International Journal of Automation and Computing
8(2), May 2011, 185-192 DOI: 10.1007/s11633-011-0572-6
Coordinated Controller Tuning of a Boiler Turbine Unit with New Binary Particle Swarm Optimization Algorithm Muhammad Ilyas Menhas1 1
Ling Wang1
Min-Rui Fei1
Cheng-Xi Ma2
Shanghai Key Laboratory of Power-Station Automation Technology, Shanghai University, Shanghai 200072, PRC 2
East China Electric Power Design Institute, Shanghai 200063, PRC
Abstract: Coordinated controller tuning of the boiler turbine unit is a challenging task due to the nonlinear and coupling characteristics of the system. In this paper, a new variant of binary particle swarm optimization (PSO) algorithm, called probability based binary PSO (PBPSO), is presented to tune the parameters of a coordinated controller. The simulation results show that PBPSO can effectively optimize the control parameters and achieves better control performance than those based on standard discrete binary PSO, modified binary PSO, and standard continuous PSO. Keywords: Coordinated control, boiler turbine unit, particle swarm optimization (PSO), probability based binary particle swarm optimization (PBPSO), controller tuning.
1
Introduction
Due to the rapid increase in the use of power for both domestic and industrial needs, it is a challenge to meet power demand with the highest reliability and efficiency. At present, the power system industry is largely relying on hydro and thermal stations. The thermal power plants are among quick and comparatively cheap peak load supporting power plants. A boiler turbine unit is a common arrangement in thermal power plants to generate electricity, where a boiler is used to produce steam that drives turbo generators to generate electricity. The control system for the boiler-turbine unit needs to meet the requirements such as those listed as follows: 1) Output power must be able to follow the demand. 2) Throttle pressure must withstand load variations. 3) Water level in the drum and steam temperature must be maintained at desired levels to prevent overheating of drums or flooding and to avoid wet steam from entering turbines or overheating of superheaters due to the excess temperature. The fuel composition in the combustion chamber must meet standards for safety and environment protection[1, 2] . The control of boiler turbine unit is a challenging task due to coupling characteristics of the system and the process time delays. Although feed-water control, temperature control, and air control systems are not strongly coupled and can be controlled separately. The throttle pressure and electric power are strongly coupled. Therefore, a fully decentralized control cannot meet the stringent control requirements, and it is necessary to incorporate a coordinated controller in thermal power plants. Hence, the boiler Manuscript received September 12, 2010; revised December 23, 2010 This work was supported by Projects of Shanghai Science and Technology Community (No. 10ZR1411800, No. 08160705900, No. 08160512100), Shanghai University “the 11th Five-Year Plan”, 211 Construction Project, and Mechatronics Engineering Innovation Group Project from Shanghai Education Commission.
turbine unit is often modeled as a two-input-two-output (TITO) system. The two inputs are boiler firing rate and governor valve position, and the two outputs are electrical power and throttle pressure. The corresponding control system is called a coordinated control system (CCS). A coordinated control system is a complex multiinput multi-output (MIMO) system. Generally, multiple proportional-integral-derivative (PID) controllers in different action modes are used to design the CCS. The PID control was first introduced in the market in 1939 and has remained the most widely used control strategy in process control. The majority of literature reveals that more than 90% of the controllers used in process industries are PID controllers and advanced versions of the PID controller. The three-term PID controllers provide efficient solutions to the control problems[3,4] . The wide application of PID control has enforced and sustained research to “get best out of PID” and “the search is on to find the next key technology or methodology for PID tuning”[5,6] . It is necessary to tune the controller to find optimal PID parameters after a process system is upgraded or a new system is installed. The most famous classical auto tuning methods include Ziegler-Nichols rules, Cohen Coon method, and relay feedback methods[7,8] . An MIMO system comprising nonlinear and coupled terms is more complex than an single-input single-output (SISO) system. The PID parameters tuning for such systems often involves trial and error methods. The trial and error methods are not only tedious but also rare to reach the optimal parameters. In order to overcome such difficulties in tuning controllers, recently, the trial and error methods have been replaced with many modern computerautomated intelligent methods, such as artificial-neuralnetworks, fuzzy-logic, and evolutionary computation[9,10] . The modern intelligent methods can search optimal parameters iteratively for a given objective function. The swarm intelligence-based and nature-inspired iterative search algorithms, like particle swarm optimization
