## Application of Multivariable Predictive Control in a Hydropower Plant

**Zohra Zidane**^{1,}, **Mustapha Ait Lafkih**^{1}, **Mohamed Ramzi**^{1}

^{1}Laboratory of Automatic and Energy Conversion (LACE), Electrical Engineering Department, Faculty of Sciences and Technology, University of Sultan Moulay Slimane

3. Multivariable Linear Quadratic Gaussian Control Algorithm

### Abstract

The hydroelectric energy 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, the 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. The simulation model of hydropower plant was constructed based on mathematical equations that summarize the behavior of the hydropower plant. The simulation model of power plant is useful in stability studies. This paper, presents the approach of Generalized Predictive Control (GPC) is applied to a multivariable model of the part turbine/generator of hydropower plant. In this study, the standard multivariable (GPC) algorithm is presented. It is then applied to achieve sets points tracking of the outputs of the plant. A Multi Input Multi Output (MIMO) model is used for control purposes. A comparative study is carried out using the named controller’s multivariable Linear Quadratic Gaussian (LQG) and multivariable (GPC) Controls. The performance of the proposed controller is illustrated by a simulation example of hydropower plant. Encouraging results are obtained that motivate for further investigations.

### At a glance: Figures

**Keywords:** Generalized Predictive Control, modeling, Linear Quadratic Gaussian Control, multivariable systems, hydropower plant

*Journal of Automation and Control*, 2013 1 (1),
pp 26-33.

DOI: 10.12691/automation-1-1-5

Received February 07, 2013; Revised November 12, 2013; Accepted December 20, 2013

**Copyright:**© 2013 Science and Education Publishing. All Rights Reserved.

### Cite this article:

- Zidane, Zohra, Mustapha Ait Lafkih, and Mohamed Ramzi. "Application of Multivariable Predictive Control in a Hydropower Plant."
*Journal of Automation and Control*1.1 (2013): 26-33.

- Zidane, Z. , Lafkih, M. A. , & Ramzi, M. (2013). Application of Multivariable Predictive Control in a Hydropower Plant.
*Journal of Automation and Control*,*1*(1), 26-33.

- Zidane, Zohra, Mustapha Ait Lafkih, and Mohamed Ramzi. "Application of Multivariable Predictive Control in a Hydropower Plant."
*Journal of Automation and Control*1, no. 1 (2013): 26-33.

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### 1. Introduction

The Hydro-electric energy is most important renewable energy in the world. It provides energy to various loads. User load requires a uniform and uninterrupted supply of input energy. The load demand varies continuously. It affects the terminal voltage and real power output at the generator terminals ^{[1, 2, 3, 4]}. The objective of the control strategy is to generate and deliver power in an interconnected system as economically and reliably as possible while maintaining the voltage and frequency within permissible limits. Hydropower plant is equipped with hydraulic turbine governor and excitation control. The errors in the terminal voltage and in the output active power, with respect to their respective references, represent the controller inputs and the generator-exciter voltage and governor-valve position represent the controller outputs. The control of real power output and the terminal voltage keeps the system in the steady state [5-10]^{[5]}.

This paper presents the application of multivariable GPC control to achieve sets points tracking of the outputs of the plant. The GPC control is one of the most favorite predictive control methods, popular in industry and also at universities. It was first published in 1987 ^{[11, 12]}. The authors wanted to find one universal method to control different systems. Multivariable GPC Control has been successfully implemented in many industrial applications, showing good performance and a certain degree of robustness. It is applicable ^{[13]} to the systems with non-minimal 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 control ^{[14, 15]} is to calculate a sequence of future control signals in such a way that it minimizes a multistage 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.

The paper is organized as follows. Section II presents the system modelling. Section III describes the designed multivariable LQG Controller. Section IV is devoted the description of multivariable GPC algorithm. In section V, the effectiveness and superiority of the proposed algorithm, is demonstrated by simulation example. Some concluding remarks end the paper.

