2. 兰州交通大学新能源学院 兰州 730070
2. School of New Energy and Power Engineering, Lanzhou Jiao Tong University, Lanzhou 730070, China
Solar thermal power generation system is affected by natural factors such as solar radiation, and also the system has strong interference. These uncertain factors directly affect the quality of solar thermal power generation. Regulating the tracking error of output oil temperature of collector is an important control objective, therefore it has practical significance to study the active fault tolerant predictive control of solar collector system in the state of fault or disturbance.
The solar collector system uses the solar radiation to heat the heat conduction oil, regulates the flow of heat conduction oil, and controls the outlet temperature of the heat conduction oil in a certain range so as to ensure the stability of the power generation. The control objective of the collection system is that the amount of the actual output is as close as possible to the amount of the desired target output. In recent years, many kinds of intelligent control algorithms are applied to the control of solar thermal power generation system [1][3]. Reference [4][7] all applied the model predictive control algorithm, where the control target was to minimize the tracking error.
Error of the outlet temperature and the single model are used to predict the model. Multimodel adaptive switching active fault tolerant control is about selecting the model with least cumulative error to determine which model of multimodels is the best match for the dynamic behavior of the system online, and continuously optimize the model parameters. The adaptive model which can be reassigned is used to compensate for the missing data in the process of modeling and to reduce the tracking error with strong disturbance or fault conditions. The algorithm has been successfully applied to other areas [8][10], the control effect is good. Based on the above analysis, the main research contents of this paper include:
1) Establish multimodel set. Collect data sets for several consecutive days to make the fuzzy Cmeans (FCM) clustering.
2) Design active fault tolerant sliding mode predictive controller. Determine the adaptive switching strategy. The inlet temperature and solar radiation are considered as disturbance, and flow of heat conduction oil is considered as controlled variable. Based on the multimodel, the adaptive model of the solar collector system is established to adapt the object and the disturbance characteristics. A model switching strategy based on the minimum cumulative error is used to select the optimal control model online.
3) The method is applied to the actual linear Fresnel power generation system, and the method is compared with [3]. The method in this paper is better than the method [3], the control precision is higher, and the time delay is shorter.
1 Dynamic Model of Solar Thermal Power Generation System 1.1 The Mathematical Model of Microsources 1.1.1 Microturbine CostR Carmona, a Spanish scholar, initially used the mathematical model [1] to describe the temperature of heat conduction oil of the solar collector [11], and then the model was used to analyze the thermal system [4][6].
$ \begin{align} & {\rho _{\rm{f}}}{C_{\rm{f}}}{A_{\rm{f}}}\frac{{{d}{T_n}(t)}}{{{{d}}t}} = {\eta _{\rm{0}}}{G_1}I(t)  {\rho _{\rm{f}}}{C_{\rm{f}}}v(t)\frac{{{T_n}(t)  {T_{n  1}}(t)}}{{\Delta x}}, \nonumber\\ & \;\;\;\;\;\;\; n = 1, \ldots, N \end{align} $  (1) 
where
Take
$ \begin{align} & {\rho _{\rm{f}}}{C_{\rm{f}}}{A_{\rm{f}}}\frac{{{\rm{d}}{T_n}(t)}}{{{\rm{d}}t}} = {\eta _{\rm{0}}}{G_1}I(t)  {\rho _{\rm{f}}}{C_{\rm{f}}}v(t)\frac{{{T_n}(t)  {T_0}(t)}}{L}, \notag\\ & \;\;\;\;\;\;\quad n = 1, \ldots, N \end{align} $  (2) 
where
The data acquisition in the linear Fresnel thermal power generation system is used to make FCM clustering analysis. In this clustering algorithm, the membership degree is used to determine the degree of each element and the measured data is classified by the method of subtraction clustering [12], [13].
