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

Abstract

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.

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Abbreviations

E 0 :

SOFC open-circuit voltage

R :

Universal gas constant

T :

Fuel cell temperature

F :

Faraday’s constant

N 0 :

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

r 0 :

Cell resistance for ohmic loss

α :

Constant coefficient

T 0 :

Standard temperature

I fc :

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

V dc :

DC capacitor voltage

V fc :

Cell output voltage

d c :

Converter duty ratio

C dc :

DC capacitor

i d, i q :

dq component of grid current

R f, L f :

Filter resistance and inductance

k d, k q :

dq component of inverter switching signal

e d, e q :

dq component of grid voltage

P grid, Q grid :

Grid active and reactive power

M F :

Mutation factor for DE

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Acknowledgements

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

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Corresponding author

Correspondence to Ashik Ahmed.

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The authors declare that they have no conflict of interest.

Appendix

Appendix

SOFC data

E0 = 1.28 V R = 8.3146 J/mol °C F = 96,487 C/mol N0 = 384
T0 = 923 °C Tref = 25 °C T2 = 25 °C ΔĤ 0 r  = − 2.4183 kJ/mol
Kh2 = 8.43e−4 kmol/(atm-s) Ko2 = 2.52e−3 kmol/(atm-s) Kh2O = 2.81e−4 kmol/(atm-s) \(\tau_{{{\text{H}}_{2} }}^{0}\) = 26.1 s
\(\tau_{{O_{2} }}^{0}\) = 2.91 s \(\tau_{{h_{2} O}}^{0}\) = 78.3 s me = 1.1 kg \(\bar{C}_{p}\) = 1e4
r0 = 0.126 ohms α = − 2870 Tin = 900 °C  

Grid data

Egrid = 1.0 pu Rf = 0 Lf = 0.7163 pu ω0 = 1.0 pu

Initial operating data

n inh2  = 1.2 mol/s n inO2  = 0.6468 mol/s Ifcref = 193.35 A Ifcpu = 0.6445 pu
Pfcpu = 0.6983 pu Qgrid = 0 Kd = 0.6 Kq = 0.4
dc = 0.3499 Vdc = 1.6667 pu id = 0.6983 pu iq = 0
Ph2 = 0.9671 atm PO2 = 0.1803 atm Ph2O = 1.3692 atm  

Base Values

Ifcb = 300 A Pfcb = 97.33 kW Sbase = Pfcb Vfcbase = 324.46 V

Linearizing constants:

