Multi-objective PID Optimization for Speed Control of an Isolated Steam Turbine using Gentic Algorithm

: This study focuses on multi-objective optimization of the PID controllers for optimal speed control for an isolated steam turbine. In complex operations, optimal tuning plays an imperative role in maintaining the product quality and process safety. This study focuses on the comparison of the optimal PID tuning using Multi-objective Genetic Algorithm (NSGA-II) against normal genetic algorithm and Ziegler Nichols methods for the speed control of an isolated steam turbine. Isolated steam turbine not being connected to the grid; hence is usually used in refineries as steam turbine, where a hydraulic governor is used for the speed control. The PID controller for the system has been designed and implemented using MATLAB and SIMULINK and the results of the design methods have been compared, analysed and conclusions indicates that the significant improvement of results have been obtained by the Multi-Objective GA based optimization of PID as much faster response is obtained as compared to the ordinary GA and Ziegler Nichols method.


INTRODUCTION
PID-Proportional, Integral and Derivative controllers alone contribute 90% of the total controllers used today (Åström and Hägglund, 2001;Zhao et al., 2011) and their simplicity of design and the ease of implementation complements their application.Optimal tuning of the PID parameters is an imperial factor, which can be formulated as an optimization problem (Krohling and Rey, 2001).
In this study, an isolated steam turbine has been considered i.e., the turbine is not connected to the grid and the speed is controlled using a hydraulic governor.The hydraulic governor regulates the steam flow in the turbine (Basu and Samiran, 2012;Ismail, 2012).For optimal operation, the system must be flexible enough to adapt with the changing conditions and regulate the process efficiently.For designing the PID controller, Ziegler Nichols frequency response has been used followed by the optimization using genetic algorithm and multi-objective genetic algorithm.The optimization has been carried out by the minimizing the objective function, stated as "Sum of the integral of the squared error and the sum of the integral of the absolute error" (Das et al., 2011).According to the results obtained, considerably better results have been obtained in case of the Multi-objective GA tuned PID controllers.

MATERIALS AND METHODS
Mathematical modelling of the steam turbine and its governing system: Steam/Gas turbine systems are centre to several industrial processes like in refineries, chemical plants, sugar industries etc. and are also termed as drive compressors and generally are of centrifugal type (Ismail, 2012).Speed and power is regulated with the governing system.The main purpose of the governing system is to keep the speed constant as the load varies.
Figure 1 illustrates the governing scheme of the turbine.The governor Valve (CV) regulates the steam flow, a Stop Valve (SV) is also provided for protection to check the accidental steam flow.Figure 2 illustrates the signal flow block diagram for the governing process.The electro-hydraulic convertor is used to convert the output electric/voltage signal to hydraulic pressure or piston position signal and the control valves are operated by the control valve servo motors.The flow of steam is proportional to the opening of the valve; hence the valve regulates the steam flow and is governed by the entire rotor system and the turbine power output and the governing system regulates the turbine mechanical power output (Murty, year).The mathematical model of the system (Fig. 3) has been taken from (Ismail, 2012) which has been obtained by obtaining the step response of the system, followed by the determination of the transfer function.The equation obtained can be obtained as a third order transfer function: Design and optimization of the PID controllers: In process control PID controllers are the most widely used controllers and they alone contribute 90% of the total controllers used today (Åström and Hägglund, 2001;Zhao et al., 2011).The general equation for a PID controller can be given as (Nise, 2004):

Tuning of PID controller using Ziegler
Ziegler Nichols tuning is the most operative of all the classical methods available.Being the third order system, the initial parameters have been estimated by using frequency response as suggested by the Ziegler Nichols (Ziegler and Nichols, 1942).But limited till the ratio of 4:1 for the first two peaks in closed loop response, leading to an oscillatory response (Goodwin et al., 2001).The parameters obtained are listed in the Table 1. Figure 4 represents the closed loop response obtained using ZN-PID parameters.

Optimization using genetic algorithm:
tuned PID controllers show an oscillatory response, so they are not fit for direct implementation for the plant.So the parameters are required to be optimized, so better parameters with least over-shoot can be obtained.The use of Genetic Algorithm optimization the PID controllers provides the advantage of its adaptability for changing constraints.The optimization of the PID controller is based upon the minimization of the integra time squared error (Corriou, 2004), ISE can be given as: Minimization of the ISE will discard the larger errors or the parameters with larger amplitude will be suppressed (Corriou, 2004).The optimization of the PID using GA focuses on obtaining the three best optimal values for (kp, ki, kd), so that it globally minimizes the objective function i.e., ISE.
The optimization of the PID controllers has been carried out using Global Optimization Toolbox and SIMULINK, with a population size of 20, scattered crossover, single side migration and roulette wheel based selection.The optimal PID gains obtained using GA are shown in Table 2 while Fig. 5 shows the closed loop response of the GA-PID controller.Figure 6 shows the plot for average and mean fitness value.

