On improved active disturbance rejection controller design for a pumped storage hydro unit

The pumped storage hydro unit (PSHU) shows bad dynamic performance because of the frequent switching of working conditions and the S-shaped characteristics of pump turbine. This paper proposes an improved fixed-time active disturbance rejection controller to improve the control performances and counteract the influences of uncertainties including modeling error and friction. A disturbance observer is designed to estimate the total uncertainty by using the information of the states and outputs, compensating the adverse effects of the total uncertainty actively by feed-forward principle as a result. The outstanding benefit of the disturbance observer lies in that it can be designed separately from the original controller and can enhance the performances of the PSHU without changing the original controller. Simulation results show that comparing with the conventional active disturbance rejection controller, the improved active disturbance rejection controller has better adaptability and higher robustness towards the system load disturbance, and the dynamic performance of the system is improved.


Introduction
Pumped storage is currently the most technologically mature, economically optimal and capable of large-scale development in the power system.It is a green, low-carbon, clean and flexible regulating power source, and has good coordination effects with wind power, solar power generation, nuclear power, thermal power, etc. Accelerating the development of pumped storage is an urgent requirement for building a new type of power system with new energy resources.Pumped storage power stations have advantages such as peak load regulation and frequency modulation, which play an important role in the stable operation of the power grid.The large-scale grid connection of new energy resources has put forward new requirements for the control and operation mode of pumped storage units.
As the core of PSHU, pump turbine governing system (PTGS) is a complex integrating water electromechanical system with multi-modular nonlinear minimum phase.The pump turbine can operate as a pump or turbine in different operations, and under certain operating conditions, it is easy to fall into the S-shaped characteristic region and the speed is extremely unstable.Due to the nonlinear characteristics of water electromechanical coupling and frequent switching of working conditions in PTGS, it seriously threatens the safe and stable operation of the unit.It is necessary to conduct in-depth research on the dynamic characteristics of PTGS.
Conventional PID control, as a linear controller, cannot adapt to the motion control of actual nonlinear systems and cannot guarantee the stability of the system under disturbances.In recent years, people have begun to attempt to apply advanced control strategies, such as variable parameter PID control [1], fuzzy control [2] and neural networks [3], to the regulation of PTGS.Compared to the above control methods, this paper proposes an active disturbance rejection controller (ADRC) to improve control quality of the system.
ADRC strategy is a model independent control algorithm, and does not require an accurate mathematical model of the controlled object.The algorithm is simple, the parameter adaptation range is wide and it can automatically detect and compensate for the disturbance of the object.The improved fixed-time state observer can quickly converge within a fixed time, shorten the transient process of observing the total disturbance of the system, and improve the anti-interference ability of the composite controller.
In summary, this article is based on the characteristics of simple structure and easy parameter adjustment of PID control, combined with the advantage of anti-interference ability of fixed-time ADRC in dealing with nonlinear systems.The combination of the two is applied to the speed control system of PTGS, in order to improve the control quality.Finally, the effectiveness of the improved fixed-time ADRC is verified through simulation.

Modeling of the pumped storage hydro unit
The governing system of a pumped storage hydro unit consists of a speed governor, a diversion system, a water pump turbine and a generator/motor.The governor adopts a microcomputer governor, which includes a microcomputer regulator and an electro-hydraulic servo system.The overall structure of the pumped storage unit governing system is shown in Figure 1.The mathematical modeling of the components of the governing system is described in detail below [4].

Modeling of the speed governor
The speed governor is the core control device of PSHU, which is the key link for the safe and stable operation of the unit.The governor consists of a microcomputer governor and an electro-hydraulic servo system.

modeling of the microcomputer governor.
Currently the control algorithm of microcomputer governor is mostly PID controller.The speed control system of the turbine has differential regulation characteristics,   is the feedback coefficient in difference regulation.The transfer function of the governor can be expressed as: where   () = () − () is the frequency deviation， r is the speed reference and x is the feedback generator unit speed; ( ) y e s is guide vane deviation.p K is the proportional gain, i K is the integral gain and d Here 1 y T is the AC servomotor response time constant, and y T is the main servomotor response time constant.

Modeling of the pump turbine
Torque t M and overboard flow q of Pump turbine are related to guide vane opening y , water head h and the generator unit speed x .The model is shown as: ( ) where x e y e h e qx e qy e qh e are the pump turbine transfer coefficients.

Modeling of the diversion system
Water diversion system is related to the flow in pressure pipes and the water head.The transfer function is described as: T is flow inertia time, f is loss coefficient of water head, r T is water hammer pressure wave time constant.

Modeling of the generator/motor
Generator/motor is the energy conversion part in PSHU, its transfer function of first-order model is expressed as: where a T is comprehensive inertia time constant of unit, g e is generator self-regulation coefficient.

Fixed-time ADRC design
According to the above modeling of the pumped storage hydro unit, based on the principle of coordinate transformation, it can be transformed into a spatial state equation [5] as: Because d is bounded, it can be observed with an observer.
Extended state observer (ESO) is the key in ADRC [6].The traditional extended state observer (ESO) is described as: An improved fixed-time ESO is designed that the disturbance estimation will be regulated in a fixed (pre-established) time.The new observer is shown as: here the parameters 1 2 3 4 , , , ( Refer to the article [7] for the observer gains , 1, 2,...8. e Z x = − to conclude that: Similarly, invoking the fixed-time ESO (8) and the uncertain system model, the dynamics of , , , e e e e can be represented as: It can be derived that the observer errors 1 2 3 4 , , , e e e e in (9) will converge in the fixed time.And there exists a fixed time T , which is independent of the initial observer error, such that when t T > , then

Conclusion
For the design of the controller of such nonlinear systems as PTGS, there are problems such as the disturbance cannot be accurately modeled and the convergence time is long, and the traditional PID controller is difficult to achieve better control performance.Considering the non-minimum phase characteristics of the controlled object, state variable reorganization is carried out.In this paper, based on the principle of ADRC, combined with fixed-time convergence of Lyapunov theory, the fixed-time ADRC is designed.The convergence time and the overshoot of the improved controller is less than the original controller design, which indicates that it is characterized by fast convergence, and at the same time, the controller has strong robustness, which improves the control performance of the controller.

Figure 2 .
Figure 2. Block diagram of the electro-hydraulic servo system transfer function.

b
denotes gain coefficient of the controller.The uncertainty term d includes unknown disturbance such as modeling error and friction.Make ( ) d f x =  a new state variable of the system.
Considering the practical application of the disturbed PTGS, it can be hypothesized that there exists a sufficiently small constant σ that d σ ≤ holds.Theorem: Considering the disturbed system ( )d f x = can be estimated precisely by the improved ESO in the fixed time T .
Based on the above designed fixed-time ESO, the improved ADRC is written as:

Table 1 .
Comparison of the tracking curves of different ADRCs.