$$\text{H}_{1} = \text{dH}_{2} \frac{{T_{2}^{4} }}{4} + c\text{H}_{2} \frac{{T_{2}^{3} }}{3} + b\text{H}_{2} \frac{{T_{2}^{2} }}{2} + a\text{H}_{2} T_{2} - \left( {d\text{H}_{2} \frac{{T^{4} }}{4} + c\text{H}_{2} \frac{{T^{3} }}{3} + b\text{H}_{2} \frac{{T^{2} }}{2} + a\text{H}_{2} T} \right)$$
$$\text{O}_{1} = d\text{O}_{2} \frac{{T_{2}^{4} }}{4} + c\text{O}_{2} \frac{{T_{2}^{3} }}{3} + b\text{O}_{2} \frac{{T_{2}^{2} }}{2} + a\text{O}_{2} T_{2} - \left( {d{\text{O}}_{2} \frac{{T^{4} }}{4} + c{\text{O}}_{2} \frac{{T^{3} }}{3} + b{\text{O}}_{2} \frac{{T^{2} }}{2} + a{\text{O}}_{2} T} \right)$$
$$W_{1} = d\text{H}_{2O} \frac{{T_{2}^{4} }}{4} + c\text{H}_{2O} \frac{{T_{2}^{3} }}{3} + b\text{H}_{2O} \frac{{T_{2}^{2} }}{2} + a\text{H}_{2O} T_{2} - \left( {d\text{H}_{2O} \frac{{T^{4} }}{4} + c\text{H}_{2O} \frac{{T^{3} }}{3} + b\text{H}_{2O} \frac{{T^{2} }}{2} + a\text{H}_{2O} T} \right)$$
$$N_{1} = \text{d}N_{2} \frac{{T_{2}^{4} }}{4} + cN_{2} \frac{{T_{2}^{3} }}{3} + bN_{2} \frac{{T_{2}^{2} }}{2} + aN_{2} T_{2} - \left( {{\text{d}}N_{2} \frac{{T^{4} }}{4} + cN_{2} \frac{{T^{3} }}{3} + bN_{2} \frac{{T^{2} }}{2} + aN_{2} T} \right)$$
$$\begin{aligned} N_{12} = {\text{d}}N_{2} \frac{{T^{4} }}{4} + cN_{2} \frac{{T^{3} }}{3} + bN_{2} \frac{{T^{2} }}{2} + aN_{2} T \hfill \\ \text{H}_{12} = {\text{dH}}_{2} \frac{{T^{4} }}{4} + c\text{H}_{2} \frac{{T^{3} }}{3} + b\text{H}_{2} \frac{{T^{2} }}{2} + a\text{H}_{2} T \hfill \\ \text{O}_{12} = {\text{dO}}_{2} \frac{{T^{4} }}{4} + c\text{O}_{2} \frac{{T^{3} }}{3} + b\text{O}_{2} \frac{{T^{2} }}{2} + a\text{O}_{2} T \hfill \\ W_{12} = {\text{dH}}_{2O} \frac{{T^{4} }}{4} + cH_{2O} \frac{{T^{3} }}{3} + b\text{H}_{2O} \frac{{T^{2} }}{2} + a\text{H}_{2O} T \hfill \\ \end{aligned}$$
$$G_{1h} = \frac{{I_{fc}^{0} N_{0} RT^{0} }}{{2000FP_{{h_{2} }}^{0} }}\quad G_{2O} = \frac{{I_{fc}^{0} N_{0} RT^{0} }}{{4000FP_{{O_{2} }}^{0} }}\quad G_{3w} = \frac{{I_{fc}^{0} N_{0} RT^{0} }}{{2000FP_{{h_{2} O}}^{0} }}$$
$$Z_{1} = \frac{1}{{\bar{C}_{p} m_{e} }}\quad M_{1} = \left( {T_{2} - T^{0} } \right)\quad T_{\text{ph}} = - M_{1} K_{{h_{2} }} \left( {H_{1} + G_{1h} } \right)Z_{1}$$