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(PSO), ant colony optimization (ACO), and differential evolution (DE), and genetic algorithm (GA) have significantly eased the task. However, some methods are easy to implement, while others are difficult. Moreover, the performance of each method is not ideal for all applications. Bhatt et al.[11] used three different optimization algorithms to identify optimal gains for controller devices in two-area multi-unit automatic generation control problem and came up with craziness-PSO showing better performance over GA and hybrid PSO algorithms. Zhu et al.[12] verified that chaotic ant swarm (CAS) can outperform GA in optimal PID controller design for an automatic voltage regulator (AVR) system. PSO algorithm is an intelligent computational method inspired by swarm intelligence and has been widely used in real-world optimization problems[13] . The PSO algorithm comprises very simple mathematics and is computationally cheaper, but it performs well. It has rapidly progressed over recent years and has many versions[13,14] . However, the standard PSO based on real numbers suffers from trapping into local optimum and premature convergence. The continuous PSO algorithm cannot solve certain optimization problems. In 1997, Kennedy and Eberkart[15] further extended the PSO to discrete domain to tackle the combinatorial optimization problems. The discrete version of PSO called discrete binary PSO (DBPSO) uses binary numbers. The binary PSO can handle binary, discrete, and certain continuous problems effectively as the binary PSO can escape from trapping into local optimum. The DBPSO algorithm is more flexible and has extensive applications. Shayeghi et al.[16] used an improved discrete PSO (DPSO) in static transmission network expansion planning (STNEP) problem. Unler and Murat[17] used DBPSO for binary classification problem. This paper is mainly focused on description of new probability based binary PSO (PBPSO) algorithm and its comparative performance analysis in identifying controller gains of coordinated controller of boiler turbine unit. The paper is organized as follows. Section 2 describes the proposed PBPSO algorithm and some variants of the existing binary PSO0 s. Section 3 presents the implementation of PBPSO for CCS. Section 4 presents comparison results. Finally, conclusions are drawn in Section 5.
2 2.1
PBPSO algorithm PSO algorithm
PSO algorithm is a population-based heuristic search algorithm and belongs to the class of swarm intelligence based algorithms. The PSO was introduced by Kennedy and Eberhart[18] by simulating social behavior of birds in 1995. It consists of a group of m particles. Each member of the group is represented by position xij = [xi1 , · · · , xid ] and velocity vij = [vi1 , · · · , vid ] in a d-dimensional search space, where x and v are vectors. The initial positions and velocities are random from a normal population u ∈ [0, 1]. All particles move in the search space to optimize an objective function f (x). Each member of the group gets a score according to the level of accuracy in reaching the target. The
score is called a fitness value. The member with the highest score is called global best. Each particle memorizes its previous best position. During the search process, if any particle gets a higher fitness value, it updates its memory and discards the previous. The process is repeated until a predefined termination criterion is reached. After any iteration, all particles update their positions and velocities to achieve better fitness value according to the following equations: k+1 k V(i,j) = w × V(i,j) + c1 × r1 (pk(i,j) − xk(i,j) )+ k c2 × r2 (g(1,j) − xk(i,j) )
(1)
k k+1 xk+1 (i,j) = x(i,j) + V(i,j)
(2)
where c1 and c2 are two acceleration coefficients, w is inertia weight, pk(i,j) is the previous best position in the history k of each particle at iteration k, g(i,j) is the best position associated with the best particle in the group at iteration k, and r1 , r2 ∈ [0, 1] are two random numbers.
