Optimal Price Zones of Electricity Markets A Mixed-Integer Multilevel Model and Global Solution Approaches V. Grimm, T. Kleinert, F . Liers, Martin Schmidt, G. Zöttl FAU Erlangen-Nürnberg, Discrete Optimization 21st Combinatorial Optimization Workshop, Aussois, 2017
Outline Motivation A Mixed-Integer Multilevel Model Solution Approaches Computational Results M. Schmidt FAU Erlangen-Nürnberg Optimal Price Zones of Electricity Markets Aussois 2017 2 · ·
Outline Motivation A Mixed-Integer Multilevel Model Solution Approaches Computational Results M. Schmidt FAU Erlangen-Nürnberg Optimal Price Zones of Electricity Markets Aussois 2017 3 · ·
Liberalized Electricity Markets Timing 1. Generation capacity investment by profit-maximizing firms 2. Spot-market trading • Energy-only market: no network considered • Sole requirement: market clearing 3. Cost-based redispatch (if required) t | T | periods of spot market generation capacity trading (firms) and redispatch expansion (firms) after each spot market (TSO) M. Schmidt FAU Erlangen-Nürnberg Optimal Price Zones of Electricity Markets Aussois 2017 4 · ·
Cost-Based Redispatch • Technically infeasible spot-market results → redispatch • Modification of traded quantities • Redispatched electricity can be transported • Objective: minimum redispatch cost M. Schmidt FAU Erlangen-Nürnberg Optimal Price Zones of Electricity Markets Aussois 2017 5 · ·
Cost-Based Redispatch • Technically infeasible spot-market results → redispatch • Modification of traded quantities • Redispatched electricity can be transported • Objective: minimum redispatch cost Price • Energy-only market: equilibrium quantity B equilibrium price C F A • Transmission constraints: Supply C transportable capacity D E Demand • Producer pays to TSO: ABDE • TSO pays to consumer: ABDF • TSO’s cost: AEF Quantity D B M. Schmidt FAU Erlangen-Nürnberg Optimal Price Zones of Electricity Markets Aussois 2017 5 · ·
Zonal Pricing • Implemented in parts of Europe, Australia, or Latin America • Market area is divided into price zones • Intra-zonal network constraints: ignored at the spot market • Inter-zonal network constraints: (partly) respected at the spot market • Bad zoning: distorted investment incentives for generation capacity leading to inefficiencies • Good zoning: congestion issues are reflected (most appropriately) in spot-market trading • Goal of the regulator: optimal configuration of price zones • Maximization of resulting social welfare M. Schmidt FAU Erlangen-Nürnberg Optimal Price Zones of Electricity Markets Aussois 2017 6 · ·
Outline Motivation A Mixed-Integer Multilevel Model Solution Approaches Computational Results M. Schmidt FAU Erlangen-Nürnberg Optimal Price Zones of Electricity Markets Aussois 2017 7 · ·
Some Notation • Transmission network: directed graph G = ( N , L ) • Scenarios/time periods: T = { t 1 , . . . , t | T | } • Node set: n ∈ N • Consumers c ∈ C n with demand d t , c ≥ 0 • Elastic demand modeled by continuous and strictly decreasing function p t , c ( d t , c ) • Generators g with production q t , g ∈ [ 0 , ¯ q g ] • Some producers may invest in generation capacity ¯ q g • Arc set: l ∈ L • Transmission lines with capacity ¯ f l • Lossless DC power flow model • Price zones Z i : parts of a partition N = Z 1 ∪ · · · ∪ Z k • k is given as input M. Schmidt FAU Erlangen-Nürnberg Optimal Price Zones of Electricity Markets Aussois 2017 8 · ·
Trilevel Market Model: Timing t specification | T | periods of spot market generation capacity of zones trading (firms) and redispatch expansion (firms) (regulator) after each spot market (TSO) M. Schmidt FAU Erlangen-Nürnberg Optimal Price Zones of Electricity Markets Aussois 2017 9 · ·
Trilevel Market Model: Model Structure max social welfare (regulator) s.t. graph partitioning with connectivity constraints max profits (competitive firms) s.t. generation capacity investment, production & demand constraints, Kirchhoff’s 1st law (inter-zonal), flow restrictions (inter-zonal) min redispatch costs (TSO) s.t. production & demand constraints, lossless DC power flow constraints M. Schmidt FAU Erlangen-Nürnberg Optimal Price Zones of Electricity Markets Aussois 2017 10 · ·