$$T_{\text{po}} = - M_{1} K_{{O_{2} }} \left( {O_{1} + G_{2O} } \right)Z_{1} \quad T_{pw} = - M_{1} K_{{h_{2} O}} \left( {W_{1} - G_{3w} } \right)Z_{1}$$
$$T_{pr} = \frac{{I_{fc}^{0} \left( {N_{0} \left( {\frac{R}{2F}\ln \frac{{P_{{h_{2} }}^{0} P_{{O_{2} }}^{{0^{0.5} }} }}{{P_{{h_{2} O}}^{0} }} - 2.52e^{ - 4} } \right) - \frac{{I_{fc}^{0} \alpha r^{0} e^{{\alpha \left( {\frac{1}{{T_{0} }} - \frac{1}{{T^{{_{0} }} }}} \right)}} }}{{T^{{0^{2} }} }}} \right)}}{1000}$$
$$T_{pt} = \dot{n}_{{n_{2} }}^{out} \left( {N_{1} + M_{1} \left( {\dot{n}_{{n_{2} }}^{out} N_{12} + \dot{n}_{{h_{2} }}^{out} H_{12} + \dot{n}_{{h_{2} O}}^{out} W_{12} + \dot{n}_{{O_{2} }}^{out} O_{12} } \right) - {\text{Tpr}} + \dot{n}_{{h_{2} }}^{out} H_{1} + \dot{n}_{{h_{2} O}}^{out} W_{1} + \dot{n}_{{O_{2} }}^{out} O_{1} } \right)Z_{1}$$
$$T_{pi} = - \left( {\frac{{N_{0} \left( {\frac{{RT^{0} }}{2F}\ln \frac{{P_{{h_{2} }}^{0} P_{{O_{2} }}^{{0^{0.5} }} }}{{P_{{h_{2} O}}^{0} }} - 2.52e^{ - 4} T^{0} + 1.2586} \right)}}{1000} + \frac{{K_{r} \Delta \hat{H}_{r}^{0} }}{500} - \frac{{I_{fc}^{0} r^{0} e^{{\alpha \left( {\frac{1}{{T_{0} }} - \frac{1}{{T^{{_{0} }} }}} \right)}} }}{500}} \right)Z_{1}$$
$$\text{H}_{t2} = {\text{dH}}_{2} \frac{{T_{2}^{4} }}{4} + c\text{H}_{2} \frac{{T_{2}^{3} }}{3} + b\text{H}_{2} \frac{{T_{2}^{2} }}{2} + a\text{H}_{2} T_{2} - \left( {{\text{dH}}_{2} \frac{{T_{\text{in}}^{4} }}{4} + c{\text{H}}_{2} \frac{{T_{\text{in}}^{3} }}{3} + b{\text{H}}_{2} \frac{{T_{\text{in}}^{2} }}{2} + a{\text{H}}_{2} T_{\text{in}} } \right)$$
$$\text{O}_{t2} = {\text{dO}}_{2} \frac{{T_{2}^{4} }}{4} + c\text{O}_{2} \frac{{T_{2}^{3} }}{3} + b\text{O}_{2} \frac{{T_{2}^{2} }}{2} + a\text{O}_{2} T_{2} - \left( {\text{dO}_{2} \frac{{T_{\text{in}}^{4} }}{4} + c\text{O}_{2} \frac{{T_{\text{in}}^{3} }}{3} + b\text{O}_{2} \frac{{T_{\text{in}}^{2} }}{2} + a\text{O}_{2} T_{\text{in}} } \right)$$
$$T_{\text{pnh}} = - (T_{\text{in}} - T_{\text{ref}} )H_{t2} Z_{1} \quad T_{\text{pno}} = - (T_{\text{in}} - T_{\text{ref}} )O_{t2} Z_{1}$$
