Whale optimization algorithm applied to load frequency control of a mixed power system considering nonlinearities and PLL dynamics

  • Dipayan GuhaEmail author
  • Provas Kumar Roy
  • Subrata Banerjee
Original Paper


In this work, an attempt has been made to implement a nature-inspired stochastic evolutionary algorithm, namely whale optimization algorithm (WOA) for exploring optimum and practical solutions of load frequency control (LFC) problem in power system. The proposed WOA mimics the ‘bubble-net feeding’ strategy of ‘humpback whales’ in the oceans. The optimization technique is individually applied to a two-area thermal power plant and two-area hydro-thermal-gas power plant with AC–DC tie-line for fine-tuning of the controller parameters. The study further houses the consequences of frequency measurement and the dynamics of a phase-locked loop (PLL) with power system nonlinearities. To establish the efficacy of WOA, the obtained results are compared with results of success history based adaptive differential evolution (SHADE), krill herd algorithm (KHA), and some other well-known control algorithms. Finally, statistical analysis is performed to affirm robustness of the proposed WOA in LFC area.


Load frequency control Whale optimization algorithm Phase-locked loop Transient responses Statistical analysis 

List of symbols

\( \xi ,\omega_{n} \)

Damping ratio and natural frequency of oscillation in rad/s

\( apf \)

Area participation factor

\( pu \)

Per unit

\( B_{1} ,B_{2} \)

Frequency bias constant of area-1 and 2, respectively

\( \dim \)

Number of control variable

\( G_{sg} \left( s \right),G_{t} \left( s \right),G_{ps} \left( s \right) \)

T.F. of governor, turbine and power system, respectively


Fitness function

\( K_{ps} \)

Gain of power system

\( k_{p} ,k_{i} ,k_{d} \)

Proportional, integral, and derivative gain, respectively

\( K_{ac} \)

Gain of AC tie-line

\( k_{f,m} ,k_{f,n} \)

Frequency control gain of area-m and area-n, respectively \( \left( {m \ne n} \right) \)

\( k_{pd} \)

Gain of phase detector

\( k_{vco} \)

Gain of voltage control oscillator (VCO)

\( n_{p} \)

Population size

\( R_{1} ,R_{2} \)

Speed regulation parameter of speed governor of area-1 and 2, respectively

\( T_{i} \)

Integral time constant in seconds

\( T_{d} \)

Time delay in seconds

\( T_{DC} \)

Time constant of HVDC line in seconds

\( T_{sg} \)

Time constant of speed governor in seconds

\( T_{t} \)

Time constant of steam turbine in seconds

\( T_{ps} \)

Time constant of power system itself in seconds

\( T_{12} \)

Synchronizing time constant of AC tie-line in seconds

\( \Delta P_{tie} \)

Tie-line power deviation in \( pu \)

\( \Delta P_{DC} \)

Power modulated by HVDC in \( pu \)

\( \Delta P_{D} \)

Load disturbance in \( pu \)

\( \Delta f_{1} ,\Delta f_{2} \)

Frequency deviation of area-1 and 2, respectively, in Hz



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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of Electrical EngineeringMotilal Nehru National Institute of TechnologyAllahabadIndia
  2. 2.Department of Electrical EngineeringKalyani Government Engineering CollegeKalyaniIndia
  3. 3.Department of Electrical EngineeringNational Institute of TechnologyDurgapurIndia

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