From: Comprehensive summary of solid oxide fuel cell control: a state-of-the-art review
Control method | Control objectives | Controller design | Parameters | Performance | Usage scenarios | Complexity | Robustness | Accuracy | |
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MPC | Jurado [85] | 1. Air or fuel flow rate | Cost function: \(\mathrm{min} J({H}_{\mathrm{p}1},{H}_{\mathrm{p}2},{H}_{\mathrm{c}}, \lambda )=\sum_{j={H}_{\mathrm{p}1}}^{{H}_{\mathrm{p}2}}{(w(k+j)-\widehat{y}(k+j))}^{2}+\lambda \sum_{j=1}^{{H}_{\mathrm{c}}}\Delta {u}^{2}(k+j-1)\) | \(\widehat{y}(k+j)\): predicted system output; \(w(k+j)\): modified setpoint; \({H}_{\mathrm{p}1}\): minimum costing horizon; \({H}_{\mathrm{p}2}\): maximum costing or prediction horizon; \({H}_{\mathrm{c}}\): control horizon; \(\lambda\): move suppression coefficient. | Meet power demand | Multivariable fuzzy Hammerstein model of SOFC | ** | ** | ** |
Sanandaji [87] | 1. Air or fuel flow rate; 2. Voltage | N. P. | N. P. | Improve load tracking capability | Linear parameter variation model of SOFC system | ** | *** | ** | |
Kupilik [88] | 1. Air or fuel flow rate | Objective function: \(J=\mathrm{minimize }{x}_{k,N}^{^{\prime}}{P}_{\mathrm{f}}{x}_{k,N}+\sum_{i=0}^{N-1}{x}_{k,i}^{T}Q{x}_{k,i}+{u}_{k,i}^{T}R{u}_{k,i}+{\Vert {Q}_{\mathrm{Y}}({y}_{k,i}-{y}_{\mathrm{ref},i})\Vert }_{2}\) | \({\mathrm{y}}_{\mathrm{ref},i}\): reference trajectory; \({P}_{\mathrm{f}}\), \(Q\), \(R\) and \({Q}_{Y}\): weighting matrix; \({x}_{k,N}\): state at time k. | Meet the change of load demand; Reduce carbon deposition | Experimental model of movable 1.5 kW SOFC system | *** | *** | *** | |
1. Air or fuel flow rate | Multilinear MPC controller: \(J\left(k\right)=\sum_{p=1}^{N}\sum_{j=1}^{{n}_{y}}{({\lambda }_{j}^{y}({y}_{j,\mathrm{ref}}\left(k+p|k\right)-{y}_{j}\left(k+p|k\right)))}^{2}+\sum_{p=0}^{{N}_{u}-1}\sum_{j=1}^{{n}_{u}}{({\lambda }_{j}^{\Delta u}\Delta {u}_{j}\left(k+p|k\right))}^{2}\) | y: controlled variable; \({y}_{\mathrm{ref}}\): reference; \(\Delta u\): increments of manipulated variables; p: Prediction horizon. | Stable output voltage | Nonlinear SOFC state model | *** | *** | ** | ||
Miaomiao [91] | 1. Current density | Performance index function: \(J\left({U}_{N},x\left(0\right)\right)=\sum_{k=0}^{N-1}\left({x}_{k}^{^{\prime}}Q{x}_{k}+{u}_{k}^{^{\prime}}R{u}_{k}\right)+{x}_{N}^{^{\prime}}P{x}_{N}\) | x: collection of the bounded parameter; Q: state weight; R: input weight; N: control domain. | Reduce computation and storage | Piece wise affine (PWA) model of SOFC | ** | *** | ** | |
Frenkel [92] | 1. Power | Control law: \({u}_{k}={e}_{1}^{\mathrm{T}}\cdot {\widetilde{u}}_{k}={e}_{1}^{\mathrm{T}}({\mathrm{S}}_{v}\cdot {\widetilde{y}}_{\mathrm{d},\mathrm{k}+1}-\mathrm{k}\cdot {\mathrm{x}}_{k})\) | \(\mathrm{k}\): control gain. | N. P. | Experimental | ** | *** | ** | |
Data-driven PC | Wang [86] | 1. Air or fuel flow rate | Cost function: \(\mathrm{min }{J}_{k}=\mathrm{min }{\left({\widehat{y}}_{k+1/L}-{r}_{k+1/L}\right)}^{T}Q\left({\widehat{y}}_{k+\frac{1}{L}}-{r}_{k+\frac{1}{L}}\right)+\Delta {u}_\frac{k}{M}^{T}{R}_{M}\Delta {u}_\frac{k}{M}+{u}_\frac{k}{M}^{T}{P}_{M}{u}_\frac{k}{M}\) | \(L\): prediction horizon; \(M\): control horizon; \({\widehat{y}}_{k+1/L}\): vector of prediction of future output; \({r}_{k+1/L}\): vector of future reference signals; \(Q\) , \({R}_{M}\) and \({P}_{M}\): weighting matrices with block diagonal structure; \({u}_{k/M}\): vector of future manipulated signal. | Verify the effectiveness of data-driven predictive control algorithm | N. P. | ** | ** | *** |