$$\begin{array}{*{20}l} {H_{y1} = K_{\text{ihy}} - \frac{{K_{\text{phy}} T^{0} }}{{T_{0} \tau_{{H_{2} }}^{0} }}} \hfill & {\text{H}_{y2} = \frac{{ - \left( {K_{\text{phy}} \left( {K_{{h_{2} }} P_{{h_{2} }}^{0} - n_{{h_{2} }}^{in0} + 2K_{r} I_{fc}^{0} } \right)} \right)}}{{K_{{h_{2} }} T_{0} \tau_{{{\text{H}}_{2} }}^{0} }}} \hfill & {{\text{H}}_{y3} = \frac{{ - \left( {2K_{\text{phy}} K_{r} T^{0} } \right)}}{{K_{{h_{2} }} T_{0} \tau_{{H_{2} }}^{0} }}} \hfill \\ {H_{y4} = \frac{{K_{\text{phy}} T^{0} }}{{K_{{h_{2} }} T_{0} \tau_{{H_{2} }}^{0} }}H_{y5} = - K_{ihy} } \hfill & {\text{O}_{x1} = K_{\text{iox}} - \frac{{K_{\text{pox}} T^{0} }}{{T_{0} \tau_{{O_{2} }}^{0} }}} \hfill & {\text{O}_{x2} = \frac{{ - \left( {K_{\text{pox}} \left( {K_{{{\text{O}}_{2} }} P_{{{\text{O}}_{2} }}^{0} - n_{{{\text{O}}_{2} }}^{{{\text{in}}0}} + K_{r} I_{\text{fc}}^{0} } \right)} \right)}}{{K_{{{\text{O}}_{2} }} T_{0} \tau_{{{\text{O}}_{2} }}^{0} }}} \hfill \\ \end{array}$$
$${\text{O}}_{x3} = \frac{{ - \left( {K_{\text{pox}} K_{r} T^{0} } \right)}}{{K_{{{\text{O}}_{2} }} T_{0} \tau_{{O_{2} }}^{0} }}{\text{O}}_{x4} = \frac{{K_{\text{pox}} T^{0} }}{{K_{{{\text{O}}_{2} }} T_{0} \tau_{{{\text{O}}_{2} }}^{0} }}$$
$$\begin{array}{*{20}l} {\text{O}_{x5} = - K_{\text{iox}} } \hfill & {I_{d1} = K_{\text{iid}} - \left( {\frac{{K_{\text{pid}} R_{f} }}{{L_{f} }}} \right)} \hfill & {I_{d2} = \omega_{0} K_{\text{pid}} } \hfill & {I_{d3} = \frac{{k_{d0} K_{\text{pid}} }}{{L_{f} }}} \hfill \\ {I_{d4} = \frac{{V_{{{\text{dc}}0}} K_{\text{pid}} }}{{L_{f} }}} \hfill & {I_{d5} = - K_{\text{iid}} } \hfill & {I_{q1} = - \omega_{0} K_{\text{piq}} } \hfill & {I_{q2} = K_{\text{iiq}} - \left( {\frac{{K_{\text{piq}} R_{f} }}{{L_{f} }}} \right)} \hfill \\ {I_{q3} = \frac{{K_{\text{piq}} k_{q0} }}{{L_{f} }}} \hfill & {I_{q4} = \frac{{V_{dc0} K_{\text{piq}} }}{{L_{f} }}} \hfill & {I_{q5} = - K_{\text{iiq}} } \hfill & {V_{1} = - \frac{{K_{pv} \left( {d_{c0} - 1} \right)}}{{\left( {C_{dc} I_{\text{fcb}} } \right)}}} \hfill \\ {V_{2} = - \frac{{k_{d0} K_{pv} }}{{C_{\text{dc}} }}} \hfill & {V_{3} = - \frac{{k_{q0} K_{pv} }}{{C_{\text{dc}} }}} \hfill & {V_{4} = K_{iv} } \hfill & {V_{5} = - \frac{{i_{d0} K_{pv} }}{{C_{\text{dc}} }}} \hfill \\ \end{array}$$

