Coupled Optimization Models for Planning and Operation of Power Systems on Multiple Scales

Transcription

1 Coupled Optimization Models for Planning and Operation of Power Systems on Multiple Scales Michael C. Ferris University of Wisconsin, Madison Computational Needs for the Next Generation Electric Grid, April 19, 2011 Ferris (Univ. Wisconsin) Coupled Opt Models Cornell, April / 26

2 The premise The next generation electric grid will be more dynamic, flexible, constrained, and more complicated. Decision processes (in this environment) are predominantly hierarchical. Models to support such decision processes must also be layered or hierachical. Optimization and computation facilitate adaptivity, control, treatment of uncertainties and understanding of interaction effects. Coupling of smaller models with well defined interfaces allows validation, understanding, and enhanced solution techniques. Ferris (Univ. Wisconsin) Coupled Opt Models Cornell, April / 26

3 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 A 1: monster Representative model is difficult decision-making to validate, timescales inflexible, in prone electric to errors. power systems presents. As an example of coupling of decisions across time scales, consider Ferris (Univ. Wisconsin) Coupled Opt Models Cornell, April / 26

4 Transmission Line Expansion Model (1) min x X π ω di ω pi ω (x) ω i N s.t. Ax b (RTO budget (and other) constraints) N: The set of all nodes X : The set of all line expansions being considered x: Amount of investment in line x X ω: Demand scenarios π ω : Probability of scenario ω occuring di ω : Demand of load node i in in scenario ω pi ω (x): Price (LMP) at load node i in scenario ω as a function of x Ferris (Univ. Wisconsin) Coupled Opt Models Cornell, April / 26

5 Ferris (Univ. Wisconsin) Coupled Opt Models Cornell, April / 26

6 Generator Expansion (2) f F : s.t. min π ω C j (y j, q y j ω ) r(h f y j ) f ω j G f j G f y j h f j G f y f 0 (budget cons) F : The set of firms G f : The set of all generators belonging to firm f ω: Demand scenarios y j : Amount of investment in generator j q j : Real power generated at bus j C j : Cost function of generator j as a function of y j and q j r: Interest Rate Ferris (Univ. Wisconsin) Coupled Opt Models Cornell, April / 26

7 Market Clearing Model (3) ω : min C j (y j, qj ω ) z,θ,q f j G f s.t. qj ω dj ω = z ij j N( pj ω ) (flow balance) i I (j) z ij = Ω ij (θ i θ j ) (i, j) A (line data) b ij (x) z ij b ij (x) (i, j) A (line capacity) u j (y j ) q j ū j (y j ) (gen capacity) z ij : q j : θ i : Ω ij : b ij (x): u j (x), ū j (x): Real power flowing along the i-j arc Real power generated at bus j Voltage phase angle at bus i Susceptance of line i-j Line capacity as a function of x Generator j limits as a function of y Ferris (Univ. Wisconsin) Coupled Opt Models Cornell, April / 26

8 Flow of information min x Transmission line expansion (1) E(f (p ω (x))) s.t. f F Generator expansion (2) = KKT (2) x, q ω y ω Market clearing (3) = KKT (3) Why isn t this just a monster model? x, y q ω, p ω (x) Ferris (Univ. Wisconsin) Coupled Opt Models Cornell, April / 26

9 Solution methods Use deriviative free method for the upper level problem (1) Models (2) and (3) form an MCP (via EMP) Solve (3) as NLP using global solver, per scenario (SNLP) Solve (KKT(2) + KKT(3)) using EMP and PATH, then repeat Alternative: Can show (due to specific problem structure that there is a (convex) NLP whose KKT conditions are that MCP Useful for theoretical analysis Resulting problem is too large for NLP solvers Can show that Gauss-Seidel/Jacobi method on problems in (2) and (3) converges in this case - decoupling makes problem tractable for large scale instances Ferris (Univ. Wisconsin) Coupled Opt Models Cornell, April / 26

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 (Univ. Wisconsin) Coupled Opt Models Cornell, April / 26

11 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 (Univ. Wisconsin) Coupled Opt Models Cornell, April / 26

12 Scenario ω 1 ω 2 ω 3 ω 4 ω 5 Probability Demand Multiplier EMP (1): Scenario q 1 q 2 q 3 q 6 q 8 ω ω ω ω ω Firm y 1 y 2 y 3 y 6 y 8 f f Ferris (Univ. Wisconsin) Coupled Opt Models Cornell, April / 26