NMPC | Huo [38] | 1. Air or fuel flow rate; 2. Voltage | Cost function: \(J=\sum_{j=1}^{{H}_{\mathrm{p}}}{[{\widehat{V}}_{\mathrm{dc}}\left(k+j\right)-{V}_{\mathrm{dcr}}(k+j)]}^{2}+\lambda \sum_{i=1}^{{H}_{\mathrm{c}}}{[{N}_{\mathrm{f}}\left(k+i\right)-{N}_{\mathrm{f}}(k+i-1)]}^{2}\) | \({\widehat{V}}_{\mathrm{dc}}\left(k+j\right)\): predicted output voltage; \({H}_{\mathrm{c}}\): control horizon; \({N}_{\mathrm{f}}\left(k+i\right)\): manipulated variable; \(\lambda\): weighting factor. | Stable output voltage | Hammerstein model of SOFC | ** | ** | ** |
Zhang [93] | 1. Air or fuel flow rate; 2. Current density | Cost function: \(\Phi =\mathrm{min }\left\{\sum_{k=T}^{T+{N}_{\mathrm{c}}}L\left({w}_{k},{v}_{k}\right)+\frac{1}{2}V\left({w}_{T+{N}_{\mathrm{c}}}\right)+\sum_{k=T}^{T+{N}_{\mathrm{c}}}(\frac{1}{2}{\eta }_{\mathrm{r},k}{O}_{\mathrm{r}}{\eta }_{\mathrm{r},k}^{T}+{O}_{\mathrm{r}}{\eta }_{\mathrm{r},k})\right\}\) | \(\mathrm{V}\left({w}_{T+{N}_{\mathrm{c}}}\right)\): penalty term; \(\eta\): relaxation factor; Qr: penalty matrix of regulator cost function. | SOFC under effective control constraints | Experimental | *** | *** | *** | |
Yang [94] | 1. Temperature | Objective function: \(J=\sum_{i=1}^{P}{q}_{i}{[y\left(k+i\right)-{y}_{\mathrm{m}}(k+i)]}^{2}+\sum_{j=1}^{M}{r}_{j}\Delta {u}^{2}(k+j-1)\) | \({q}_{i}\) and \({r}_{j}\): weight coefficient; P: prediction horizon; M: control horizon; \(y\left(k+i\right)\): actual output; \({y}_{\mathrm{m}}(k+i)\): model predictive output. | Effectively control temperature of SOFC stack | Improved TS fuzzy model of SOFC stack | ** | *** | ** | |
Murshed [95] | 1. Voltage | Objective function: \(\mathrm{min }J=\sum_{i=1}^{N}\left[{\Vert \widehat{x}\left(k+\frac{i}{k}\right)-{x}_{\mathrm{ref}}\Vert }_{Q}^{2}+{\Vert u\left(k+\frac{i}{k}\right)-{u}_{\mathrm{ref}}\Vert }_{R}^{2}+{\Vert \Delta u\left(k+\frac{i}{k}\right)\Vert }_{S}^{2}\right]+{W}_{\mathrm{indirect}}[\sum {\dot{n}}_{i,E}{\int }_{{T}_{\mathrm{ref}}}^{{T}_{\mathrm{AE}}}{C}_{p,i}(T)\mathrm{d}T+{\dot{n}}_{i,E}\frac{1-r}{r}{\int }_{{T}_{\mathrm{ref}}}^{{T}_{\mathrm{B}}}{C}_{p,i}(T)\mathrm{d}T]\) | Q, R, S and \({W}_{\mathrm{indirect}}\): weighting matrices; \({T}_{\mathrm{ref}}\): minimum temperature; \({T}_{\mathrm{AE}}\), \({T}_{\mathrm{B}}\): exit temperatures; \({\dot{n}}_{i,E}\): flow rate. | Improve the operation efficiency of the system | Experimental | *** | *** | *** | |
Bhattacharyya [96] | 1. Air or fuel flow rate; 2. Voltage | Optimization formula: \(\mathrm{min}{ \Vert {W}_{1}(\widehat{y}-\xi )\Vert }_{2}+{\Vert {W}_{2}u\Vert }_{2}\) | \({W}_{1}\): weighting matrix; \({W}_{2}\): move suppression matrix. | Meet the step change of load | Experimental | *** | ** | ** | |
