Optimal Design of Proportional–Integral Controllers for Grid-Connected Solid Oxide Fuel Cell Power Plant Employing Differential Evolution Algorithm

  • Ashik AhmedEmail author
  • Md. Shahid Ullah
  • Md. Ashraful Hoque
Research Paper


This paper proposes the application of differential evolution (DE) algorithm for the optimal tuning of proportional–integral controller designed to improve the small signal dynamic response of a grid-connected solid oxide fuel cell (SOFC) system. The small signal model of the study system is derived and considered for the controller design as the target here is to track small variations in SOFC load current. The proposed proportional–integral (PI) controllers are incorporated in the feedback loops of hydrogen and oxygen partial pressures, grid current dq components and dc voltage with an aim to improve the small signal dynamic responses. The controller design problem is formulated as the minimization of an eigenvalue-based objective function where the target is to find out the optimal gains of the PI controllers in such a way that the discrepancy between the obtained and desired eigenvalues is minimized. Eigenvalue and time domain simulations are presented for both open-loop and closed-loop systems. To test the efficacy of DE over other optimization tools, the results obtained with DE are compared with those obtained by particle swarm optimization (PSO) algorithm and invasive weed optimization (IWO) algorithm. Three different types of load disturbances are considered for the time domain-based results to investigate the performances of different optimizers under different sorts of load variations. Moreover, nonparametric statistical analyses, namely one-sample Kolmogorov–Smirnov (KS) test and paired sample t test, are used to identify the statistical advantage of DE algorithm over the other two. The presented results suggest the supremacy of DE over PSO and IWO in finding the optimal solution.


Grid-connected solid oxide fuel cell Differential evolution algorithm Small signal model Eigen-value based objective function Synchronously rotating dq reference frame 

List of Symbols


SOFC open-circuit voltage


Universal gas constant


Fuel cell temperature


Faraday’s constant


Number of cells in series

\(P_{{{\text{H}}_{2} }}\)

Hydrogen partial pressure

\(P_{{{\text{O}}_{2} }}\)

Oxygen partial pressure

\(P_{{{\text{H}}_{2} {\text{O}}}}\)

Water vapor partial pressure


Cell resistance for ohmic loss


Constant coefficient


Standard temperature


Cell output current

\(\dot{n}_{{\text{H}_{2} }}^{\text{in}}\)

Hydrogen inlet flow rate

\(\dot{n}_{{{\text{O}}_{2} }}^{\text{in}}\)

Oxygen inlet flow rate

\(\dot{n}_{{\text{H}_{2} \text{O}}}^{\text{in}}\)

Water vapor outlet flow rate


DC capacitor voltage


Cell output voltage


Converter duty ratio


DC capacitor

id, iq

dq component of grid current

Rf, Lf

Filter resistance and inductance

kd, kq

dq component of inverter switching signal

ed, eq

dq component of grid voltage

Pgrid, Qgrid

Grid active and reactive power


Mutation factor for DE



The authors wish to express their acknowledgement for the support from EEE Department, Islamic University of Technology, Bangladesh in completing this work.

Compliance with Ethical Standards

Conflict of interest

The authors declare that they have no conflict of interest.


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Copyright information

© Shiraz University 2019

Authors and Affiliations

  • Ashik Ahmed
    • 1
    Email author
  • Md. Shahid Ullah
    • 2
  • Md. Ashraful Hoque
    • 1
  1. 1.EEE DepartmentIslamic University of TechnologyGazipurBangladesh
  2. 2.EEE DepartmentDaffodil International UniversityDhakaBangladesh

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