13 Scenario ω 1 ω 2 ω 3 ω 4 ω 5 Probability Demand Multiplier EMP (2): Scenario q 1 q 2 q 3 q 6 q 8 ω ω ω ω ω Firm y 1 y 2 y 3 y 6 y 8 f f Ferris (Univ. Wisconsin) Coupled Opt Models Cornell, April / 26

14 Coupling collections of (sub)-models with well defined (information sharing) interfaces facilitates: appropriate detail and consistency of sub-model formulation (each of which may be very large scale, of different types (mixed integer, semidefinite, nonlinear, variational, etc) with different properties (linear, convex, discrete, smooth, etc)) ability for individual subproblem solution verification and engagement of decision makers ability to treat uncertainty by stochastic and robust optimization at submodel level and with evolving resolution ability to solve submodels to global optimality (by exploiting size, structure and model format specificity) (A monster model that mixes several modeling formats loses its ability to exploit the underlying structure and provide guarantees on solution quality) Ferris (Univ. Wisconsin) Coupled Opt Models Cornell, April / 26

15 Extended Mathematical Programs Optimization models improve understanding of underlying systems and facilitate operational/strategic improvements under resource constraints Problem format is old/traditional min x f (x) s.t. g(x) 0, h(x) = 0 Extended Mathematical Programs allow annotations of constraint functions to augment this format. Developing interfaces and exploiting hierarchical structure using computationally tractable algorithms will provide overall solution speed, understanding of localized effects, and value for the coupling of the system. Ferris (Univ. Wisconsin) Coupled Opt Models Cornell, April / 26

16 Nash Equilibria Nash Games: x is a Nash Equilibrium if x i arg min x i X i l i (x i, x i, q), i I x i are the decisions of other players. Quantities q given exogenously, or via complementarity: 0 H(x, q) q 0 empinfo: equilibrium min loss(i) x(i) cons(i) vifunc H q Key difference: optimization assumes you control the complete system Equilibrium determines what activities run, and who produces what Ferris (Univ. Wisconsin) Coupled Opt Models Cornell, April / 26

17 Supply function equilibria OPF(α): min y energy dispatch cost (y, α) s.t. conservation of power flow at nodes Kirchoff s voltage law, and simple bound constraints α are (given) price bids, parametric optimization Ferris (Univ. Wisconsin) Coupled Opt Models Cornell, April / 26

18 Supply function equilibria OPF(α): min y energy dispatch cost (y, α) s.t. conservation of power flow at nodes Kirchoff s voltage law, and simple bound constraints α are (given) price bids, parametric optimization Leader(ᾱ i ): max αi,y,λ firm i s profit (α i, y, λ) s.t. 0 α i ˆα i y solves OPF(α i, ᾱ i ) Note that objective involves multiplier from OPF problem Ferris (Univ. Wisconsin) Coupled Opt Models Cornell, April / 26

19 Supply function equilibria OPF(α): min y energy dispatch cost (y, α) s.t. conservation of power flow at nodes Kirchoff s voltage law, and simple bound constraints α are (given) price bids, parametric optimization Leader(ᾱ i ): max αi,y,λ firm i s profit (α i, y, λ) s.t. 0 α i ˆα i y solves OPF(α i, ᾱ i ) Note that objective involves multiplier from OPF problem Leader(ᾱ i ): max αi,y,λ firm i s profit (y, λ, α) s.t. 0 α i ˆα i y, λ solves KKT(OPF(α i, ᾱ i )) This is an example of an MPCC since KKT form complementarity constraints Ferris (Univ. Wisconsin) Coupled Opt Models Cornell, April / 26

20 Multi-player EPEC and security constraints (ᾱ 1, ᾱ 2,..., ᾱ m ) is an equilibrium if ᾱ i solves Leader(ᾱ i ), i (Nonlinear) Nash Equilibrium where each player solves an MPCC Ferris (Univ. Wisconsin) Coupled Opt Models Cornell, April / 26

21 Multi-player EPEC and security constraints (ᾱ 1, ᾱ 2,..., ᾱ m ) is an equilibrium if ᾱ i solves Leader(ᾱ i ), i (Nonlinear) Nash Equilibrium where each player solves an MPCC MPCC is hard (lacks a constraint qualification) Nash Equilibrium is PPAD-complete (Chen et al, Papadimitriou et al) In practice, also require contingency (scenario) constraints imposed in the OPF problem Leader/follower game: Stackleberg Supply chains with market leader Ferris (Univ. Wisconsin) Coupled Opt Models Cornell, April / 26

