Start-up process optimization of pumped-storage unit with Tilt–Integral

Abstract

Focused on the hydraulic–mechanical coupled transient behaviors of the pumped-storage unit (PSU), this paper discusses the theoretical implementation and the parameter optimization of tilt–integral–derivative (TID) controller in turbine start-up processes. A novel learning backtracking search algorithm (LBSA) is adapted to optimize the control parameters, and a comprehensive index combining the relative deviation of the rotational speed and the relative deviation of the volute’s water pressure is designed as the objective function. A nonlinear mathematical model which can reflect hydraulic and mechanical dynamic characteristics of the pumped-storage governing system (PSGS) is established for the simulation of the transient period. Simulation experiments of two typical operating conditions are carried out under no-load condition. The simulation results show that the TID controller attaches significance to suppressing the amplitude of water hammer pressure and damping the hydraulic fluctuations. It will also restrain the trend of state trajectories oscillating back and forth in the unstable “S” shaped regions.

Keywords

Nonlinear control control systems design hydraulics optimal control evolutionary algorithms

Introduction

Nowadays, power grid structure is rapidly upgraded in the context of “carbon neutrality” and “carbon peak,” and an increasing number of clean and renewable energy is integrated into the power system, especially wind and photovoltaic energy (Sureshkumar and Ponnusamy, 2020); however, they are intermittent and unstable, which will lead to the volatility of the entire power system. Pumped-storage power station (PSPS) can effectively compensate the volatility and intermittency of new energy generation owing to its immense regulated storage capacity. Thus, PSPS inevitably shoulder the responsibility of regulations such as peak load modulation, frequency modulation, phase modulation, and energy storage. These tasks result in frequent switching of working states of pumped-storage unit (PSU). It is essential for PSU to improve the rapidity and stability in start-ups and shutdowns.

Start-up process of PSU generally refers to the starting procedure on turbine mode, where the movement strategy of guide vane determines the transient state of PSU in both mechanical and hydraulic aspects. In general, the strategy of guide vane control in start-up mainly consists of two stages: in the first stage, a direct guide vane opening is implemented until the rotational speed reaches to a given value; in the second stage, a closed-loop control is used to track the rated speed. If the guide vane opens inappropriately, such as its no-load guide vane opening sets too high, the rotational speed of the unit will increase at a fast rate and reach the rated speed early, causing a great overshoot of rotational speed and subsequent multiple oscillations. To guarantee the stability and security of the PSPS, scholars worldwide have paid countless effort in the research of start-up process optimization of PSU.

In most engineering projects, classical proportional–integral–derivative (PID)-type controllers are applied in the closed-loop control to govern the turbine units. Researchers have focused on the parameter tuning and optimization of PID controller and made considerable progress. Many evolutionary algorithms have been proposed to solve the parameter identification problem in turbine regulation system, such as genetic algorithm (GA) (Tapia et al., 2020), gravitational search algorithm (GSA) (Chen et al., 2014), particle swarm optimization (PSO) (Chen et al., 2019), and other improved algorithms (Lei et al., 2021; Rezghi et al., 2020). While the start-up scheme is optimized, the dynamic response indices related to the rotational speed such as the overshoot, the settling time, and the oscillation count are often considered in the objective functions. The impact of water hammer pressure on the rotational components and conduit system is constantly neglected, whereas improper value of hammer pressure will lead to severe shock and wear on the hydraulic and mechanical components, reducing the stability and life cycle of the system. It is necessary to take the suppression of water pressure fluctuation into consideration in the optimization objective.

Fractional-order PID (FOPID) has outstanding adjustment ability among numerous controllers (Erol, 2020; Estakhrouiyeh et al., 2018; Li et al., 2017b). Gomaa Haroun and Yin-Ya (2019) proposed a novel hybrid intelligent FOPID combined with a fuzzy FOPID controller for load frequency control (LFC). Zhang et al. (2019) designed a FOPID controller for a PSU; the proposed controller could improve the control quality of PSGS under multiple water heads. Xu et al. (2016) proposed an adaptively fast fuzzy FOPID (AFFFOPID) controller. The dynamic responses of start-up condition were obtained to make comparison among AFFFOPID controller and traditional PID and FOPID controllers. It has revealed that AFFFOPID performs better in overshoot and steady-state error aspects. Although the research results shown above prove that FOPID has better performance and reliability than conventional PID controller, FOPID controller has two extra parameters more than the PID controller; that is to say, while FOPID controller enables more degrees of freedom for parameter tuning, it also makes the synthesis of the controller more complex (Li et al., 2017b) and will slow down the optimization process.