### 2. System Modeling

**List of Symbols**

: Stator voltage in d-axis and q-axis circuit

: Terminal voltage

: Transient EMF in the quadratic axis of the generator

: Stator–rotor mutual reactance

: Field voltage

: Field resistance

: Self reactance of field winding

: Exciter input

: Rotor angle

: Mechanical power

: Water power

: Inertia constant

: Rotor speed of the generator

: Angular frequency of the infinite bus bar

: Mechanical damping torque coefficient

: Damping torque coefficient due to damper windings

: Real power output at the generator terminals

: Exciter time constant

: Governor valve time constant

: Turbine time constant

: Governor input

: Governor valve position

: Valve constant

: Total d-axis synchronous reactance between the generator and the infinite busbar

: Total q-axis synchronous reactance between the generator and the infinite busbar

: Total d-axis transient reactance including the generator and the infinite busbar

: d-axis transient open-circuit time constant

: Reactance of the transformer

: Reactance of the transmission line

: Reactance of the system

The block diagram of the sample controlled power system is shown in Figure 1 that comprises a hydraulic turbine driving a synchronous generator which is connected to an infinite bus via a step-up transformer and a transmission line. The output real power P_{t} and terminal voltage V_{t} at the generator terminals are measured and fed to the controller. The outputs of the controller (system control inputs) are fed into the generator-exciter and governor-valve. In the simulation studies described here, the nonlinear equations of the synchronous generator are represented by a third-order nonlinear model based on park’s equations. The hydraulic turbine, governor valve and exciter are each represented by a first order model. The model equations are as follows [16-24]^{[16]}:

**Fig**

**ure**

**1.**Controlled sample hydropower system

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The Mechanical equations

The rotor speed of the generator is given by:

(1) |

The mechanical equation of the motion is as follows:

i.e

(2) |

Where,

The electrical generator dynamics equations

(3) |

The electrical equations (assumed )

(4) |

(5) |

(6) |

(7) |

(8) |

(9) |

(10) |

(11) |

More details about power system modelling can be seen in ^{[20, 21, 22, 23]}.

Using the above equations, we can express P_{t}(t) as

(12) |

In terms of the state variables and , the equation (2) becomes

(13) |

In terms of the state variables and the equation (3) becomes

(14) |

Where,

The governor valve equation is given by

(15) |

The exciter equation defined by

(16) |

The turbine equation

(17) |

In terms of the state variables and the equations (15,16,17) written as follow:

(18) |

(19) |

(20) |

Defining the state variables vector then the equations above can be written in the key form:

(21) |

The output y_{1}, y_{2} may be expressed in terms of these state variable by

(22) |

(23) |

Where,

(24) |

(25) |

**2.1. Linear Model of Synchronous Generator**

A linear Multi-Input Multi-output (MIMO) model of the generator system is required to design a controller for such system. It is derived from the system nonlinear model by linearizing the nonlinear equation (13) and the nonlinear equation (14) around a specific operating point. The linear state-space model is derived next where the variables shown represent small displacements around the selected operating point.

(26) |

Where,

， et are the Jacobian matrices of partial derivatives of f and g respectively to X and U evaluated at the point.

The linear state-space model defined by

(27) |

Where,

The matrices A, B, C and D have the form:

Where,

Expressions for parameters K1, K2, K3, K4, K5, K6, K7, K8, K9, K10, K11, K12, K13 and K14 are given in Appendix.

**2.2. State Space to Transform Function Conversion**

Consider the state equation (27). We may take its Laplace transform and rearrange it as follows:

(28) |

If we combine this with the transform of the output equation: , we get

Or, equivalently

(29) |

In the Control Systems Toolbox, the command converts the state equation to a transfer function for i^{th} input.

### 3. Multivariable Linear Quadratic Gaussian Control Algorithm

Let us, consider the following process model ^{[24]}:

(30) |

Where,

is the output vector

is the input vector

is a sequence of independent random vectors with zero mean value and finite covariance matrix

is the backward shift operator such that

To this model, we can associate the companion block state representation in the observable form by:

(31) |

Where:

is the system state

The problem is to found a control vector by state feedback that minimizing the following criterion:

(32) |

is the control horizon

is the reference vector sequence

is a symmetric semi definite positive matrix

is a symmetric definite positive matrix

The solution is:

(33) |

Where,

(34) |

(35) |

(36) |

is the Riccati matrix

(37) |

Remark

The solution is an explicit form of the state variables. But they are not available. Therefore a state observer is necessary.

