Simulation Studies of Adaptive Predictive Control for Small Hydro Power Plant

Small hydro power is one of the most important renewable energy in the world. It does not encounter the problem of population displacement and is not as expensive as solar or wind energy. However, small hydro electrical generating units are usually isolated from the grid network; thus, they require control to maintain of constant the power for any working conditions. This paper presents a flow control approach for the speed control of hydro turbines. Power can be controlled by controlling the amount volume of water running into turbine. In this study, the adaptive predictive control is designed to control a flow for the automatic control of s mall hydro power plants. The standard Generalized Predictive Control (GPC) algorithm is presented. The Adaptive Generalized Predictive Control is then applied to achieve set point tracking of the output of the plant. A Single Input Single Output (SISO) model is used for control purposes. The model parameters are estimated on-line using an identification algorithm based on Recursive Least Squares (RLS) method. The performance of the proposed controller is illustrated by a simulation example of Small hydro power plant. Obtained results have shown better characteristics concerning both set point tracking and disturbance robustness for adaptive predictive control.


Introduction
Small hydro power was one of the earliest small scale renewable energy technologies to be developed, and is still an important source of energy today. It has the potential to produce an important share of power, with a low p rice, mo re than solar or wind power. Small hydro elect rical power plants are usually built in remote commun ities, as they use the river's flow in the mountains. User loads require a uniform and an uninterrupted supply of input energy. In addition, small hydro powers are often isolated fro m grid networks. Thus, they require control to maintain of constant the power for any wo rking conditions. Power can be controlled by controlling the amount volu me of water running into turbine to produce just the necessary power [1][2][3][4][5][6][7][8]. It is well known that the mathematical model is very crucial for a control system design. For a small hydro power, there are many models to represent the mach ine behavior with a good accuracy. However, the parameters of the model are also important because the mathematical model cannot prov ide a co rrect behav io r wit hout co rrect parameters in the model. Therefore, the parameters can be determined by identification technique. This paper presents the application of Adaptive Generalized Pred ictive Control (GPC) to achieve set point tracking of the output of the plant. The Generalized Predictive Control (GPC) is one of the most favorite predictive control methods, popular in industry and also at universities. It was first published in 1987 [9], [10]. The authors wanted to find one universal method to control different systems. GPC has been successfully implemented in many industrial applications, showing good performance and a certain degree of robustness. It is applicable [3] to the systems with non-min imal phase, unstable systems in open loop, systems with unknown or varying dead time, systems with unknown order and nonlinear systems approximated by linear models.
The basic idea of GPC [12], [13] is to calcu late a sequence of future control signals in such a way that it min imizes a mu ltistage cost function defined over a prediction horizon. The index to be optimized is the expectation of a quadratic function measuring the distance between the predicted system output and some reference sequence over the horizon plus a quadratic function measuring the control effort. The predictive model is carried out based on the solving Diophantine equations.
In the present paper the Adaptive Generalized Predict ive Control method is designed to control a small hydro power plant and a Single Input Single Output (SISO) model is used for control purposes. The model parameters are estimated On-line using an identification algorith m based on Recursive Least Squares method. It is proved in the paper that in spite of important variations of the plant output; the developed adaptive structure maintains high level of performances (tracking, d isturbance robustness and overshoot, cancellation of oscillat ion). The paper is organized as fo llo ws. Sect ion II presents the Generalized Predict ive Control algorith m. Sect ion III is devoted the description of the adaptive control algorith m. In section IV, the effectiveness and superiority of the adaptive system, is demonstrated by simulat ion examp le. So me concluding remarks end the paper

