Optimal operation of transmission power networks by using improved stochastic fractal search algorithm


This paper presents the application of an improved stochastic fractal search algorithm (ISFSA) for optimizing five single objectives of optimal power flow (OPF) problem and satisfying all constraints consisting of operating limits of electric components, power balance and load voltage magnitude limits. The proposed ISFSA is formed by implementing three improvements on the conventional stochastic fractal search algorithm (SFSA). The first improvement cancels one ineffective formula but keeps another one in diffusion process. The second improvement selects some worst solutions in the first update and some best solutions in the second update for producing new solutions. In the third improvement, a proposed technique is applied for carrying out the update processes. Comparisons of obtained results from three standard IEEE power systems indicate that the proposed method is superior to SFSA in terms of optimal solution quality, execution speed as well as success rate. The performance comparisons with other existing methods available in previous studies also lead to the conclusions that the proposed method can reach lower generation fuel cost, smaller total power losses, less amount of emission, better voltage profile and faster execution process. As a result, it can be recommended that the proposed ISFSA should be used for OPF problem in high-voltage power system field.

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\(F_{i}\) :

Fuel cost function of the ith thermal unit

\(\phi_{i} ,\phi_{j}\) :

Phase angles of voltage at the ith bus and the jth bus

\(a_{fi} ,b_{fi} ,c_{fi} ,d_{fi} ,e_{fi}\) :

Fuel cost coefficients of the ith thermal unit

\(a_{fim} ,b_{fim} ,c_{fim}\) :

Fuel cost coefficients of the fuel type m of the ith thermal unit

\(a_{ei} ,b_{ei} ,c_{ei} ,d_{ei} ,e_{ei}\) :

Emission function coefficients of the ith thermal unit

\(FF_{s,j}\) :

Fitness function of new solution s at the jth diffusion

\(FF_{s}^{new}\) :

Fitness function of the new solution s

\(FF_{s}\) :

Fitness function of the sth retained solution

\(FF_{average}\) :

Average fitness function of the whole population

\(G_{ij} ,B_{ij}\) :

Conductance and susceptance of a branch connecting the ith bus and the jth bus

\(K_{1} ,K_{2} ,K_{3} ,K_{4} ,K_{5}\) :

Penalty factors

\(N_{fs}\) :

Number of fuel sources

N VPZi :

Number of violated power zones of the ith thermal unit

\(N_{bus}\) :

Number of buses in considered system

\(N_{lb}\) :

Number of load buses

\(N_{tb}\) :

Number of transformer buses

\(N_{cb}\) :

Number of compensator buses

\(N_{tl}\) :

Number of transmission lines in the considered power system

\(N_{di}\) :

Maximum number of diffusion

\(N_{ps}\) :

Population size

\(P_{i}^{\hbox{min} } ,P_{i}^{\hbox{max} }\) :

Lower and upper limitations of real power of the ith thermal unit

\(P_{i}\) :

Real power output of the ith thermal unit

\(P_{im}^{\hbox{min} } ,P_{im}^{\hbox{max} }\) :

Lowest and the highest generations of the ith thermal unit corresponding to the mth fuel type

\(P_{loadi} ,Q_{loadi}\) :

Real and unreal power of load at the ith bus

\(P_{{i,VPZ_{j} }}^{\hbox{min} } ,P_{{i,VPZ_{j} }}^{\hbox{max} }\) :

Lower and upper bounds of the jth violated power zone of the ith thermal unit

\(Q_{sci}^{\hbox{min} } ,Q_{sci}^{\hbox{max} }\) :

Minimum and maximum reactive power output of the capacitor banks at the ith bus

\(Q_{i}^{\hbox{min} } ,Q_{i}^{\hbox{max} }\) :

Lower and upper limitations of reactive power of the ith thermal unit

\(Q_{i} ,V_{i}\) :

Currently working unreal power and voltage magnitude of the ith thermal unit

\(rand_{s,j}\) :

Random number for the solution s at the jth diffusion

\(S_{br}^{\hbox{max} }\) :

Maximum apparent power flow of the brth transmission line

\(Sol_{s,j}^{new}\) :

The sth new solution at the jth diffusion

\(T_{i}^{\hbox{min} } ,T_{i}^{\hbox{max} }\) :

Minimum and maximum setting of tap changer at the ith bus

\(V_{i}^{\hbox{min} } ,V_{i}^{\hbox{max} }\) :

Lower and upper limitations of voltage magnitude of the ith thermal unit

VPZj :

The jth violated power zone

\(V_{li}^{\hbox{min} } ,V_{li}^{\hbox{max} }\) :

Lower and upper bounds of operation voltage of the ith bus

Iter, NIt :

Current iteration and the maximum number of iterations

Pro s :

Ratio of rank of the sth solution to population size

N cv :

Number of control variables

βs, εs :

Random number within 0 and 1 for the sth solution

ε :

Random number within 0 and 1


Emission improvement level


Fuel cost


Fuel cost improvement level


Optimal power flow


Total power losses


Total power loss improvement level


Voltage deviation


Voltage deviation improvement level


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See Tables 20, 21, 22, 23 and 24.

Table 20 Optimal solutions obtained by the proposed method for case 1 of IEEE 30-bus power system
Table 21 Optimal solutions obtained by the proposed method for case 2 of IEEE 30-bus power system
Table 22 Optimal solutions obtained by the proposed method for case 3 of IEEE 30-bus power system
Table 23 Optimal solutions obtained by the proposed method for IEEE 57-bus power system
Table 24 Optimal solution obtained by the proposed method for IEEE 118-bus power system

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Nguyen, T.T., Nguyen, T.T., Duong, M.Q. et al. Optimal operation of transmission power networks by using improved stochastic fractal search algorithm. Neural Comput & Applic 32, 9129–9164 (2020). https://doi.org/10.1007/s00521-019-04425-0

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  • Stochastic fractal search
  • Diffusion process
  • Update process
  • Optimal power flow
  • Objective function
  • Operating limitations