2.2
Discrete binary (DBPSO) algorithm
In 1997, Kennedy and Eberhart introduced a discrete version of the PSO. Their aim was to tackle combinatorial optimization problems with this version of the PSO algorithm. In DBPSO, particle positions are represented by binary bits of appropriate length. In DBPSO, velocity update equation (1) is used to update the pseudo-probability of any bit of particle position, while the actual probability S ∈ [0, 1] is determined by a sigmoid function as S=
1 . (1 + exp(−V ))
(3)
Now, a comparison between S and a random number r ∈ [0, 1] will determine a bit as ( 1, if r 6 s x= (4) 0, else. Kennedy and Eberhart successfully optimized some functions with the DBPSO algorithm. However, the optimization performance of DBPSO was not satisfactory.
2.3
Modified discrete (MBPSO) algorithm
binary
PSO
In 2004, Shen et al.[19] proposed a modified discrete binary PSO (MBPSO). In MBPSO, the updating formulas are given by the following equations: xij (old), if 0 6 vij 6 a 1 pij , if a < vij 6 (1 + a) (5) xij (new) = 2 1 g , (1 + a) < v 6 1. if ij ij 2 MBPSO replaces all original equations of basic PSO and gives entirely new updating formulas. In MBPSO, there are no acceleration coefficients or inertia weight. The initial position bits are determined randomly. The velocity is constrained in the interval u ∈ [0, 1]. The update is performed according to (5), where a is a predefined static probability fixed as a constant value in the range [0, 1], usually
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0.5. Here, a comparison between v(i, j) and a randomly produced value r ∈ [0, 1] determines the bit of the particle position according to (5).
2.4
Probability based discrete binary PSO algorithm
The PBPSO algorithm is reworking on DBPSO algorithm. It is a hybrid approach that integrates the concept of probability optimization and particle swarm optimization. The PBPSO algorithm differs from DBPSO as follows: 1) Both the velocity and position update rules in (1) and (2) of the continuous PSO algorithm are preserved. 2) The position update rule in (2) of continuous PSO algorithm is considered to update the pseudo-probability of the actual position bit. 3) The estimate of actual probability of a particle position bit x(i,j) to be 0 or 1 is determined using a new linear estimator P , as shown below:
P =
x − Xmin Xmax − Xmin
(6)
where x is the probability, and it is constrained in the interval [Xmin , Xmax ]; similarly, the velocity variable V is constrained in the interval [Vmin , Vmax ] as in DBPSO. P is the actual probability and is confined in the interval [0, 1]. Xmax and Xmin are two constants and Xmax = −Xmin and Vmax = −Vmin . P ∈ R, x ∈ R, and V ∈ R. The initial binary bits to represent each dimension of particle position vector are not random. Instead, in PBPSO, these bits are determined by performing a comparison between P of relation (6) and a random number r ∈ [0, 1] as ( 1, if rand 6 P xb = (7) 0, else. 4) The initial velocities are set to zero. 5) The initial pseudo probabilities are also set to zero. Further, a mutation operation is introduced to maintain the population diversity. The mutation operation will change a bit with a very small probability. A random number r ∈ [0, 1] is generated, and if r 6 m, then a bit state of x(i,j) can be changed from 1 to 0 or 0 to 1. Here, m is a predefined mutation probability. In PBPSO, the non-linear sigmoid function of DBPSO algorithm as in (3) has been replaced by a new linear estimator P , as in (6). Because of velocity constriction, the algorithm inherently possesses some mutation probability. The modifications have significantly improved performance of the algorithm. The whole procedure of the PBPSO algorithm is explained as follows: Step 1. Initialization This step comprises setting of population size, PSO parameters, like acceleration coefficients, inertia weight, number of binary bits to represent the position vector of each