Step 1: Determine the number of categories
Step 2: The fuzzy membership degree
$ \begin{align} {u_{ij}} = \begin{cases} {\left[{\sum\limits_{k = 1}^C {\frac{{{{\left\ {{x_i}{\upsilon _j}} \right\}^{\frac{2}{{m1}}}}}}{{{{\left\ {{x_i}{\upsilon _k}} \right\}^{\frac{2}{{m  1}}}}}}} } \right]^{  1}}, & \left\ {{x_i}  {\upsilon _k}} \right\ \ne 0\\ 1, & \left\ {{x_i}  {\upsilon _k}} \right\ = 0, ~~{k} = j\\ 0, & \left\ {{x_i}  {\upsilon _k}} \right\ = 0, ~~{k} \ne j. \end{cases} \end{align} $  (3) 
Step 3: Use (4) to calculate the center of each category.
$ \begin{align} {\upsilon _j} = \frac{{\sum\limits_{i = 1}^n {u_{ij}^m{x_i}} }}{{\sum\limits_{i = 1}^n {u_{ij}^m} }}. \end{align} $  (4) 
Step 4: The target value is calculated according to (5) to determine whether the values meet the target value or not. If the values meet the target value, the clustering is end. Otherwise, return Step 2.
$ \begin{align} J = \sum\limits_{i = 1}^n {\sum\limits_{j = 1}^C {{{({u_{ij}})}^m}\left\ {{x_i}  {\upsilon _j}} \right\} }. \end{align} $  (5) 
In this paper, DB is used as an evaluation index of the classification [14]. The smaller the index value is, the better the clustering effect is. In this paper, for 3500 sets of actual power generation data
The clustering centers of the 6 types are respectively:
$ \begin{align*} & {\upsilon _1} = (\begin{array}{*{20}{c}} {236.6} & {196.0} & {10.15} & {780.5} \end{array})\\ & {\upsilon _2} = (\begin{array}{*{20}{c}} {244.4} & {203.0} & {10.03} & {890.9} \end{array})\\ & {\upsilon _3} = (\begin{array}{*{20}{c}} {250.1} & {210.0} & {10.20} & {916.6} \end{array})\\ & {\upsilon _4} = (\begin{array}{*{20}{c}} {253.6} & {213.4} & {10.32} & {933.0} \end{array})\\ & {\upsilon _5} = (\begin{array}{*{20}{c}} {259.7} & {219.6} & {10.18} & {935.5} \end{array})\\ & {\upsilon _6} = (\begin{array}{*{20}{c}} {273.6} & {232.9} & {10.04} & {953.2} \end{array}). \end{align*} $ 
The above classification data results considered the inlet oil temperature, solar radiation and flow rate of conduction heat oil as input, and outlet temperature as output. In order to overcome the shortcomings of the least squares due to its poor correction ability, forgetting factor recursive least square method is adopted [9], [15], [16]. The controlled auto regressive (CAR) model expressed by formula (6) is used to identify the parameters.
