As shown in Fig. (1), c₀ and y_n represent the unit speed control instructions and the output signal of PID controller, respectively; x is the rotation speed feedback control signal; e is the error signal generated by unit speed input and output command signal; y is the guide vane opening signal. Hydraulic turbine governor is composed of electro-hydraulic servo system and the controller which adopts parallel PID regulation law.

The transfer function of PID controller is described as follows [1S. Wei, Simulation of hydraulic turbine regulation system., Huazhong University of Science and Technology Press: Wuhan, 2011., 24H. Fang, L. Chen, and Z. Shen, "Application of an improved PSO algorithm to optimal tuning of PID gains for water turbine governor", Energy Convers. Manage., vol. 52, pp. 1763-1770, 2011.
[http://dx.doi.org/10.1016/j.enconman.2010.11.005] ].

(1)

where K_P is the proportion adjustment coefficient, K_I is the integral adjustment coefficient (s ^{- 1}), K_D is the differential adjustment factor (s), s is the Laplace operator and T_n represents the differential time constant, usually, T_n=0.1*K_D.

As the actuator of water turbine, servomechanism can turn the controller’s output into hydraulic signal. And then this signal is enhanced gradually through the guiding device, auxiliary servomotor and main reaction servomotor for producing enough power to drive the turbine guide vane, so as to adjust the speed and power of hydro-turbine generator unit sets (HGU). The transfer function of the electro-hydraulic servo system is expressed as [1S. Wei, Simulation of hydraulic turbine regulation system., Huazhong University of Science and Technology Press: Wuhan, 2011., 24H. Fang, L. Chen, and Z. Shen, "Application of an improved PSO algorithm to optimal tuning of PID gains for water turbine governor", Energy Convers. Manage., vol. 52, pp. 1763-1770, 2011.
[http://dx.doi.org/10.1016/j.enconman.2010.11.005] ]:

(2)

where T_y is the relay device related time constant (s).

The water flow and the aqueduct wall of water diversion system are assumed to be rigid if the fluctuation is small, the transfer function is [1S. Wei, Simulation of hydraulic turbine regulation system., Huazhong University of Science and Technology Press: Wuhan, 2011., 24H. Fang, L. Chen, and Z. Shen, "Application of an improved PSO algorithm to optimal tuning of PID gains for water turbine governor", Energy Convers. Manage., vol. 52, pp. 1763-1770, 2011.
[http://dx.doi.org/10.1016/j.enconman.2010.11.005] ]:

(3)

where T_w is inertial time constant for water (s), q_t(s) and h_t(s) are the Laplace transform of water flow q_t(t) and effective water head h_t(t), respectively.

WTRS system is a complicated system with serious nonlinearity, it's necessary to simplify it and ignore some secondary factors when we build the mathematical model. And it seems impossible that there exists a simulated experiment which can completely simulate the actual working process. But only when the system suffers small fluctuation of dynamic process (i.e. the disturbance and the parameter variation are small in the system), the system can be thought as a linear one.

In this case, the mathematical model of hydraulic turbine can be expressed as [1S. Wei, Simulation of hydraulic turbine regulation system., Huazhong University of Science and Technology Press: Wuhan, 2011., 24H. Fang, L. Chen, and Z. Shen, "Application of an improved PSO algorithm to optimal tuning of PID gains for water turbine governor", Energy Convers. Manage., vol. 52, pp. 1763-1770, 2011.
[http://dx.doi.org/10.1016/j.enconman.2010.11.005] ]:

(4)

where e_y, e_h, e_x, e_qy, e_qh, and e_qx are the turbine transfer coefficients of controlled objects, m_t is the generated water torque.

In this part, the simplified first-order generator model is adopted and the transfer function is deduced as:

(5)

(6)

where m_g(s) is the Laplace transform of load disturbance torque m_g(t), T_a is the generator inertial time constant (s), e_g is the generator load adjustment coefficient, e_n is the controlled system self-regulation coefficient.

When the guide vane opening changes quickly, e.g. it decreases rapidly, due to the effect of inertia of water, flow velocity increases with the decreasing of guide vane opening, which will make the turbine torque increase. In the consideration of this phenomenon, e_qx, the transmission coefficient of discharge to speed, is introduced into the typical model to form the WTRS mathematical model with respect to water flow inertia effect which is shown in Fig. (2).

