Active–disturbance–rejection–control and fractional–order–proportional–integral–derivative hybrid control for hydroturbine speed governor system

Abstract

A novel approach named active–disturbance–rejection–control (ADRC) and fractional–order–proportional–integral–derivative (FOPID) hybrid control scheme is proposed for hydroturbine speed governor system, which is based on ADRC and FOPID control methods. By combining the advantages of ADRC and FOPID controllers, the proposed ADRC–FOPID hybrid control scheme can actively reject the unpredictable disturbance, even with random noises, and can be adapted to the nonlinearities as well as unknown dynamics of hydroturbine speed governor system. The control performances of ADRC–FOPID, ADRC, FOPID as well as conventional proportional–integral–derivative (PID) controllers have been compared. And ADRC–FOPID has been proved to be an effective control scheme for hydroturbine speed governor system.

Keywords

Active disturbance rejection control fractional-order control hybrid control proportional–integral–derivative control hydroturbine speed governor

Introduction

Hydropower is an important and vital renewable energy resource, which converts the energy in flowing water into electricity. Usually, a typical hydropower plant consists of water tunnel, penstock, surge tank, hydraulic turbine, speed governor, generator, and electrical network. However, a hydropower plant is a complex system in which hydrodynamics and mechano-electric dynamics are all involved in such a nonlinear dynamic system.¹ Moreover, hydroturbine speed governor systems are multi-parameter, high-dimension complex systems with nonlinear, time-varying, and non-minimum phase characteristics. Although in modern hydropower plant, the conventional proportional–integral–derivative (PID) controller is widely applied in hydroturbine speed governor system,^2,3 it still needs a much more effective and reliable control law for hydroelectric power plant.⁴ Many control methodologies have been reported for improving the performance of hydroturbine speed governor systems in the past decades. Chen et al.⁵ applied an intelligent integral control strategy by fuzzy logic. Zhang and Wang⁶ proposed a neuron model-free control method for hydroturbine speed governor. Djukanovic et al.⁷ presented neural network (NN) coordinated control for both exciter and governor for low head power plant. Gui et al.⁸ proposed a new nonlinear robust control strategy for a hydroturbine governor based on differential geometry theory and the nonlinear robust control principle. Jing et al.⁹ proposed a control strategy for the regulation of a hydroelectric turbine that is achieved using a discontinuous controller and incorporating a three-state valve and fuzzy logic for the reliability and the static and dynamic performance of the system. Mahmoud et al.¹⁰ designed a new fuzzy logic controller to control a hydropower plant that has several hydraulically coupled turbines. Wang et al.¹¹ presented a kind of hybridized control system based on genetic algorithms and fuzzy NNs to control hydroelectric generating units. Eker¹² presented a robust single-input multi-output design approach for speed control of hydro-turbines. Munoz-Hernandez and Jones¹³ applied generalized predictive control (GPC) to a multivariable model of the Dinorwig pumped-storage hydroelectric power station. Natarajan¹⁴ designed and analyzed a robust PID controller for a hydraulic turbine generator governor using a frequency-response technique. Husek¹⁵ presented a PID controller design procedure for a hydraulic turbine generator governor meeting sensitivity margin specifications.

However, in reality, the highly nonlinear characteristics of hydroturbine speed governor system make it difficult to obtain an accurate model. On the other hand, fuzzy logic and artificial NN is usually quite complex and take a long time to develop. In addition, these solutions are not portable, that is, the control algorithms cannot be easily adjusted and reused for a different problem.

