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A Search Space Reduction Strategy and a Mathematical Model for Multistage Transmission Expansion Planning with \(N-1\) Security Constrains

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Abstract

This paper proposes a linear disjunctive model to solve a multistage transmission expansion planning problem (MTEP) considering \(N - 1\) security constraints. The use of a disjunctive linear model guarantees finding the optimum solution of the problems using existing classical optimization methods. For large-scale systems, when finding the optimum or even high-quality solutions of the MTEP problem is not possible in polynomial time, a search space reduction methodology (SSRM) is proposed. By using SSRM, it is possible to obtain very high-quality solutions or in most cases the optimum solution of the MTEP problem. The \(N-1\) security constraint indicates that the transmission system must be expanded in such a way that, despite the outage of a system line (a pre-defined set of contingencies), the system continues to operate properly. The model was implemented using a modelling language for mathematical programming (AMPL) and solved using the CPLEX, which is a commercial solver. The IEEE 24-bus, Colombian 93-bus, and Bolivian 57-bus systems are used to evaluate and show the performance of the proposed mathematical model and the search space reduction strategy.

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Abbreviations

\(\alpha _t\) :

Discount factor used to calculate the net present value of the investment at stage \(t\)

\(c_{ij}\) :

The investment cost of transmission lines in corridor \(ij\)

\(w_{ij,y,t}\) :

Binary variable used to model candidate line \(y\) in corridor \(ij\), stage \(t\)

\(f_{ij,y,c,t}\) :

Active power flow of candidate line \(y\) in corridor \(ij\), scenario \(c\) and stage \(t\)

\(f_{ij,c,t}^0\) :

Active power flow of existing lines in the base topology in corridor \(ij\), scenario \(c\) and stage \(t\)

\(g_{i,c,t}\) :

Active power generation in bus \(i\), contingency \(c\) and stage \(t\)

\(d_{i,t}\) :

Active power demand in bus \(i\) and stage \(t\)

\(n_{ij}^{0}\) :

Number of existing lines in the base topology in corridor \(ij\)

\(N_{ij,c}^{\text{ cont }}\) :

Elements of contingency matrix \(N^{\text{ cont }}\). For scenario \(c\), it is equal 1 if an outage for a line in corridor \(ij\) occurs; if equal 0 the line is in normal operation

\(\theta _{i,c,t}\) :

Voltage phase angle in bus \(i\), scenario \(c\) and stage \(t\)

\(x_{ij}\) :

Reactants of transmission line in corridor \(ij\)

\(\overline{f}_{ij,c}\) :

Maximum active power flow for a line in corridor \(ij\) and scenario \(c\)

\(M\) :

Enough large number used in disjunctive model

\(\overline{g}_{i,t,c}\) :

Maximum active power generation in bus \(i\), scenario \(c\) and stage \(t\)

\(\overline{n}_{ij}\) :

Maximum number of lines that can be added to the corridor \(ij\)

\(\text{ ref }\) :

Reference bus

\(\varOmega _b\) :

Set of buses

\(C\) :

Set of operation scenarios, where \(C\in C^{0}\cup C^{1}\cup C^{2}\) contains three different scenarios: the base case scenario (without contingency in lines) \(C^{0}\), the set of contingency scenarios in existing lines \(C^{1}\) and the set of contingency scenarios in candidate lines \(C^{2}\). Note that each scenario \(c\in C\) represents an operation state of the system

\(T\) :

Set of stages

\(\varOmega _l\) :

Set of corridors

\(Y\) :

Set of candidate transmission lines

\(E_t\) :

Set of solutions in stage \(t\)

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Acknowledgments

This work was supported by FAPESP and CAPES.

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Authors and Affiliations

  1. Departamento de Matemática, Universidade do Estado de Mato Grosso (UNEMAT), Sinop, Mato Grosso, Brazil

    Emivan F. da Silva

  2. Electrical and Computer Engineering Department, Carnegie Mellon University, Pittsburgh, PA, USA

    Mohsen Rahmani

  3. Departamento de Engenharia Elétrica, Universidade Estadual Paulista “Júlio de Mesquita Filho” (UNESP), Ilha Solteira, São Paulo, Brazil

    Marcos J. Rider

Authors
  1. Emivan F. da Silva
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  2. Mohsen Rahmani
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  3. Marcos J. Rider
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Correspondence to Marcos J. Rider.

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da Silva, E.F., Rahmani, M. & Rider, M.J. A Search Space Reduction Strategy and a Mathematical Model for Multistage Transmission Expansion Planning with \(N-1\) Security Constrains. J Control Autom Electr Syst 26, 57–67 (2015). https://doi.org/10.1007/s40313-014-0154-2

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