# Optimal Placement of Series Capacitive Compensation in Transmission Network Expansion Planning

Article

## Abstract

This paper proposes a novel mathematical model to solve the static and multistage transmission network expansion planning problems considering the optimal placement of series capacitive compensation (SCC) devices. This model jointly examines the construction of new transmission lines, transformers, and the optimal placement of SCC devices to minimize the total investment cost while satisfying the demand requirements in the planning horizon. The problem is formulated as a mixed-integer nonlinear programming problem, which is solved using a high-performance hybrid genetic algorithm. Simulations done in four electrical systems (IEEE 24-bus, South Brazilian 46-bus, North-Northeast Brazilian 87-bus, and Colombian 93-bus) show that inclusion of SCC devices in the planning model results in lower investment cost and a better redistribution of power flows through transmission components.

## Keywords

Transmission network expansion planning Optimal placement of series capacitive Mixed-integer nonlinear programming problem High-performance hybrid genetic algorithm

## Sets

$$\varOmega _\mathrm{b}$$

Set of all network buses

$$\varOmega _\mathrm{l}$$

Set of all transmission right-of-way

$$\varOmega _\mathrm{s}$$

Set of all transmission right-of-way with SCC devices

$$\varOmega _\mathrm{t}$$

Set of stages considered in the multistage planning

## Constants

$$c_{ij}$$

Cost of a circuit added in the right-of-way $$i-j$$ (US$) $$c_{ij}^\mathrm{scc}$$ Cost of SCC installed in the right-of-way $$i-j$$ (US$)

d

Demand vector with elements of the $$d_{i}$$ type corresponding to the demand at bus i (MW)

$$d_{i}$$

Maximum demand in the bus i (MW)

$$d_{i}^{t}$$

Maximum demand in the bus i at stage t (MW)

$$\overline{f}_{ij}$$

Maximum power flow per circuit $$i-j$$ (MW)

$$\overline{g}_{i}$$

Maximum generation in the bus i (MW)

I

Annual discount rate

k

Iteration counter

$$k_\mathrm{ps}$$

Population size

$$k_\mathrm{sr}$$

Selection rate

$$k_\mathrm{mr}$$

Mutation rate (%)

$$k_\mathrm{of}$$

Maximum number of iterations without improvement in the objective function

m

Iteration counter without an improvement in the objective function

$$n_\mathrm{l}$$

Total number of circuits in the network

$$n_\mathrm{l}^\mathrm{scc}$$

Total number of lines with SCC devices in the network

$$n_{ij}^\mathrm{o}$$

Number of circuits in the base electrical network

$$\overline{n}_{ij}$$

Maximum number of circuits in the right-of-way $$i-j$$

$$\alpha$$

Penalty parameter associated with load shedding (US$/MW) $$\gamma _{ij}$$ Susceptance of circuit $$i-j$$ (p.u.) $$\delta _\mathrm{inv}^{t}$$ Present value of the investment (US$)

$$\delta _\mathrm{oper}^{t}$$

Present value of operation cost (US\$)

## Variables

$$\mathbf{B}$$

Susceptance matrix of the initial network and candidate circuits (p.u)

$$f_{ij}$$

Power flow in the right-of-way $$i-j$$ (MW)

$$\mathbf{g}$$

Real power generation vector with elements $$g_{i}$$ corresponding to the generation in bus i (MW)

$$n_{ij}$$

Number of circuits added in the right-of-way $$i-j$$

$$n_{ij}^{t}$$

Number of circuits added in the right-of-way $$i-j$$ at stage t

$$r_{i}$$

Artificial generation at bus i (MW)

$$\mathbf{r}$$

Artificial generators vector with elements $$r_{i}$$ corresponding to the artificial generation in bus i (MW)

$$x_{ij}^\mathrm{total}$$

Reactance of the compensated circuit (p.u)

$$x_{ij}$$

Reactance of the line without compensation (p.u.)

$$x_{ij}^\mathrm{scc}$$

Reactance associated with the SCC device (p.u.)

$$\beta _{ij}$$

Percentage (between 0 and 1) of the cost of installing SCC devices in the right-of-way $$i-j$$

$$\gamma _{ij}^\mathrm{total}$$

Total susceptance of the compensated right-of-way $$i-j$$ (p.u.)

$$\varvec{\varTheta }$$

Phase angle vector of the bus voltage with elements $$\theta _{i}$$ corresponding to the angle of the voltage in bus i (radians)

$$\lambda _{ij}$$

Percentage (between 0 and 1) of the reactive compensation in the right-of-way $$i-j$$

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© Brazilian Society for Automatics--SBA 2019

## Authors and Affiliations

• Luis A. Gallego
• 1
• Lina P. Garcés
• 2
• Javier Contreras
• 3
1. 1.Department of Electrical EngineeringLondrina State UniversityLondrinaBrazil
2. 2.Electrical, Mechanical and Computer Engineering SchoolFederal University of GoiásGoiâniaBrazil
3. 3.E.T.S. de Ingenieros IndustrialesUniversity of Castilla–La ManchaCiudad RealSpain