Recently, more and more researchers have reported their achievements related to the theory and application of fractional-order controller in many fields of science and engineering,¹⁶ especially to enhance the performance of conventional PID controller using the concept of fractional calculus, where the orders of integral and derivative are non-integer.¹⁷ Fractional-order proportional–integral–derivative (FOPID or $P I^{λ} D^{μ}$ ) controller, where $λ$ and μ are the integrating and derivative orders proposed by Podlubny,¹⁸ is an extension of conventional PID controller. FOPID controller is characterized by five parameters, that is, proportional gain, K_p, integrating gain, K_i, derivative gain, K_d, integrating order, $λ$ , as well as derivative order, $μ$ . In the past years, FOPID controller has been applied in torsional system’s backlash vibration suppression control by Ma and Hori.¹⁹ Monje et al.²⁰ proposed a tip position control of a single-link lightweight flexible manipulator. Tang et al.²¹ and Zamani et al.¹⁷ reported the designing of FOPID controller for an automatic voltage regulator (AVR) system, respectively. Petras²² introduced fractional-order feedback control of an armature-controlled DC motor utilizing a constant field current. Luo et al.²³ used the describing function method and Bode plots analysis for the ultra low-speed position tracking performance with the designed fractional-order controller. Melicio et al.²⁴ proposed fractional-order controllers for the variable-speed operation of wind turbines with permanent-magnet synchronous generator (PMSG)/full-power converter topology. Debbarma et al.^25,26 used firefly algorithm for the simultaneous optimization of the gains and other parameters of I^λD^μ controller and governor speed regulation parameters for automatic generation control. Sondhi and Hote²⁷ applied FOPID controller for load-frequency-control (LFC). Because the extra real parameters λ and μ are involved, FOPID provides more flexibility and gives more possibility to realize the desired control performance.

As a nonlinear control technology, active disturbance rejection control (ADRC), first proposed by Han in 1995,^28,29 aims to design controller for nonlinear uncertain system, which learns from the advantage of traditional PID controller. ADRC has the unique characteristics of model independence and it can actively reject both internal and external disturbances.³⁰ The basic idea of ADRC is using an extended state observer (ESO) to track or estimate the plant dynamics and unknown disturbance in real time and dynamically compensate for it. Recently, ADRC has been applied in various engineering fields. Sun and Gao³¹ presented the design and implementation of an advanced digital controller for a 1-kW H-bridge DC–DC power converter based on the active disturbance rejection concept. Li et al.³² proposed to design an ADRC control algorithm for the antenna pointing control of a large flexible satellite system. Zheng et al.³³ presented to drive the axis of a micro-electro-mechanical systems (MEMS) gyroscope. Xia et al.³⁴ presented ADRC controller for multivariable systems with time delay. Wu and Chen³⁵ presented a new fast tool servo-control method for noncircular turning process (NCTP). Vencent et al.³⁶ described the application of a control technique known as ADRC to superconducting radio frequency (RF) cavities. Su et al.³⁷ developed a highly robust ADRC controller to implement high-precision motion control of permanent-magnet synchronous motors. Huang et al.³⁸ designed a first-order ADRC scheme for ALSTOM gasifier. Guo and Jin³⁹ adopted ADRC and the sliding mode control (SMC) approaches to deal with the stabilization of an Euler–Bernoulli beam system. Liang et al.⁴⁰ applied ADRC to the control of superheated steam temperature in large-capacity generation units as ADRC-PID cascade control. Dong et al.⁴¹ proposed a novel design, which is based on the concept of ADRC, of a robust decentralized LFC algorithm for an inter-connected three-area power system. Goforth et al.⁴² proposed ADRC for hysteretic systems with unknown characteristics. Madonski et al.⁴³ applied of a special case of an ADRC in governing a proper realization of basic limb rehabilitation trainings. Zhao and Gao⁴⁴ proposed a novel modification of ADRC for time-delay systems. Alonge et al.⁴⁵ proposed the ADRC control methodology for motion control systems with induction motor.

Usually, the nonlinearities and unknown dynamics of hydroturbine speed governor system as well as unpredictable external load disturbances always have an unexpected influence on the control performance, therefore, to achieve better control performance is still a challenging problem. Inspired by the literatures mentioned above, a novel approach named ADRC-FOPID hybrid control scheme is proposed for hydroturbine speed governor system, which is designed through a combined ADRC and FOPID control law by exploiting the unique disturbance estimation and compensation concept. By actively rejecting disturbance, ADRC-FOPID can effectively deal with nonlinear, time-varying dynamic, as well as other uncertainties of hydroturbine speed governor system. The computer simulation results show that ADRC-FOPID yields satisfactory system responses even under the conditions with random noises.