Another fractional-order controller named tilt–integral–derivative (TID) is strongly linked to FOPID but possesses fewer tune parameters (Merrikh-Bayat, 2017). It gives adequate consideration to the flexibility of the optimization process while providing sufficient robustness. TID scheme was first provided in a patent (Lurie, 1994), and its physical structure is similar to that of general PID controller. The difference is that the proportional component of the PID controller is replaced by a tilted component with a transfer function $s^{(- 1 / ε)}$ , where ε can be any value except zero, but preferably is a real number from 2 to 3 (Lurie, 1994). Specifically, as compared to classical PID controller, the TID controller allows for simpler tuning, better disturbance rejection ratio, and less influence of plant parameter variations on closed-loop response. Khamari et al. (2020) employed TID controller for automatic generation control (AGC) under deregulated environment. Simulation results indicated the superiority of proposed approach with TID in the aspects of settling times, peak over shoots, and under shoots compared to other different conventional controllers. Morsali et al. (2017) utilized TID controller to regulate the gate-controlled series capacitor (GCSC) for LFC and tuned the coefficients of TID controller combining with the modified group search optimization (MGSO) method. Kumar Sahu et al. (2016) designed an improved TID with filter for LFC of multi-area power systems. It was faster for system with the proposed TID controller to suppress the oscillations in area frequency and tie-line power than that with other PID controllers. In Oshnoei et al. (2021), a neoteric controller based on the combination of FOPID and TID controllers was proposed as AGC loop controller. Then, the dynamic responses of the proposed controller were compared with various controlling schemes under different scenarios. Sain et al. (2016) used TID controller in magnetic levitation (Maglev) system and GA was utilized to optimize the coefficients of the controller. They also modified the structure of TID into integral-tilted-derivative (I-TD) and improved the performance in terms of maximum overshoot, gain margin, and phase margin. TID controller shows the superiority in LFC, AGC, and other engineering fields, but there are few studies of TID control on the transient process in the field of hydroelectric generation.

In this paper, TID controller is applied for the speed governing control scheme of a PSGS simulation platform referring to the parameters and structure of an actual PSPS in central China. An improved backtracking search algorithm (IBSA) is employed to optimize the parameters of the TID controller, while the combination of the error integral related to rotational speed and the error integral related to hydraulic pressure is chosen as the optimization objective. By further optimization, TID controller shows its superior performance in suppressing water pressure fluctuation and alleviating the frequency of oscillations phenomenon in “S” characteristic area.

The rest of the paper is organized as follows: The mathematical simulation model of PSGS with TID controller is constructed in section “Mathematical model of PSGS.” The optimization process of start-up law with an improved BSA is introduced in section “Optimization approach.” A case study is carried out and the unit’s performance of the optimized start-up process led by the TID controller is discussed and analyzed in section “Optimization results and analysis.” The conclusions are presented in section “Conclusion.”

Mathematical model of PSGS

PSU can realize the energy conversion by pumping water when power demand is low and release water to drive the turbine when power demand is high. In this process, the governor plays a crucial role. It regulates the frequency and output in turbine mode as well as controls the sequence in the pumping direction (Zheng et al., 2019). By changing the guide vane opening, the unit can manipulate the discharge flowing through the turbine runner to adjust output, thus to adapt to different working conditions. While PSU starts up, its operation condition varies in a wide range, where the dynamic characteristics cannot be accurately reflected through the linear model. Hence, the characteristic method, as the most common calculation method to solve the hydraulic transition process of conduit system, is studied and applied to establish the start-up model in this paper. Different from commonly used models built by algebraic differential equations, the nonlinear model of PSGS using the characteristic method can accurately describe the hydraulic and mechanical transient characteristics of the system under a wide range of operating conditions. A nonlinear model of PSGS is established as a simulation platform, whose structure is illustrated in Figure 1. It briefly includes the upstream and downstream reservoirs, surge tanks, pressure water pipelines, and pump turbine generating unit.

Figure 1.

Structural schematic diagram of a PSGS.

Model of the conduit system with characteristic method

The basic motion equation and continuous equation for unsteady flow in pressure pipeline are described as formulas (1) and (2) (Zheng et al., 2022). The characteristic method transforms the original partial differential equations into ordinary differential equations in the range of the characteristic line $dx / dt = \pm a$ ; thus, the simplified characteristic line equations and compatibility equations are obtained as formulas (3) and (4)

Motion equation : g \frac{\partial H}{\partial x} + V \frac{\partial V}{\partial x} + \frac{\partial V}{\partial t} + \frac{f}{2 D} V | V | = 0

(1)

Continuous equation : \frac{\partial H}{\partial t} + V (\frac{\partial H}{\partial x} + \sin α) + \frac{a^{2}}{g} \frac{\partial V}{\partial x} = 0

(2)

C^{+} : {\begin{matrix} \frac{dH}{dt} + \frac{a}{gA} \frac{dQ}{dt} + \frac{af}{2 gD A^{2}} Q | Q | = 0 \\ \frac{dx}{dt} = a \end{matrix}

(3)

C^{-} : {\begin{matrix} \frac{dH}{dt} - \frac{a}{gA} \frac{dQ}{dt} - \frac{af}{2 gD A^{2}} Q | Q | = 0 \\ \frac{dx}{dt} = - a \end{matrix}

(4)

where H is the piezometric head, V is the velocity of flow, H and V are both the functions of conduit length x and time t, respectively, D is the pipeline diameter, A is the sectional area of pipeline, f is the friction coefficient, Q is the water discharge, a is the velocity of water hammer wave, α is the angle between the pipe and the horizontal direction, which is usually very small, and Vsinα term is ignored in equations (3) and (4).

As depicted in Figure 1, PSGS’s conduit system consists of four equivalent pipes, including the water diversion tunnel, the penstock, the draft tube, and the tailrace tunnel. The characteristic method is used in four pipes. It is similar in the treatment of inner nodes in four pipelines. The difference is the treatment of their boundary nodes because of their different hydraulic structures at both ends.

