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See discussions, stats, and author profiles for this publication at: https://www.researchgate.net/publication/225102466 Comparative performance study of QN and LM algorithms in predictive control for NNARX-identiﬁed model of hydro-power plant Article in Engineering with Computers · June 2007 DOI: 10.1007/s00366-006-0024-z CITATIONS READS 2 7,012 4 authors, including: S P Singh Ajay Singh Raghuvanshi Jawaharlal Nehru Krishi Vishwavidyalaya National Institute of Technology Raipur 82 PUBLICATIONS 1,557 CITATIONS 66 PUBLICATIONS 410 CITATIONS SEE PROFILE All content following this page was uploaded by Ajay Singh Raghuvanshi on 04 June 2014. The user has requested enhancement of the downloaded file. SEE PROFILE Engineering with Computers (2007) 23: 71–78 DOI 10.1007/s00366-006-0024-z O R I GI N A L A R T IC L E Nand Kishor Æ S. P. Singh Æ A. S. Raghuvanshi P. R. Sharma Comparative performance study of QN and LM algorithms in predictive control for NNARX-identified model of hydro-power plant Received: 11 January 2005 / Accepted: 26 January 2006 / Published online: 30 August 2006 Springer-Verlag London Limited 2006 Abstract In this paper input-constrained predictive control strategy for NNARX (neural network non-linear auto-regression with exogenous signal) model of hydro-turbine is presented. The input (gate position) and output (turbine power) data are generated by means of dynamic plant model. The collected data are utilized to develop the NNARX model of the plant. Then NNbased predictive control (NNPC) scheme is applied to control the turbine power. The control cost function (CCF) includes the squared diﬀerence between the model predicted output and desired response and a weighted squared change in the control signal. The CCF is minimized with both Quasi-Newton and Levenberg– Marquardt iterative algorithms. To demonstrate the suitability of the strategy, the plant has been simulated on two diﬀerent reference signals. Keywords Cost function Æ Hydro-plant Æ Identiﬁcation Æ Neural network Æ Predictive control List of symbols surge ; H turbine H head available in riser of surge tank, turbine (p.u.) penstock tunnel ; H head loss in tunnel, penstock H (p.u.) turbine ; U surge velocity of water in tunnel, tunnel ; U U turbine, surge tank (p.u.) noload velocity of water at no load (p.u.) U ftunnel, fpenstock head loss coefﬁcient in tunnel, penstock (p.u.) Csurge storage constant of surge tank (s) devia;tunnel head deviation in tunnel (p.u.) H devia;penstock head deviation in penstock (p.u.) H S. P. Singh Department of Electrical Engineering, Motilal Nehru National Institute of Technology, Allahabad, India N. Kishor (&) Æ A. S. Raghuvanshi Æ P. R. Sharma Department of Electrical Engineering, Royal Bhutan Institute of Technology, Phuentsholing, Bhutan E-mail: nand_research@yahoo.co.in Zp Tw Tep Twc L U0 g H0 Dx Gopening Dturbine Again hydraulic surge impedance of penstock (=Tw/Tep) water time constant in penstock (s) ð¼ LU0 =gH0 Þ; Tw varies with load elastic time constant of penstock (s) water time constant in tunnel (s) length of penstock velocity of water in penstock at rated head (p.u.) acceleration due to gravity rated turbine head (p.u.) deviation of rotor speed (p.u.) gate opening (p.u.) damping factor or coefﬁcient in p.u. torque/p.u. speed deviation turbine gain 1 Introduction A hydro-turbine is non-linear, non-stationary system whose characteristics vary signiﬁcantly with unpredictable load. The turbine model considered in the design of the governor plays an important role in the eﬃcient operation of the power plant. At present, linear control theory based PID controller ﬁnds its application in the power plant. Numerous methods for PID tuning are reported in the literature [1]. To realize parameter optimization of PID controller, an orthogonal test strategy is adapted in [2–5] for hydro-turbine control application. In this approach, control performance index is deﬁned, which depends on control parameters Kp, Ki and Kd. Each of these parameters is considered under various levels as discrete variable. An optimization algorithm is developed to search for better control parameters in the neighboring space of the present ones. Lansberry and Wozniak [6] have used genetic algorithm (GA) approach 72 for optimal governor tuning. The work investigates the GA as one possible means of adaptively optimizing the gains of proportional-plus-integral governors. Yamamoto et al. [7] and Cheng et al. [8] present the concept of intelligent tuning of PID controller. The former authors have