The rest of the paper is organized as follows. Section “FOPID controller” briefly describes FOPID controller. Section “ADRC” describes ADRC controller. Section “PID, FOPID, ADRC, and ADRC-FOPID controllers for hydroturbine speed governor system” presents conventional PID controller, FOPID controller, ADRC controller, and ADRC-FOPID hybrid control strategy for hydroturbine speed governor system, respectively. Section “Performance tests for hydroturbine speed governor system” gives the comparison results of computer simulations on an actual hydroturbine speed governor system for ADRC-FOPID, ADRC, FOPID, and conventional PID controllers. Section “Conclusion” finally summarizes the main contributions.

FOPID controller

In general, FOPID controller is an extension of conventional PID controller with fractional calculus. Laplace domain notion is applied to describe the fractional integrodifferential operation. Actually, FOPID controller is an infinite dimensional linear filter, and finite dimensional approximation of FOPID controller should be utilized in a proper range of frequency of practical interest.⁴⁶

Usually, the transfer function of FOPID controller can be simply described as

G_{F O P I D} (s) = K_{p} + \frac{K_{i}}{s^{λ}} + K_{d} s^{μ}

(1)

where s is Laplace operator, λ and μ are the non-integer order of integrator and differentiator, respectively, and can be any real numbers. Compared to conventional PID controller, the adjustable parameters of non-integer order of integrator and differentiator provide more flexibility and more possibility to realize the desired control performance. Especially, if $λ = μ = 1$ , the structure of FOPID controller turns to be conventional PID controller in parallel structure. Actually, the performance of the control system can be drastically affected by these two extra parameters. For example, with the increase in the values of λ or μ, the overshoot and settling time become smaller. However, too small or large values of λ or μ yield undesirable response, or even make the control system unstable. Hamamci presented a solution to the problem of stabilizing a given fractional dynamic system using fractional-order PI^λ and PI^λD^μ controllers, which is practically useful in the analysis and design of fractional-order control systems.^47,48 The parameters of FOPID controller must be well tuned for better control performance. In this paper, the FOPID controller has been optimized by minimizing the time domain performance criterion of integral of time multiplied absolute error (ITAE) of turbine speed relative deviation with particle swarm optimization (PSO) algorithm.¹⁷

ADRC

The ADRC consists of three main components: tracking differentiator (TD), ESO, and nonlinear controller (NC).^29–31 TD provides the desired transient and differential trajectory of set values, ESO estimates the system state variables and the extended state variables, that is, the disturbances of the system, and then NC generates the control law with the nonlinear form. ADRC only needs to know the orders rather than the explicit mathematical model of the controlled plant. The key point of ADRC is directly estimating the system dynamics and the total disturbances which are extended as a new system state in real time using an ESO and then compensating for them.

TD

Considering in a second-order plant, $v (t)$ is the input signal to be differentiated, then the TD is

{\begin{cases} {\dot{x}}_{1} = x_{2} \\ {\dot{x}}_{2} = - r s i g n (x_{1} - v (t) + \frac{x_{2} | x_{2} |}{2 r}) \end{cases}

(2)

where r is an adjustable and positive parameter that can be selected accordingly to speed up or slow down the transient profile, and $s i g n (\cdot)$ denotes the standard sign function. It provides the fastest tracking of $v (t)$ and its derivative subject to the acceleration limit of r.