System Matrices:

$$A = \left[ {\begin{array}{*{20}c} {\frac{{ - T^{0} }}{{T_{0} \tau_{{{\text{H}}_{2} }}^{0} }}} & 0 & 0 & { - \frac{{\left( {K_{{h_{2} }} P_{{h_{2} }}^{0} - n_{{h_{2} }}^{in0} + 2I_{fc}^{0} K_{r} } \right)}}{{K_{{h_{2} }} T_{0} \tau_{{{\text{H}}_{2} }}^{0} }}} & { - \frac{{2K_{r} T^{0} }}{{K_{{h_{2} }} T_{0} \tau_{{H_{2} }}^{0} }}} & 0 & 0 & 0 \\ 0 & {\frac{{ - T^{0} }}{{T_{0} \tau_{{{\text{O}}_{2} }}^{0} }}} & 0 & { - \frac{{\left( {K_{{h_{2} O}} P_{{h_{2} O}}^{0} - 2I_{\text{fc}}^{0} K_{r} } \right)}}{{K_{{h_{2} O}} T_{0} \tau_{{h_{2} O}}^{0} }}} & { - \frac{{K_{r} T^{0} }}{{K_{{O_{2} }} T_{0} \tau_{{O_{2} }}^{0} }}} & 0 & 0 & 0 \\ 0 & 0 & {\frac{{ - T^{0} }}{{T_{0} \tau_{{h_{2} O}}^{0} }}} & { - \frac{{\left( {K_{{h_{2} O}} P_{{h_{2} O}}^{0} - 2I_{fc}^{0} K_{r} } \right)}}{{K_{{h_{2} O}} T_{0} \tau_{{h_{2} O}}^{0} }}} & {\frac{{2K_{r} T^{0} }}{{K_{{h_{2} O}} T_{0} \tau_{{h_{2} O}}^{0} }}} & 0 & 0 & 0 \\ {T_{\text{ph}} } & {T_{\text{po}} } & {T_{\text{pw}} } & {T_{\text{pt}} } & {T_{\text{pi}} } & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & {\frac{{ - \Delta I_{\text{fc}} }}{{T_{\text{el}} }}} & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & { - \frac{{R_{f} }}{{L_{f} }}} & {\omega_{0} } & {\,\frac{{k_{d0} }}{{L_{f} }}} \\ 0 & 0 & 0 & 0 & 0 & { - \omega_{0} } & { - \frac{{R_{f} }}{{L_{f} }}} & {\frac{{k_{q0} }}{{L_{f} }}} \\ 0 & 0 & 0 & 0 & { - \frac{{\left( {d_{c0} - 1} \right)}}{{\left( {C_{\text{dc}} I_{\text{fcb}} } \right)}}} & { - \frac{{k_{d0} }}{{C_{\text{dc}} }}} & { - \frac{{K_{q} }}{{C_{\text{dc}} }}} & 0 \\ \end{array} } \right]$$
$$B = \left[ {\begin{array}{*{20}c} {\frac{{T^{0} }}{{T_{0} K_{{h_{2} }} \tau_{{{\text{H}}_{2} }}^{0} }}} & 0 & 0 & {T_{\text{pnh}} } & 0 & 0 & 0 & 0 \\ 0 & {\frac{{T^{0} }}{{T_{0} K_{{{\text{O}}_{2} }} \tau_{{{\text{O}}_{2} }}^{0} }}} & 0 & {T_{\text{pno}} } & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & {\frac{{V_{{{\text{dc}}0}} }}{{L_{f} }}} & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & {\frac{{V_{{{\text{dc}}0}} }}{{L_{f} }}} & 0 \\ 0 & 0 & 0 & 0 & 0 & { - \frac{{I_{{{\text{fc}}0}} }}{{\left( {C_{\text{dc}} I_{\text{fcb}} } \right)}}} & { - \frac{{i_{d0} }}{{C_{\text{dc}} }}} & { - \frac{{i_{q0} }}{{C_{\text{dc}} }}} \\ \end{array} } \right]^{\,T}$$
$$B_{\text{sys}} = \left[ {\begin{array}{*{20}c} 0 & 0 & 0 & 0 & {\frac{1}{{T_{el} }}} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & {{\text{H}}_{y5} } & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & {{\text{O}}_{x5} } & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & {I_{d5} } & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & {I_{q5} } & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & {V_{7} } \\ \end{array} } \right]^{\,\,T}$$
