Abstract
A novel approach for solving CC-OPF problems is proposed, where the uncertainties are evaluated based on polynomial approximate method and Cornish–Fisher expansion. Both the objective function and constraints are reformulated in the form of polynomials, and then the uncertainties can be evaluated directly, where Cornish–Fisher expansion is used to handle the chance constraints. After the original probabilistic model is reformed into a deterministic optimization model, the traditional optimization methods can be implemented, e.g., interior point method. Case studies on IEEE 9-bus system and 118-bus system show that this new method can efficiently solve the CC-OPF modeled by AC load flow equations.
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Abbreviations
- \(\alpha \) :
-
The confidence level of chance constraints
- \(\beta ^N_\alpha (x)\) :
-
The quantile point of standard normal distribution with probability \(\alpha \)
- \(\beta _\alpha (x)\) :
-
The quantile point \(\beta _\alpha (x)\) for probability \(\alpha \)
- \(\hat{\mathbf {c}}_i\) :
-
The unknown coefficients to be determined
- \(\hat{c}\) :
-
The monomial coefficient of the target function in the form of polynomial approximation
- \(\kappa _n\) :
-
The nth cumulant
- \(\langle a,b\rangle \) :
-
The projection operator defined as \(\int \int a\cdot b\,d\mathbf {u}d\varvec{\Xi }\)
- \(\mathbf {u}\) :
-
The array of decision variables, where \(\mathbf {u}=\{u_i\}_{i=1}^{N_u}\)
- \(\mathbf {x}\) :
-
The array of system states, where \(\mathbf {x}=\{x_i\}_{i=1}^{N_x}\)
- \(\mathbf {x}^*\) :
-
The approximate solutions of \(\mathbf {x}\)
- \(\hbox {Pr}{[}\cdot {]}\) :
-
The probability function
- \(\omega (x)\) :
-
The probability density function (PDF) of x
- \(\varPhi _i\) :
-
The polynomial basis from \(\{\varPhi _i\}_i^N\)
- \(\varXi \) :
-
The array of random variables, where \(\varXi =\{\xi _i\}_{i=1}^{N_\xi }\)
- \(A(\cdot )\) :
-
The equality constraints, e.g., load flow equations
- \(C_G(\cdot )\) :
-
The cost function
- \(E{[}\cdot {]}\) :
-
The expectation operator
- \(E_n^C\) :
-
The nth central moment
- \(E_n^R\) :
-
The nth raw moment
- \(f(\mathbf {u})\) :
-
The monomial expression in terms of \(\mathbf {u}\)
- k :
-
The index of polynomial basis to be projected on, where \(k=1,\ldots ,N\)
- N :
-
The dimension of polynomial basis
- \(N_\xi \) :
-
The dimension of random variables
- \(N_u\) :
-
The dimension of decision variables
- \(N_x\) :
-
The dimension of system states
- \(N_{\kappa }\) :
-
The maximum order of cumulants
- R :
-
The residual of A with substitution of polynomial approximation
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Funding
Funding was provided by Postdoctoral Research Foundation of China (Grant No. 2018M642430), Technology Research Project of State Grid Jiangsu Electric Power Co., LTD. (Grant No. SGTYHT/16-JS-198).
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Cai, Y., Wang, L., Zhou, J. et al. Chance-Constrained OPF Based on Polynomials Approximation and Cornish–Fisher Expansion. Iran J Sci Technol Trans Electr Eng 44, 1357–1367 (2020). https://doi.org/10.1007/s40998-020-00316-6
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DOI: https://doi.org/10.1007/s40998-020-00316-6