Fig. (2) Mathematical model of WTRS with respect to water flow inertia effect.

In PSO algorithm, the population is initialized to a group of random particles and each particle has its own speed and position which are represented as v_i^t = (v_i1^t, v_i2^t, …, v_id^t) and x_i^t = (x_i1^t, x_i2^t, …, x_id^t), respectively. Thereinto, i = 1, 2, ..., m, where m is the number of particles; d = 1, 2, ..., D, D is the dimensions of individual particle. The search process in a D-dimensional space for a swarm composed of m particles is formed. During this process, the quality of each particle is evaluated by the fitness function; the position of particle is affected primarily by two factors. One is p_best, the particle’s personal best position that is represented as p_i^t = (p_i1^t, p_i2^t, …, p_id^t), and the other is g_best, global historical best position, which can be represented as p_g^t = (p_g1^t, p_g2^t,…, p_gd^t). Particle’s velocity and position updating formulas are shown below.

(7)

(8)

(9)

where t denotes the current iteration; w^t is the inertia weight, normally, w_max = 0.9, w_min = 0.4; t_max is the number of maximum generation; c₁ and c₂ are two constants, which are the learning weights following individual optimal and global optimal, respectively; r₁^t and r₂^t are two random numbers drawn from the interval (0, 1).

In the standard PSO, the inertia weight decreases linearly during the process of search. However, by this linear decreasing inertia weight strategy, it is hard, if not impossible, to accurately reflect the optimization process for the following two deficiencies.

1. At the beginning of iterations, a larger value of weight coefficient contributes to strengthen global exploring ability of particles, but while the particles are very close to the global optimal, the larger speed may lead them to deviating from the global optimal and straying from the correct direction, thus reducing the search accuracy.
2. In the late iterations, all particles being gathered nearby the optimal value, it’s a smaller inertial factor that makes particles to do fine search around the optimal scope. But at this moment, the smaller speed caused by the smaller inertia weight may make solutions trend to be identity and trap in local optimal.

Therefore, it's clear that there exists the deficiency about the choice of the inertia weight based on the number of iterations in standard PSO. Therefore, in order to improve the performance, the inertia weight should be dynamically, nonlinearly changed to get better dynamics of balance between global and local search capabilities [21Y. Shi, and R.C. Eberhart, "Fuzzy adaptive particle swarm optimization", Congr. Evol. Comput, vol. 1, pp. 101-106, 2001.]. In this paper, the fuzzy thought is introduced into PSO algorithm to solve the problem of selecting inertia weight in different stages of searching process.

In FPSO algorithm, the inertia weight is influenced by two factors, one is the current linearly decreasing inertia weight, and the other is the normalized current best performance evaluation (NCBPE) [21Y. Shi, and R.C. Eberhart, "Fuzzy adaptive particle swarm optimization", Congr. Evol. Comput, vol. 1, pp. 101-106, 2001.]. In the minimization optimization problem, NCBPE can be calculated as Eq. (10).

(10)

where CBPE represents the current best performance evaluation of particles, CBPE_max is the non-optimal CBPE at the beginning of interations, CBPE_min is the estimated minimum at the end of interations.

Combining NCBPE and linearly decreasing inertia weight w^t to modify the weight coefficient w_c^t in FPSO. w_c^t can nonlinearly, dynamically change to keep a tradeoff between the particle global exploring ability and local search ability. That is to say, NCBPE and current linearly decreasing inertia weight w^t are selected as the inputs of the fuzzy control and the revised inertia weight w_c^t is the output. In the Table 2, the inputs and output are defined to have the same fuzzy sets: small (S), medium (M) and large (L). The membership functions and the critical parameters of three variables are list in the Table 1.

Table 1
The border of membership functions.