The discrete form is

{\begin{cases} x_{1} (t + 1) = x_{1} (t) + h_{0} x_{2} (t) \\ x_{2} (t + 1) = x_{2} (t) + h_{0} f h a n (x_{1} (t + 1) - v (t), x_{2} (t), r, h_{0}) \end{cases}

(3)

where h₀ is the sampling time, in seconds. And the function $f h a n (x_{1}, x_{2}, r, h_{0})$ is

{\begin{cases} d = r h_{0} \\ d_{0} = h_{0} d \\ y_{0} = x_{1} + h_{0} x_{2} \\ a_{0} = \sqrt{d^{2} + 8 r | y_{0} |} \\ a = {\begin{array}{l} x_{2} + \frac{a_{0} - d}{2} s i g n (y_{0}), \begin{matrix} | y_{0} | > d_{0} \end{matrix} \\ x_{2} + \frac{y_{0}}{h_{0}}, \begin{matrix} | y_{0} | \leq d_{0} \end{matrix} \end{array} \\ f h a n = {\begin{array}{l} - r s i g n (a), \begin{matrix} | a | > d \end{matrix} \\ - r \frac{a}{d}, \begin{matrix} | a | \leq d \end{matrix} \end{array} \end{cases}

(4)

ESO

ESO is a nonlinear observer designed to estimate the total disturbance as an augmented state of the system. Considering a second-order control system as

{\begin{cases} {\dot{x}}_{1} = x_{2} \\ {\dot{x}}_{2} = f (x_{1}, x_{2}, ω) + b u \\ y_{1} = x_{1} \end{cases}

(5)

where x₁ is the system coordinate, x₂ is the first-order derivative of x₁, y₁ is the system output, b is a constant, u is the control output, w is the disturbance, and function f represents the generalized disturbance which includes the unknown system dynamics and the total disturbances.

The third-order ESO of the plant (5) is constructed as

{\begin{cases} e = z_{1} - y_{1} \\ {\dot{z}}_{1} = z_{2} - β_{1} g_{1} (e) \\ {\dot{z}}_{2} = z_{3} - β_{2} g_{2} (e) + b u \\ {\dot{z}}_{3} = - β_{3} g_{3} (e) \end{cases}

(6)

where e is the estimation error; $β_{1}$ , $β_{2}$ , and $β_{3}$ are the observer gains; g₁, g₂, and g₃ are the nonlinear functions; z₁ and z₂ are the observations of the tracking state variables, that is, the estimation of the output y₁ and ${\dot{y}}_{1}$ , respectively; and z₃ is the estimated variable of total internal and external disturbances.

The discrete form of ESO is

{\begin{cases} e (t) = z_{1} (t) - y_{1} (t), \\ f e (t) = f a l (e (t), α_{1}, δ), \\ f e_{1} (t) = f a l (e (t), α_{2}, δ) \\ z_{1} (t + 1) = z_{1} (t) + h_{0} (z_{2} (t) - β_{1} e (t)) \\ z_{2} (t + 1) = z_{2} (t) + h_{0} (z_{3} (t) - β_{2} f e (t) + b u (t)) \\ z_{3} (t + 1) = z_{3} (t) - h_{0} β_{3} f e_{1} (t) \end{cases}

(7)

The nonlinear function $f a l (e, α, δ)$ is

f a l (e, α, δ) = {\begin{cases} \frac{e}{δ^{1 - α}}, \begin{matrix} | e | \leq δ \end{matrix} \\ {| e |}^{α} s i g n (e), \begin{matrix} | e | > δ \end{matrix} \end{cases}

(8)

where both $α$ and $δ$ are positive parameters.

The structure of ADRC controller is shown in Figure 1, where $v_{1}$ and $v_{2}$ are the tracking signal and differential signal of $v$ , respectively; $u_{0}$ is the output of NC; and $b_{0}$ is a design parameter.

Figure 1.

The structure of a second-order ADRC controller.

The inputs of NC, that is, $e_{1}$ and $e_{2}$ , are combined nonlinearly to produce the control output. If NC is still applied with PID control law, then the final ADRC control law is

u = K_{p} (v_{1} - z_{1}) + \frac{K_{i}}{s} (v_{1} - z_{1}) + K_{d} (v_{2} - z_{2}) - \frac{z_{3}}{b_{0}}

(9)

The parameters of TD, ESO, and NC can be set to obtain the desired performance of the controller.