Multi objective optimization using GA:
oscillatory response has been obtained by Ziegler Nichols and Genetic Algorithm, so the parameters are not optimum for the direct implementation, their organized optimization must be carried out so that an un-oscillatory response can be obtained.of the PID controllers using Multi-Objective Genetic Algorithm aims at improving the objective the both the objectives used by obtaining an optimal Pareto solution.In this study, two objective functions have been used F1 (ISE) and F2 (IAE): Tuning of PID controller using Ziegler-Nichols: Ziegler Nichols tuning is the most operative of all the classical methods available.Being the third order system, the initial parameters have been estimated by using frequency response as suggested by the Ziegler-1942).But this method is limited till the ratio of 4:1 for the first two peaks in closed loop response, leading to an oscillatory response 2001).The parameters obtained are listed in the Table 1. Figure 4 represents the closed loop PID parameters.
ptimization using genetic algorithm: As the ZN tuned PID controllers show an oscillatory response, so they are not fit for direct implementation for the plant.So the parameters are required to be optimized, so shoot can be obtained.The use of Genetic Algorithm optimization the PID provides the advantage of its adaptability for changing constraints.The optimization of the PID controller is based upon the minimization of the integral , ISE can be given Minimization of the ISE will discard the larger errors or the parameters with larger amplitude will be .The optimization of the PID using GA focuses on obtaining the three best , so that it globally minimizes the objective function i.e., ISE.
The optimization of the PID controllers has been on Toolbox and , with a population size of 20, scattered crossover, single side migration and roulette wheel based selection.The optimal PID gains obtained using GA are shown in Table 2 while Fig. 5 shows the closed troller.Figure 6 shows the plot for average and mean fitness value.ulti objective optimization using GA: Since an oscillatory response has been obtained by Ziegler Nichols and Genetic Algorithm, so the parameters are not optimum for the direct implementation, their organized optimization must be carried out so that an oscillatory response can be obtained.Optimization Objective Genetic Algorithm aims at improving the objective function of by obtaining an optimal Pareto solution.In this study, two objective functions d F2 (IAE): ( )    The implementation of the system and its optimization has been carried out in SIMULINK using Global Optimization Toolbox.Population size of 45 with adaptive feasible mutation function and selection of individuals on the basis of tournament with a tournament size considered.Figure 7 shows the closed loop response of the system with Mobj-GA PID controller.The optimized PID parameters are shown The implementation of the system and its optimization has been carried out in MATLAB and using Global Optimization Toolbox.Population size of 45 with adaptive feasible mutation function and selection of individuals on the basis of of 2 has been considered.Figure 7 shows the closed loop response of GA PID controller.The shown in Table 3.In    4 shows the various solutions obtained by the optimization using Multi-Objective GA.

RESULTS AND DISCUSSION
The designing and implementation of the PID control closed loops has been carried out in and SIMULINK.From the closed loop response shown in figures above, ZN and GA gives an oscillatory response and the response obtained after the multi objective optimization using genetic algorithm reflects the minimum overshoot and settling times.So the PID parameters obtained by the multi objective optimization are perfect for the implementation for the process and also ensures better stability and process safe 10 shows the compared response of the ZN, GA and Mobj-GA and the Table 5 shows the compared numerical results obtained.Figure 11 shows the compared graphical representation of the results obtained in Table 5.
Compared closed loop response of the ZN, GA and Fig. 8a, distance between members of each generation for average Pareto the change in distance measure with previous generations and Fig. 9 shows the Pareto front obtained between the two objective functions.Table 4 shows the various solutions obtained Objective GA.

RESULTS AND DISCUSSION
The designing and implementation of the PID control closed loops has been carried out in MATLAB .From the closed loop response shown in figures above, ZN and GA gives an oscillatory response and the response obtained after the multi optimization using genetic algorithm reflects the minimum overshoot and settling times.So the PID parameters obtained by the multi objective optimization are perfect for the implementation for the process and also ensures better stability and process safety.Figure 10 shows the compared response of the ZN, GA and GA and the Table 5 shows the compared numerical results obtained.Figure 11 shows the compared graphical representation of the results

CONCLUSION
The use of multi objective optimization using genetic algorithms for tuning the PID controllers offer significantly improved response of the speed control of the isolated steam turbine.The response obtained is lesser oscillatory with significantly reduced overshoot percentage of 14 from 62.7% (ZN) and better settling time as compared to the response obtained by Ziegler Nicholas and GA tuned PIDs but rise time valves for GA-PID are better but the over-shoot percentage is 3 times higher as compared to Mobj-PID controller response.Thus Mobj-GA PID controllers offers improved stability and better process safety.

Fig. 6 :
Fig. 6: Plot for beat and mean fitness of individuals First objective function ISE i.e., Integral Square Error tries to minimize the larger amplitudes by the larger errors while second objective function IAE i.e., Integral Absolute Error minimizer the smaller errors; thus forcing the solution towards the The optimization uses II algorithm boosts attaining the best fitness value using controlled elitist genetic algorithm.It also favours increasing the diversity of the population which prevents the algorithm from being struck in a local solution.Diversity of solutions is controlled by the elite ile elitism is controlled by Pareto fraction and Pareto front also bounds the

Fig. 7 :
Fig. 7: Closed loop response of the Mobj GA PID controllers

Fig. 10 :
Fig. 10: Compared closed loop response of the ZN, GA and Mobj-PID controllers

Fig. 8a ,
Fig. 8a, distance between members of each generation is shown, Fig. 8b gives the plot spread, which is the change in distance measure with respect to the previous generations and Fig Pareto front obtained between the two objective functions.Table4shows the various solutions obtained by the optimization using Multi-Objective GA.

Table 1 :
PID parameters obtained by Ziegler Nichols PID gains Kp Ki Kd

Table 2 :
PID parameters obtained by genetic algorithm

Table 1 :
PID parameters obtained by Ziegler Nichols

Table 3 :
PID parameters obtained by multi objective GA PID gains Kp Ki Kd

Table 4 :
Different set of solutions obtained while optimizing

Table 3 :
PID parameters obtained by multi objective GA

Table 5 :
Comparison of the results