discussed the use of adaptive and learning control scheme—which is neural network techniques—while the latter authors have presented an improved dynamic performance of the intelligent PID controller over the conventional PID. The developed intelligent PID controller is based on an anthromorphic intelligence. Fangtong et al. [9] introduced the dynamic modeling of hydro-turbine generating set as a single machine with region load using recursive least square estimation algorithm. Also, non-linear model with NN structure 3-2-1 was discussed in the study [10]. The non-linear simulation of hydro-turbine governing system based on NN is described in [11]. Two three-layered perceptrons NN1 and NN2 of structures 1-4-1 and 2-12-2, respectively, are considered. The former structure forms a nonlinear relation between servomotor stroke and guide vane opening while the latter structure between guide vane, speed eﬃciency and discharge. The modeling of dynamic interaction between the gate (input) and turbine mechanical power (output) is important as it regulates the operation of the governor action. It is diﬃcult to obtain a non-linear simulation model of the hydro-turbine. The conventional characteristic curves of hydro-turbine do not provide suﬃcient insight into non-linear simulation. A great deal of attention has been drawn towards its linearized modeling. However, detailed hydraulic system model, which should include compressible and elastic eﬀects of penstock, is required to ensure eﬀective control. These eﬀects represent a delay e2sTe in the hydraulic structure, which is irrational. As a result a dynamic interaction between the hydraulic system and the electrical system exists. Also while ensuring an eﬀective control action, it becomes necessary to use a reduced order turbine–penstock model including delay eﬀects, especially of long penstock in hydro-power plant. In recent years, many diﬀerent types of NN-based control systems have attracted increased interest. NN is either trained to operate as a controller or a conventional control scheme is utilized with NN model of the plant/process. An example of the latter is the NN-based predictive control (NNPC) strategy. With NNPC, a neural network model is developed to identify the nonlinear dynamics of the plant and a predictive control strategy is implemented, being the sequence of control actions computed iteratively by solving the control cost function (CCF) [12]. An important feature of predictive control is the ability to handle constraints of actuated variables and internal variables [13]. This paper presents predictive control scheme adapted for hydro-plant NNARX (neural network non-linear auto-regression with exogenous signal) model. A feedforward multilayer perceptron neural network (MLPNN) with 15 hidden neurons is proposed for identiﬁcation. The CCF in non-linear predictive control is minimized iteratively by both Quasi-Newton and Levenberg-Marquardt approaches. The minimization is computed assuming input constraints as (1) umin=¥ and umax=¥ and (2) umin=0 and umax=1.0. An appropriate selection of control parameters is illustrated. The predictive control technique based on identiﬁed NNARX turbine model is investigated on two diﬀerent set-point reference signals (turbine power). 2 Basic power plant equations As shown in Fig. 1, when the turbine-generator unit is started, water from the reservoir ﬂows into the tunnel and then to a high-pressure conduit called penstock. The water from the penstock ﬂows into a scroll casing, which distributes it evenly on the runner blades, mounted on the common shaft with the generator. This results in electro-mechanical conversion of power. The wicket gate is operated by means of the governor to regulate the ﬂow as a function of variable electrical power. The sudden closure of gates leads to pressure wave set-up moving upstream in the penstock and increase in pressure at turbine input (water hammer). The pressure wave occurrence is due to penstock-wall elasticity and water compressibility properties. In the case of hydro-plant layout having a long penstock, the travel time of wave is signiﬁcant [14]. The riser in the surge tank helps to suppress water hammer eﬀect and the tank acts to store/ supply water temporarily. Practically, the water levels in the riser and tank remain the same. The dynamic equations of hydraulic and mechanical/electrical systems in power plant are described [15]. Dynamics of tunnel: Water ﬂow through the tunnel subjected to frictional forces leads to head loss given as tunnel ¼ ftunnel U tunnel jU tunnel j: H ð1Þ The dynamics