PID, FOPID, ADRC, and ADRC-FOPID controllers for hydroturbine speed governor system

PID and FOPID controllers for hydroturbine speed governor system

The block diagram of a typical hydroturbine speed governor system with conventional PID control law is shown in Figure 2, which consists of conventional PID controller, electro-hydraulic servo system, hydroturbine system, generator, and load.^1–4 As seen in Figure 2, c is turbine speed relative deviation set point, x is turbine speed relative deviation, y $y$ is wicket gate servomotor stroke relative deviation, q is flow rate relative deviation, h is water pressure relative deviation, m_t is turbine torque relative deviation, and m_g0 is load torque relative deviation, all in per unit. K_p, K_i, and K_d are the proportional, integral, and derivative gains of PID controller, respectively. T_n is derivative filter time constant which is usually set as $T_{n} = 0.1 * K_{d}$ . T_w is water starting time, T_a is generator unit mechanical time, all in seconds. e_x, e_qx, e_y, e_qy, e_h, e_qh are partial derivatives of hydroturbine, and e_g is load self-regulation factor.

Figure 2.

Typical block diagram of hydroturbine governor system with conventional PID control law.

FOPID controller for hydroturbine speed governor system is simply to replace conventional PID controller with the transfer function (1) in Figure 1.

ADRC and ADRC-FOPID controllers for hydroturbine speed governor system

Ignoring e_qx, the differential equations of hydroturbine speed governor system without PID control law is

{\begin{cases} \frac{d x}{d t} = \frac{e_{x} - e_{g}}{T_{a}} x + \frac{e_{y}}{T_{a}} y + \frac{e_{h}}{T_{a}} h - \frac{1}{T_{a}} m_{g 0} \\ \frac{d y}{d t} = \frac{1}{T_{y}} u - \frac{1}{T_{y}} y \\ \frac{d h}{d t} = \frac{e_{q x} (e_{g} - e_{x})}{e_{q h} T_{a}} x + (\frac{e_{q y}}{e_{q h} \cdot T_{y}} - \frac{e_{q x} e_{y}}{e_{q h} T_{a}}) y \\ - (\frac{e_{q x} e_{h}}{e_{q h} T_{a}} + \frac{1}{e_{q h} T_{w}}) h - \frac{e_{q y}}{e_{q h} T_{y}} u + \frac{e_{q x}}{e_{q h} T_{a}} m_{g 0} \end{cases}

(10)

Select three new state variables x₁, x₂, and x₃, where

{\begin{array}{l} x_{1} = x \\ x_{2} = \dot{x} = \frac{e_{x} - e_{g}}{T_{a}} x + \frac{e_{y}}{T_{a}} y + \frac{e_{h}}{T_{a}} h - \frac{1}{T_{a}} m_{g 0} \\ \begin{array}{l} x_{3} = \overset{\cdot\cdot}{x} = \frac{{(e_{x} - e_{g})}^{2}}{T_{a}^{2}} x + \frac{e_{y} e_{q h} (e_{x} - e_{g}) + e_{q y} T_{a}^{2} - e_{y} e_{q h} T_{a}}{e_{q h} T_{a}^{2} T_{y}} y \\ + \frac{e_{h} e_{q h} (e_{x} - e_{g}) T_{w} - T_{a}^{2}}{e_{q h} T_{w} T_{a}^{2}} h - \frac{e_{x} - e_{g}}{T_{a}^{2}} m_{g 0} + \frac{e_{y} e_{q h} - e_{q y} T_{a}}{e_{q h} T_{y} T_{a}} u \end{array} \end{array}

(11)

Let $c_{1} = e_{q h} T_{a} T_{y} T_{w}$ , $e_{n} = e_{g} - e_{x}$ , then

{\begin{cases} a_{1} = - e_{n} / c_{1} \\ a_{2} = - (e_{n} T_{y} + e_{q h} T_{w} + T_{a}) / c_{1} \\ a_{3} = - (e_{n} e_{q h} T_{y} T_{w} + T_{a} T_{y} + e_{q h} T_{a} T_{w}) / c_{1} \\ b = e_{y} / c_{1} \\ b_{1} = - (e_{q y} e_{h} - e_{q h} e_{y}) T_{w} / c_{1} \end{cases}