Lee [97] | 1. Air or fuel flow rate; 2. Voltage | Performance index: \(J\left(k,k+1\right)\triangleq x{\left(k|k\right)}^{T}\mathcal{z}x\left(k|k\right)+u{\left(k|k\right)}^{T}\gamma u\left(k|k\right)+V\left(k+1|k\right)\) | \(V\left(k+1|k\right)\): terminal cost; x: state vector; u: control input; \(\mathcal{z}\)>0, \(\gamma\)>0. | Improve load tracking capability | Sector bounded nonlinear model of SOFC system | ** | *** | ** | |
Wu [27] | 1. Air or fuel flow rate | Control input: \({u}_{k}\left(i\right)={u}_{k-1}\left(i\right)+{K}_{\mathrm{P}}{e}_{k-1}\left(i\right)+{K}_{\mathrm{D}}{\dot{e}}_{k-1}\left(i\right) i=1,\dots ,{N}_{\mathrm{c}}\) | \({N}_{\mathrm{c}}\): control window; k: iteration index; \({e}_{k-1}\left(i\right)\): tracking error; \({\dot{e}}_{k-1}\left(i\right)\): error difference; \({K}_{\mathrm{P}}\) and \({K}_{\mathrm{D}}\): learning gains. | Extend system life | Experimental model of SOFC system | **** | *** | **** | |
Awryńczuk [28] | 1. Air or fuel flow rate | Cost function: \(J\left(k\right)=\sum_{p=1}^{N}{({V}_{\mathrm{dc}}^{\mathrm{sp}}\left(k+p|k\right)-{\widehat{V}}_{\mathrm{dc}}\left(k+p|k\right))}^{2}+\lambda \sum_{p=0}^{{N}_{\mathrm{u}}-1}{(\Delta {q}_{\mathrm{f}}\left(k+p|k\right))}^{2}\) | \(\lambda\): weighting coefficient; \({\widehat{V}}_{\mathrm{dc}}\left(k+p|k\right)\): predicted values; \({V}_{\mathrm{dc}}^{\mathrm{sp}}\left(k+p|k\right)\): set-point values; \({N}_{\mathrm{u}}\): control horizon. | Meet fuel use restrictions; Good quality control | Continuous time model of SOFC system | *** | **** | **** | |
GPC | Deng [98] | 1. Current density | Cost function: \(J=E\left\{\sum_{j=1}^{N}{(\widehat{y}\left(k+j\right)-{y}_{\mathrm{r}}(k+j))}^{2}+\sum_{j=1}^{M}\lambda (j){(\Delta u(k+j-1))}^{2}\right\}\) | E{ }: expectation operator; N: maximum costing horizon; M: control horizon; \(\lambda (j)\): control weighting sequence; \(\widehat{y}\left(k+j\right)\): optimum ahead prediction; \({y}_{\mathrm{r}}(k+j)\): future reference trajectory. | Improve load tracking capability; Extended stack life | Fractional order model of SOFC | *** | *** | ** |
Jiang [99] | 1. Air or fuel flow rate; 2. Temperature | Objective function: \(J=E\left\{\sum_{j=1}^{{N}_{2}}{[\widehat{y}\left(k+j\right)-{y}_{\mathrm{r}}(k+j)]}^{2}+\sum_{j=1}^{{N}_{\mathrm{u}}}{\eta }_{j}{[\Delta U(k+j-1)]}^{2}\right\}\) | E{ }: expectation operator; \({\eta }_{j}\): control weighting sequence which limits the amplitude of the control sequence; \({N}_{2}\): maximum costing horizon; Nu: control horizon; \(\widehat{y}\left(k+j\right)\): predictive system output at the (k+j) th instance; \({y}_{\mathrm{r}}(k+j)\): reference value at the (k+j) th instance. | Improve system response speed and stability; Improve anti-interference ability | TS fuzzy model of SOFC system | *** | *** | *** | |
Pohjoranta [32] | 1. Temperature | Objective function: \(J\left({N}_{1},{N}_{2},{N}_{3}\right)=\sum_{j={N}_{1}}^{{N}_{2}}{\Vert \widehat{y}\left(t+j|t\right)-w(t+j)\Vert }_{R}^{2}+\sum_{j=1}^{{N}_{3}}{\Vert u\left(t+j-1\right)-u(t+j-2)\Vert }_{Q}^{2}\) | \(\widehat{y}\left(t+j|t\right)\): optimum j-step-ahead prediction of the output y; \({N}_{1}\), \({N}_{2}\): prediction horizon; \({N}_{3}\): control horizon; \(w(t+j)\): set point trajectory; R and Q: positive definite weight matrices. | Extended service life | Simulink model of 10 kW SOFC system | *** | *** | *** | |