The schematic diagram of numerical solution principle of the characteristic method is shown in Figure 2. In space, the conduit system is divided into N sections. Each node is represented by the serial number i, i = 1, 2, …, N + 1. The length of the section from the ith to (i + 1)th node is Δx and the corresponding sampling interval ΔT is described as Δx/a. The water head and discharge of ith node at each sampling moment are calculated according to the corresponding states of adjacent (i − 1)th and (i + 1)th nodes at the previous sampling moment. Their recursive equations of relationship are shown in the following formulas

{\begin{matrix} {H_{i, t}}_{+ T} = {C_{Pi, t}}_{+ T} - B Q_{i, t + T} \\ {H_{i, t}}_{+ T} = {C_{Mi, t}}_{+ T} + B Q_{i, t + T} \\ {C_{Pi, t}}_{+ T} = H_{i - 1, t} + B Q_{i - 1, t} - R | Q_{i - 1, t} | Q_{i - 1, t} \\ {C_{Mi, t}}_{+ T} = H_{i + 1, t} - B Q_{i + 1, t} + R | Q_{i + 1, t} | Q_{i + 1, t} \end{matrix}

(5)

where the parameters ${C_{Pi, t}}_{+ T} = H_{i - 1, t} + B Q_{i - 1, t} - R | Q_{i - 1, t} | Q_{i - 1, t}$ , ${C_{Mi, t}}_{+ T} = H_{i + 1, t} - B Q_{i + 1, t} + R | Q_{i + 1, t} | Q_{i + 1, t}$ , B and R are constants, $B = a / gA$ , $R = f Δ x / 2 gD A^{2}$ , and t refers to any sampling time.

Figure 2.

Numerical solution principle of characteristic method.

Boundary treatments in characteristic model

When dealing with boundaries between different components and pipelines, only one compatibility equation can be used, so we need to give another equation, that is, boundary condition. The boundaries are treated as border nodes in the characteristic model. In this paper, the boundary conditions of reservoirs, surge tanks, and pump turbine are considered.

Upstream and downstream reservoirs

The water levels of upstream and downstream reservoirs are considered to be constant since the change of water levels in a short time is very small and can be ignored. The junction of upstream reservoir and diversion tunnel is regarded as a border node, which is numbered as i = 1, as shown in Figure 3(a), and the water level and discharge of upstream reservoir boundary are expressed as follows

{\begin{matrix} H_{1, t + T} = H_{u} \\ Q_{1, t + T} = \frac{H_{1, t + T} - C_{M 1, t + T}}{B} \end{matrix}

(6)

where H_u refers to the water level of upstream reservoir, $C_{M 1, t + T} = H_{2, t} - B Q_{2, t} + R | Q_{2, t} | Q_{2, t}$ .

Figure 3.

(a) Structural schematic representation of the upstream reservoir boundary and (b) structural schematic representation of the downstream reservoir boundary.

Figure 3(b) illustrates the characteristic line model of downstream reservoir boundary. The junction between the tailrace tunnel and downstream reservoir is regarded as a border node, which is numbered as i = N + 1. The water level and discharge of downstream reservoir boundary are expressed as follows

{\begin{matrix} H_{N + 1, t + T} = H_{d} \\ Q_{N + 1, t + T} = \frac{C_{PN + 1, t + T} - H_{N + 1, t + T}}{B} \end{matrix}

(7)

where H_d refers to the water level of downstream reservoir, $C_{PN + 1, t + T} = H_{N, t} + B Q_{N, t} - R | Q_{N, t} | Q_{N, t}$ .

Surge tanks

The surge tank is the key component of hydraulic structure of hydropower station. Figure 4 illustrates the structural schematic representation of the surge tank. The left and right sides of the surge tank are connected with the end and the beginning of two adjacent pipe segments, respectively. The water level of the surge tank node is equal to the water levels of the left and right boundary nodes (node1 and node2), that is, H₁ = H₂ = H_s, respectively. The characteristic line equations in the C⁺ direction of the left pipeline and the C⁻ direction of the right pipeline are taken as boundary equations, expressed as follows

{\begin{matrix} H_{1, t + T} = {C_{P 1, t}}_{+ T} - B Q_{1, t + T} \\ H_{2, t + T} = {C_{M 2, t}}_{+ T} + B Q_{2, t + T} \end{matrix}

(8)

where $C_{P 1, t + T} = H_{3, t} + B Q_{3, t} - R | Q_{3, t} | Q_{3, t}$ and $C_{M 2, t + T} = H_{4, t} - B Q_{4, t} + R | Q_{4, t} | Q_{4, t}$ .

Figure 4.

Structural schematic representation of the surge tank.

Pump turbine

Pump turbine is a rotating machine that converts kinetic energy into mechanical energy, whose structure is shown in Figure 5. Ignoring the flow loss through the internal water guide mechanism of the turbine, the discharge through the turbine is equal to the discharge at the two adjacent points, that is, Q_T,t+T = Q_5,t+T = Q_6,t _+T. The characteristic line equations in the C⁺ direction of node 5 and the C⁻ direction of node 6 are expressed as follows

{\begin{matrix} H_{5, t + T} = {C_{P 5, t}}_{+ T} - B Q_{5, t + T} \\ H_{6, t + T} = {C_{M 6, t}}_{+ T} + B Q_{6, t + T} \end{matrix}

(9)

where $C_{P 5, t + T} = H_{7, t} + B Q_{7, t} - R | Q_{7, t} | Q_{7, t}$ , $C_{M 6, t + T} = C_{M 8, t} - B Q_{8, t} + R | Q_{8, t} | Q_{8, t}$ , besides, the working head of pump turbine $H_{T, t + T} = H_{5, t + T} - H_{6, t + T}$ .

