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A scalable solution framework for stochastic transmission and generation planning problems

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Abstract

Current commercial software tools for transmission and generation investment planning have limited stochastic modeling capabilities. Because of this limitation, electric power utilities generally rely on scenario planning heuristics to identify potentially robust and cost effective investment plans for a broad range of system, economic, and policy conditions. Several research studies have shown that stochastic models perform significantly better than deterministic or heuristic approaches, in terms of overall costs. However, there is a lack of practical solution techniques to solve such models. In this paper we propose a scalable decomposition algorithm to solve stochastic transmission and generation planning problems, respectively considering discrete and continuous decision variables for transmission and generation investments. Given stochasticity restricted to loads and wind, solar, and hydro power output, we develop a simple scenario reduction framework based on a clustering algorithm, to yield a more tractable model. The resulting stochastic optimization model is decomposed on a scenario basis and solved using a variant of the Progressive Hedging (PH) algorithm. We perform numerical experiments using a 240-bus network representation of the Western Electricity Coordinating Council in the US. Although convergence of PH to an optimal solution is not guaranteed for mixed-integer linear optimization models, we find that it is possible to obtain solutions with acceptable optimality gaps for practical applications. Our numerical simulations are performed both on a commodity workstation and on a high-performance cluster. The results indicate that large-scale problems can be solved to a high degree of accuracy in at most 2 h of wall clock time.

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Notes

  1. The referenced report (Chupka et al. 2008) estimates that $298B in transmission and $582B in distribution investments are needed by 2030.

  2. A common rule is to recommend investments in transmission lines that are part of the optimal investment plan for most or all of the scenario-specific solutions (Munoz et al. 2014).

  3. For instance, Jin et al. (2014) reports that although the generation investments recommended by planning models with and without unit commitment are different, the generation fleet selected by a model based strictly on economic dispatch yields a total system cost that is only 0.02 % higher than that resulting from a model based on unit commitment.

  4. The selection of disjuntive parameters is extremely important to ensure a tight mixed-integer formulation. Here we follow the approach described in Munoz et al. (2014).

  5. Although permitting and construction lead times for renewable generation technologies and small conventional generators could be much shorter in the US, transmission and large generation projects can take up to 10 years to complete.

  6. Modeling AC power flows in a long-term investment planning model is an ongoing subject of research.

  7. In reality, the capacity value of some renewable resources decreases as the amount of installed capacity of the resource increases. For instance, in a system where most of the peak-load hours occur in the late afternoon, when the probabilities of lost load are highest and when solar resources are often fully available, the first few MWs of solar will directly reduce the need of power supply from peaking generation units. In that case, solar resources should be assigned a capacity value that is close to their rated capacity (i.e., \(ELCC \approx 1\)). However, higher penetrations levels of solar could result in a shift of peak net load hours to the early evening, when the resource is no longer available. In this second case, the marginal contribution of an additional MW of solar capacity towards a reduction of the peak net load (and a reduction of the probability of lost load) is nearly zero (i.e., \(ELCC \approx 0\)) (Mills and Wiser 2012). In practice, electric power utilities rely on estimates of ELCC from historical data for new power plants. In some cases, these values are updated year by year based on the actual contribution of the resources towards reductions of peak net loads (Mills and Wiser 2012). Munoz and Mills (2015) propose a new method to account for the reduction in the capacity value of solar PV within investment planning models at higher penetration levels, but its implementation is beyond the scope of this paper.

  8. The percentage of variance captured relative to the full dataset is measured as the ratio of the between-cluster variance and the total variance.

  9. Note that the problem \(min~\sum _{s \in \varOmega } c_s x_s\), subject to \(x_s \ge 0\) can be re-written as \(min~\sum _{s \in \varOmega } u_s\), subject to \(u_s \ge c_s x_s\) and \(u_s, x_s \ge 0\). Jensen’s inequality is directly applicable to the latter.

  10. We invoke the RINS heuristic every 100 nodes in the MILP branch and cut tree.

  11. Histograms of generation and transmission capital costs per MW are shown in Figs. 3 and 4 in the Appendix 9.

  12. Note that the lower bound obtained using the linear relaxation is global and it does not depend on the number of scenarios used to find a trial investment plan. Therefore, the more scenarios of clustered hours considered in the linear relaxation, the tighter the lower bound.

  13. We are implicitly assuming that the 500-scenario solutions are extremely close to the true optimal. We justify this assumption by the small optimality gap attained with that experiment.

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Acknowledgments

This material is based upon work supported by the U.S. Department of Energy, Office of Science, Office of Advanced Scientific Computing Research, Applied Mathematics program under contract number KJ0401000 through the Project “Multifaceted Mathematics for Complex Energy Systems”. Sandia National Laboratories is a multi-program laboratory managed and operated by Sandia Corporation, a wholly owned subsidiary of Lockheed Martin Corporation, for the U.S. Department of Energy’s National Nuclear Security Administration under Contract DE-AC04-94-AL85000.

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  1. Analytics Department, Sandia National Laboratories, P.O. Box 5800, MS 1326, Albuquerque, NM, 87185-1326, USA

    Francisco D. Munoz & Jean-Paul Watson

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  1. Francisco D. Munoz
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Correspondence to Francisco D. Munoz.

Appendix

Appendix

1.1 Cost data

Fig. 3
figure 3

Histogram of generation capital costs per MW

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Fig. 4
figure 4

Histogram of transmission capital costs per MW

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Munoz, F.D., Watson, JP. A scalable solution framework for stochastic transmission and generation planning problems. Comput Manag Sci 12, 491–518 (2015). https://doi.org/10.1007/s10287-015-0229-y

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