Table 2 Chronological summary of PID control

Traditional PID

Aguiar [39]

1. Current density disturbance;

2. Air flow rate

N. P.

N. P.

Improve load tracking capabilit

A dynamic model

**

**

**

Li [36]

1. Fuel flow rate;

2. Voltage

\(K_{{\text{P}}} = - \frac{{{\text{cos}}\left( {\psi - \theta _{{\text{p}}} } \right)}}{{M_{{\text{p}}} }}\)

\(K_{{\text{I}}} = - \frac{{w'_{{\text{c}}} {\text{sin}}\left( {\psi - \theta _{{\text{p}}} } \right)}}{{M_{{\text{p}}} }}\)

N. P.

Maintain output voltage;

Maintain fuel utilization

A dynamic model

**

**

**

Sorrentino [61]

1. Air or fuel rate

N. P.

N. P.

Effective temperature control

One-dimensional steady-state model of planar SOFC

**

**

**

Chaisantikulwat [37]

1. Hydrogen concentration

The controller output:

\(u(s)={K}_{\mathrm{c}}(1+\frac{1}{{\tau }_{\mathrm{I}}s}+{\tau }_{\mathrm{D}}s)e(s)\)

\(e(s)\): error;

\({K}_{\mathrm{c}}\): the controller gain;

\({\tau }_{\mathrm{I}}\): integral time;

\({\tau }_{\mathrm{D}}\): derivative time.

Maintain constant voltage

A dynamic model

**

***

**

Hajimolana [62]

1. Temperature;

2. Pressure

The control system consists of two fully decentralized PI controllers.

N. P.

Improve anti-interference ability

A dynamic compartmental model

**

**

**

Komatsu [51]

1. DC power output;

2. Cell operating temperature;

3. Fuel utilization factor;

4. Steam-to-carbon ratio.

N. P.

N. P.

Improve system operation efficiency

A dynamic model

***

**

**

Cheng [63]

1. Air or fuel rate.

N. P.

N. P.

Strong anti-interference ability

A SOFC system model based on BP neural network

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**

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Vreko [64]

1. Air or fuel rate;

2. Temperature

\(u\left(t\right)={K}_{\mathrm{P}}(e\left(t\right)+\frac{1}{{T}_{\mathrm{I}}}\underset{0}{\overset{t}{\int }}(e\left(\tau \right)-\frac{u\left(\tau \right)-{u}_{\mathrm{r}}(\tau )}{{K}_{\mathrm{P}}})\mathrm{d}\tau )\)

\({u}_{\mathrm{r}}\left(t\right)=\left\{\begin{array}{c}u\left(t\right), automatic mode\\ {u}_{\mathrm{m}}, manual mode\end{array}\right.\)

\({u}_{\mathrm{lim}}\left(t\right)=\left\{\begin{array}{c}{u}_{\mathrm{min}}, if u\left(t\right)<{u}_{\mathrm{min}}\\ u\left(t\right), if {u}_{\mathrm{min}}\le u(t)\le {u}_{\mathrm{max}}\\ {u}_{\mathrm{max}}, if u\left(t\right)>{u}_{\mathrm{max}}\end{array}\right.\)

\({T}_{\mathrm{I}}\): integral time constant;

\(e\left(t\right)\): control error;

\({u}_{\mathrm{r}}\): controller output.

Improve system robustness

Experimental model of 2.5 kW SOFC system

**

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Kupecki [65]

1. Air or fuel rate;

2. Current

Control strategy composed of 13 PID controllers.

N. P.

Maintain temperature; Increase stack power

Experimental model of 1kW SOFC system

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**

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Singh [66]

N. P.

N. P.

N. P.

Reduce rise time and stabilization time

Experimental

**

***

**

Zhang [67]

1. Air or fuel rate;

2. Current

N. P.

N. P.

Improve system efficiency and achieve fast tracking of output power

Experimental model of 5kW SOFC system

**

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***

Decentralized PID

Sendjaja [40]

1. Air or fuel rate

\({K}_{\mathrm{c}}=\frac{1}{{K}_{\mathrm{P}}}\frac{{\tau }_{1}}{{\tau }_{\mathrm{c}}+\widetilde{\theta }}; {\tau }_{\mathrm{I}}=\mathrm{min }\left({\tau }_{1},4\left({\tau }_{\mathrm{c}}+\widetilde{\theta }\right)\right); {\tau }_{\mathrm{D}}={\tau }_{2}\)

\({K}_{\mathrm{c}}\): controller gain;

\({\tau }_{\mathrm{I}}\): integral time;

\({\tau }_{\mathrm{D}}\): derivative time;

\({\tau }_{\mathrm{c}}\): desired closed-loop time constant;

\(\widetilde{\theta }=\theta +{T}_{\mathrm{s}}/2\) with \({T}_{\mathrm{s}}\) being the sampling time.