Figure 5.

Structural schematic representation of the pump turbine.

Furthermore, apart from the characteristic model of the conduit system, the torque and discharge of the pump turbine are also calculated by means of the complete characteristic curves which reflect the static characteristic of the pump turbine obtained from the prototype. Li et al. (2017a) and Zhou et al. (2017) have given the detailed calculation processes of the torque and discharge, which are omitted here for space cause.

Model of the TID controller

The motivation for TID control is from the consideration of the so-called theoretically optimal loop response due to Bode. This compensator is herein referred to as a “Tilt” compensator, as it provides a feedback gain as a function of frequency which is tilted or shaped with respect to the gain/frequency of a conventional or positional compensation unit (Xue and Chen, 2002), as shown in Figure 6(a) and (b). The object of TID is to provide an improved feedback loop compensator having the advantages of the conventional PID compensator, but providing a response which is closer to the theoretically optimal response determined by Bode as illustrated in Figure 6(c).

Figure 6.

(a) Bode diagram of the transfer functions of the feedback control system of PID, (b) Bode diagram of the transfer functions of the feedback control system of TID, and (c) graphical illustration of the ideal Bode loop response for a feedback control compensator.

The structure of TID controller is similar to that of PID controller. The only difference is that the proportional component in PID controller is replaced by a tilted component having a transfer function characterized by l/s^(1/ε) in TID controller (Lurie, 1994), as shown in Figure 7(a) and (b). Generally, the transfer function of PID and TID controller can be formulated as

{\begin{matrix} G (s)_{PID} = K_{P} + \frac{K_{I}}{s} + K_{D} s \\ G (s)_{TID} = \frac{K_{t}}{s^{1 / ε}} + \frac{K_{I}}{s} + K_{D} s \end{matrix}

(10)

where K_P, K_I, K_D, and K_t represent the proportion, integral, differential coefficients of PID controller, and tilt gain of the fractional integral component, respectively. While ε is the tilt fractional component and it perfectly ranges from 2 to 3, this component provides stronger robustness for the TID controller because it simplifies the tuning process, improves the anti-interference ability of the system, and enhances the system’s robustness when facing to the uncertainty of system parameters (Ahmed et al., 2022).

Figure 7.

Structure of controller: (a) structure of PID controller and (b) structure of TID controller.

Rotor equations of the generator-motor

A simplified first-order differential equation model is adopted to describe the dynamic process of the rotor, as shown in equation (11)

T_{a} \frac{dn}{dt} + e_{n} (n - n_{r}) = m_{t} - m_{g}

(11)

where T_a is the inertial time constant of the generator-motor, e_n is the adjusting coefficient of the generator-motor, n denotes the relative speed, and n_r denotes the rated speed; moreover, m_t represents the relative turbine torque deviation and m_g represents the relative damping torque deviation.

Model of the servo system

The servo system is the actuator of the governor, whose main function is to adjust the main servomotor stroke according to the governor control signal u, thus to control the opening size of the opening angle of active wicket gate y. Briefly, the servo system mainly consists of the amplifying element, the main distribution valve, the main servomotor, and other nonlinear elements. Taking the existent of the dead zone, speed limit, and output saturation into consideration, the transfer function block diagram of the servo system is shown in Figure 8. Here, k is the amplification coefficient and T_y1 and T_y are the time constants of the main distribution valve and the main servomotor, respectively.

Figure 8.

Transfer function block diagram of servo system.

Optimization approach

Since backtracking search algorithm (BSA) was first proposed by Civicioglu (2013) in 2013, it has rapidly attracted scholars’ attention and research. Gradually, BSA shows a remarkable effect in compensating defects of some algorithms, such as oversensitive to the parameters, long time-consuming computation, and premature convergence (Civicioglu, 2013). The mutation strategy and crossover strategy of BSA are novel and efficient, resulting in good performance in global search ability and convergence speed. It is adoptable to various numerical optimization problems. The most unique thing is that the BSA owns a memory of historical populations, which means it can generate reliable solutions utilizing both previous and current generations, thus enhancing the population diversity (Zaman and Gharehchopogh, 2022).

Nevertheless, there are still a few problems with the BSA. First, when the historical information and the current information of BSA tend to be identical, the algorithm is prone to fall into the state of “premature.” In other words, it will weaken the capacity for avoiding the local optimality (Chen et al., 2017). Then, BSA lacks favorable guidance because of its randomness, making the convergence speed slow (Wang et al., 2015).

At present, scholars have carried out a large number of studies on BSA, and the improvement of BSA mainly focused on four aspects: improving the initialization strategy, optimizing the propagation operator, improving the design of algorithm parameters, and embedding local search mechanism. To make full use of the whole search space, Yuan et al. (2016) adopted the orthogonal design method to replace the original random initialization mechanism, as well as designed two chaotic mappings to improve the section-I and mutation operators of BSA, respectively. Zhao et al. (2014) proposed an improved BSA method which mixed three mutation operators to deal with constrained optimization problems. The algorithm adopted mutation operators of differential evolution algorithm and breeding GA in the latter part of iteration, which relatively improved the convergence speed and increased the diversity of the population. The current research on parameter design of BSA mainly focused on the adaptive variation control parameter F. Scholars ameliorated F mainly from two aspects: (a) controlled the fluctuation range of F; (b) designed F with strong adaptive ability (Su et al., 2016; Zhao et al., 2014). To improve the local mining capacity of BSA, the local search mechanism was proposed. Yuan et al. (2015) embedded the local search mechanism and combined the improved BSA with another algorithm to solve the problem of hydrothermal generator set.

