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In this paper, an improved cascade control strategy is presented for hydroturbine speed governors. Different from traditional proportional-integral-derivative (PID) control and model predictive control (MPC) strategies, the performance index of the outer controller is constructed by integrating the entropy and mean value of the tracking error with the constraints on control energy. The inner controller is implemented by a proportional controller. Compared with the conventional PID-P and MPC-P cascade control methods, the proposed cascade control strategy can effectively decrease fluctuations of hydro-turbine speed under non-Gaussian disturbance conditions in practical hydropower plants. Simulation results show the advantages of the proposed cascade control method.

With the high rate of growth of electricity consumption, hydropower plants have been playing a significant role in peak load and frequency regulation of the electric system. Control of hydroturbine speed is economically essential in terms of guaranteeing stability and improving efficiency. Nevertheless, it is challenging because of the complex speed adjusting process characterized by nonlinearity, uncertainty and various disturbances. The principal control target of hydroturbine speed governors, when various disturbances are imposed and/or the set point changes, is to keep the rotor speed within a proper range by tuning the influent flow of the penstock so that the mechanical transmission torque and the generator torque can arrive at a new balance.

Researchers have made a lot of efforts to develop some advanced control strategies for controlling hydroturbine speed governors. In order to be closer to the actual conditions, more and more studies have focused on nonlinear systems [

The above advanced strategies have shown their potential use in controller design for hydroturbine speed governing systems. However, when random disturbances are considered, these methods will not achieve ideal control performance. Actually, in the hydroturbine speed governing process, disturbances can cause fluctuations of rotor speed besides variations of load [

Fortunately, stochastic distribution control theory has been established to deal with the stochastic systems with non-Gaussian noises [

A hydroturbine is a rotary engine that takes energy from moving water. Flowing water is directed onto the blades of a turbine runner, creating a force on the blades. Since the runner is spinning, the force acts through a distance (force acting through a distance is the definition of work). In this way, energy is transferred from the water flow to the turbine. In the energy conversion process, the inlet total head of turbine, fluctuates of flow and water hammer effect would affect the stability of the control system.

The transfer function between the inlet total head

The turbine model can be obtained by considering the effects of water hammer, head loss caused by friction and inelastic penstock [

The synchronous generator model can be expressed by [

The servo motor model is described by [

Carriers’ position and inlet water flow disturbances involved in hydroelectric turbines should be paid high attention, since they can cause fluctuations of rotor speed besides variations of load. These disturbances are usually non-Gaussian, which makes the hydroelectric turbine model be a linear, non-Gaussian stochastic dynamic system. Denoting the target hydro-turbine speed as

Minimum entropy based hydro-turbine speed cascade control systems.

The main purpose of this paper is to design the outer controller such that the tracking error is minimized both in magnitude and randomness, which means that the PDF of tracking error should be made as sharp and narrow as possible near zero. Since the tracking error

In the improved minimum entropy criterion (6),

It can be observed from Equations (7) and (8) that the PDF of tracking error should be estimated in order to calculate the criterion (6). As a nonparametric method, the histogram-based estimation approach [

Substituting the histogram PDF Equation (9) into Equation (7), the nonparametric estimator for Renyi’s entropy can be obtained:

The mean value of tracking error

The design object of the minimum entropy controller is to make the performance index

Denote

According to Equations (13) and (14), the performance index (6) can be rewritten as:

The optimal control input

Then, the recursive sub-optimal control law can be obtained as follows:

In order to analyze the stability of the proposed system, the model of the equivalent plant in outer loop should be formulated first.

Denote the transfer functions of inner proportional controller and sensor 2 in

Since the tuned value of

Therefore, the transfer function of the equivalent plant in outer loop can be formulated by:

Since cascade control can reject disturbances introduced in the inner loop,

In order to analyze the closed-loop stability of the hydro-turbine speed control system, the increment form of Equation (21) is formulated as:

Motivated by the analytical method in [

Using the condition (16), we have:

Substituting Equation (24) into Equation (22), we have:

Denote

Finally, the stability condition of the outer closed-loop system is:

The proposed cascade control method is applied to regulate the hydroturbine rotation speed. In this simulation, the transfer functions of the penstock hydraulic servo system, hydroturbine with elastic water hammer effect and synchronous generator are chosen as,

The outer controller is the given optimization controller using minimum entropy criterion and the inner controller is a conventional proportional controller

The hydroturbine rotation speed is governed at a steady state before 500 s, the target rotating speed increases from 500 rpm to 510 rpm at 500 s and lasts forever. The responses of the hydroturbine rotation speed with three different controllers are shown in

Distributions of disturbances _{1} and _{2}.

The responses of the outer controllers are shown in

Responses of the hydroturbine rotation speed.

Responses of the outer controller.

Performance index.

Information potential and entropy of the tracking error.

PDF of the tracking error.

PDFs at typical instants.

This paper investigates the minimum entropy-based cascade control problem for hydroturbine speed control systems with non-Gaussian disturbances. The performance index of a closed-loop control system consists of entropy, mean value of tracking error and control energy constraint. In this regard, the task of control is to design the control algorithm so that the established criterion is minimized. The histogram-based estimation approach is adopted to obtain the nonparametric estimation of the tracking error PDF, and then the whole performance index can be obtained. By minimizing the improved minimum entropy criterion, an incremental control law is eventually formulated. Comparative simulation results show that the presented method can achieve better speed control performance in dealing with non-Gaussian disturbances. Future work incorporates considering nonlinearities exist in the hydroturbine speed control systems and doing experiments on a real turbine.

This work was supported by National Basic Research Program of China under Grant (973 Program 2011 CB710706) and China National Science Foundation under Grant (60974029). These are gratefully acknowledged. The authors wish to thank Ting Zhang for relevant simulations. We also wish to thank Guolian Hou and Fang Fang for their constructive suggestions when the paper was revised.

Jianhua Zhang conceived the project. Mifeng Ren, Di Wu, Jianhua Zhang and Man Jiang searched relevant literatures. Mifeng Ren and Jianhua Zhang carried out the theoretical derivation. Di Wu provided the simulation results. Mifeng Ren, Jianhua Zhang and Di Wu analyzed the simulation results and wrote the paper. Correspondence and requests for materials should be addressed to Jianhua Zhang.

The authors declare no conflict of interest.

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