23 Simple electricity system optimization problem SSP: s.t. min j T C j(v(j)) i H V i(x(i)) i H U i (u(i)) + j T v(j) d, x(i) = x 0 (i) u(i), i H u(i), v(j), x(i) 0. u(i) water release of hydro reservoir i H v(j) thermal generation of plant j T production function U i (strictly concave) converts water release to energy water level reservoir i H is denoted x(i) C j (v(j)) denote the cost of generation by thermal plant V i (x(i)) to be the future value of terminating the period with storage x (assumed separable) Ferris (Wisconsin) MOPEC SSC 18 / 24

24 SSP equivalent to CE Thermal plants solve The hydro plants i H solve TP(j): max πv(j) C j (v(j)) s.t. v 1 (j) 0. HP(i): max πu i (u(i)) + V i (x(i)) s.t. x(i) = x 0 (i) u(i) u(i), x(i) 0. Perfectly competitive (Walrasian) equilibrium is a MOPEC CE: u(i), x(i) arg max HP(i), i H, v(j) arg max TP(j), j T, 0 ( i H U i (u(i)) + j T v(j)) d π 0, Ferris (Wisconsin) MOPEC SSC 19 / 24

25 GAMS/EMP: Stochastic programming tools GAMS has extended mathematical programming tools to build models of models Given the core model, can annotate parameters as random variables Automatically solves expected value problem Can solve using deterministic equivalent or specialized solvers (including Bender s decomposition, importance sampling (DECIS), etc) Also allows for a variety of new constructs (such as risk measures and chance constraints) R ω [c 0 (x) + ] T p ωt (q ωt + qωt) + c 1 (q ωt + + qωt) t=0 Ferris (Wisconsin) MOPEC SSC 20 / 24

26 Two stage problems TP(j): max π 1 v 1 (j) C j (v 1 (j))+ R ω [π 2 (ω)v 2 (j, ω) C j (v 2 (j, ω))] s.t. v 1 (j) 0, v 2 (j, ω) 0, for all ω Ω. HP(i): max π 1 U i (u 1 (i))+ R ω [π 2 (ω)u i (u 2 (i, ω)) + V i (x 2 (i, ω))] s.t. x 1 (i) = x 0 (i) u 1 (i) + h 1 (i), x 2 (i, ω) = x 1 (i) u 2 (i, ω) + h 2 (i, ω), for all ω Ω, u 1 (i), x 1 (i) 0, u 2 (i, ω), x 2 (i, ω) 0, for all ω Ω. Ferris (Wisconsin) MOPEC SSC 21 / 24

27 Results Suppose every agent is risk neutral and has knowledge of all deterministic data, as well as sharing the same probability distribution for inflows. SP solution is same as CE solution Using coherent risk measure (weighted sum of expected value and conditional variance at risk), 10 scenarios for rain 1 High initial storage: risk-averse central plan (RSP) and the risk-averse competitive equilibrium (RCE) have same solution (but different to risk neutral case) 2 Low initial storage: RSP and RCE are very different. Since the hydro generator and the system do not agree on a worst-case outcome, the probability distributions that correspond to an equivalent risk neutral decision will not be common. 3 Extension: Construct MOPEC models for trading risk Ferris (Wisconsin) MOPEC SSC 22 / 24

28 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 (Wisconsin) MOPEC SSC 23 / 24

29 Conclusions Optimization helps understand what drives a system Collections of models needed for specific decisions Modern optimization within applications requires multiple model formats, computational tools and sophisticated solvers EMP model type is clear and extensible, additional structure available to solver Uncertainty is present everywhere We need not only to quantify it, but we need to hedge/control/ameliorate it Stochastic MOPEC models capture behavioral effects (as an EMP) Policy implications addressable using Stochastic MOPEC Extended Mathematical Programming available within the GAMS modeling system Ferris (Wisconsin) MOPEC SSC 24 / 24