Risky Capacity Equilibrium Models with Incomplete Risk Tradin

Risky Capacity Equilibrium Models with Incomplete Risk Trading Daniel Ralph (Cambridge Judge Business School) Andreas Ehrenmann (CEEMR, Engie) Gauthier de Maere (CEEMR, Enngie) Yves Smeers (CORE, U catholique de Louvain) Paper available at https://www.eprg.group.cam.ac.uk/eprg-working-paper-1720/ MES W02 INI Cambridge, Mar 2019

Capacity equilibria under risk aversion RN competitive capacity equilibrium problem Risk averse competitive capacity equilibrium problem Electricity capacity expansion is kind of stochastic equilibrium Today, agents invest in physical capacity & financial hedges Agents behaviours are Risk Averse (Risk Neutrality is special case) Open Loop not strategic about how others react Tomorrow, spot market is uncertain Scenarios for different fuel & C prices, weather (demand) etc. This talk: Competitive spot market under Perfect Competition Paper also covers: Cournot spot market market power

RN competitive capacity equilibrium problem Risk averse competitive capacity equilibrium problem Why study stochastic equilibria under risk? Has anyone observed risk aversion in electricity ops. or planning? What about trading of contracts in electricity industry? Is that purely arbitraging, or, instead, risk management? At least one Associate Editor is skeptical:... why is it relevant to study risk aversion in electricity markets? Is there any evidence that decision makers in these markets are risk-averse? In practice, energy firms trade in risk: forward contracts on fuel long term contracts on price financial contracts to represent physical asset swaps

Outline RN competitive capacity equilibrium problem Risk averse competitive capacity equilibrium problem 1 RN competitive capacity equilibrium problem Risk averse competitive capacity equilibrium problem 2 Risk trading Risky capacity equilibrium problems 3 Risky capacity equilibrium models and existence 4

RN competitive capacity equilibrium problem Risk averse competitive capacity equilibrium problem Notation for competitive spot market in scenario ω Stage 1: Given (i) capacity x, (ii) spot market scenario ω, the spot equilibrium is given by: Genco optimises production Y ω given capacity x, price P ω ( ) min C ω Yω Pω Y ω s.t. 0 Y ω x Y ω where C ω (y) := convex cost of producing quantity y Retailer optimises consumption Q ω given price P ω min Qω P ω Q ω U ω (Q ω ) s.t. Q ω 0 where U ω : IR IR is concave utility of consumption Price of electricity P ω clears spot market given quantities: 0 Y ω Q ω P ω 0

RN competitive capacity equilibrium problem Risk averse competitive capacity equilibrium problem Classical Risk Neutral (RN) competitive capacity equilibrium Suppose agents are Risk Neutral via probability Θ = (Θ ω ) Ω ω=1 Stage 0: Invest in capacity x with convex investment cost I(x) Stage 1: Stochastic spot market, in Y ω and Q ω, as above Genco sets investment x & views production Y = (Y ω ) ω via [ ( ) ] I(x) + E Θ C ω Yω Pω Y ω min x,y s.t. x X, 0 Y ω x for all ω This is standard 2 stage stochastic program with recourse Retailer sets consumption Q ω in each scenario ω as before Price of electricity P ω clears spot market in each scenario ω

RN competitive capacity equilibrium problem Risk averse competitive capacity equilibrium problem Risk averse competitive capacity equilibrium problem Stage 0: Risk averse investment in plant capacity x Stage 1: Stochastic spot market sets Y ω, Q ω for each ω Genco invests in capacity x & risk W G, plans production Y via ( ( ) ) I(x) + r G C ω Yω Pω Y ω min x,y s.t. x X, W G W, 0 Y ω x for all ω Retailer trades risk W R, plans consumption Q via ( ( ) ) min r R P ω Y ω U ω Qω Q,W R s.t. 0 Q ω for all ω Price of electricity P ω clears spot market in each scenario using coherent risk measures r G or r R not expectations

Outline Risk trading Risky capacity equilibrium problems 1 RN competitive capacity equilibrium problem Risk averse competitive capacity equilibrium problem 2 Risk trading Risky capacity equilibrium problems 3 Risky capacity equilibrium models and existence 4

Risk Trading to hedge investments Risk trading Risky capacity equilibrium problems Risk trading means buying a set of financial securities / products / contracts W = (W ω ) that pay out depending on ω. Eg, take future prices P = (P ω ) and Genco s future production Y = (Y ω ) as fixed Genco s risked cost is Z G = ( C ω ( Yω ) Pω Y ω ) Genco wants W G such that r G (Z G W G ) r(z G ) What hedges W G & W R are available to Genco & Retailer? What do such contracts cost?

More about risk markets, I. Risk trading Risky capacity equilibrium problems Risk trading in power markets Forward commodity contracts but not beyond several years Forward electricity price contracts are less liquid Financial transmission rights Bets on emergence of a global carbon price are not traded Hence look at incomplete case, especially for Long Term risk Bets cannot be bought on all outcomes in IR Ω Only contracts W W, a subspace of IR Ω, are available Theoretical complete case W = IR Ω is risk nirvana

More about risk markets, II. Risk trading Risky capacity equilibrium problems Suppose Then Genco s risked cost is Z G = ( ( ) ) C ω Yω Pω Y ω ( ) Retailer s risked cost is Z R = (P ) ω Y ω U ω Qω Price of risk is some P r IR Ω, e.g., W W costs P r [W R ] := ω P r ωw ω Genco optimizes risk: min W G W Pr [W G ] + r G (Z G W G ) Retailer, likewise: min W R W Pr [W R ] + r R (Z R W R ) P r clears risk market: W G + W R = 0 [Arrow 1964] P r is a PDF related to risk sets for r R, r G [R-Smeers 2015]