particle, initial velocities of particles, initial positions of particles, bounds on velocity and the pseudo-probability, and search boundaries. For example, let us consider a swarm with m particles in a d-dimensional search space, where each direction is represented by a binary bit either 0 or 1. The initial actual position vector, initial velocity, and initial pseudo-probability associated with i-th particle can be given as Xb(i,d) = [xb(i,1) , xb(i,2) , · · · , xb(i,d) ]
(8)
V(i,d) = [v(i,1) , v(i,2) , · · · , v(i,d) ] = [0, 0, · · · , 0]
(9)
x(i,d) = [x(i,1) , x(i,2) , · · · , x(i,d) ] = [0, 0, · · · , 0]
(10)
where i = 1, · · · , m is the index of the particle, and j = 1, · · · , d is the index of binary bit sequence or dimension. Step 2. Generating initial candidate solutions In order to generate the initial Xb bits of the actual particle position vector, a random number r ∈ [0, 1] is generated and compared with P as ( 1, if rand 6 P(i,j) xb(i,j) = (11) 0, else. Because the initial value of x is 0, according to (3), P = 0.5, the initial Xb bits have an equal probability to be either 1 or 0. Step 3. Initial fitness computing The initial fitness values of the swarm comprising m particles are computed on objective function F (x). Step 4. Setting initial personal best (Pbest ) and global best (gbest ) The initial personal best is the current position of the particle and the corresponding fitness value while the initial global best is position of the best particle in the swarm. Both positions and associated fitness values are recorded. Step 5. Update of pseudo-probability x and velocity V Now, an update of velocity V and pseudo-probability x is done according to following equations: k+1 k V(i,j) = w × V(i,j) + c1 × r1 (pkxb(i,j) − xkb(i,j) )+
c2 × r2 (gxkb(1,j) − xkb(i,j) ) k k+1 xk+1 (i,j) = x(i,j) + V(i,j)
(12) (13)
Step 6. Generation of new candidate solutions After updating V and X, new candidate solutions are generated with new probability for a bit to take the value. Step 7. Fitness computing on successive iterations The new candidate solutions are evaluated on the fitness function F (x). Step 8. Renewal of personal best (Pbest ) and global best (gbest ) On successive iterations, renew of personal best and global best is performed by replacing previous personal best and global best with new personal best and new global best of a better fitness value. Step 9. Termination of the algorithm
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The algorithm terminates if stopping criterion is reached. The flow chart diagram of the PBPSO algorithm is given in Fig. 1.
or Gc (s) = Kp +
Ki + Kd s s
(15)
where Kp , Ti , and Td represent the proportional gain, integral time, and derivative time, respectively. 1) The proportional term provides a control action proportional to the error and reduces the rise time. 2) The integral term reduces the steady-state error by performing an integral control action and eliminates the steady-state error. 3) The derivative term improves the stability of the system and reduces the overshoot by improving the transient response.
3.2
PBPSO-based coordinated controller structure of a boiler turbine unit
The schematic diagram of a coordinated controller for a boiler turbine unit can be considered, as in Fig. 2, where P is the process model, D is the decoupler for achieving diagonal dominance to eliminate the coupling of the process, µ is the governor valve position, β is the boiler firing rate, P and N are the desired throttle pressure and output power, while Po and No are the actual throttle pressure and output power.
3.3
Process model
Consider the following transfer function model of a boiler turbine unit at any operating point[1] . Fig. 1
3 3.1
Flow chart diagram of the PBPSO algorithm
PBPSO implementation PID controller
The three term continuous PID controller transfer function can be expressed as · ¸ 1 Gc (s) = Kp 1 + + Td s (14) Ti s
Fig. 2
P (s) = 4.2471(3.4s + 1) (100s + 1)(20s + 1)(10s + 1) 0.224 (100s + 1)(20s + 1)
3.224s(3.4s + 1) (100s + 1)(10s + 1) . 0.19(20s + 1) − (100s + 1) (16)
As the system requires decoupling to achieve diagonal dominance, a steady-state decoupler for the above process model is given as
PBPSO based coordinated control structure of boiler turbine unit with a compensator
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" −1
D = [P (0)]
# 0.2355 0.2776
=
0 −5.2632
.