In this paper, the teaching–learning-based optimization (TLBO) was used. A learning operator of TLBO was embedded in the proposed improved BSA. The learning BSA (LBSA) mutated with a random probability by combining the contemporary optimal information with historical information. Remaining individuals learn the knowledge of the contemporary optimal individual, the worst individual, and other random individuals. Then, it obtained renewable population. The proposed LBSA can promote BSA’s global performance by means of modifying the mutation process.

Objective function

In the start-up process of the units, we should not only pay attention to the stationarity and rapidity of the rise of rotational speed but also consider the safety and stability of units and conduit system. Hoping that the time from receiving the boot instruction to the speed reaching the rated speed is short and the transition should be as smooth as possible. The time integral of the relative deviation of the rotational speed is taken into account in the objective function. Hoping that the dynamic stress of conduit and runner blade should be as small as possible in the closed-loop regulation process. The time integral of the relative deviation of the hydraulic pressure at the inlet of volute is taken into account as another term in the objective function. Finally, a comprehensive index combining the relative deviation value of the rotational speed and the relative deviation value of the water pressure is proposed as the objective function, written as follows

J = ω_{1} * \int_{0}^{t_{u}} | \frac{n (t) - n_{r}}{n_{r}} | dt + ω_{2} * \int_{0}^{t_{u}} | \frac{H_{vol} (t) - H_{vol_average}}{H_{vol_average}} | dt

(12)

where t_u is the upper limit of integration time; n(t) and n_r are the actual value and the rated value of the rotational speed, respectively; H_vol(t) and H_{vol_average} are the actual value and the average value of the water hammer pressure at the inlet of volute, respectively; $H_{vol_average} = \int_{0}^{t_{u}} H_{vol} (t) / t_{u}$ ; and ω₁ and ω₂ are the weight coefficients, in this paper, both ω₁ and ω₂ are taken as 0.5, regarding the suppression of hydraulic fluctuation is as equally important as the steady and rapid speed rise.

Decision variables

In this paper, the control law of one-stage guide vane opening and closed-loop TID control is accepted, as shown in Figure 9. When the guide vane opens at a constant speed and reaches the maximum no-load opening limitation y_c, the opening of the guide vane stays stable until the rotational speed comes up to 90% of the rated value. Then, the governor will switch to the closed-loop control mode, that is, TID controller will be put in to track the rated speed. The parameters of the control law and servo system are determined in Table 1. Where the start time refers to the moment that the PSU begins to run in the simulation, the open-loop rise time represents the time interval from the moment guide vane begins to open to the moment it reaches y_c. The relative limitation denotes the maximum no-load opening limitation. In this study, the open-loop rise time and the relative limitation are set as 6 and 0.2 through the trial-and-error method, respectively.

Figure 9.

The control law of guide vane opening.

Table 1.

Simulation parameter settings in the start-up process.

Guide vane opening			Servomotor
Start time(seconds)	Open-looprise time (seconds)	Relativelimitation(pu)	Maximum speed (pu/s)	Minimum speed (pu/s)
10	6	0.2	8.296 × 10⁻³	−4.978 × 10⁻³

For TID controller, K_t, K_I, K_D, and ε are determined as the decision variables. Equation (13) shows the expression of decision variables

θ = (K_{t}, K_{I}, K_{D}, ε)

(13)

Constraint conditions

In practice, the varying ranges of optimization variables always have some limitations, and we need to treat them as constraints. Here, we mainly consider two types of constraints.

(1) Constraints on decision variables θ

θ \in [θ_{l}, θ_{u}]

(14)

where θ_l and θ_u are the lower and upper limits of θ, respectively.

(2) Constraints on settling time t_a

t_{a} \leq T_{u}

(15)

where t_a indicates the minimum time to achieve and stabilize within the range of rated speed deviation and T_u represents the upper limit of t_a.

Optimization steps of LBSA

In general, the LBSA majorly has five steps: population initialization, selection-I, improved mutation, crossover, and selection-II.

Population initialization

In the process of initialization, the LBSA randomly produces initial population P and historical generation population (oldP), as shown in equation (16)

P_{i, j} : U (lo w_{j}, u p_{j}), old P_{i, j} : U (lo w_{j}, u p_{j})

(16)

where $i \in [1, 2, 3, \dots, N]$ and N refers to the population number; $j \in [1, 2, 3, \dots, D]$ and D refers to the dimension of the population; low_j and up_j represent the lower and upper bounds of the jth dimension components, respectively; and U is a random uniform distribution function.

Selection-I

The selection-I is the iterative starting point of the algorithm, where oldP is first updated through the rule in equation (17). Then, equation (18) will be utilized to sort individual positions randomly

if a < b, then oldP : = P

(17)

oldP : = peimuting (oldP)

(18)

where $: =$ is the update operator and a and b are two uniformly distributed numbers generated between 0 and 1. Permuting function is used to randomly disrupt the order of individuals. It can be seen from equation (17) that a new oldP is selected from the previous n generations of historical population oldP and the previous generation population P, thus forming a unique function of memorizing the previous generation populations. Meanwhile, the probability of selecting oldP becomes smaller and smaller but the probability of selecting P remains unchanged at 0.5 as the number of iterations increases.

