A minimax cogeneration commitment chosen on training scenarios had the largest regret on held-out scenarios

CheAI Research Inc. | Research article | June 14, 2023

Keywords: unit commitment; combined heat and power; robust optimization; minimax; out-of-sample regret

Abstract

1. Introduction

A cogeneration site must commit units before demand and prices are known. A commitment optimized for one scenario may perform poorly in another, and robust optimization offers protection against a declared set of scenarios [1, 2]. The protection, however, is guaranteed only for the scenarios in the set. Whether it extends to scenarios outside the set is an empirical question that is rarely tested in dispatch studies.

This study tests it directly on a small benchmark, scoring each policy first on the scenarios it was chosen from and then on scenarios withheld from all policies.

2. Methods

2.1 Benchmark

A site with 3 backpressure steam-turbine units decides, before the hour, which units to commit. Once committed, each unit's steam and power output is dispatched to meet steam and electricity demand at minimum cost, with any electricity shortfall purchased from the grid. Unit heat-to-fuel and power-to-heat ratios are taken from the EPA Catalog of CHP Technologies [3]; each unit can turn down to a declared minimum load. The scenarios (Table 1) are engineering assumptions and describe no real site, tariff or market.

Table 1. Benchmark scenarios.

2.2 Policies

The nominal policy minimizes cost in the base scenario alone. The scenario-weighted policy minimizes the average cost over the 4 training scenarios. The minimax policy minimizes the worst training-scenario cost. Each is a mixed-integer linear program solved to optimality.

2.3 Evaluation and verification

Each policy's commitment was fixed and dispatched in the 2 held-out scenarios, and its regret computed as its cost minus the cost of the best commitment for that scenario. An exhaustive enumeration of all 8 commitments, solved in extended precision independently of the optimization library, reproduces the nominal optimum to a relative error of 0 and the minimax optimum to 0; 4 commitments are feasible in every training scenario.

3. Results

3.1 Training performance

Table 2. Commitment, objective and regret of the three policies.

The policies optimize different objectives, so their own objective values are not directly comparable; the worst-case column places all three on a common basis (Table 2). The nominal and weighted policies commit the same two units. Minimax commits all three, lowering the worst training cost by 2.31 USD/h at a cost of 0.051 USD/h in the base scenario. Every feasible commitment includes the largest unit, and for each the worst case is the high-steam scenario, where the feasible commitments span 53.07 USD/h, 1.9 percent of cost (Figure 1). The nominal and minimax commitments, the two lowest in that scenario, differ by only 2.31 USD/h.

3.2 Out-of-sample performance

On the held-out scenarios, a moderate case and a high-gas-price case, minimax has the largest maximum regret, 31.46 USD/h against 29.76 USD/h for the nominal and weighted commitment, and the highest regret in both scenarios (Figure 2). The commitment with the best worst case in training performed worst on the scenarios it was not chosen from. The largest regret of any policy is 1.3 percent of the scenario cost.

4. Discussion

Finite-set minimax protects against the scenarios it is given; nothing in the formulation extends that protection to other scenarios. On this benchmark the protection was small in training and reversed out of sample. The stakes were also small: once the largest unit is committed, the choice among the remaining units changes cost by at most 2.2 percent in any training scenario. A dispatch study reporting one optimal commitment for one set of prices answers a narrower question than a site manager faces. Where scenarios are uncertain, a robust commitment should be validated on scenarios outside its training set before it is relied on.

5. Limitations

• Scenarios are declared without probabilities; minimax over a finite set is not a guarantee over an uncertainty set, and scenario weights are not probabilities.
• Units are linear, with published ratios rather than vendor curves, and without part-load efficiency, start-up cost, ramp limits or steam pressure levels.
• Exports earn no revenue, and emissions are not considered.
• The exhaustive reference is exact only at the declared three-unit size.

6. Conclusions

A minimax commitment reduced the worst training-scenario cost by 2.31 USD/h but had the largest regret on held-out scenarios. Robust commitments should be evaluated out of sample before use.

References

1. Ben-Tal A, El Ghaoui L, Nemirovski A. Robust Optimization. Princeton University Press, Princeton, 2009.
2. Kouvelis P, Yu G. Robust Discrete Optimization and Its Applications. Kluwer Academic Publishers, Dordrecht, 1997.
3. U.S. Environmental Protection Agency, Combined Heat and Power Partnership. Catalog of CHP Technologies. Washington, DC, 2017.

Related studies

• Optimal dispatch of a modeled cogeneration site: a deterministic dispatch study
• Price sensitivity of the optimal dispatch: how one site's optimum moves with prices
• Steam header transients: another limitation of single-point dispatch