of head variation in the tunnel is devia;tunnel ¼ Twc dUtunnel : ð2Þ H dt Dynamics of surge tank: Riser (surge tank) head: surge ¼ 1:0 H devia;tunnel ; H H Z tunnel surge dt: surge ¼ 1 U H Csurge ð3Þ ð4Þ Dynamics of penstock: The water ﬂow in the penstock being subjected to friction causes a head loss given as penstock ¼ fpenstock U 2 ; ð5Þ H turbine Hdevia;penstock ¼ Zp tanh Tep s Uturbine : ð6Þ The available head at the turbine inlet is surge H penstock H devia;penstock : Hturbine ¼ H ð7Þ 73 Reservoir Surge tank fsigmoid ðUðtÞÞ ¼ Tunnel ð10Þ while the closely related hyperbolic tangent function is deﬁned as Generator Penstock Hydro-turbine ftanh ðUðtÞÞ ¼ 2fsigmoid ðUðtÞÞ 1: ð11Þ These functions being non-linear thus describe the nature of the NN. Cybenko [17] and Funahashi [18] have proved that a MLPNN with a single hidden layer of sigmoidal or hyperbolic tangent function can approximate any continuous function. In the present study, tanh is used as activation function for neurons in hidden layer. A typical block diagram of NNARX model for identiﬁcation is shown in Fig. 2. Turbine gate Fig. 1 A general layout of hydro-power plant Mechanical power: The mechanical power developed is noload Þ Pdamping ; turbine ðU turbine U Pmech ¼ Again H Pdamping ¼ Dturbine Gopening Dx: 1 ; 1 þ e/ðtÞ ð8Þ ð9Þ The term Pdamping represents the damping eﬀect due to friction and is proportional to the rotor speed deviation and the gate opening. This term is important in the modeling of the turbine for generator start-ups and other emergency situations. Non-linear models of power plant are used in study involving large perturbations, which normally occur during an islanding, load rejection and system restoration conditions. In the present work, the aim is to identify the NNARX model showing the turbine power Pmech dynamics with the random variation in gate position and develop a predictive controller to track the given reference signals. 3 NNARX model identification Over the last one decade, NN has been considered a promising approach in system identiﬁcation. A blackbox non-linear mathematical model that relates the inputs and outputs of any system may be developed with NN. This approach does not need an exact mathematical model of the system. Narendra and Parthasarathy [16] proposed the use of neural network in conjunction with system theory to develop realizable models. With the availability of such NN model, various control techniques may be applied for designing an eﬃcient controllable system. A MLPNN structure is developed to model the nonlinear dynamic relationship between the gate position and turbine mechanical power. A feedforward MLPNN is formed with layers of neurons between the input and output layers called hidden layers. The hidden layer neurons act as connecting element between neurons of intermediate layers. The inputs to each neuron are combined along with a bias (if any) and the neuron produces an output if the sum of inputs exceeds a threshold value of activation function. Generally, the sigmoid function/hyperbolic tangent function is used as activation function. The sigmoid function is given as 3.1 Methodology The dynamic model represented by (1)–(9) is simulated to generate the data. The hydro-power plant parameters used in the study are mentioned in Appendix. The absolute value of pseudorandom binary signal is applied to the input to represent the variation of gate position. And the corresponding turbine mechanical power is computed. The collected data (input and output) are divided into two sets, one for training the NN and the other for validation. Thus an input–output model is established, resulting in black-box identiﬁcation approach. The hydro-power plant can be represented in discrete input and output form by the identiﬁcation structure: ^y ðk; hÞ ¼ f ðUðk; hÞÞ þ nðkÞ: ð12Þ The NNARX regressor vector is expressed as UðkÞ ¼ ½yðk 1Þ; . . . ; yðk na Þ; uðk nk Þ; . . . ; uðk nb nk þ 1ÞT ; ð13Þ where na, nborder of the model structure nk delay in the input signal nn=[ na, nb, nk ] is the NNARX model structure. y (k−1) y (k−na) u(k−nk) NNARX model structure (Feedforward multilayer neural network) u(k−nk − nb + 1) Fig. 2 A typical non-linear neural network ARX structure ŷ(k) 74 The identiﬁcation cost function (ICF) formulated as the mean square error on a training data is minimized with respect to network weights. This is deﬁned by J¼ N2 X ½rðk þ iÞ ^y ðk þ iÞ2 þ i¼N1 Nu X q½Duðk þ i 1Þ2 i¼1 ð15Þ N N 1 X 1 X VN ðh; ZeN Þ ¼ ½yðtÞ ^y ðtjhÞ2 ¼ n2 ðtjhÞ; 2N t¼1 2N t¼1 ð14Þ where N number of training data samples Ze Na vector which contains the system output and regression vector U (t), =[y(t)U (t)] The parameter estimate ^ h ¼ arg min VN ðh; ZeN Þ: In the development of NNARX model structure, the NN System Identiﬁcation Toolbox for MATLAB developed by Nørgaard [19] is utilized. 