The state observer is given by:

(38) |

Where

(39) |

### 4. Multivariable Generalized Predictive Control Algorithm

When considering regulation about a particular operating point, even a non-linear plant generally admits a locally-linearized model ^{[11, 12]} given by the equation (30).

The objective of the GPC control is the output to follow some reference signal taking into account the control effort. This can be expressed in the following cost function:

(40) |

Where:

is the prediction horizon.

is the initial horizon.

is the control horizon.

is the output reference.

is the output weighting factor.

is the control weighting factor.

The control objective is to compute at each time t, control inputs that minimize the quadratic criterion for this there are two cases:

**Case ****1****.**** **

Let us first build j-step ahead predictors with following Diophantine equation:

(41) |

Where:

The polynomial matrices and are uniquely defined by: and j.

Using equation (30) and (40) we obtain:

(42) |

The optimal predictor at time t is given by:

(43) |

Where:

Defining then the equation above can be written in the key vector form:

(44) |

Note that: so that one way to computing is simply to consider the Z-transform plant’s step-response and to take the first j terms and therefore for j=0, 1, 2 …< i independent of the particular G polynomial ^{[11]}.

The matrix G is then lower-triangular of dimension

From the definitions above of the vectors and with:

(45) |

The expectation of the cost-function of (4) can be written as follow:

(46) |

The solution, minimizing the criterion can be explicitly found, using:

(47) |

it follows that:

(48) |

Note that the first element of is so that the current control u(t) is given by:

(49) |

**Case 2.**** **

It is possible to reduce computational burden by imposing a constant control input vector after a fixed horizon.

In this case the vector and the matrix G become:

Where,

### 5. Simulation and Discussion

To demonstrate the effectiveness of the above presented multivariable GPC Control algorithm, the result are presented and compared with those of the multivariable LQG control. The simulation results are obtained by using Matlab Toolbox.

Initial condition (operating point) for the non linear system:

The hydropower plant model is as follow:

Where:

The simulation has been done with respect to the following considerations:

Parameters of the GPC controller

Parameters of the LQG controller

The reference is chosen as a square wave.

Simulations were carried out to verify the advantages of using multivariable GPC control in this application.

In the figures above, it can be observed the comparative results between multivariable GPC Control and multivariable LQG control.

**Fig**

**ure**

**2.**Power output P

_{t}

The output real power P_{t}, the exciter input voltage U_{e}, the terminal voltage V_{t} and governor valve position U_{g}, under GPC and LQG controls are shown, respectively, in Figures 2, 3, 4 and 5. Best performance is characterized by best tracking, robustness, lower or no over/undershoots less or no oscillations. Based on this, for Pt response, GPC shows the best response whereas LQG shows the worse one with a bigger non minimum phase undershoot, which is eliminated by using GPC control. For V_{t} response GPC control produces the best response in terms of tracking, and overshoot, cancellation of oscillation. The LQG response has a very high overshoot, which is eliminated by using GPC control.

**Fig**

**ure**

**3.**Exciter input U

_{e}

**Fig**

**ure**

**4.**Terminal voltage V

_{t}

**Fig**

**ure**

**5.**Governor Input Ug

### 6. Conclusion

In this paper, a Multivariable Generalized Predictive Controller and Multivariable Linear Quadratic Gaussian controller were designed for a sample power system comprising a water turbine driving a synchronous generator. From the simulation results, it is clear that, the GPC control exhibits better performance for P_{t} and V_{t} responses than the LQG control.

### Appendix

Expressions for parameters K1, K2, K3, K4, K5, K6, K7, K8, K9, K10, K11, K12, K13 and K14 in the system model are:

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