Generalized Predictive Control Algorithm
The GPC scheme [14] can be seen in Figure 1. It consists of the plant to be controlled, a reference model that specifies the desired performance of the plant, a linear model o f the plant, and the Cost Function Minimizat ion (CFM) algorithm that determines the input needed to produce the plant's desired performance. The GPC algorith m consists of the CFM block.
The GPC system starts with the input signal, r(t), which is presented to the reference model. This model produces a tracking reference signal, w(t) that is used as an input to the CFM block. The CFM algorith m produces an output, which is used as an input to the plant. Between samples, the CFM algorith m uses this model to calculate the next control input, u(t+1), fro m predictions of the response from the plant's model. Once the cost function is minimized, this input is passed to the plant. This algorith m is outlined below. When considering regulation about a particular operating point, even a non-linear plant generally ad mits a locally-linearized model [9] and [10]: Where, A and B are polynomials in the backward shift If the plant has a non-zeros dead-time the lead ing elements of the polynomial In literature w(t) has been considered to be a moving average form: is uncorrelated random sequence, and combining with (1) we obtain the CARMA (Controlled Autoregressive Moving Average): The objective of the GPC control is the output y(t) to follow some reference signal y*(t) taking into account the control effort. This can be expressed in the following cost function: locally-linearized model [9] and [10]: Where: h p is the prediction horizon. h i is the initial horizon. h c is the control horizon. y*(t) is the output reference. R is the output weighting factor. Q is the control weighting factor. The control objective is to co mpute at each time t, control inputs that minimize the quadratic criterion for this there are t wo cases: Let us first build j-step ahead predictors with following Diophantine equation: Using equation (1) and (5) we obtain: The optimal pred ictor, given measured output data up to time t and given u(t+i) for i>1, is clearly: then the equation above can be written in the key vector form: Where the vectors are all so that one way to computing j G is simply to consider the Z-transform p lant's step-response and to take the first j terms and therefore j j i g g = for j=0, 1, 2 …< i independent of the particular G polynomial [9].
The matrix G is then lower-triangular of dimension

 
Note that if the plant dead time d > 1 the first d-1 rows of the G will be null, but if instead h i is assumed to be equal to d the leading element is non-ze ro [1].
Fro m the definitions above of the vectors and with: The expectation of the cost-function of (4) can be written as follow:  (10) it follows that: so that the current control u(t) is given by:

Adaptive Control Algorithm
The adaptive controller which is proposed here is indirect controller. To estimate the unknown system parameters The following (RLS) algorith m has been using:

Simulation and Discussion
In order to illustrate the behavior of the above presented Adaptive Generalized Predictive Control, the simu lation results of the small hydro power p lant model obtained by using on-line identification technique, are given. The model is chosen as follows [15][16][17]: Several experiments have been carried out to determine a suitable control model order an appropriate sample time for control. A tried order model (na=3, nb=2, delay=0) sampled at 1 second gave a reasonable description of a s mall hydro power using on-line pilot plant dynamics.
The simu lation has been done with respect to the following considerations: • The sampling time T=1 • The plant model structure na=3, nb=2 and delay=0 • The reference is chosen as a square wave First the non-disturbed small hydro power system is controlled by Adaptive Generalized Predictive Controller to track the set point. The tracking response is shown in figure 2 where we seen that the tracking performance is successfully achieved.
Then, a step disturbance of 4 percent is injected in the small hydro power output. This quantity has been included in the output response as a fo rcing term to represent unmeasured disturbances. The system response is given in figure 4, where the tracking performance is achieved successfully and the effect of disturbance is well rejected.
Then, the small hydro power output is affected by a random d isturbance as shown in figure 6. It can be seen that the system response follows the reference with less oscillation.
Finally, the plant was subjected to a step disturbance coupled with a Gaussian stochastic disturbance, as shown in figure 8, where the tracking performance is achieved successfully. It can be observed that the Adaptive Generalized Predict ive Control shows better characteristics concerning the variances of the plant output and the control input.
The behavior of the model's parameters is shown in figures 3, 5, 7 and 9.

Conclusions
Since Morocco has plenty of hydrological resources, it was aimed to contribute to the economical way to use water energy via imp roving Sma ll Hydro Po wer pilot Plant. This paper, include better solution in term of efficiency and modern approach. The Adaptive Generalized Pred ictive Control strategy was proposed to maintain the power constant for any working conditions. The prediction model is designed using the Diophantine equations solution. It has proved that, even with important variations of the plant output, the developed adaptive structure maintains a high level of performances, in terms of tracking, disturbance robustness and overshoot, cancellation of oscillation.
Thanks to using modern control, the productivity of the power system can be augmented; the machine and power plants are to be longer lived. Also, the pollution and CO 2 emissions can be reduced.
Finally, the efficiency of the proposed controller will be checked on an experimental test bench as soon as possible.