3.2 Training parameter The collected data set is divided into two sets, one for training and the other for validation of the model. During network training, the hidden layer weights were initialized automatically. The learning rate g=1e 4 while the momentum coeﬃcient a=0. The network training is terminated when either the maximum error gradient is less than 1e 4 or when the number of iteration exceeds 500. A Gauss–Newton based Levenberg– Marquardt method is utilized for minimization of meansquare error criteria (VN). 3.3 Optimal network structure The non-linear model representations have great diﬃculty in selection of appropriate model order. To select an optimum NN structure, the inﬂuence of variation in vectors: past input, past output data and number of hidden layer neurons (HLNs) on ICF is studied. After a few simulation trials, the NN conﬁguration, HLNs, H=15 and nn=[ 1,10,1 ] has been determined. and identiﬁed NNARX model, it is possible to calculate the optimal control strategy for a non-linear turbine control. The term r(k+i) is the required reference turbine power, ^y ðk þ iÞ the predicted NN model output, Du(k+i 1) the controlled gate position, N1 and N2 the minimum and maximum prediction horizons, respectively, Nu the control horizon and q a control penalty factor. The control approach uses a receding horizon strategy. The basic concept of the receding horizon control is to solve an above deﬁned (15) optimization problem for a ﬁnite future at a current time and to implement the ﬁrst optimal control input as the current control input. At each sample time k, minimizing CCF for the selected values of the control design parameters [ N1, N2, Nu, q ] calculates the vector u=[ Du(k),Du(k+1),...,Du(k+Nu 1) ]. The performance of predictive control is largely aﬀected by these parameters. N1 is usually set to a value 1 and N2 is set to deﬁne the prediction horizon (i.e. the number of time-steps in the future for which the plant response is recursively predicted). 4.1 Control law With identiﬁed NNARX plant model, an iterative search method is applied to minimize the CCF. The Quasi-Newton and Levenberg–Marquardt algorithm is used in the calculation of suitable controlled gate position [21]. The minimization is computed with input constraints as (1) umin= ¥ and umax=¥ and (2) umin=0 and umax=1.0. 4.2 Selection of control parameters In order to determine the appropriate value of {N1, N2, Nu, q }, the inﬂuence of control penalty factor q and maximum prediction horizon N2 on CCF is analyzed in the following sections. 4.2.1 Quasi-Newton 4 Non-linear predictive control Taking advantage of the recognized universal approximation properties of NNs, a non-linear plant in NNARX form is obtained using feedforward MLPNN. Based on the neural model, a predictive control strategy is implemented, being the sequence of control actions computed by solving an optimization problem iteratively. By the methodology adapted using a weighted sum of quadratic CCF [20]: With input constraints as umin= ¥ and umax=¥ it is observed after several trails that a minimum value of CCF is obtained at q=0.7, q=0.8, q=0.9 and q=1.0 with least horizon values of N1=1,N2=10 and Nu=1. This is presented in Fig. 3a. Next the same is determined with variation of maximum prediction horizon N2. Figure 3b illustrates a minimized CCF value for N1=1,N2=12,Nu=1 and q=0.7. Selecting umin=0 and umax=1.0 as input constraint the CCF surface is illustrated in Fig. 4. The appropriate parameters are 75 Fig. 3 The CCF optimization (umin= ¥ and umax=¥) using QN algorithm. a Eﬀect of control penalty factor. b Eﬀect of maximum prediction horizon Fig. 4 The CCF optimization (umin=0 and umax=1.0) using QN algorithm Fig. 6 The CCF optimization (umin=0 and umax=1.0) using LM algorithm observed as: N1=1,N2=12,Nu=1 and q=1.0 to q=0.09. with least horizon values of N1=1,N2=10 and Nu=1 is obtained. This is shown in Fig. 5a. And Fig. 5b presents an optimized value for N1=1,N2=16,Nu=2 and q=0.9. The CCF surface with variation in control parameters selecting umin=0 and umax=1.0 is illustrated in Fig. 6. The parameters are appropriately determined as N1=1,N2=12,Nu=1 and q=1.0 to q=0.09. 