Risk trading Risky capacity equilibrium problems Risky competitive capacity equilibrium problem Stage 0: Risk averse capacity investment & risk trading Stage 1: Stochastic spot market. Genco invests in capacity x & risk W G, plans production Y Retailer trades risk W R, plans consumption Q Price of electricity P ω clears spot market in each scenario Price of risk P r clears risk market

Risk trading Risky capacity equilibrium problems Risky competitive capacity equilibrium problem Stage 0: Risk averse capacity investment & risk trading Stage 1: Stochastic spot market. Genco invests in capacity x & risk W G, plans production Y via ( min I(x) + P r ( ) ) [W G ] + r G C ω Yω Pω Y ω W Gω x,y,w G s.t. x X, W G W, 0 Y ω x for all ω Retailer trades risk W R, plans consumption Q via ( min P r ( ) ) [W R ] + r R P ω Y ω U ω Qω WRω Q,W R s.t. W R W, 0 Q ω for all ω Price of electricity P ω clears spot market in each scenario Price of risk P r clears risk market: W G + W R = 0

Risk trading Risky capacity equilibrium problems Small example of RN competitive capacity equilibria One producer, annualized plant CAPEX I = 90 /kw, MC C = 60 /MWh over τ = 8760 hours pa, Ω = 5 scenarios RN case uses uniform distribution: Θ = (1/5,..., 1/5) IR 5 RN Genco solves min Ix + τ E [ ] Θ (C Pω )Y ω s.t. 0 Yω x. x,y Retailer has quadratic utility U ω (Q ω ) = A ω Q ω B 2 Q2 ω Linear demand intercepts: (A ω ) = (300, 350, 400, 450, 500) RN Retailer solves [ ] min Θ Q P ω Q ω A ω Q ω + B 2 Q2 ω. RN competitive capacity equilibrium: x = 389M W with Scenario ω 1 2 3 4 5 E Θ [ ] Q [MWh] 240 290 340 389 389 330 P [ /MWh] 60 60 60 61 111 70 Invest. margin [ /kw] -90-90 -90-84 354 0

Risk trading Risky capacity equilibrium problems Small example of risky competitive capacity equilibrium models Compare RN equilibrium to equilibria in three risky cases where Dimen. W is 0 (no risk trading), 1 (partial trading) or 5 (complete): Utilities r G, r R are good deal coherent risk measures Traded Risk Mean Mean Mean Mean Products Capacity Welfare Quantity Price Invest. margin (RN) 389 491 330 70 0 0 products 339 376 309 90 176 1 product 348 387 313 87 144 Complete 349 388 314 87 142 This relates to a theme pursued by Philpott, Zakeri et al on whether apparent market imperfections are due to either strategic (anti-competitive) action of Gencos, or their risk attitudes.

Outline 1 RN competitive capacity equilibrium problem Risk averse competitive capacity equilibrium problem 2 Risk trading Risky capacity equilibrium problems 3 Risky capacity equilibrium models and existence 4

Risky capacity equilibrium models and existence of equilibria We contribute model formulation and existence of solutions in several areas Risk Traded risk Problem Existence attitude Spot market products type Neutral Compet. - Optimization Averse Compet. Complete Optimization Averse Compet. Incomplete Equilibrium Neutral Cournot - Nash game Averse Cournot Complete Nash game Averse Cournot None Nash game Averse Cournot 1 or more Equilibrium

Existence of risky capacity equilibria in incomplete cases Partial risk trading, where {0} W IR Ω is under-studied: This talk: https://www.eprg.group.cam.ac.uk/eprg-working-paper-1720/ [Kok 2018], PhD, U. Auckland [Ibada et al 2017], Mathematical Programming [de Maere et al 2017], Energy Policy [de Maere et al 2013], Mathematical Programming Our approach differs from others based on Risky Design Games [R-Smeers, SIOPT 2015]. This adapts the classical RN model so that every agent chooses its own RN view

Reformulation for Theory and Computation: Incomplete Risk Neutral subproblem Each agent is Risk Neutral via different probability. Given that only the Genco makes capacity investments, we only see Θ G here: Genco sets investment x & views production Y = (Y ω ) ω via [ ( ) ] I(x) + E ΘG C ω Yω Pω Y ω min x,y s.t. x X, 0 Y ω x for all ω Retailer sets consumption Q ω in each scenario ω as before Price of electricity P ω clears spot market in each scenario ω THIS CAN BE SOLVED BY OPTIMISATION

Where do Θ G and Θ R come from? Risk pricing subproblem Given Z G = ( ) ) ( ( ) (C ) ω Yω Pω Y ω, Z R = P ω Y ω U ω Qω, [R-Smeers 15] says that the probabilities Θ G, Θ R comprise a marginal risk equilibrium for Z G, Z R : There are W G, W R W such that W G + W R = 0 and Θ G r G ( ZG W G ) Θ R r R ( ZR W R ) Θ G W = Θ R W = P r, price of risk Dual formulations are standard complementarity (for cv@r) or cone complementarity (for Good Deal risk measures) conditions

Reformulation for Theory and Computation: Suggests Alternating Method Step i. Solve following by optimization; Go to Step ii. Genco sets investment x & views production Y = (Y ω ) ω via [ ( ) ] I(x) + E ΘG C ω Yω Pω Y ω min x,y s.t. x X, 0 Y ω x for all ω Retailer sets consumption Q ω in each scenario ω Price of electricity P ω clears spot market in each scenario ω Step ii. Solve following by complementarity; Go to Step i. Prices of risk Θ G, Θ R that clear risk market

Concluding thoughts Risk management is business as usual in the power industry Power companies use financial products to hedge risk Risk aversion is not an imperfection... But incompleteness of financial markets is. Incomplete risky capacity equilibrium problems need analytical and computational development