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Chance-constrained economic dispatch with renewable energy and storage

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Abstract

Increasing penetration levels of renewables have transformed how power systems are operated. High levels of uncertainty in production make it increasingly difficulty to guarantee operational feasibility; instead, constraints may only be satisfied with high probability. We present a chance-constrained economic dispatch model that efficiently integrates energy storage and high renewable penetration to satisfy renewable portfolio requirements. Specifically, we require that wind energy contribute at least a prespecified proportion of the total demand and that the scheduled wind energy is deliverable with high probability. We develop an approximate partial sample average approximation (PSAA) framework to enable efficient solution of large-scale chance-constrained economic dispatch problems. Computational experiments on the IEEE-24 bus system show that the proposed PSAA approach is more accurate, closer to the prescribed satisfaction tolerance, and approximately 100 times faster than standard sample average approximation. Finally, the improved efficiency of our PSAA approach enables solution of a larger WECC-240 test system in minutes.

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Author information

Authors and Affiliations

  1. Department of Systems and Industrial Engineering, University of Arizona, Tucson, AZ, 85721, USA

    Jianqiang Cheng

  2. Sandia National Laboratories, Livermore, CA, 94551, USA

    Richard Li-Yang Chen, Habib N. Najm, Ali Pinar & Cosmin Safta

  3. Sandia National Laboratories, Albuquerque, NM, 87185, USA

    Jean-Paul Watson

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  1. Jianqiang Cheng
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  3. Habib N. Najm
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  4. Ali Pinar
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  5. Cosmin Safta
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Corresponding author

Correspondence to Jianqiang Cheng.

Additional information

This work was funded by the Laboratory Directed Research and Development (LDRD) program of the Sandia National Laboratories. Sandia National Laboratories is a multimission laboratory managed and operated by National Technology & Engineering Solutions of Sandia, LLC, a wholly owned subsidiary of Honeywell International Inc., for the U.S. Department of Energys National Nuclear Security Administration under contract DE-NA0003525. The views expressed in the article do not necessarily represent the views of the U.S. Department of Energy or the United States Government. This research has also been supported in part by the Bisgrove Scholars program (sponsored by Science Foundation Arizona).

Appendix A

Appendix A

We consider a simplified 6-bus system illustrated in Fig. 4. In this system, there are two conventional generators with capacity 10 and 20 (with cost 5 and 1 per unit, respectively). There are two wind farms with independent uniformly distributed capacity on the intervals [0, 20] and [0, 40]. There are two load buses with loads 9 and 4 units, respectively, and five transmission lines with capacities of 5 and 10 units. We choose a confidence parameter \(\alpha =0.19\) for constraint (9f) (chance constraint) and \(\beta =50\%\) for constraint (9e). Accordingly, we have the following optimization problem.

$$\begin{aligned} \min _{\varvec{f,p}}\quad&5p_1+p_2&\end{aligned}$$
(19a)
$$\begin{aligned} \text {s.t.} \quad&f_1-p_{w_1}=0,\; f_1-f_2=9,\end{aligned}$$
(19b)
$$\begin{aligned}&f_2-f_3+p_1=0,\; f_3-f_4+p_{w_2}=0,\end{aligned}$$
(19c)
$$\begin{aligned}&f_4-f_5=4,\; f_5+p_2=0,\end{aligned}$$
(19d)
$$\begin{aligned}&|f_i|\le 5, i=1,3\;,|f_i|\le 10, i=2,4, 5\end{aligned}$$
(19e)
$$\begin{aligned}&0\le p_1\le 10, 0\le p_2\le 20, p_{w_1}, p_{w_2}\ge 0\end{aligned}$$
(19f)
$$\begin{aligned}&p_{w_1}+p_{w_2}\ge 6.5, \end{aligned}$$
(19g)
$$\begin{aligned}&\mathbb P \{p_{w_1} \le \tilde{p}_{w_1}, p_{w_2} \le \tilde{p}_{w_2}\; \} \ge 0.81 \end{aligned}$$
(19h)

which is equivalent to the following problem:

$$\begin{aligned} \min _{\varvec{p}}\quad&5p_1+p_2&\end{aligned}$$
(20a)
$$\begin{aligned} \text {s.t.} \quad&|p_{w_1}|\le 5, |p_{w_1}-9|\le 10, |p_{w_1}-9+p_1|\le 5, \end{aligned}$$
(20b)
$$\begin{aligned}&|p_{w_1}-9+p_1+p_{w_2}|\le 10\end{aligned}$$
(20c)
$$\begin{aligned}&|p_{w_1}-13+p_1+p_{w_2}|\le 10\end{aligned}$$
(20d)
$$\begin{aligned}&p_{w_1}-13+p_1+p_{w_2}+p_2=0\end{aligned}$$
(20e)
$$\begin{aligned}&0\le p_1\le 10, 0\le p_2\le 20, p_{w_1}, p_{w_2}\ge 0\end{aligned}$$
(20f)
$$\begin{aligned}&p_{w_1}+p_{w_2}\ge 6.5, \end{aligned}$$
(20g)
$$\begin{aligned}&\mathbb P \{ p_{w_1} \le \tilde{p}_{w_1}, p_{w_2} \le \tilde{p}_{w_2}\; \} \ge 0.81 \end{aligned}$$
(20h)

The optimal solution of problem (20) is \(\varvec{p}=(p_1,p_2,p_{w_1},p_{w_2})\) \(=\) (2.532, 3.968, 1.468, 5.032) with the optimal value 16.628. Without constraint (20g), the optimal solution is \(\varvec{p}=(0.2, 9,3.8,0)\) with the optimal value 10. Upon further investigation, we found that constraint (20g) is necessary because of transmission constraints. Without transmission limit, it is always the case that the objective function is a non-increasing function of the total penetration of wind energy.

Fig. 4
figure 4

Simplified power grid

Full size image

Additionally, we consider three cases: \(\varvec{p}_1=(4, 1.5, 0, 7.5)\), \(\varvec{p}_2=(3, 3.5,1, 5.5)\) and \(\varvec{p}_3=(2, 5, 2, 4)\). Without constraint (20g), all three solutions are feasible solution to problem (20) with cost 21.5, 18.5 and 15, respectively. Although \(\varvec{p}_3\) has the smallest cost among them, it does not satisfy constraint (20g). Moreover compared with solution \(\varvec{p}_1\), \(\varvec{p}_2\) has a smaller cost but it also has a lower wind penetration. Thus, we can conclude that (1) constraint (20g) is not redundant and (2) maximum dispatch of (free) wind does not necessitate a lower cost for general power systems, due to the transmission limitations.

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Cheng, J., Chen, R.LY., Najm, H.N. et al. Chance-constrained economic dispatch with renewable energy and storage. Comput Optim Appl 70, 479–502 (2018). https://doi.org/10.1007/s10589-018-0006-2

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