4.2.2 Levenberg–Marquardt With umin= ¥ and umax=¥ using Levenberg–Marquardt algorithm, the optimization at q=0.9 and q=1.0 Fig. 5 The CCF optimization (umin= ¥ and umax=¥) using LM algorithm. a Eﬀect of control penalty factor. b Eﬀect of maximum prediction horizon 76 Fig. 7 The QN-based predictive control of set-point trajectory (reference-1), turbine output and control action. a umin= ¥ and umax=¥. b umin=0 and umax=1.0 Fig. 8 The QN-based predictive control of set-point trajectory (reference-2), turbine output and control action. a umin= ¥ and umax=¥. b umin=0 and umax=1.0 5 Simulation results A sequence of reference tests are conducted to investigate initial start-up and set-point following. The identiﬁed NNARX turbine model with control design parameters as summarized in Table 1 is simulated on two diﬀerent set-point step reference signals. The model output with NN-based predictive controller solved by QN algorithm is illustrated in Fig. 7 (reference-1). It is seen that a good set-point tracking is achieved for all steps with both types of input constraints. The response shows a non-minimum phase characteristic of hydro- turbine. And the control signal is maintained within a modest magnitude. Keeping the same control parameters, simulation results on another desired output of diﬀerent amplitudes step function (reference-2) is shown in Fig. 8. A poor set-point tracking is optimized with umin= ¥ and umax=¥ as input constraint. Also the gate position variation does not remain in prescribed limits. On the other hand the result obtained with umin=0 and umax=1.0 exhibits a close and oﬀset-free reference tracking and gate position variation within limits. Next the controller performance using LM optimization approach is demonstrated in Figs. 9 and 10. An Fig. 9 The LM-based predictive control of set-point trajectory (reference-1), turbine output and control action. a umin= ¥ and umax=¥. b umin=0 and umax=1.0 77 Fig. 10 The LM-based predictive control of set-point trajectory (reference-2), turbine output and control action. a umin= ¥ and umax=¥. b umin=0 and umax=1.0 Table 1 The control design parameters for predictive control approach Algorithm Quasi-Newton Levenberg–Marquardt Control parameters Input constraint: umin= ¥ and umax=¥ Input constraint: umin=0 and umax=1.0 N1=1,N2=12,Nu=1,q=0.7 N1=1,N2=16,Nu=2, q=0.9 N1=1,N2=12,Nu=1, q=1.0 N1=1,N2=12,Nu=1, q=1.0 output response without overshoot and oscillation has been obtained as similar to QN algorithm based strategy. However, the gate position variation is oscillatory to some extent as can be seen in Fig. 9a. And in practice, this would cause undue actuator wear. However, the CCF optimized with umin=0 and umax=1.0 results in good performance as illustrated in Fig. 9b. When the CCF is optimized on reference-2 with umin= ¥ and umax=¥ as input constraint, a poor setpoint tracking and gate position variation beyond the prescribed limits are obtained. This is illustrated in Fig. 10a. Thus suitable controlled gate position has not been found to make the plant output follow the setpoints. The performance on second input constraint as presented in Fig. 10b is in close resemblance to reference-1. This suggest the controllers are capable to follow adequately well the set-point trajectory, regardless of set-points values and diﬀerence in level between each reference. 6 Conclusion Based on the identiﬁed turbine NNARX model, this paper presented a non-linear predictive control scheme. The CCF was solved using Quasi-Newton and Levenberg–Marquardt algorithms with input constraints as (1) umin= ¥ and umax=¥ and (2) umin=0 and umax=1.0. Also the study was carried out to show the inﬂuence of control parameters on CCF. A relatively good set-point tracking was shown with CCF having input constraint umin= ¥ and umax=¥ minimized using QN approach. On the other hand there was no signiﬁcant diﬀerence between the two optimization algorithms in set-point tracking performance optimized with input constraint umin=0 and umax=1.0. Thus both the approaches’ oﬀer as an eﬀective computational method to drive the NNARX model output follows the set-points. 7 Appendix Parameters of the system studied: Zp (Tw/Tep)=6.335; Tw=2.23 s; Tep=0.352 s; Twc=39.15 s; Csurge=240 s; ftunnel=0.0475 p.u.; noload ¼ 0:13 p.u.; D fpenstock=0.089 p.u.; Again=1.67; U x=0.05; Dturbine=2.0. References 1. Cominos P, Munro N (2002) PID controllers: recent tuning methods and design to speciﬁcation. IEE Proc Control Theory Appl 149:46–53 2. Luqing YE, Shouping WEI, Zhaohui LI, Malik OP, Hope GS, Hancock GC (1990) An intelligent self-improving control strategy and its microprocessor-based implementation to a hydro-turbine governing system. 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