Several rules need to be considered in the process of rule’s setting, which are as followings and as Table 2:

1. During the initial stage of iterations, at the moment that the value of w^t is larger, if NCBPE is also large, particles are deemed to be far away from the optimal value, and then particles need to speed up and do a comprehensive exploration in the global scope; on the contrary, if NCBPE is medium at this moment, it means that the distance between the particles and optimal value is medium, w_c^t should be set to an appropriate value; and if NCBPE is small, it represents the particles are close to the optimal value and w_c^t should be given a smaller value to avoid the speed too fast at the same time.
2. During the middle stage of iterations, the value of w^t is medium, and if NCBPE is large or medium at this point, w_c^t should be taken to a medium value; contrarily, if NCBPE is small, it means the particles are very close to the optimal value and w_c^t should be set to a small value, which contributes to increase the possible of fine searching near the optimal solutions.
3. During the late stage of iterations, w^t is given a small value. At this moment, if NCBPE is large or medium, w_c^t should be taken to a large value to speed up for the sake of finding the optimal value; rather, if NCBPE value is small, w_c^t should be set to a medium value.

Table 2
The fuzzy control rules.

PSO can fast convergence in the early iterations, however convergence speed gradually slows down along with iteration process and the loss of diversity leads particles to tend to be identity; hence, they are easy to fall into local optimal at late iterations. In order to maintain the diversity of population and enhance the global search ability of particles, in this article, the crossover thought in genetic algorithm (GA) is introduced into FPSO, increasing the diversity of swarm to let it step out the local optimal through the genetic exchange between adjacent particles.

Combining the fuzzy thought and crossover thought, an improved fuzzy particle swarm optimization (IFPSO) algorithm is proposed, whose update formulas of particle's velocity and position are as below.

(11)

(12)

(13)

where,

where alfa and rand(n) are two random numbers within the interval (0, 1). P_c is the crossover probability, which will affect the convergence efficiency. A larger one will increase the possibility that particles generate new individuals, which renders convergence process more efficient. However, if the crossover probability is too large, the optimal value that has been found may be lost, generally P_c =0.8. v_max is the boundary value of speed. x_max and x_min are upper limit and lower limit, respectively.

By the means of above two strategies added in the standard PSO, the proposed IFPSO can be summarized as the following.

1. Randomly initialize the particles position and velocity and set the various parameters of population.
2. Calculate the fitness, and update p_best and g_best.
3. Evaluate the value of NCBPE according to Eq. (10), and set the fuzzy control parameters and fuzzy rules.
4. Update the particle velocity and position according to Eqs. (11) and (12).
5. Judge whether the velocity and position are beyond the boundary. The velocity and position are limited within the range, if they are against the boundary.
6. Calculate particle fitness again, if the current particle fitness is better than p_best, then replace p_best with current fitness; If the current global optimal value is superior to global optimal, then replace g_best with the current global optimal.
7. If iterations have reached to the maximum, stop searching and output the global optimal, otherwise, return to step 3.

Fitness is the particle quality evaluation standard, with which we can seek to find the optimized PID values. The optimized controller should make the system obtaining good performance, such as faster response speed, smaller overshoot and lower settling time, in the process of dynamic response and make the steady-state error converging to zero finally. The sum of adjusting time with another number which is the integral of sum of system error absolute value and the squared governor output is taken as the fitness function of the proposed algorithm. Moreover, once the system appears overshoot, the overshoot part will be taken as a part of fitness function.

(14)

(15)

where y_n (t) is the output of PID controller; t_r is regulating time; p₁, p₂, p₃ and p₄ are the weight coefficients, and p₄ is much larger than p₁; T_s represents the simulation time.

In this paper, the two WTRS systems, as illustrated in Figs. (1 and 2), are tested in simulation experiments, which are set as case 1 and case 2, respectively. And the required data of WTRS systems are listed in Table 3. The parameters setting of the proposed algorithm is shown in Table 4. In Table 5, the values of CBPE_max and CBPE_min for two cases are listed.

Table 3
Parameters of two different types of WTRS systems.

Table 4
Parameters setting for IFPSO.

Table 5
Values of CBPE_max and CBPE_min for two cases.

In order to verify the performance of the proposed algorithm when it is applied to optimize the turbine governor parameters, the simulation experiments for hydro-generator units under 5% and 10% frequency disturbance are carried out. The step disturbances of given speed with corresponding amplitudes are employed to excite the systems. The average best fitness of IFPSO and PSO are compared in Figs. (3, 5, 7, 9), both of which are repeated for 50 trials for overcoming the randomness of the heuristic algorithms. In addition, the traces of turbine speed deviation obtained by IFPSO, standard PSO, ZN algorithm and fuzzy PID algorithm are compared in Figs. (4, 6, 8, 10), respectively. Some results considering best fitness (Best F), the standard deviation of Best F (STDEV), overshoot (Overshoot) and settling time (t_s) are listed in Tables 6 and 7.