Improved mutation

In the basic BSA, the mutation is the process of perturbing population P and obtaining mutate population M, as shown in equation (19). Only the historical population oldP and the current population P participate in generating the new position. Some evolutionary algorithms have shown that we can speed up the convergence of the algorithm by tracking the best position of individual. So the learning guidance of the best individual is added in the LBSA’s mutation to improve the learning capacity of the algorithm, as shown in equation (20)

M = P + F * (oldP - P)

(19)

M = P + F * (0.5 * (oldP - P) + 0.5 * rand (\cdot) * (Teacher - P))

(20)

In equations (19) and (20), F is the mutation control parameter which controls the amplitude of the mutation search direction matrix (oldP−P) (Erol, 2020), Teacher represents the best population of the current generation, and randi(ã) is a uniformly distributed random integer function. In both the mutation process of BSA and LBSA, the contemporary population P learns from the previous population oldP based on mutation factor F to achieve the effect of mutation and the trail population takes advantage of its historical generations’ experiences. The great difference is that the LBSA brings the learning guidance in the updating process to increase the diversity of population, further to endow the algorithm with powerful global search capability.

Crossover

The process of crossover is relatively precise. It can be divided into two steps. In the first step, a mapping matrix map of N×D size with initial element values of 0 is generated, and two strategies are adopted to update the mapping matrix map with equal probability, as shown in equation (21). In the second step, the positions of intersecting individual elements in population P are determined according to the generated matrix map, and then, such individual elements in P are exchanged with the corresponding position elements in population M so as to obtain the final trail population T, as shown in equation (22). In short, the crossover is controlled by a matrix map of 0–1. When the element in map is 1, the corresponding element in M is assigned to population T. Otherwise, the corresponding element in P is assigned to the population T

{\begin{matrix} {map}_{i, u (1 : ceil (mixrate * rand * D))} = 1 | u \\ = permuting (1, 2, \dots, D) if c < d \\ {map}_{i, randi (D)} = 0 otherwise \end{matrix}

(21)

Ti, j = {\begin{matrix} M_{i, j} & if ma p_{i, j} = 1 \\ P_{i, j} & otherwise \end{matrix}

(22)

where c and d are two uniformly distributed numbers generated between 0 and 1, ceil(·) is a top integral function, mixrate is the cross probability parameter, and randi(·) is a uniformly distributed random integer function. The cross probability parameter controls the number of elements of individuals that will mutate in the trial by using mixrate (mixrate *rand*D). Two predefined strategies are randomly used to define BSA’s map. The first strategy uses mixrate (equation (21), line 1). The second strategy allows only one randomly chosen individual to mutate in each trial (equation (21), line 2).

Boundary control will be carried out at the end of cross over process on the individuals in the final population T. If there were boundary-crossing elements in the final population, these elements would be re-generated using equation (16).

Selection-II

In the selection-II, individual fitness plays a decisive role. By comparing the fitness value of the counterpart individuals in the population P and the trail population T, the individuals with better fitness are selected and the new population P is generated, as shown in equation (23)

P_{i} = {\begin{matrix} T_{i} & fitness (T_{i}) < fitness (P_{i}) \\ P & else \end{matrix}

(23)

The population P is renewed through the step of selection-II, and then, it returns to the step of selection-I to participate in the next iteration until the termination condition is satisfied, outputting the optimal solution.

Optimization results and analysis

Based on the above, the start-up process at two working conditions in the turbine mode has been simulated, one condition was at normal head H_t = 545 m and the other was at the low head H_t = 525 m. H_t = 545 m is close to PSU’s rated working head where the unit is expected to operate. Similarly, H_t = 525 m is the minimum head that the unit is allowed to work, at which the rotational speed of the unit will swing and lead to the rapid deterioration of the control quality. That is why these two working heads have been selected. They could completely represent two typical working conditions of the PSU.

Basic parameters

In section “Mathematical model of PSGS,” a simulation model was carried out to reflect the dynamic and hydraulic characteristics of the PSU. The actual parameters of a real PSPS in central China have been used to complete the model for simulation in this part. The detailed parameters in PSGS are listed in Table 2. The parameter settings of four equivalent pipes mentioned in section “Mathematical model of PSGS” are listed in Table 3, and the boundaries of decision variables mentioned in section “Optimization approach” are shown in Table 4. In Table 4, θ₁ refers to the decision variables of PID controller and θ₂ refers to the decision variables of TID controller. Here, the control parameters of PID were optimized to be taken as the control group.

Table 2.

Parameters in PSGS for simulation.

Parameters	Values
Rated head (m)	540
Rated discharge (m³/s)	62.09
Rated rotation speed (r/min)	500
Rated mechanical torque (N m)	5.845 × 10⁶
Maximum water head (m)	565
Minimum water head (m)	525
100% guide vane opening (°)	20.6
Rated power output (MW)	306
Power frequency (Hz)	50
Time constant of generator inertia (s)	10.503
Self-adaptive coefficient (pu)	0.05

PSGS: pumped-storage governing system.

Table 3.

Parameters of four equivalent pipes.

No.	Pipe name	From end	To end	Diameter (m)	Length (m)	Loss efficient
1	Diversion tunnel	Upstream reservoir	Upstream surge tank	6.20	444.23	0.015
2	Penstock	Upstream surge tank	Turbine	4.37	983.55	0.026
3	Draft tube	Turbine	Tailrace surge tank	4.30	170.4	0.010
4	Tailrace tunnel	Tailrace surge tank	Downstream reservoir	6.58	1065.2	0.015

Table 4.