$$A_{\text{sys}} = \left[ {\begin{array}{*{20}c} {\frac{{ - T^{0} }}{{T_{0} \tau_{{{\text{H}}_{2} }}^{0} }}} & 0 & 0 & { - \frac{{\left( {K_{{h_{2} }} P_{{h_{2} }}^{0} - n_{{h_{2} }}^{\text{in0}} + 2I_{\text{fc}}^{0} K_{r} } \right)}}{{K_{{h_{2} }} T_{0} \tau_{{{\text{H}}_{2} }}^{0} }}} & { - \frac{{2K_{r} T^{0} }}{{K_{{h_{2} }} T_{0} \tau_{{H_{2} }}^{0} }}} & 0 & 0 & 0 & {\frac{{T^{0} }}{{T_{0} K_{{h_{2} }} \tau_{{H_{2} }}^{0} }}} & 0 & 0 & 0 & 0 \\ 0 & {\frac{{ - T^{0} }}{{T_{0} \tau_{{{\text{O}}_{2} }}^{0} }}} & 0 & { - \frac{{\left( {K_{{O_{2} }} P_{{O_{2} }}^{0} - n_{{O_{2} }}^{{{\text{in}}0}} + I_{\text{fc}}^{0} K_{r} } \right)}}{{K_{{O_{2} }} T_{0} \tau_{{O_{2} }}^{0} }}} & { - \frac{{K_{r} T^{0} }}{{K_{{O_{2} }} T_{0} \tau_{{O_{2} }}^{0} }}} & 0 & 0 & 0 & 0 & {\frac{{T^{0} }}{{T_{0} K_{{O_{2} }} \tau_{{O_{2} }}^{0} }}} & 0 & 0 & 0 \\ 0 & 0 & {\frac{{ - T^{0} }}{{T_{0} \tau_{{h_{2} O}}^{0} }}} & { - \frac{{\left( {K_{{h_{2} O}} P_{{h_{2} O}}^{0} - 2I_{\text{fc}}^{0} K_{r} } \right)}}{{K_{{h_{2} O}} T_{0} \tau_{{h_{2} O}}^{0} }}} & {\frac{{2K_{r} T^{0} }}{{K_{{h_{2} O}} T_{0} \tau_{{h_{2} O}}^{0} }}} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ {T_{\text{ph}} } & {T_{\text{po}} } & {T_{\text{pw}} } & {T_{\text{pt}} } & {T_{pi} } & 0 & 0 & 0 & {T_{\text{pnh}} } & {T_{\text{pno}} } & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & { - \frac{1}{{T_{\text{el}} }}} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & { - \frac{{R_{f} }}{{L_{f} }}} & {\omega_{0} } & {\frac{{k_{d0} }}{{L_{f} }}} & 0 & 0 & {\frac{{V_{{{\text{dc}}0}} }}{{L_{f} }}} & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & { - \omega_{0} } & { - \frac{{R_{f} }}{{L_{f} }}} & {\frac{{k_{q0} }}{{L_{f} }}} & 0 & 0 & 0 & {\frac{{V_{{{\text{dc}}0}} }}{{L_{f} }}} & 0 \\ 0 & 0 & 0 & 0 & { - \frac{{\left( {d_{c0} - 1} \right)}}{{\left( {C_{\text{dc}} I_{\text{fcb}} } \right)}}} & { - \frac{{k_{d0} }}{{C_{\text{dc}} }}} & { - \frac{{k_{q0} }}{{C_{\text{dc}} }}} & 0 & 0 & 0 & { - \frac{{i_{d0} }}{{C_{\text{dc}} }}} & { - \frac{{i_{q0} }}{{C_{\text{dc}} }}} & { - \frac{{I_{{{\text{fc}}0}} }}{{\left( {C_{\text{dc}} I_{\text{fcb}} } \right)}}} \\ {H_{y1} } & 0 & 0 & {H_{y2} } & {H_{y3} } & 0 & 0 & 0 & {H_{y4} } & 0 & 0 & 0 & 0 \\ 0 & {O_{x1} } & 0 & {O_{x2} } & {O_{x3} } & 0 & 0 & 0 & 0 & {O_{x4} } & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & {I_{d1} } & {I_{d2} } & {I_{d3} } & 0 & 0 & {I_{d4} } & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & {I_{q1} } & {I_{q2} } & {I_{q3} } & 0 & 0 & 0 & {I_{q4} } & 0 \\ 0 & 0 & 0 & 0 & {V_{1} } & {V_{2} } & {V_{3} } & {V_{4} } & 0 & 0 & {V_{5} } & {V_{6} } & {V_{7} } \\ \end{array} } \right]$$

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Ahmed, A., Shahid Ullah, M. & Ashraful Hoque, M. Optimal Design of Proportional–Integral Controllers for Grid-Connected Solid Oxide Fuel Cell Power Plant Employing Differential Evolution Algorithm. Iran J Sci Technol Trans Electr Eng 43, 999–1019 (2019). https://doi.org/10.1007/s40998-019-00207-5

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Keywords

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