$ \begin{align} y(k) = {{\bf{\varphi }}^T}(k)\hat\theta (k). \end{align} $  (6) 
where,
$ \begin{align*} & y(k + 1) = [{y_1}(k + 1), {y_2}(k + 1), {y_3}(k + 1), \\ & \;\;\quad {y_4}(k + 1), {y_5}(k + 1), {y_6}(k + 1){]^{{T}}} \\[3mm] & \hat\theta = \left[{\begin{array}{*{20}{c}} {{a_{11}}} & {{a_{12}}} & {{a_{13}}} & {{a_{14}}}\\ {{a_{21}}} & {{a_{22}}} & {{a_{23}}} & {{a_{24}}}\\ {{a_{31}}} & {{a_{32}}} & {{a_{33}}} & {{a_{34}}}\\ {{a_{41}}} & {{a_{42}}} & {{a_{43}}} & {{a_{44}}}\\ {{a_{51}}} & {{a_{52}}} & {{a_{53}}} & {{a_{54}}}\\ {{a_{61}}} & {{a_{62}}} & {{a_{63}}} & {{a_{64}}} \end{array}} \right]\nonumber \\[3mm] & {{ {\varphi }}^T}(k + 1) = \left[{\begin{array}{*{20}{c}} {{y_1}(k)} & {{u_1}(k)} & {{T_{\rm out}}_1(k)} & {{I_1}(k)}\\ {{y_2}(k)} & {{u_2}(k)} & {{T_{\rm out2}}(k)} & {{I_2}(k)}\\ {{y_3}(k)} & {{u_3}(k)} & {{T_{\rm out3}}(k)} & {{I_3}(k)}\\ {  {y_4}(k)} & {{u_4}(k)} & {{T_{\rm out4}}(k)} & {{I_4}(k)}\\ {  {y_5}(k)} & {{u_5}(k)} & {{T_{\rm out5}}(k)} & {{I_5}(k)}\\ {  {y_6}(k)} & {{u_6}(k)} & {{T_{\rm out6}}(k)} & {{I_6}(k)} \end{array}} \right].\nonumber \end{align*} $ 
Initial value
$\begin{align} \begin{cases} {y_1}(k + 1) = 0.9217{y_1}(k) + 0.3011{u_1}(k)\\ \;\;\quad +~ 0.0701{T_{\rm in}}_1(k) + 0.0015{I_1}(k)\\[2mm] {y_2}(k + 1) = 0.9501{y_2}(k) + 0.2126{u_2}(k)\\ \;\;\quad +~ 0.0631{T_{\rm in2}}(k) + 0.0041{I_2}(k)\\[2mm] {y_3}(k + 1) = 0.9438{y_3}(k) + 0.6421{u_3}(k)\\ \;\;\quad +~ 0.0763{T_{\rm in}}_3(k) + 0.0039{I_3}(k) \\[2mm] {y_4}(k + 1) = 0.9573{y_4}(k) + 0.434{u_4}(k)\\ \;\;\quad +~ 0.0614{T_{\rm in4}}(k) + 0.0032{I_4}(k) \\[2mm] {y_5}(k + 1) = 0.9680{y_5}(k) + 0.3171{u_5}(k)\\ \;\;\quad +~ 0.0549{T_{\rm in5}}(k) + 0.0036{I_5}(k) \\[2mm] {y_6}(k + 1) = 0.9702{y_6}(k) + 0.4021{u_6}(k)\\ \;\;\quad +~ 0.0593{T_{\rm in6}}(k) + 0.0031{I_6}(k) \end{cases} \end{align} $  (7) 
where
According to the results of the optimal clustering center, the outlet temperature is considered as the center to analyze the error of classification. Fig. 1 shows the effect of deviation of each class of data from the center point of the cluster. In different temperature and cross sectional output data, outlet temperature is also affected by the inlet temperature and solar radiation. Therefore, the system can be considered as a multi disturbance system.
Predictive control includes prediction model, rolling optimization and feedback correction. Under the condition of system fault or disturbance, the multimodel adaptive switching active fault tolerant control can update the control law based on the optimal predictive model online, which results in stable operation of the closedloop system. In the process of rolling optimization, update control law online, and through the feedback of correction, the error is corrected online. Fig. 2 is the structure of a multimodel active fault tolerant controller of solar thermal power generation.