Improve load tracking capability

Benchmark nonlinear dynamic model of SOFC

**

***

**

Fuzzy PID

Marzooghi [68]

1. Voltage;

2. Current

Fuzzy PI controller with super capacitor is proposed.

N. P.

Improve performance under transient disturbances

Experimental model of 480-kW SOFC system

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**

Adaptive PID

Xu [69]

1. Voltage

Adaptive constrained controller:

\(u(t)=\mathrm{Sat}\{(u(k-1)+\mathrm{Sat}\{({u}_{\mathrm{c}}(k)-u(k-1)), T{\dot{q}}_{\mathrm{fmin}},T{\dot{q}}_{\mathrm{fmax}}\}){\overline{q} }_{\mathrm{fmin}}, {\overline{q} }_{\mathrm{fmax}}\}\)

T: sampling time;

\(\mathrm{Sat}(a,b,c)=\left\{\begin{array}{c}b, a\le b \\ a, b<a<c\\ c, a\ge c \end{array}\right.\)

Maintain fuel utilization

A dynamic model

***

***

***

Robust PID

Cao [70]

1. Air or fuel rate.

\({u}_{{\dot{N}}_{{\mathrm{H}}_{2}}}=\frac{n\cdot \delta {I}_{\mathrm{st}}}{2F\cdot {r}_{\mathrm{FU}}}\)

\({u}_{{\dot{N}}_{\mathrm{air}},\mathrm{FF}}=\frac{n\cdot \delta {I}_{\mathrm{st}}\cdot {\mathrm{AE}}_{\mathrm{SV},\mathrm{opt}}}{4F\cdot {X}_{{\mathrm{O}}_{2}}}\)

\({u}_{{\dot{N}}_{\mathrm{air}},\mathrm{by},\mathrm{FF}}=\frac{n\cdot \delta {I}_{\mathrm{st}}\cdot {\mathrm{AE}}_{\mathrm{SV},\mathrm{opt}}\cdot {\mathrm{BP}}_{\mathrm{SV},\mathrm{opt}}}{4F\cdot {X}_{{\mathrm{O}}_{2}}}\)

\(\delta {I}_{\mathrm{st}}\): stack current error;

\({r}_{\mathrm{FU}}\): control objective

of fuel utilization (FU);

\({\mathrm{AE}}_{\mathrm{SV},\mathrm{opt}}\) and \({\mathrm{BP}}_{\mathrm{SV},\mathrm{opt}}\): optimization at different stack Voltage operating points

Improve system reliability

Experimental model of kW SOFC system

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Cheng [71]

1. Air or fuel rate;

2. Temperature

\(\Delta u={K}_{\mathrm{s}}\mathrm{sign}(T-{T}^{\mathrm{ref}})+{K}_{\mathrm{P}}(T-{T}^{\mathrm{ref}})+{K}_{\mathrm{I}}\int (T-{T}^{\mathrm{ref}})\mathrm{d}t+{K}_{\mathrm{D}}\frac{\mathrm{d}}{\mathrm{d}t}(T-{T}^{\mathrm{ref}})\)

\({K}_{\mathrm{s}}={\left(\frac{T-{T}^{\mathrm{ref}}}{2}\right)}^{2}\)

\(\Delta u\): control variables;

\(T\): controlled variables;

\({T}^{\mathrm{ref}}\): reference values of controlled variables;

\({k}_{\mathrm{s}}\): speed of the sliding mode control.

Ensure temperature safety; Maintain efficient operation

Experimental model of 5 kW cross-flow SOFC system

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iPI-ASMC

Abbaker [8]

1. Voltage

Total output of controller:

\({u}_{\mathrm{c}}(t)=\frac{1}{\widetilde{\alpha }}[{\dot{y}}_{\mathrm{d}}(t)-\widehat{\xi }(t)-{\xi }_{\mathrm{u}}+\lambda e(t)+\mu s+\eta (t)\varphi (s)]\)

\(\lambda\) and \(\mu\): positive constant;

\(e(t)\): tracking error.

Improve dynamic respons

Experimental

****

****

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PID based on Intelligent Algorithm

Zhang [72]

PID parameters

Fitness function:

\(J=\underset{0}{\overset{{t}_{\mathrm{s}}}{\int }}\left|e(t)\right|\mathrm{d}t\)

t: time;

t_s: integral upper limit time;

e(t): battery SOFC control error.

Improve operational reliability

A fractional PID parameter optimization model of SOC control

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