An Iterative Algorithm for Profit Maximization by Market Equilibrium Constraints
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1 An Iterative Algorithm for Profit Maximization by Market Equilibrium Constraints Andrés Ramos Mariano Ventosa Michel Rivier Abel Santamaría 14 th PSCC Seville, 25 June 2002
2 Outline 1. Introduction 2. Model Description 3. Comparison with an MCP Approach 4. Case Study 5. Conclusions An Iterative Algorithm for Profit Maximization by Market Equilibrium Constraints - 2
3 Introduction Two main approaches to model market equilibrium: Cournot (firms compete only in quantities) Supply function equilibrium (in quantities and prices) A great number of medium and long term oligopoly market models are based on Cournot equilibrium An Iterative Algorithm for Profit Maximization by Market Equilibrium Constraints - 3
4 Outline 1. Introduction 2. Model Description 3. Comparison with an MCP Approach 4. Case Study 5. Conclusions An Iterative Algorithm for Profit Maximization by Market Equilibrium Constraints - 4
5 Model Description Based on a detailed production cost model Detailed representation of operation constraints Decision variables are generator output levels Determine the cheapest commitment and operation of the hydrothermal system Market equilibrium among firms represented by a set of additional constraints. By these constraints each strategic agent maximizes its profit (revenues minus costs) Determine the output level for each strategic firm Advantage: use of any available production cost model (PCM) Drawback: market equilibrium constraints depend on SMP, not directly obtained from a PCM An Iterative Algorithm for Profit Maximization by Market Equilibrium Constraints - 5
6 Production cost model with market equilibrium constraints Objective Function Operating Constraints Market Constraints Minimization of: Subject to: Total Generation Variable Costs + + Costs of non accepted demand bids Interperiod Constraints (Maintenance scheduling, fuel purchase, hydro scheduling, seasonal pumping) Intraperiod Constraints (Generation-demand balance, thermal unit commitment, generation limits) Market Equilibrium Constraints Price equation Market Share Constraints An Iterative Algorithm for Profit Maximization by Market Equilibrium Constraints - 6
7 Utility function Market equilibrium obtained by maximizing consumer s and producer s surplus Price Demand Curve Supply Curve Cleared Price Consumer s surplus Producer s surplus Variable costs Output Non-served demand costs Market Equilibrium Quantity An Iterative Algorithm for Profit Maximization by Market Equilibrium Constraints - 7
8 Market equilibrium constraints Objective Function Minimisation of: Subject to: Total Generation Variable Costs + + Costs of non accepted demand bids Equilibrium Constraints Marginal revenues equal marginal cost for each Operating strategic firm Interperiod Constraints Constraints Marginal cost of each firm as a function of its committed Intraperiod units Constraints System Marginal Price equation The price is a function of the served demand Market Market Share Equilibrium Constraints Constraints Market Constraints The market Price share of equation each leader in quantity Constraints must be equal to its market share objective Market Share Constraints An Iterative Algorithm for Profit Maximization by Market Equilibrium Constraints - 8
9 Iterative computation of system marginal price (SMP) Marginal cost (MC) of each firm i not directly calculated by the PCM due to discrete commitment decisions P i output of firm i Introduce minimum MC i (P) i Solve the market equilibrium constraints model P i SMP MCi( Pi) demand slope Compute new MC i (P) i * DIFF = MC i (P) i *- MC i (P) i Eliminate MC i (P) i DIFF 0 NO YES An Iterative Algorithm for Profit Maximization by Market Equilibrium Constraints - 9
10 Producer surplus Producer Surplus Market Equilibrium Iter 4 Iter 3 Iter 2 Iter 1 Output An Iterative Algorithm for Profit Maximization by Market Equilibrium Constraints - 10
11 Iterative algorithm for hydro scheduling of strategic firms 1. Obtain an initial market equilibrium by solving the optimization problem 2. Select a load level L1 with high SMP where the market equilibrium constraints are not binding for some strategic company 3. Select a load level L2 with lower SMP where the same company can decrease its production 4. Find a fringe company that can decrease its production in load level L1 and simultaneously increase in load level L2 5. Exchange the hydro generation of the strategic and fringe companies among load levels taking into account the technical hydro constraints (i.e., maximum and minimum output, maximum and minimum reserve levels, etc.) and any other constraint (i.e, firm market share) 6. If two load levels with different SMP can be selected go to step 2. In other case go to end An Iterative Algorithm for Profit Maximization by Market Equilibrium Constraints - 11
12 Outline 1. Introduction 2. Model Description 3. Comparison with an MCP Approach 4. Case Study 5. Conclusions An Iterative Algorithm for Profit Maximization by Market Equilibrium Constraints - 12
13 Comparison with a MCP approach Advantages of PCM with MEC approach Realistic modeling of the electric system (e.g., binary unit commitment variables) Robust and efficient solution methods (e.g., Branch and bound for MIP) Constraints must be linear Convergence not guaranteed but obtained in practice Advantages of MCP approach Compact problem formulation Only continuous variables Possibility of introducing nonlinear constraints Optimality guaranteed and solution uniqueness with linear constraints Slow solution method and depending of the initial value An Iterative Algorithm for Profit Maximization by Market Equilibrium Constraints - 13
14 Outline 1. Introduction 2. Model Description 3. Comparison with an MCP Approach 4. Case Study 5. Conclusions An Iterative Algorithm for Profit Maximization by Market Equilibrium Constraints - 14
15 Case study Time scope divided into 3 periods with 3 load levels for weekdays subperiod and 2 load levels for weekend subperiod 18 thermal units and 6 hydro units from 3 different firms SMP rises from 4.86 pta/kwh in cost minimization to 5.16 pta/kwh in market equilibrium Total profit increases in a 5 % while the energy produced decreases in a 10 % An Iterative Algorithm for Profit Maximization by Market Equilibrium Constraints - 15
16 System marginal price Cost Minimization Market Constraints An Iterative Algorithm for Profit Maximization by Market Equilibrium Constraints - 16
17 Outline 1. Introduction 2. Model Description 3. Comparison with an MCP Approach 4. Case Study 5. Conclusions An Iterative Algorithm for Profit Maximization by Market Equilibrium Constraints - 17
18 Conclusions Market equilibrium under an oligopoly based on the Cournot conjecture, represented by a production cost model plus market equilibrium constraints Iterative algorithm to calculate the system marginal cost Iterative algorithm to refine hydro scheduling for strategic firms Resulting MIP problem solved by robust, efficient and reliable solvers An Iterative Algorithm for Profit Maximization by Market Equilibrium Constraints - 18
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