Boundaries of decision variables.

Variables	Boundaries	Values
θ₁	L₁	(0, 0, 0)
	U₁	(1, 1, 1)
θ₂	L₂	(0, 0, 0, 2)
	U₂	(1, 1, 1, 3)

Start-up process in normal head condition

When operating under the normal condition, the PSGS shows its relevantly stability in all aspects. The control parameters of TID controller have been optimized based on the improved BSA and its governing capabilities have been analyzed. Here, we focused on the state variables related to the rotational speed and the water hammer pressure in the conduit system. Three state variables, that is, overshoot, settling time, and oscillation number of rotational speed and the values associated with the objective function in start-up process, were obtained. To be fair, the control parameters of PID were also optimized by the same method and they were taken as a control group. As shown in Table 5, the settling time is defined as the time span from the beginning of the simulation to the moment the speed reaches and stabilizes within ±1% speed error. For convenience, B is utilized to represent the value of $\int_{0}^{t_{u}} | (n (t) - n_{r}) / n_{r} | dt$ and H_errn is utilized to represent the value of $\int_{0}^{t_{u}} | (H_{vol} (t) - H_{vol_average}) / H_{vol_average} | dt$ , and J refers to the value of objective function. Subsequently, the curves of three PSGS state variables in the transient process are depicted in Figure 10.

Table 5.

The values of state variables and objective function at H_t = 545 m.

	Overshoot (pu)	Settling time (seconds)	Oscillation number of rotational speed	B	H_errn	J
TID	6.3%	19.76	1	763.80	28.74	396.27
PID	4.5%	20.08	2	759.17	38.49	398.83

TID: tilt–integral–derivative; PID: proportional–integral–derivative.

Figure 10.

The transient process of crucial state variables at normal head H_t = 545 m: (a) transient process of turbine rotational speed, (b) transient process of water hammer pressure at the inlet of volute, and (c) transient process of water hammer pressure at the outlet of draft tube.

As shown in Figure 10(a)–(c), the transient process curves of three state variables controlled by PID and TID completely overlap between 0 and 16 seconds, which confirms to the simulation situation that the start time and open-loop parameter settings of start-up process controlled by two controllers are consistent. As can be seen from the enlarged portion in Figure 10(a), compared to the rotational speed dominated by PID controller, the rotational speed dominated by TID controller increases at a faster speed since the beginning of closed-loop control process and takes a shorter time (19.76 seconds) to stabilize, but it has a larger overshoot (6.3%). As shown in Figure 10(b), the fluctuation of water hammer pressure at the inlet of the volute controlled by PID is obviously more severe than that controlled by TID, detailed as the following: (1) The maximum value of the water hammer pressure at the inlet of the volute in the closed-loop PID control process is almost 20 m larger than that in the closed-loop TID control process, the minimum value of the water hammer pressure at the inlet of the volute in the closed-loop PID control process is nearly 20 m smaller than that in the closed-loop TID control process. (2) The oscillation number (4 times) of water hammer pressure in the PID control process is larger than that (2 times) in the TID control process. (3) The time when the water hammer pressure tends to stabilize is longer in the PID control process. The information unfold in Figure 10(c) is similar to that shown in Figure 10(b).The advantages of TID control process can be clearly shown in the oscillation times and stabilize time of water hammer pressure.

It can be concluded in Figure 10(a)–(c) that the proposed TID control scheme achieves better damping property. The performance revealed in Figure 10(a)–(c) also corresponds to the values associated with the objective function in the table. In the cost of a little higher overshoot, the hydraulic oscillation times and the amplitude of water hammer pressure are suppressed as much as possible so as to alleviate the impact and abrasion of hydraulic fluctuation on the conduit system as well as to improve the smoothness and rapidness of the start-up transient process. To sum up, the transient process commanded by TID controller can further improve overall performances of the PSGS in this condition.

Start-up process in low head condition

A simulation experiment under the low working head H_t = 525 m was carried out to further analyze the superiority of the TID controller in the start-up process. The type of control performance indexes obtained in the simulation are the same as that in normal head condition, as shown in Table 6. Similarly, the transient curves of three state variables in low head condition are illustrated in Figure 11.

Table 6.

The values of state variables and objective function at H_t = 525 m.

	Overshoot (pu)	Settling time (seconds)	Oscillation number of rotational speed	B	H_errn	J
TID	4.5%	13.8	0	763.64	27.56	395.60
PID	3.5%	20.48	3	765.04	38.38	401.71

TID: tilt–integral–derivative; PID: proportional–integral–derivative.

Figure 11.

The transient process of crucial state variables at low head H_t = 525 m: (a) transient process of turbine rotational speed, (b) transient process of water hammer pressure at the inlet of volute, and (c) transient process of water hammer pressure at the outlet of draft tube.

On the whole, the situations are much more complicated under the low head condition. In Figure 11(a)–(c), when only the PID scheme curves are observed, the rotational speed and the hydraulic pressure at the inlet(outlet) of the volute(draft) tube take more time to stabilize and go through more oscillations compared with the corresponding curves in Figure 10(a)–(c), which is in line with the situation that the PSU will operate unsteadily under the working condition of low head. It also verifies the great influence of the working heads in the transient period. However, the TID controller can alleviate the instability of the unit operation and enhance part of the damping capacity to suppress the oscillation.

