# Stochastic economic analysis of FACTS devices on contingent transmission networks using hybrid biogeography-based optimization

## Abstract

Flexible AC transmission systems (FACTS) devices have advantages of enhancing AC system controllability and stability, increasing power transfer capability and relieving congestion. Finding the best sizing and siting of them is obligatory to obtain the maximum benefit. In this paper, the optimum planning is done for determining the suitable sizing and siting of the FACTS devices during their lifetime span. The proposed planning approach is implemented to the networks, which are suffering from the contingency to demonstrate that how proper sizing and siting of the FACTS devices are able to deal with the existing problems in such networks. In fact, the maximum social welfare, reducing load shedding cost and construction cost of the new branches besides of the technical issues such as voltage improvement are the main concerns of the present work. To accomplish these aims accurately and reduce the computation time, a hybrid approach, consisting of mathematical and heuristic methods, is proposed for solving the proposed planning problem. The mentioned algorithm is a combination of biogeography-based optimization (BBO) as a heuristic algorithm and nonlinear programming as a mathematical approach. Furthermore, since the stochastic natures of renewable energy sources and load variations contribute to the optimum decisions, the stochastic formulation has been considered in the planning problem using the efficient point estimation (\( 2m + 1 \)) scheme. Finally, the proposed planning approach is tested on different test systems, namely IEEE 14-bus, IEEE 57-bus and IEEE 300-bus, in order to verify its effectiveness from a different point of views.

## Keywords

Biogeography-based optimization (BBO) Nonlinear programming FACTS Contingency management## List of symbols

*l*Index for load

*n, m*Indices for node

*g*Index for generator

*h*Index for hour

*y*Index for year

*s*Index for scenario

*f*Index for FACTS devices

*j*Index for uncertain variable in (\( 2m + 1 \)) scheme

- \( \omega \)
Index for habitat

- \( \varphi \)
Index for SIV in each habitat

- iter
Index for iteration

- NPV
Index for net present value of the variable

- FACTS
State with FACTS devices

- NOFACTS
State without FACTS devices

- exp
Superscript for expected value of the variable

- Max
Superscript for maximum amount of the variable

- Min
Superscript for minimum amount of the variable

- new
Superscript for updated variable

- BBO
Biogeography-based optimization

## Parameters

- \( A,B \)
Shape and scale of the Weibull distribution

- \( P_{\text{wr}} \)
Rated power of the wind

*a*,*b*,*c*Coefficients of bid function of the interruptible loads

- \( v_{\text{i}} \)
Cut-in speed

- \( v_{\text{r}} \)
Rated speed

- \( v_{\text{o}} \)
Cut-out speed

- \( N_{\text{TCSC}} \)
Number of the TCSC

- \( N_{\text{SVC}} \)
Number of the SVC

- \( N_{\text{scen}} \)
Number of the scenarios

- \( n_{G} \)
Number of the generators

- \( n_{l} \)
Number of the load

- \( n_{\text{lifetime}} \)
Lifetime of the FACTS devices

- rate
Rate of interest

- \( {\text{price}}^{\text{shed}} \)
Price of the load shedding ($/MW)

- \( L_{\text{int}} \)
Set of the interruptible load

- \( \alpha \)
Fixed coefficient in BBO algorithm

## Variables

- \( v \)
Velocity of wind

- \( P_{w} \)
Wind output power

- \( P_{g} \)
Generator output active power (MW)

- \( Q_{g} \)
Generator output reactive power (MW)

- \( P^{\text{load}} \)
Active load in each node (MW)

- \( Q^{\text{load}} \)
Reactive load in each node (MW)

- \( P^{\text{fc}} \)
Forecasted active power in each node (MW)

- \( P^{\text{shed}} \)
Active shed power in each node (MW)

- \( C_{\text{Gen}} \)
Cost function of the diesel generator

- \( {\text{Cost}}_{\text{TCSC}} \)
Investment cost of the TCSC per kVAr

- \( {\text{Cost}}_{\text{SVC}} \)
Investment cost of the SVC per kVAr

- \( {\text{Cost}}_{\text{Install,FACTS}} \)
Total investment cost of the FACTS devices

- Ben
The obtained benefit of the TRANSCO

- \( S_{\text{SVC}} \)
SVC capacity in MVAr

- \( S_{\text{TCSC}} \)
TCSC capacity in MVAr

- Bid
Interruptible load bid

*V*Voltage magnitude of each node

- \( I^{\text{line}} \)
Line current

- prob
Probability of each scenario

- lmp
Locational marginal price ($/MW)

- \( \chi \)
Location of the particular concentration in (\( 2m + 1 \)) scheme

- \( \xi \)
Standard deviation of the particular variable

- \( \lambda \)
Standard control moment of the uncertain variable

*w*Weight of the calculated concentration

*x*Decision variable in each habitat in BBO algorithm

- \( \mu_{\text{BBO}} \)
Emigration rate in BBO algorithm

- \( \lambda_{\text{BBO}} \)
Immigration rate in BBO algorithm

## Function

- \( f_{\text{wind}} \)
Probability density function of the wind speed

- \( f_{\text{load}} \)
Probability density function of the load

- \( F(.) \)
Active power balance

- \( G(.) \)
Reactive power balance

- \( N(.) \)
Normal distribution

- \( f \)
Probability distribution density of particular uncertain variable

## Notes

### Compliance with ethical standards

### Conflict of interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

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