(17)
Z
Experiments and simulation results
3.4.1 Experimental conditions The experiments were carried out using Matlab, Pentium 3.06 GHz, and 2 GB of RAM. Each algorithm was tested for 8 000 (200×40)fitness evaluations under identical conditions. 3.4.2 Simulation model The simulation model of the coordinated controller for the process given by (16) was developed using Matlab. 3.4.3 Parameter settings for algorithms As optimization, performance of any algorithm depends on parameter setting and inappropriate parameter values may yield poor results. In this particular case, to identify the best parameters for each of the algorithms used for comparison with PBPSO, simulation experiments were repeated independently with each algorithm with different parameter values to determine the most optimal parameters. The parameter values of various algorithms are listed in Table 1, where n is the population size and N is the number of total iterations. Table 1
t
|yd1 − yout1 |dt + 0
The decoupler D is placed in series with P so that G = P D is diagonal, then two PID controllers PID1 and PID2 are tuned to meet the required control performance.
3.4
Z
t
f= Z
|yd2 − yout2 |dt+ 0
Z
t
t
|yd3 − yout3 |dt + 0
|yd4 − yout4 |dt
where yd1 = yd4 = 1 and yd2 = yd3 = 0. In order to perform a bounded search, parameter search boundaries were specified, as shown below: (
0 6 Kp1 6 3, 0 6 Ki1 6 3, 0 6 Kd1 6 3 0 6 Kp2 6 3, 0 6 Ki2 6 3, 0 6 Kd2 6 3
where, Kp1 , Ki1 , Kd1 and Kp2 , Ki2 , Kd2 | {z } | {z } PID1
PID2
The best PID controller parameters and associated IAE values among 20 simulation experiments with each algorithm were summarized in Table 2. The transient performances were shown in Fig. 3. From Table 2, Figs. 3 and 4, it can be seen that the PBPSO algorithm yields superior results in comparison to PSO, DBPSO, and MBPSO both in terms of transient performance requirements and IAE values.
Parameter values of algorithms
Algorithms
c1
c2
w
n
N
Vmax
Xmax
PBPSO
2.0
2.0
0.7
40
200
50
50
DBPSO
2.0
2.0
0.7
40
200
50
−
MBPSO
−
−
−
40
200
PSO
1.5
1.5
0.7
40
200
As in (5) 0.3
−
3.4.4 Objective function In optimization, a performance criterion or an objective function returns an optimal solution. Therefore, the time integral performance index and the integral of the absolute error (IAE) were chosen as the performance criteria: Z t IAE = |yd − y|dt. (18)
(a) Transient performances of output power
0
3.4.5 Experiments and simulation results After setting algorithmic parameters, each variable of the PID controller was set as a 16 bit binary string. Therefore, a total of 96 bit were used to encode each candidate solution comprising six variables for two PID controllers. These candidate solutions were decoded to their corresponding decimal equivalents using standard binary to decimal conversion rules. The resultant decimal values were passed to both PID controllers simultaneously to evaluate the fitness of each candidate solution to minimize the following objective function:
(19)
0
(b) Transient performances of throttle pressure Fig. 3 Transient performances
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International Journal of Automation and Computing 8(2), May 2011 Table 2
Performance criterions
Modes
Algorithms
PID parameters obtained with various algorithms Controller parameters Kp1
Ki1
Kd1
Kp2
Ki2
Kd2
Fitness
IAE
PID
PSO
3
0.0229
2.5522
3
3
3
85.3014
IAE
PID
DBPSO
2.8350
0.0230
0.0663
2.5303
2.9221
0.0008
81.3047
IAE
PID
MBPSO
2.9988
0.0241
0.1079
2.9999
2.9290
0.0306
80.8404
IAE
PID
PBPSO
3
0.0227
0.014
2.9993
3
0.0694
77.6731
it is quite obvious from the fitness plot provided in Fig. 4 that the PBPSO method gives good results both in terms of IAE values and convergence speed. Fig. 5 shows the convergence behavior of PID parameters during 200 iterations. It is observed that PBPSO determines the true optimal controller gains. PSO, DBPSO, and MBPSO trapped in local optima in many cases, while PBPSO has shown robustness in escaping from local optimum.
