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An implementation of improved harmony search algorithm for scenario-based transmission expansion planning

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Abstract

In this paper a new scenario-based framework is presented for transmission expansion planning (TEP) under normal and N–1 conditions. The proposed framework takes into account cost of network losses, cost of the transmission circuits and substations in the optimization process as objective functions, while considers short-term and also long-term constraints under normal and N–1 conditions as problem constraints. The proposed model is a non-convex optimization problem having a non-linear mixed-integer nature. A new improved harmony search algorithm (IHSA) is used in order to obtain the final optimal solution. The IHSA is a recently developed optimization algorithm which imitates the music improvisation process. In this process, the harmonists improvise their instrument pitches searching for the perfect state of harmony. The newly planning methodology has been demonstrated on the well-known Garver’s 6-bus test system and a real life network of south Brazilian electric power grid in order to demonstrate the feasibility and capabilities of the proposed algorithm. The detailed results of the case studies are presented and thoroughly analyzed. The obtained TEP results illustrate the sufficiency and profitableness of the newly developed method in expansion planning when compared with other methods.

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Abbreviations

\(ALC_s\) :

Yearly losses cost in scenario \(s\)

\(BW(Gen)\) :

Band width for any generation

\(BW_{\min }\) :

The minimum BW

\(BW_{\max }\) :

The maximum BW

\(c\) :

Index of the outage state of any facility

\(CC_{ij}\) :

Cost of a circuit that may be added to the corridor \(i-j\)

\(CS_k\) :

Cost of substation \(k\) that may be added

\(CCS_s\) :

Annual capital cost of substation \(k\) that may be added

\(D^s\) :

Demand vector in scenario \(s\)

\(e\) :

Index of the expanded network adequacy (in year)

\(F^s\) :

Active power matrix with elements \(f_{ij}\) in scenario \(s\)

\(F_c^s\) :

Active power matrix with elements \(f_{ij}\) in scenario \(s\) and outage state \(c\)

\(f_{ij}^s\) :

Active power flow in corridor \(i-j\) in scenario s [MW]

\(f_{ij,c}^s\) :

Active power flow in corridor \(i-j\) in scenario s and outage state \(c\) [MW]

\(\overline{f_{ij}}\) :

Thermal capacity in each line [MW]

\(g_u^s\) :

Active generation of generator \(u\) in scenario \(s\) [MW]

\(g_{u,c}^s\) :

Active generation of generator \(u\) in scenario \(s\) and outage state \(c\) [MW]

\(\underline{g_u},\overline{g_u}\) :

Lower and upper bound for the generator \(u\) [MW]

\(G^s\) :

Generation vector in scenario \(s\)

\(G_c^s\) :

Generation vector in scenario \(s\) and outage state \(c\)

\(Gen\) :

Number of generation in harmony search algorithm

\(i,j\) :

Index of buses

\(I_{ij}\) :

Flow current of corridor \(i-j\) (that load growth after the expansion has changed and therefore it’s time dependent)

\(IC_s\) :

Annual capital investment cost of transmission lines in scenario \(s\)

\(k\) :

Index of substations

\(L\) :

The number of decision variables

\(M\) :

The number of equality constraints

\(n_{ij}^s\) :

Number of new circuits added to the corridor \(i-j\) in scenario \(s\)

\(n_{ij,c}^s\) :

Number of new circuits added to the corridor \(i-j\) in scenario \(s \) and outage state \(c\)

\(n_{ij}^0\) :

Number of initial circuits in corridor \(i-j\)

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

Maximum number of new branches which can be existed to the corridor \(i-j\)

\(N\) :

The number of inequality constraints

\(p_s\) :

Occurrence degree of scenario \(s\)

\(PAR(Gen)\) :

The PAR for each generation

\(PAR_{\min }\) :

The minimum PAR

\(PAR_{\max }\) :

The maximum PAR

\(r_b^s\) :

Unsupplied load at bus \(b\) in scenario \(s\)

\(r_{b,c}^s\) :

Unsupplied load at bus \(b\) in scenario s and outage state \(c\)

\(R_{ij}\) :

Resistance of corridor \(i-j\)

\(s\) :

Index of scenario

\(S^s\) :

Branch-node incidence matrix in scenario \(s\)

\(S_c^s\) :

Branch-node incidence matrix in scenario s and outage state \(c\)

\(T\) :

Lifetime of any facility

\(TEC\) :

Total expansion cost

\(u\) :

Index of generators

\(U\) :

Set of all generators

\(v\) :

Interest rate

\(y_{i0}^s\) :

Shunt admittance at bus \(i\) in scenario \(s\)

\(y_{i0,c}^s \) :

Shunt admittance at bus \(i\) in scenario \(s\) and outage state \(c\)

\(y_{ij}^0 \) :

Initial admittance of corridor \(i-j\)

\(Y\) :

Bus admittance matrix of system

\(\alpha \) :

A factor to turn unsupplied load to cost ($US/MW)

\(\gamma _{ij}^s\) :

Total susceptance of circuits in corridor \(i-j\) in scenario \(s\)

\(\gamma _{ij,c}^s \) :

Total susceptance of circuits in corridor \(i-j\) in scenario \(s\) and outage state \(c\)

\(\Delta \) :

Set of all substations

\(\theta \) :

Phase angle of each bus

\(\lambda ^s\) :

Transmission line loading in scenario \(s\)

\(\lambda _c^s\) :

Transmission line loading in scenario \(s\) and outage state \(c\)

\(\lambda _{max}\) :

Maximum of transmission line loading index

\(\mu _{Loss}\) :

Losses coefficient (that variation hostelry of load than peak condition to modeling)

\(\xi \) :

Set of the transmission network expansion planning variables

\(\underline{\xi _l},\overline{\xi _l}\) :

The lower and upper bounds for each decision variables

\(\Pi (\xi )\) :

Transmission network expansion planning indices

\(\tau _{ij}^s\) :

New circuit admittance of corridor \(i-j\) in scenario \(s\)

\(\tau _{ij,c}^s\) :

New circuit admittance of corridor \(i-j\) in scenario \(s\) and outage state \(c\)

\(\Phi m\) :

Equality constraints

\(\chi _{MWh}\) :

Cost of any Megawatt-hour ($US/MWh)

\(\Psi _n\) :

Inequality constraints

\(\Omega _b\) :

Set of all busses network

\(\Omega _c\) :

Set of all outage states

\(\Omega _e\) :

Expanded network adequacy (in year)

\(\Omega _s\) :

Set of all scenarios

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Correspondence to Mojtaba Shivaie.

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Communicated by P. Ducang.

Appendices

Appendix A: Formation of the substations

In the TEP process is generally supposed that substations do not need expansion. The determination technique for expansion cost of substations (i.e.,\(\,\, {\textit{CS}}_k )\) with complete details can be found in Shayeghi et al. (2008). Also, according to mentioned reference, in this work the network formation of the substations for Garver’s 6-bus test system is given in Table 17.

Table 17 Formation of substations for Garver’s 6-bus test system

Appendix B: The 46-buses South Brazilian network data

The network data for the 46-buses south Brazilian network test system such as corridors and parameters of network structure are presented in Table 18. Also, data of the generation and load parameters are given in Table 19.

Table 18 Line data for 46-buses south Brazilian network
Table 19 Generation and load parameters for 46-buses south Brazilian network

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Shivaie, M., Ameli, M.T. An implementation of improved harmony search algorithm for scenario-based transmission expansion planning. Soft Comput 18, 1615–1630 (2014). https://doi.org/10.1007/s00500-013-1167-7

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