A cheaper-gas year flips the optimal steam dispatch into a regime that emits a fifth more

CheAI Research Inc. | Research article | October 9, 2026

Keywords: sensitivity analysis; cogeneration dispatch; natural gas prices; operating regimes; interval arithmetic

Abstract

1. Introduction

A dispatch optimum computed at one set of prices is a point estimate. Prices of gas and electricity vary substantially between years and regions, and a sensitivity analysis is needed to know whether the optimum, and the emissions that follow from it, are robust to that variation. For cogeneration systems the question is sharpened by the structure of the problem: optima often lie on equipment limits, so they may jump between limits rather than move continuously.

This study varies each price across its cited range, identifies the resulting operating regimes, and tests where the proof of exactness used at reference prices remains valid.

2. Methods

2.1 Configuration

The analysis uses the coupled configuration of the modeled site, in which the steam and cogeneration system also supplies a 6.99 MW chlor-alkali cell house. Its reference import, 11.32 MW, and emissions therefore exceed those of the uncoupled dispatch study by the cell house's demand.

2.2 Price ranges

Natural gas varies between 3.72 and 7.04 USD/GJ on a higher-heating-value basis, the lowest and highest annual US industrial prices of 2021 to 2025 [1], around the reference value of 4.50. Purchased electricity varies between 0.0718 and 0.0832 USD/kWh. The export price varies between 0.0336 and 0.0504 USD/kWh, an assumed band, since no public source reports a site's export price. Each price moves alone while the others remain at their reference values, and prices are constant across hours.

2.3 Optimization and certification

At each price point the hourly dispatch was optimized with IDAES, Pyomo and Ipopt [2, 3, 4]. Exactness at reference prices rests on a certificate: an interval-arithmetic proof that cost falls with combined turbine flow along the feasible boundary, is then strictly concave, then rises, so that each face's minimum lies at one of its two vertices. The certificate was re-evaluated at every price point. Where it failed, every candidate face was solved with checked local optimization and the best result taken. Emissions use the eGRID2023 CAMX annual factor [5], grossed up for grid losses. These grid emissions comprise Scope 2 and the grid-loss share, which the GHG Protocol assigns to Scope 3 [6].

3. Results

3.1 Cost sensitivity

Optimal cost changes by -490.28 and +1,356.07 USD/h at the low and high gas prices, and by -36.21 and +85.68 USD/h at the low and high purchase prices (Figure 1). The export price changes cost by less than a cent per hour, because export remains zero at both ends; that bar reflects solver round-off only.

3.2 Three operating regimes

Every optimum lies on a vertex of the feasible set, and across these prices only three vertices are optimal (Table 1). In the balanced regime the larger boiler runs at its rating, both turbines at their flow limits and the condensing tail near its minimum. In the maximum-condensing regime both boilers run at their ratings and the surplus passes through the condensing tail, 60.1 t/h, generating most of the site's electricity. In the minimum-condensing regime the tail falls to its minimum and the site purchases more power.

Table 1. Optimal dispatch at each price point.

Cheaper gas and more expensive grid power act in the same direction: onsite condensing power becomes the cheaper source of electricity, and the optimizer generates as much as the boilers allow. At the low gas price import falls from 11.32 to 2.62 MW, grid emissions from 2,301 to 534 kg per hour, and Scope 1 rises from 27,745 to 35,747 kg per hour (Figure 2). Total emissions rise by 20.7 percent, because condensing power carries far more carbon than the grid import it displaces. The high gas price has the opposite effect, reducing total emissions by 2.3 percent.

The regime change corresponds to a crossing in the cost of two candidate vertices (Figure 3). At reference prices the balanced vertex is cheaper; as gas becomes cheaper or grid power more expensive, the maximum-condensing vertex cost falls faster and crosses below it. Near the crossing, a small price change produces a large change in emissions.

3.3 Validity of the certificate

The certificate held at 4 of 7 price points. At the high gas price one of its premises is false rather than merely unproven: the slope required to be negative is positive at all 401 of 401 sample points, from 10.48 down to 9.61 USD/h per t/h on the larger boiler's face. At the low purchase price the slope changes sign within the segment, from +0.261 to -0.292, and is non-negative at 192 samples. At the low gas price a negative control that should not be provable was proved. Results at these three points are labelled as best-found local optima throughout.

4. Discussion

Prices change the structure of the optimum, not only its cost. Because the regimes differ sharply in emissions, a dispatch study reporting a single optimum at one set of prices can understate how emissions respond to market conditions. Proofs of exactness are likewise conditional on the inputs at which they are evaluated and must be re-established when those inputs change.

The price basis of the modeled site deserves particular attention. The site is placed on the California grid but priced on US industrial gas. California industrial gas prices were between 8.9 and 12.9 USD/GJ on a higher-heating-value basis in every year from 2021 to 2025 [1], above the entire tested range. Within the tested range, more expensive gas favors the minimum-condensing regime; the dispatch at California prices was not computed and is not extrapolated here.

5. Limitations

• Prices vary one at a time; correlated movements of gas and power prices are not tested.
• Prices are constant across hours; hourly price shape is not represented.
• Emissions use the CAMX annual-average factor, not the marginal factor of the hours affected by a regime change.
• The site is modeled; the regimes are properties of this configuration, not of cogeneration sites in general.

6. Conclusions

The cost-optimal dispatch of the modeled site moves between three discrete regimes as prices vary, and cheaper gas raises total emissions by 20.7 percent. The certificate of exactness failed at 3 of 7 price points. Dispatch results should be reported with their price sensitivity, and certificates re-evaluated at each set of inputs.

References

1. U.S. Energy Information Administration. Natural Gas Prices: Price of Natural Gas Delivered to Industrial Consumers, by State. Washington, DC.
2. Lee A, Ghouse JH, Eslick JC, et al. The IDAES process modeling framework and model library: flexibility for process simulation and optimization. Journal of Advanced Manufacturing and Processing, 3(3):e10095, 2021.
3. Bynum ML, Hackebeil GA, Hart WE, et al. Pyomo: Optimization Modeling in Python, 3rd edition. Springer, Cham, 2021.
4. Wachter A, Biegler LT. On the implementation of an interior-point filter line-search algorithm for large-scale nonlinear programming. Mathematical Programming, 106(1):25-57, 2006.
5. U.S. Environmental Protection Agency. Emissions and Generation Resource Integrated Database, eGRID2023 Revision 2. Washington, DC.
6. World Resources Institute and World Business Council for Sustainable Development. GHG Protocol Scope 2 Guidance. Washington, DC, 2015.

Related studies

• Optimal dispatch of a modeled cogeneration site: the balanced regime at reference prices
• Chlor-alkali and steam system coupling: the local optima the certificate resolved
• Robust cogeneration commitment: commitment under uncertain prices
• Inventory and avoided emissions: the hourly marginal factor