(12)

The differential equations of hydroturbine speed governor system can be transformed to

{\begin{cases} {\dot{x}}_{1} = x_{2} \\ {\dot{x}}_{2} = x_{3} \\ {\dot{x}}_{3} = a_{1} x_{1} + a_{2} x_{2} + a_{3} x_{3} + b_{1} \dot{u} + b u \\ y = x_{1} \end{cases}

(13)

Take $b_{1} \dot{u}$ as disturbance, and then, system (13) can be transformed to

{\begin{cases} {\dot{x}}_{1} = x_{2} \\ {\dot{x}}_{2} = x_{3} \\ {\dot{x}}_{3} = f (x_{1}, x_{2}, x_{3}, t) + b u \\ y = x_{1} \end{cases}

(14)

The discrete form of TD for hydroturbine speed governor system is as same as equation (3).

The ESO for hydroturbine speed governor system is

{\begin{cases} e = z_{1} - x_{1} \\ {\dot{z}}_{1} = z_{2} - β_{1} e \\ {\dot{z}}_{2} = z_{3} - β_{2} g (e) \\ {\dot{z}}_{3} = z_{4} - β_{3} g (e) + b u \\ {\dot{z}}_{4} = - β_{4} g (e) \end{cases}

(15)

An improved style of the nonlinear function $g (e)$ , that is, $f a l (e, α, δ)$ , is

f a \lg (e, α, δ) = {\begin{cases} - {(- e)}^{α}, e < - δ \\ \frac{α - 1}{2} δ^{α - 3} e^{3} + \frac{3 - α}{2} δ^{α - 1} e, \\ - δ \leq e \leq δ \\ e^{α}, e > δ \end{cases}

(16)

Select $α_{1} = 0.5, α_{2} = 0.25, α_{3} = 0.125, δ = 0.1$ , then, ESO is

{\begin{cases} e = z_{1} - x_{1} \\ {\dot{z}}_{1} = z_{2} - β_{1} e \\ {\dot{z}}_{2} = z_{3} - β_{2} f a \lg (e, 0.5, 0.1) \\ {\dot{z}}_{3} = z_{4} - β_{3} f a \lg (e, 0.25, 0.1) + b u \\ {\dot{z}}_{4} = - β_{4} f a \lg (e, 0.125, 0.1) \end{cases}

(17)

Finally, the ADRC control law which NC is conventional PID control law is

u = K_{p} (c_{1} - z_{1}) + \frac{K_{i}}{s} (c_{1} - z_{1}) + K_{d} (c_{2} - z_{2}) - \frac{z_{4}}{b_{0}}

(18)

The structure of ADRC controller which NC of ADRC is conventional PID control law for hydroturbine speed governor system is shown in Figure 3, where c₁ and c₂ are the turbine speed relative deviation set point, in per unit, as well as its derivative, respectively.

Figure 3.

Block diagram of ADRC with conventional PID control law for hydroturbine speed governor system.

In reality, the highly nonlinear characteristics of hydroturbine speed governor system make it difficult to obtain an accurate model. Usually, the nonlinearities and unknown dynamics of hydroturbine speed governor system as well as unpredictable external load disturbances always have an unexpected influence on the control performance.

Since FOPID and ADRC are well-known nonlinear control technologies, and FOPID controller is widely used in the areas of modeling and control, which provides more flexibility of nonlinear characteristics and gives more possibility to achieve control objectives, therefore, in this paper, FOPID controller is applied to construct the NC of ADRC to obtain the proposed ADRC-FOPID controller for hydroturbine speed governor system, finally. Actually, the ADRC-FOPID controller is simply to replace conventional PID control law by FOPID control law in NC, which combines the advantages of ADRC and FOPID controller. The structure of ADRC-FOPID is shown in Figure 4, and the final control law is

u = K_{p} (c_{1} - z_{1}) + \frac{K_{i}}{s^{λ}} (c_{1} - z_{1}) + K_{d} s^{μ - 1} (c_{2} - z_{2}) - \frac{z_{4}}{b_{0}}

(19)

Figure 4.

