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886 Faten Semadi, Vincent Valles and Jose Luis Gonzalez Barrios −1 ⎡ k11z −d 0 ⎤ 11 ⎢ z −1 ⎥ −1 ⎢1+ T11z G ' ( z ) = ⎥ (12) −1 −1 ⎢ k 21z k − − ⎥ 21 22 z d d 22 ⎢ z z −1 −1 ⎥ ⎣1+ T 1+ T 21z 22 z ⎦ As it is clear from the structure of the identified model, the water temperature is almost not related to the outlet oil temperature (G12=0), and this weak relation can be modeled as a disturbance to the system. The other components are simply modeled with a first order system with a time varying time delay. This structure of the model can be used for the GPC synthesis. B. Design of multivariable GPC controller The controller design consists of three steps. A) prediction model determination. B) Objective function assignment and C) control law calculation. The prediction model for the system can be derived from the equation (12), in which the model is rewritten in the form of : − 1 −1 e( t) A ( q ) y( t) = B( q ) u( t −1) + (13) Δ In which u (t) is control signal and y (t) is the process output as the vector of oil and water −1 −1 −1 temperatures. Moreover, e (t) is measurement noise with zero mean and Δ = 1− q . A ( q ) , B ( q ) are the polynomial matrices with degree n A and n B respectively. The Objective function to be minimized has the form of : ∑ ∑[ ] ∑[ ] = = = ⎭ ⎬⎫ m N ⎧ 2 N u 2 2 (14) J = ⎨ri y i ( t + j) − wi ( t + j) + λ Δu( t + j −1) i 1 ⎩ j 1 j 1 Or in matrix form: T T J = y − w R y − w + λ Δu Δ (15) [ ] [ ] u In which N is the maximum of prediction horizon and 2 N is the control horizon. λ is the u penalty coefficient and R is the weighting matrix of errror signal. In order to generate the control signal, the future outputs of the system is predicted by the following equation: ˆy = GΔ u+ f (16) In which, f is the free response of the system and G includes the step response parameters. The optimal solution for the control signal to minimize the cost function (15), while preserving closed loop stability is calculated from the following equation. [ ] T 1 T λ − Δ u= I 0 � 0 ⎡ ⎣G RG+ I⎤ ⎦ RG ( w− f) (17) C. Parameter Tuning in Multivariable GPC The tuning of the controller parameters is mostly based on experience, and the simulation of the closed loop response. The designer has the freedom to tune either the cost function weighting, or change the disturbance dynamics, observer dynamics, the desired trajectory and finally the prediction and control Horizons. More penalizing λ on the control effort and R on the tracking error will reduce the control effort. The structure of the model is fixed in this method and only the noise levels can be assigned to −1 −1 −1 tune the performance. However, from the inherent integrator form of D ( z ) = ( 1 − z ) A( z ) forces the error of the closed loop system to a step disturbane to converge asymptotically to zero [8, 11]. IV. Closed-Loop Simulation Result The designed controller for the system has the parameters 1 2 functions are tuned through the simulation to: N = 6, N = 37, N = 3 and the weighting u
Modeling and Temperature Controller Design for Yazd Solar Power Plant 887 Q i ⎡1 = ⎢ ⎣0 0 ⎤ ⎥, 1⎦ R i ⎡3. 5 = ⎢ ⎣0 0⎤ 0. 2 ⎥ ⎦ Moreover, the time delays for the oil and water is calculated from the design condition (NDI=900 w/m 2 ) and the system pipe length to constant values of 35 and 50 times the sampling time, respectively. The time constant of the system model is identified to be ten seconds. The closed loop response of the system using GPC controller is compared to that using PI controller with the controller gains: kp = 0.04, ki = 0.007 . The variation of solar radiation is considered as a constant disturbance and illustrated in figure (3). In this simulation the solar radiation is changed from 900 to 800 w/m 2 and its effect on the oil and water temperature is illustrated in figure (3). Due to the larger time delays in oil path compared to that in the water path, the water temperature output shows a faster response. However, both oil and water temperatures are rejecting the effect of disturbance with PI and GPC controllers. Although the overall performance of two controllers are relatively desirable the coupling of two quantities have greater impact on the PI controller compared to that on GPC design. Figure 3: The effect of solar radiation on the response 405 400 395 390 385 380 375 370 PI Effect of Variation in NDI 365 100 120 140 160 180 200 time 220 240 260 280 In figure (4) the effect of measurement noise is illustrated on the response. GPC controller is effectively rejecting the noise effect, due to its predictive nature. Figure (5) illustrates the effect of leakage in oil and water pipes. In this case it is assumed that the oil leakage occurs on the path before entering the solar field and the water leakage occurs before entering the heat exchanger. Due to this failure in the system the temperature of oil and water is observed just after the leakage happens. However, the controller is able to regulate the temperature, in spite of the failure. GPC
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