5
Fig. 4
4
Convergence characteristics/fitness plot
Comparison results
In all cases, under identical conditions, the proposed PBPSO algorithm yields the best optimization results, and
Conclusions
In this paper, firstly, a new binary PSO, PBPSO, is proposed. The proposed PBPSO algorithm is compared with three former PSO algorithms of PID coordinated controller tuning of boiler turbine unit. The comparative study through a series of simulation experiments have shown that PBPSO method is more effective in identifying true optimal gains in comparison to the continuous PSO, DBPSO, and MBPSO.
(a)
(b)
(c)
(d)
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(e)
(f)
Fig. 5
Convergence behaviors of PID parameters
It is also observed that PBPSO algorithm executes faster than MBPSO and DBPSO due to the new estimator P that reduces computation time by eliminating the exponential term in DBPSO. The MBPSO has shown somehow better optimization ability in comparison to DBPSO and PSO, but it also traps in local optimum. PBPSO has proven to be a robust method in terms of both convergence speed and consistency. The convergence curves of PID parameters reveal that the PBPSO algorithm possesses the required characteristics of an efficient optimization algorithm, such as exploration, exploitation, jumping out, and convergence. Though the optimal gains identified by PBPSO algorithm confirmed the satisfactory performance, however an alternate multi-objective cost function might be more suitable in identifying more appropriate set of optimal gains.
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[6] P. Marsh. Turn on, tune in, where can the PID controllers go next. New Electron, vol. 31, no. 4, pp. 31–32, 1998.
[15] J. Kennedy, R. C. Eberhart. A discrete binary version of the particle swarm optimization algorithm. In Proceedings
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[16] H. Shayeghi, M. Mahdavi, A. Bagheri. An improved DPSO with mutation based on similarity algorithm for optimization of transmission lines loading. Energy Conversion and Management, vol. 51, no. 12, pp. 2715–2723, 2010. [17] A. Uncler, A. Murat. A discrete particle swarm optimization method for feature selection in binary classification problems. European Journal of Operational Research, vol. 206, no. 3, pp. 528–539, 2010. [18] J. Kennedy, R. C. Eberhart. Particle swarm optimization. In Proceedings of IEEE International Conference on Neural Networks, IEEE, Perth, Australia, vol. 4, pp. 1942–1948, 1995. [19] Q. Shen, J. H. Jiang, C. X. Jiao, G. L. Shen, R. Q. Yu. Modified particle swarm optimization algorithm for variable selection in MLR and PLS modeling: QSAR studies of antagonism of angiotensin II antagonists. European Journal of Pharmaceutical Sciences, vol. 22, no. 2–3, pp. 145–152, 2004. Muhammad Ilyas Menhas received the B. Sc. degree in electrical engineering from University of Azad Jammu & Kashmir Pakistan in 2002. He has been an assistant engineer in AJK Electricity Department and a visiting lecturer at Centre for Computer Science and Information Technology University College Kotli, Azad Jammu & Kashmir, Pakistan. Currently, he is a Ph. D. candidate at School of Mechatronics and Automation Shanghai University, Shanghai, PRC. His research interests include intelligent optimization algorithms and automatic control.
E-mail:
[email protected] (Corresponding author)
Ling Wang received the B. Sc. and Ph. D. degrees in control theory and control engineering from East China University of Science and Technology, PRC in 2002 and 2007, respectively. He is an associate professor at School of Mechatronics and Automation, Shanghai University, PRC. His research interests include intelligent optimization algorithms and automatic control. E-mail:
[email protected]
Min-Rui Fei received the Ph. D. degree in control theory and control engineering from Shanghai University, PRC in 1997. Since 1998, he has been a professor at School of Mechatronics and Automation, Shanghai University. He is a vice chairman of Chinese Association of System Simulation, a standing director of China Instrument & Control Society, and director of Chinese Artificial Intelligence Association. His research interests include intelligent control, networked control system, and wireless sensor networks. E-mail:
[email protected] Cheng-Xi Ma received the M. Sc. degree in control theory and control engineering from School of Mechatronics and Automation, Shanghai University, PRC in 2010. He is an employee of East China Electric Power Design Institute, PRC. His research interests include optimization and automatic control. E-mail:
[email protected]