Block diagram of ADRC-FOPID for hydroturbine speed governor system.

Performance tests for hydroturbine speed governor system

An actual hydropower plant in China, named Chencun Hydropower Plant, is used as an example to verify the performances of the proposed ADRC-FOPID, ADRC, FOPID, and conventional PID controllers. The required data of this hydroturbine speed governor system are listed in Table 1. The parameters of TD and ESO of ADRC and ADRC-FOPID are shown in Table 2.

Table 1.

Parameters of hydroturbine speed governor system.

T_y	T_a	T_w	e_x	e_qx	e_y	e_qy	e_h	e_qh	e_g
0.3	5.72	0.83	−0.88	−0.03	1.40	1.23	0.35	0.13	0.0

Table 2.

Parameters for ADRC and ADRC-FOPID.

β ₁	β ₂	β ₃	β ₄	r	h ₀	b ₀
100	300	1000	1	10	0.01	1.14

ADRC: active disturbance rejection control; FOPID: fractional-order proportional–integral–derivative.

Actually, any effective controller can yield a good step response that will result in a good performance criterion in time domain, which include overshoot M_p (in per unit), settling time t_s (in s), and steady state error (in per unit).

First, a 10% step frequency disturbance (speed increasing) test was performed to compare the performances of ADRC and conventional PID controllers. In this test, $c = 10 %$ and $m_{g 0} = 0.0$ . The traces of comparison results without uniformly distributed random noise as well as with 3% uniformly distributed random noise are shown in Figure 5. The gains and comparison results without uniformly distributed random noise of ADRC and PID controllers are shown in Table 3. From Figure 5 and Table 3, it can be seen that without uniformly distributed random noise, PID controller can yield a pretty well response. The overshoot M_p is 1.2% and settling time t_s is 5.5 s; ADRC controller yields a slightly better response than PID controller, that is, M_p is 0.1% and t_s is 5.5 s, the same as PID controller. However, with 3% uniformly distributed uniform distribution random noise, the response of PID controller appears apparent oscillation. However, the response of ADRC controller is stable and there is no oscillation. Obviously, ADRC controller achieves better performance than PID controller in frequency disturbance test.

Figure 5.

Comparison results of ADRC and PID under 10% frequency disturbance: (a) without random noise and (b) with 3% random noise.

Table 3.

Gains and comparison results for PID and ADRC controllers.

Disturbance	Controller	K_p	K_i	K_d	M_p	t_s
Frequency	PID	4.11	0.47	2.21	1.2	5.5
Frequency	ADRC	2.89	0.38	1.01	0.1	5.5
Load	PID	5.61	1.95	3.09	3.3	4.5
Load	ADRC	5.33	1.88	2.83	3.3	4.5

PID: proportional–integral–derivative; ADRC: active disturbance rejection control.

Second, a 10% step load disturbance (load rejection) test was performed. In this test, $c = 0.0$ and $m_{g 0} = - 10 %$ . The traces of comparison results without uniformly distributed random noise as well as with 3% uniformly distributed random noise are shown in Figure 6. The gains and comparison results without uniformly distributed random noise of ADRC and PID controllers are still shown in Table 3. From Figure 6 and Table 3, it can be seen that without uniformly distributed random noise, the performance of ADRC controller is the same as that of PID controller, that is, M_p is 3.3% and t_s is 4.5 s. However, with 3% uniformly distributed random noise, the response of PID controller shows significant oscillation again and becomes unstable. However, for ADRC controller, no obvious oscillation appeared and the transient is stable with low level overshoot and fast response. Again, ADRC controller achieves better performance than PID controller in load disturbance test, obviously, because of the strong ability of active disturbance rejection.