22 Transmission switching Opening lines in a transmission network can reduce cost But that is infeasible A feasible dispatch Total Cost: $20/MWh x 100 MWh 200 MW generated +$40/MWh x 100 = $6,000/h Capacity limit: 100 MW $20/MWh 100 MW generated Capacity limit: 100 MW $20/MWh 133 MW 67 MW 67 MW 33MW 200 MW load 100 MW generated 33MW 200 MW load $40/MWh $40/MWh 67 MW (a) Infeasible due to line capacity 9 (b) Feasible dispatch 10 Need to use expensive generator due to power flow characteristics and capacity limit on transmission line Ferris (Univ. Wisconsin) Coupled Opt Models Cornell, April / 26

23 The basic model min g,f,θ c T g generation cost s.t. g d = Af, f = BA T θ A is node-arc incidence θ L θ θ U bus angle constraints ḡ L g ḡ U generator capacities f L f f U transmission capacities with transmission switching (within a smart grid technology) we modify as: min g,f,θ s.t. c T g g d = Af θ L θ θ U ḡ L g ḡ U either f i = (BA T θ) i, f L,i f i f U,i if i closed or f i = 0 if i open Use EMP to facilitate the disjunctive constraints (several equivalent formulations, including LPEC) Ferris (Univ. Wisconsin) Coupled Opt Models Cornell, April / 26

24 But who cares? Why aren t you using my *********** algorithm? (Michael Ferris, Boulder, CO, 1994) Ferris (Univ. Wisconsin) Coupled Opt Models Cornell, April / 26

25 But who cares? Why aren t you using my *********** algorithm? (Michael Ferris, Boulder, CO, 1994) Show me on a problem like mine Must run on defaults Must deal graciously with poorly specified cases Must be usable from my environment (Matlab, R, GAMS,...) Must be able to model my problem easily EMP provides annotations to an existing optimization model that convey new model structures to a solver NEOS is soliciting case studies that show how to do the above, and will provide some tools to help Ferris (Univ. Wisconsin) Coupled Opt Models Cornell, April / 26

26 Stochastic competing agent models (with Wets) Competing agents (consumers, or generators in energy market) Each agent maximizes objective independently (utility) Market prices are function of all agents activities Additional twist: model must hedge against uncertainty Facilitated by allowing contracts bought now, for goods delivered later 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 move to system optimal solutions from equilibrium (or market) solutions Ferris (Univ. Wisconsin) Coupled Opt Models Cornell, April / 26

27 The model details: c.f. Brown, Demarzo, Eaves Each agent maximizes: u h = s ( π s κ l c α h,l h,s,l ) Time 0: Time 1: p 0,l c h,0,l + q k z h,k p 0,l e h,0,l l k l p s,l c h,s,l l l p s,l D s,l,k z h,k + l k p s,l e h,s,l Additional constraints (complementarity) outside of control of agents: 0 h z h,k q k 0 0 h d h,s,l p s,l 0 Ferris (Univ. Wisconsin) Coupled Opt Models Cornell, April / 26

28 Stochastic programming and risk measures SP: min c x + R[d y] s.t. Ax = b T (ω)x + W (ω)y(ω) h(ω), for all ω Ω, x 0, y(ω) 0, for all ω Ω. Annotations are slightly more involved but straightforward: Need to describe probability distribution Define (multi-stage) structure (what variables and constraints belong to each stage) Define random parameters and process to generate scenarios Can also define risk measures on variables Automatic reformulation (deterministic equivalent), solvers such as DECIS, etc. Ferris (Univ. Wisconsin) Coupled Opt Models Cornell, April / 26

29 Additional techniques requiring extensive computation Chance constraints: Prob(T i x + W i y i h i ) 1 α - can reformulate as MIP and adapt cuts Use of discrete variables (in submodels) to capture logical or discrete choices Optimization of simulation or noisy functions Robust or stochastic programming Decomposition approaches to exploit underlying structure identified by EMP Nonsmooth penalties and reformulation approaches to recast problems for existing or new solution methods Conic or semidefinite programs - alternative reformulations that capture features in a manner amenable to global computation Ferris (Univ. Wisconsin) Coupled Opt Models Cornell, April / 26

30 Conclusions 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 Extended Mathematical Programming available within the GAMS modeling system Able to pass additional (structure) information to solvers Embedded optimization models automatically reformulated for appropriate solution engine Exploit structure in solvers Extend application usage further Ferris (Univ. Wisconsin) Coupled Opt Models Cornell, April / 26