As seen from Figure 11(a), the unit speed controlled by TID steps into the stable region at t = 13.8 seconds, much fewer than the settling time (20.48 seconds) of speed controlled by PID, but the overshoot of TID scheme is 4.5%, a little bit higher than the overshoot (3.5%) of PID scheme. In Figure 11(b) and (c), it is easy to see that the oscillation period of the hydraulic pressure with PID controller at both the inlet of the volute and outlet of the draft tube lasts for a long time, at least 20 seconds longer than that with PID controller in normal head condition, but TID scheme stabilizes the oscillation quickly, achieving a relatively nice damping effect. The oscillation times of water hammer pressure are closely related to the oscillation of rotational speed. Take the hammer pressure curves shown in Figure 11(b) as the example. When the closed-loop process is controlled by TID, the oscillation number of pressure hammer is 2 times, much fewer than the oscillation number (more than 5 times) of pressure hammer controlled by PID, just like the oscillation number of rotational speed, PID scheme’s is 3 times, TID scheme’s is none. In conclusion, the improvement effect of TID controller is much more obvious under low head condition.

Impact of “S” characteristic area in the pump turbine

The pump turbine has an inherent feature called “S” character because of its reversible design. In the “S” characteristic area, the unit tends to switch back and forth between the conditions of turbine working, turbine braking, and the anti-pump working. A constant unit speed perhaps corresponds to three various flow values (Wang et al., 2017). The existence of “S” characteristic area will lead to the instability of operation such as severe speed fluctuation and frequent hammer pressure oscillation. It mainly occurs in the turbine mode. When under no-load operation, as the working head decreases, the rotational speed will swing at a smaller value and result in a wider swing range, increasing the possibility of entering the unstable zone. Hence, much attention is paid to the unit’s performance in “S” characteristic area in the start-up process.

Figure 12(a) and (b) shows the flow and torque transient process trajectory curves under no-load operation dominated by PID and TID controllers, and at the normal working head H_t = 545 m, it is not hard to see that the flow and torque of PID scheme fall into chaotic status in the “S” characteristic area more easily, and they oscillate back and forth more violently and rapidly. By comparison, the flow and torque curves controlled by TID suppress the oscillation to a large extent.

Figure 12.

State trajectories in start-up processes at normal working head H_t = 545 m: (a) Q₁₁–N₁₁ curve and (b) M₁₁–N₁₁ curve.

Figure 13(a) and (b) shows the flow and torque transient process trajectory curves dominated by PID and TID controllers at the low working head H_t = 525 m. Compared to the state trajectories under normal working condition, the intensity of the oscillation at low head increases significantly, which is consistent with the reality that the unit is more likely to descent into chaos and produce more violent and complicated vibration under the condition of low head. But in general, TID controller restrains the trend of state trajectory plunging into chaos and greatly weakens the frequency of oscillation back and forth, ensuring the orderliness and safety of the system start-up process.

Figure 13.

State trajectories in start-up processes at low working head H_t = 525 m: (a) Q₁₁–N₁₁ curve and (b) M₁₁–N₁₁ curve.

Conclusion

In this paper, TID controller is proposed in the pumped-storage governing system to improve the robustness and stability of the start-up process. As a hydro-electric-mechanical coupling comprehensive regulation system, PSGS has inherent nonlinearity and uncertainty. To verify the governing ability of TID controller, the nonlinear simulation model of PSGS based on an actual station in China is established, which can reflect both hydraulic and mechanical dynamic characteristics of the system. An improved algorithm LBSA is applied to be the intelligent optimizer, and the objective function combines both the absolute integral of the relative error of the rotational speed and the absolute integral of the relative error of the volute water hammer pressure. Subsequently, the start-up process with TID controller is optimized under no-load condition at two different working heads. The simulation results indicate that the scheme with TID controller can further improve overall performances of the PSGS. It plays a significant role in suppressing the hydraulic oscillation times and the amplitude of water hammer pressure in the whole system. The governing system with TID controller alleviates the impact and wear of hydraulic fluctuation on the conduit system as well as improves the transient behaviors of the start-up process. Furthermore, the proposed scheme with TID controller will restrain the trend of state trajectory plunging into chaos and greatly weaken the frequency of oscillations back and forth in “S” characteristic area. In future, a comprehensive comparison with other FOPID controllers will be incorporated as a scope of further research. We will further explore the detailed similarities and differences between TID controller and other FOPID controllers.

Footnotes

Appendix A

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

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work is supported by the National Natural Science Foundation of China (grant no. 52009096 and no. 52179088), the Fundamental Research Funds for the Central Universities (grant no. 2042022kf1022) and China Postdoctoral Science Foundation (grant no. 2022T150498).

ORCID iD

Qijuan Chen

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Start-up process optimization of pumped-storage unit with Tilt–Integral–Derivative controller

Abstract

Keywords

Introduction

Mathematical model of PSGS

Model of the conduit system with characteristic method

Boundary treatments in characteristic model

Upstream and downstream reservoirs

Surge tanks

Pump turbine

Model of the TID controller

Rotor equations of the generator-motor

Model of the servo system

Optimization approach

Objective function

Decision variables

Constraint conditions

Optimization steps of LBSA

Population initialization

Selection-I

Improved mutation

Crossover

Selection-II

Optimization results and analysis

Basic parameters

Start-up process in normal head condition

Start-up process in low head condition

Impact of “S” characteristic area in the pump turbine

Conclusion

Footnotes

Appendix A

Declaration of conflicting interests

Funding

ORCID iD

References