Abstract
This paper concerns the assessment of two methods for convex relaxation of the short-term hydrothermal scheduling problem. The problem is originally formulated as a mixed integer programming problem, and then approximated using both Lagrangian and Linear relaxation. The two relaxation methods are quantitatively compared using a realistic data description of the Northern European power system, considering a set of representative days. We find that the Lagrangian relaxation approximates system operational costs in the range 55-81% closer to the mixed integer programming problem solution than the Linear relaxation. We show how these cost gaps vary with season and climatic conditions. Conversely, the differences in both marginal cost of electricity and reserve capacity provided by the Lagrangian and Linear relaxation are muted.
Highlights
T HE use of fundamental optimization and simulation models for forecasting system operational costs and the marginal costs of electricity (MCE) is a well-established practice in many power markets [1], [2]
The major novel contribution in this work lies in the quantitative assessment of the differences in operational costs, MCE and MCR obtained by the Linear Relaxation (LIR) and Lagrangian Relaxation (LR) convex relaxation methods when solving realistic and large-scale1 instances of
In terms of operational costs, the LR method was in the range 55-81% closer than the LIR method to the mixed integer programming problem (MIP) solution
Summary
Conversion between m3/s and Mm3; Bounds on discharge, in m3/s; Upper discharge bound for segment n, in m3/s; Bounds on bypass for station h, in m3/s; Bounds on volume for reservoir h, in Mm3; Bounds on generation for station h, in MW; Bounds on generation for unit g, in MW; Upper bound on thermal generation for unit g and segment m, in MW; Max. ramping on discharge, in m3/s/h; Max. ramping on generation, in MW/h; Max. ramping on HVDC cable l, in MW/h;. No-load cost for unit g, in e; Marginal cost for unit g, in e/MWh; Start-up cost of units and stations, in e; Shut-down cost of units and stations, in e; Cost of curtailment in area a, in e/MWh; Marginal value of demand, in e/MWh; Inflow to reservoir h, in Mm3; Price inelastic demand (in area a), in MW; Spinning up/down-regulation reserve requirement in area a, in MW; Non-spinning up-regulation reserve requirement in area a, in MW; Wind power in area a, in MW; Energy equivalent for station h, discharge segment n or N =|N |, in MW/m3/s; Coefficient for Benders cut c, in e/Mm3; Right-hand side for Benders cut c, in e; Direction of line l, seen from area a, {−1, 1}; PTDF from area a on line l; Max. flow on line l, in MW; Lagrangian multiplier, where ∗ reflects subscripts P, F +, F −, S+, in e/MWh; Lagrangian multiplier for cut c, fraction; Mismatches in relaxed constraints, in MW; Convergence tolerance used in Lagrangian relaxation.
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