1st Level: Specification of Price Zones Maximization of total social welfare � d red t , c � � � � � � � c inv q new c var g q red g ¯ ψ 1 := p t , c ( ω ) d ω − + g t , g 0 g ∈ G new t ∈ T n ∈ N c ∈ C n n ∈ N t ∈ T g ∈ G all n n subject to graph partitioning with multi-commodity flow connectivity constraints M. Schmidt FAU Erlangen-Nürnberg Optimal Price Zones of Electricity Markets Aussois 2017 11 · ·
1st Level: Specification of Price Zones Maximization of total social welfare � d red t , c � � � � � � � c inv q new c var g q red g ¯ ψ 1 := p t , c ( ω ) d ω − + g t , g 0 g ∈ G new t ∈ T n ∈ N c ∈ C n n ∈ N t ∈ T g ∈ G all n n subject to graph partitioning with multi-commodity flow connectivity constraints � x n , i = 1 n ∈ N i ∈ [ k ] � z n , i = 1 i ∈ [ k ] n ∈ N z n , i ≤ x n , i n ∈ N , i ∈ [ k ] � m i a ≤ Mx n , i n ∈ N , i ∈ [ k ] a ∈ δ out n � � m i m i a − a ≥ x n , i − Mz n , i n ∈ N , i ∈ [ k ] a ∈ δ out a ∈ δ in n n M. Schmidt FAU Erlangen-Nürnberg Optimal Price Zones of Electricity Markets Aussois 2017 11 · ·
2nd Level: Capacity Investment & Spot Market Economic Assumption: Perfect Competition • No market power; otherwise multiple equilibria (Zöttl, 2010) • Mathematically “necessary” assumption • Commonly used in electricity market literature: Boucher, Smeers (2001), Daxhelet, Smeers (2007), Grimm, Martin, S., Weibelzahl, Zöttl (2016) M. Schmidt FAU Erlangen-Nürnberg Optimal Price Zones of Electricity Markets Aussois 2017 12 · ·
2nd Level: Capacity Investment & Spot Market Economic Assumption: Perfect Competition • No market power; otherwise multiple equilibria (Zöttl, 2010) • Mathematically “necessary” assumption • Commonly used in electricity market literature: Boucher, Smeers (2001), Daxhelet, Smeers (2007), Grimm, Martin, S., Weibelzahl, Zöttl (2016) Objective Profit (= total social welfare) maximization � d spot t , c � � � � � � � g q spot c inv q new c var g ¯ ψ 2 := p t , c ( ω ) d ω − + g t , g 0 g ∈ G new t ∈ T n ∈ N c ∈ C n n ∈ N t ∈ T g ∈ G all n n M. Schmidt FAU Erlangen-Nürnberg Optimal Price Zones of Electricity Markets Aussois 2017 12 · ·
2nd Level: Capacity Investment & Spot Market Zonal version of Kirchhoff’s first law d spot � d spot q spot � q spot n ∈ N , t ∈ T = t , c , = t , n t , n t , g g ∈ G all c ∈ C n n � x n , i d spot � x n , i q spot D i Q i i ∈ [ k ], t ∈ T t = t , n , t = t , n n ∈ N n ∈ N � ( 1 − x n , i ) x m , i f spot F in i , t = i ∈ [ k ], t ∈ T t , l l =( n , m ) ∈ L � x n , i ( 1 − x m , i ) f spot F out = i ∈ [ k ], t ∈ T i , t t , l l =( n , m ) ∈ L D i t + F out = Q i t + F in i ∈ [ k ], t ∈ T i , t i , t Nonlinearities can be linearized M. Schmidt FAU Erlangen-Nürnberg Optimal Price Zones of Electricity Markets Aussois 2017 13 · ·
2nd Level: Capacity Investment & Spot Market Flow restrictions on inter-zonal lines − ¯ ≤ ¯ f l − ( 1 − y l ) M ≤ f spot f l + ( 1 − y l ) M l ∈ L , t ∈ T t , l Demand and production bounds 0 ≤ d spot t ∈ T , n ∈ N , c ∈ C n t , c 0 ≤ q spot q new t ∈ T , n ∈ N , g ∈ G new ≤ τ ¯ g n t , g 0 ≤ q spot q ex t ∈ T , n ∈ N , g ∈ G ex ≤ τ ¯ g n t , g M. Schmidt FAU Erlangen-Nürnberg Optimal Price Zones of Electricity Markets Aussois 2017 14 · ·
3rd Level: Cost-Based Redispatch Minimize redispatch costs � d spot t , c � � � � � � t , g − q spot c var g ( q red ψ 3 := p t , c ( ω ) d ω + t , g ) d red t ∈ T n ∈ N c ∈ C n t ∈ T n ∈ N g ∈ G all t , c n subject to lossless DC power flow model: • Kirchhoff’s 1st law � � � � d red f red q red f red n ∈ N , t ∈ T t , c + = t , g + t , l , t , l l ∈ δ out g ∈ G all l ∈ δ in c ∈ C n n n n • Kirchhoff’s 2nd law f red = B l ( θ t , n − θ t , m ), l = ( n , m ) ∈ L , t ∈ T t , l n = 0 , t ∈ T θ t ,ˆ • Flow capacities − ¯ ≤ ¯ f l ≤ f red f l , l ∈ L , t ∈ T t , l M. Schmidt FAU Erlangen-Nürnberg Optimal Price Zones of Electricity Markets Aussois 2017 15 · ·
Model Discussion 1st Level MIQP with graph partitioning and multi-commodity flow model 2nd Level MIQP; no “genuine” 2nd level integers 3rd Level QP with lossless DC power flow model max ψ 1 ( W 2 , W 3 ) s.t. ( W 1 , X 1 ) ∈ Ω 1 Level 1 max ψ 2 ( W 2 ) Level 2 s.t. ( W 2 , X 1 ) ∈ Ω 2 min ψ 3 ( W 2 , W 3 ) Level 3 s.t. ( W 2 , W 3 ) ∈ Ω 3 M. Schmidt FAU Erlangen-Nürnberg Optimal Price Zones of Electricity Markets Aussois 2017 16 · ·
Outline Motivation A Mixed-Integer Multilevel Model Solution Approaches Computational Results M. Schmidt FAU Erlangen-Nürnberg Optimal Price Zones of Electricity Markets Aussois 2017 17 · ·
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