1. Air or fuel flow rate | Objective function: \(J=E\left\{\sum_{j=1}^{{N}_{2}}{\left[\widehat{y}\left(k+j\right)-{y}_{\mathrm{r}}(k+j)\right]}^{2}+\sum_{j=1}^{{N}_{\mathrm{u}}}{\eta }_{j}{\left[\Delta U(k+j-1)\right]}^{2}\right\}\) | E{ }: expectation operator; \({\eta }_{j}\): control weighting sequence; \({N}_{2}\): maximum cost horizon; \({N}_{\mathrm{u}}\): control horizon; \(\widehat{y}\left(k+j\right)\): predictive system output; \({y}_{\mathrm{r}}(k+j)\): reference value. | Improve load tracking capability; Extend system life | Simulink model of 5kW SOFC system | *** | **** | *** | ||
Boubaker [102] | 1. Air or fuel flow rate | N.P. | N.P. | Improve response speed | Dynamic model of SOFC system. | *** | *** | *** | |
CMPC | Li [53] | 1. Voltage | Performance index: \(J\left(k\right)=\sum_{i=0}^{N-1}[{\left(x\left(k+i+\frac{1}{k}\right)-\overline{x }\right)}^{T}\mathrm{Q}\left(x\left(k+i+\frac{1}{k}\right)-\overline{x }\right)+R\Delta {q}_{\mathrm{f}}^{2}(k+i/k)]+\psi (x\left(k+N+\frac{1}{k}\right)-\overline{x })\) | N: prediction horizon; \(x\left(k+i+\frac{1}{k}\right)\): predicted state; \(\overline{x }\): equilibrium value of the state vector; k: current time. | Successfully handle control and control motion constraints; Satisfactory closed-loop performance is obtained | Dynamic model of SOFC system | *** | *** | ** |
Spivey [103] | 1. Air or fuel flow rate; 2. Temperature; 3. Pressure | Objective function: \(\mathrm{min} J=\frac{1}{2}{\left(y-{y}_{\mathrm{ref}}\right)}^{T}Q\left(y-{y}_{\mathrm{ref}}\right)+\frac{1}{2}\Delta {u}^{T}R\Delta u+\frac{1}{2}{\xi }^{T}V\xi\) | y: vector of controlled variables at all prediction time steps; \({y}_{\mathrm{ref}}\): reference trajectory; \(Q\), \(R\) and \(V\): weight matrices; \(\Delta u\): change in manipulated variables between each control time step; \(\xi\): slack variables. | Extended service life | A dynamic, quasi-two-dimensional model for a high-temperature tubular SOFC combined with ejector and pre-reformer models | *** | *** | ** | |
Fuzzy MPC | Wu [14] | 1. Air or fuel flow rate | Preference index: \(\mathrm{min }J={\Vert \frac{1}{2}(\left|Y\left(k\right)-{Y}_{\mathrm{h}}\left(k\right)\right|+\left|Y\left(k\right)-{Y}_{\mathrm{l}}\left(k\right)\right|-\left|{Y}_{\mathrm{h}}\left(k\right)-{Y}_{\mathrm{l}}\left(k\right)\right|)\Vert }_{Q}^{2}+{\Vert \Delta U(k)\Vert }_{R}^{2}+{\Vert \Delta Y(k)\Vert }_{W}^{2}\) | \({Y}_{\mathrm{h}}\left(k\right)\): higher limit sequence; \({Y}_{\mathrm{l}}\left(k\right)\): lower limit sequence; \(\Delta Y(k)\): increment of output prediction; Q, R and W: weight coefficient matrix | Stable output voltage; Stable fuel utilization | Fuzzy model with region tracking for SOFC system | *** | *** | *** |
AMPC | Liu [104] | 1. Voltage | Objective function: \(\mathrm{min}J\left(x\left(t\right),U\left(t\right),{N}_{P},{N}_{C}\right)\) \(s.t.\left\{\begin{array}{c}x\left(t+1\right)=Ax\left(t\right)+{\mathrm{B}}_{\mathrm{u}}u\left(t\right)+{\mathrm{B}}_{\mathrm{d}}d\left(t\right),\\ \left(t=t,t+1,\dots ,t+{N}_{p}-1\right); \\ y\left(t\right)=Cx\left(t\right); \\ \frac{2{K}_{r}I}{0.9}\le u\left(t\right)\le \frac{2{K}_{r}I}{0.7} \end{array} \right.\) | N.P. | Improve output voltage tracking ability and dynamic performance | Dynamic model of SOFC system | *** | **** | *** |