In [17][19], a single model sliding mode predictive control is designed for the high speed aircraft. In [18], a multimodel sliding mode predictive control is designed for the disturbance signal. Active fault tolerant predictive controller is designed for highspeed train based on [19]. In the literature, the designed controller according to different system has achieved good control effect. By referring to the above documents, this article designed the active fault tolerant control in solar thermal applications. Consider the following uncertain discrete linear systems
$ \begin{align} y(k + 1)  {{{A}}}y(k) = {{{B}}}u(k) + \xi (k) \end{align} $  (8) 
where
Define the switching function: set the reference command signal
$ \begin{align} e(k) = y(k)  {y_r}(k). \end{align} $  (9) 
Define the linear switching function:
$ \begin{align} s(k) = {\sigma ^{T}}e(k) \end{align} $  (10) 
where
$ \begin{align} s(k + 1) = {\sigma ^{T}}e(k + 1). \end{align} $  (11) 
The predicted sliding mode surface is
$ \begin{align} s(k + p) = & \ {\sigma ^{T}}{A^p}y(k)\nonumber\\ & + \sum\limits_{i = 1}^p {{\sigma ^{T}}{A^{p  i}}[} Bu(k + pi)\nonumber\\ & + \xi (k + pi)]  {\sigma ^{T}}{y_r}(k + p). \end{align} $  (12) 
Use the difference value between actual switching function output value
$ \begin{align} {{\tilde s}_p}(k + p) = s(k + p) + {\zeta _p}[s(k){s_p}(k  {k p})] \end{align} $  (13) 
where
Take the commonly reaching rate as the reference trajectory.
$ \begin{align} \begin{cases} {s_r}(k + p) = \mu {s_r}(k + p  1) + \eta {\rm sgn} ({s_r}(k + p  1))\\ {{s_r}(k) = s(k)} \end{cases} \end{align} $  (14) 
where
Define the performance index
$ \begin{align} J= \sum\limits_{i = 1}^N {{{({s_r}(k + i)  \tilde s(k + i))}^2}} + \sum\limits_{j = 0}^{M  1} {{\lambda _j}{u^2}(k + j)} \end{align} $  (15) 
where
If the output vector is expressed as:
$ \begin{align} & S = {[s(k + 1), \ldots, s(k + N)]^{T}}\;\;\nonumber \\ & {S_r} = {[{s_r}(k + 1), \ldots, {s_r}(k + N)]^{T}}\nonumber \\ & \tilde S = {[{\tilde s_{}}(k + 1), \ldots, \tilde s(k + N)]^{T}}\;\;\;\nonumber \\ & \bar S = {[s(k){s_p}(k\left {k1} \right.), \ldots, s(k){s_p}(k\left {k  N} \right.)]^{T}}\notag\\ & \ \ = {[\bar S(1), \ldots, \bar S(N)]^{T}}\notag\\ & U = {[u(k), \ldots, u(k + M1)]^{T}}\notag\\ & F = {[{\sigma ^{T}}A, \ldots, {\sigma ^{T}}{A^N}]^{T}}\; \nonumber \\ & \Xi = {\rm diag}\{ {\zeta _1}, \ldots, {\zeta _N}\} \notag\\ & \Lambda = {\rm diag}\{ {\lambda _1}, \ldots, {\lambda _M}\}\notag\\ & {{\tilde Y}_r} = {[{{\tilde y}_r}(k), \ldots, {{\tilde y}_r}(k + N 1)]^{T}} \nonumber \end{align} $ 
$ \begin{array}{l} G = \left[{\begin{array}{*{20}{c}} {{\sigma ^T}B} & 0 & \cdots & 0\\ {{\sigma ^T}AB} & {{\sigma ^T}B} & \cdots & 0\\ \vdots & \vdots & \cdots & {{\sigma ^T}B}\\ \vdots & \vdots & \ddots & \vdots \\ {{\sigma ^T}{A^{N2}}B} & {{\sigma ^T}{A^{N3}}B} & \cdots & {\sum\limits_{i = 0}^{NM  1} {{\sigma ^T}{A^i}B} }\\ {{\sigma ^T}{A^{N  1}}B} & {{\sigma ^T}{A^{N  2}}B} & \cdots & {\sum\limits_{i = 0}^{N  M} {{\sigma ^T}{A^i}B} } \end{array}} \right]\\ P = \left[{\begin{array}{*{20}{c}} {{\sigma ^T}} & 0 & \cdots & 0\\ {{\sigma ^T}A} & {{\sigma ^T}} & \cdots & 0\\ \vdots & \vdots & \ddots & \vdots \\ {{\sigma ^T}{A^{N1}}} & {{\sigma ^T}{A^{N2}}} & \cdots & {{\sigma ^T}} \end{array}} \right] \end{array} $ 
then (12) and (13) can be represented as a vector form, which can be expressed by (16) and (17).