Figure 6.

Comparison results of ADRC and PID under 10% load disturbance: (a) without noise and (b) with 3% random noise.

Finally, the comparison results of ADRC-FOPID, ADRC, and FOPID controllers for a 10% step load disturbance test are presented. The traces of comparison results without uniformly distributed random noise, with 3% uniformly distributed random noise as well as with 10% uniformly distributed random noise are shown in Figure 7. The gains and comparison results without uniformly distributed random noise of ADRC-FOPID, ADRC, and FOPID controllers are listed in Table 4. The gains of ADRC-FOPID controller are same as those of FOPID controller, the parameters of ESO and TD of ADRC-FOPID controller are same as those of ADRC controller, which are listed in Table 2, except that $b_{0} = 0.1$ . From Figure 7 and Table 4, it can be found that without uniform distribution random noise, ADRC-FOPID achieves the best performance, that is, M_p is 2.7% and t_s is 3.7 s. FOPID controller generates lower overshoot but slightly longer settling time than that of ADRC controller, that is, M_p is 2.9% and t_s is 4.6 s. With 3% uniformly distributed random noise, ADRC-FOPID controller still achieves the best performance. There are almost no apparent oscillations appeared for all of the three controllers, and the responses are stable. However, with 10% uniformly distributed random noise, the response of FOPID is unstable with significant oscillation. The responses of ADRC-FOPID and ADRC controllers also show oscillation but the performance of ADRC-FOPID controller is relatively superior to that of ADRC controller in terms of settling time and overshoot. Moreover, Figure 8 shows the estimation errors of ESO of ADRC-FOPID controller, the minor difference between system variables and the estimations show that ADRC-FOPID controller achieves significantly better errors estimation performance.

Figure 7.

Comparison results of ADRC-FOPID, FOPID, and ADRC under 10% load disturbance: (a) without random noise, (b) with 3% random noise, and (c) with 10% random noise.

Table 4.

Gains for FOPID, ADRC, and ADRC-FOPID controllers.

Controller	K_p	K_i	K_d	λ	μ	M_p	t_s
FOPID	6.12	1.98	3.92	1.01	1.15	2.9	4.6
ADRC	5.33	1.88	2.83	N/A	N/A	3.3	4.5
ADRC-FOPID	6.12	1.98	3.92	1.01	1.15	2.7	3.7

ADRC: active disturbance rejection control; FOPID: fractional-order proportional–integral–derivative.

Figure 8.

Estimation errors of ESO of ADRC-FOPID.

Conclusion

In this paper, a novel approach based on ADRC-FOPID hybrid control strategy has been successfully applied for hydroturbine speed governor system, which combines the advantages of ADRC and FOPID controllers. ADRC-FOPID has strong ability of active disturbance rejection and more flexibility of nonlinear characteristics of hydroturbine speed governor system, which resolves the contradiction between fast response and small overshoot. The control performance of ADRC-FOPID is compared to ADRC, FOPID, and conventional PID controllers. The computer simulation results show that ADRC controller outperforms PID controller even with random noise because of the strong ability of active disturbance rejection. Without random noise, the best performance is of ADRC-FOPID. FOPID controller generates similar dynamic response to ADRC with relatively lower overshoot but slightly longer settling time. Moreover, under the condition of random noise, the overshoot of FOPID is still smaller than that of ADRC. Furthermore, the performance of ADRC-FOPID controller is still relatively superior to that of FOPID and ADRC controller.

It can be concluded that the proposed ADRC-FOPID control scheme has good capabilities for rejecting disturbances and performs pretty well compared to ADRC, FOPID, and conventional PID controllers. ADRC-FOPID controller is more tolerant of nonlinear dynamics and uncertainties of hydroturbine speed governor system. It can actively estimate the influence of the external disturbance and noise and then compensate for it during each sampling period. Finally, ADRC-FOPID is an effective control scheme for hydroturbine speed governor system.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

This work was partially supported by the 111 Project under Grant No.B14022.

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