Fig. (3) Average best fitness under 5% frequency disturbance (case 1).

Fig. (4) Speed deviation under 5% frequency disturbance (case 1).

Fig. (5) Average best fitness under 10% frequency disturbance (case 1).

Fig. (6) Speed deviation under 10% frequency disturbance (case 1).

Fig. (7) Average best fitness under 5% frequency disturbance (case 2).

Fig. (8) Speed deviation under 5% frequency disturbance (case 2).

Fig. (9) Average best fitness under 10% frequency disturbance (case 2).

Fig. (10) Speed deviation under 10% frequency disturbance (case 2).

Table 6
Simulation results of case 1 under frequency disturbance.

From the profile of average best fitness, it is obvious that,compared with standard PSO, IFPSO acquires better convergence results based on the proposed fitness function with faster convergence speed. The final average best fitness obtained by IFPSO is smaller than that obtained by PSO, which means that the particles are easier to jump out from local optimal to find the better global optimal within the global range at the late iterations of the search process by using IFPSO.

Table 6 shows that, for the typical WTRS system model (case 1), whether under 5% or 10% frequency disturbance conditions, the system dynamic performance obtained by IFPSO algorithm outperforms than those by other three algorithms, which presents a shorter settling time and smaller overshoot, although, sometimes the system has a slower response speed when under IFPSO based tuning. Table 7 shows the compared results of the aforementioned four algorithms for the WTRS with respect to water flow inertia effect (case 2). It is found that IFPSO-based strategy obtains the best performance in terms of settling time and overshoot.

Table 7
Simulation results of case 2 under frequency disturbance.

From the above it is observed that, when compared with PSO algorithm, ZN algorithm and fuzzy PID algorithm, the proposed IFPSO can obtain better overshoot and settling time, thereby justifying the rationality and superiority of the proposed algorithm for optimization tuning of WTRS systems under frequency disturbance conditions.

In order to assess the validity and robustness of the proposed algorithm, load disturbance condition is also considered. Different step disturbances of load are adopted to excite systems. The experiments for systems under 5% and 10% load disturbance conditions are performed within 20 s and repeated 50 times.

From the compared results for fitness function convergence situations of IFPSO and PSO shown in Figs. (11, 13, 15, 17), we can find that, the final average best fitness value obtained by IFPSO is smaller than that by PSO. It illustrates that IFPSO possesses superior ability in obtaining higher-quality optimal value with faster convergence speed compared with PSO algorithm in the searching process and it is capable of stepping out the local optimal.

Fig. (11) Average best fitness under 5% load disturbance (case 1).

Fig. (12) Speed deviation under 5% load disturbance (case 1).

Fig. (13) Average best fitness under 10% load disturbance (case 1).

Fig. (14) Speed deviation under 10% load disturbance (case 1).

Fig. (15) Average best fitness under 5% load disturbance (case 2).

Fig. (16) Speed deviation under 5% load disturbance (case 2).

Fig. (17) Average best fitness under 10% load disturbance (case 2).

Figs. (12, 14, 16, 18) show the speed deviation transient process of the two WTRS systems by using the optimal solutions of IFPSO algorithm and other three optimization approaches. Obviously, from these figures, it can be found that with the regulation of IFPSO based controller, the fluctuation caused by the load disturbance is fast weakened and gradually disappear, and then achieve the steady state as desired soon.

Fig. (18) Speed deviation under 10% load disturbance (case2).

Some experimental simulation results under different load conditions are listed in Tables 8 and 9. It is seen that compared with the other three approaches, the proposed IFPSO gets a great improvement in terms of settling time with smaller overshoot.

Table 8
Simulation results of case 1 under load disturbance.

Table 9
Simulation results of case 2 under load disturbance.

Besides, under all of the running conditions, the overshoot level obtained by IFPSO based controller is lower than those obtained by other three algorithms based controllers. Obviously, the lower overshoot results in the better performance of WTRS, therefore the IFPSO based controller is confirmed to be more suitable for the tuning of WTRS system.