$ \begin{array}{l} & S = Fe(k) + GU + P{\tilde Y_r}\;\; \end{array} $  (16) 
$ \begin{align} & \tilde S = S + \Xi \bar S.\;\;\;\; \end{align} $  (17) 
So, the performance index expressed by(15) can be written as a vector form expressed by (18).
$ \begin{align} J = {(H  GU)^{T}}(H  GU) + \Lambda {U^{T}}U \end{align} $  (18) 
where
Let
$ \begin{align} U = {({G^{T}}G + \Lambda)^{  1}}{G^{T}}H. \end{align} $  (19) 
The switching strategy is to select the optimal control model online [18] for the state of system under the condition of unknown fault or disturbance, and the active fault tolerant predictive control is carried out through the rolling optimization and feedback correction.
At the
$ \begin{align} {J_i}(k) = \alpha e_i^2(k) + \beta \sum\limits_{j = 1}^L {{\theta ^j}e_i^2(k  j)} \end{align} $  (20) 
where
Under the condition of system failure or disturbance, the active fault tolerant control can update the control law according to the optimal control model online selection, which can stabilize the closed loop system [19]. Active fault tolerant control is realized by the adaptive model switching.
3.5.2 Adaptive Model SwitchingThe adaptive model can be used to obtain a faster convergence rate according to the different conditions. According to the model switching strategy, if the initial parameters of the model are the optimal model parameters of the multimodel set, the optimal model can be selected from the current optimal model set and the parameters can be updated with (21). Otherwise, the initial value of the model is reassigned to the minimum cumulative error model parameters, and then the model parameters are updated online through (21). According to the model switching strategy, the obtained optimal model is the adaptive model, which can guarantee the stability of the system and the accuracy of tracking. Parameters are updated as follows [9], [19].
$ \begin{align} \begin{cases} \hat \theta (k) = \hat \theta (k  1) + K(k)[y(k){\varphi ^T}(k)\hat \theta (k 1)]\\[1mm] K\left( k \right) = \dfrac{{P\left( {k  1} \right)\varphi \left( k \right)}}{{\lambda + {\varphi ^T}\left( k \right)P\left( {k  1} \right)\varphi \left( k \right)}}\\[3mm] P(k) = \dfrac{1}{\lambda }\left[IK(k){\varphi ^T}(k)\right]P(k  1) \end{cases} \end{align} $  (21) 
where
In the process of simulation analysis, system model selects (2). Parameters of the linear Fresnel power generation demonstration project in the west of China are selected and the outlet oil temperature of conducting oil on August 5, 2015 is taken as the target curve. Controlled variable is the flow of heat conduction oil. Inlet oil temperature and solar radiation are disturbances which can be measured. The flow range of heat conduction oil is 3 (l/s)12 (l/s).
The average variance of two kinds of simulation results is analyzed and calculated. MSE = 1.53264 in Fig. 5 and MSE = 0.43276 in Fig. 8.
It can be seen from Fig. 7 and Fig. 10 that multimodel active faulttolerant sliding mode predictive control is better than that of [3].
5 ConclusionIn this paper, we collected the data of 3500 sets of solar thermal power generation sets, classified them, and set up the mathematical model. A faulttolerant sliding mode predictive controller is designed, which can reduce error, improve the system's robustness and antiinterference ability. The adaptive prediction model can be used to reduce the error caused by loss of data, the disturbance and fault.
From Fig. 5 and Fig. 8, it can be seen that the method proposed in this paper has higher accuracy and shorter lag time, and has the ability to improve the robustness and convergence rate of the solar thermal power generation system.
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