Genetic algorithm-based fuzzy-PID control methodologies for enhancement of energy efficiency of a dynamic energy system

Energy Conversion and Management 52 (2011) 725–732

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Energy Conversion and Management journal homepage: www.elsevier.com/locate/enconman

Genetic algorithm-based fuzzy-PID control methodologies for enhancement of energy efficiency of a dynamic energy system G. Jahedi, M.M. Ardehali * Energy Research Center, Department of Electrical Engineering, Amirkabir University of Technology (Tehran Polytechnic), 424 Hafez Ave, Tehran, Iran

a r t i c l e

i n f o

Article history: Received 5 December 2009 Accepted 30 July 2010 Available online 24 August 2010 Keywords: Energy Efficiency Fuzzy logic Genetic algorithm Control PID

a b s t r a c t The simplicity in coding the heuristic judgment of experienced operator by means of fuzzy logic can be exploited for enhancement of energy efficiency. Fuzzy logic has been used as an effective tool for scheduling conventional PID controllers gain coefficients (F-PID). However, to search for the most desirable fuzzy system characteristics that allow for best performance of the energy system with minimum energy input, optimization techniques such as genetic algorithm (GA) could be utilized and the control methodology is identified as GA-based F-PID (GA–F-PID). The objective of this study is to examine the performance of PID, F-PID, and GA–F-PID controllers for enhancement of energy efficiency of a dynamic energy system. The performance evaluation of the controllers is accomplished by means of two cost functions that are based on the quadratic forms of the energy input and deviation from a setpoint temperature, referred to as energy and comfort costs, respectively. The GA–F-PID controller is examined in two different forms, namely, global form and local form. For the global form, all possible combinations of fuzzy system characteristics in the search domain are explored by GA for finding the fittest chromosome for all discrete time intervals during the entire operation period. For the local form, however, GA is used in each discrete time interval to find the fittest chromosome for implementation. The results show that the global form GA–F-PID and local form GA–F-PID control methodologies, in comparison with PID controller, achieve higher energy efficiency by lowering energy costs by 51.2%, and 67.8%, respectively. Similarly, the comfort costs for deviation from setpoint are enhanced by 54.4%, and 62.4%, respectively. It is determined that GA–F-PID performs better in local from than global form. Ó 2010 Elsevier Ltd. All rights reserved.

1. Introduction Features such as embedding experiential knowledge make the artificial intelligence (AI) based approaches ideal for control of dynamic energy systems [1]. As there exist inherent nonlinearities and disturbances that are nearly impossible to account for, the operators experience accumulated over time can be critically useful in operating an energy system where the input of energy for heating and cooling is intended to achieve a set goal by the equipment. Aside from energy costs of operating energy system, where conversion of energy from one form to another is intended to heat or cool the built environment, there is a cost associated with maintaining thermal comfort temperature setpoint. The significance of maintaining thermal comfort temperature and its relation with energy consumption has been examined and it is estimated that 0.08% increase in performance of personnel at the workplace is compensated with 15% increase in the energy consumption [2]. Because thermal comfort of occupants is a function of surrounding temperatures, it is anticipated that the use of optimized fuzzy logic * Corresponding author. Tel.: +98 21 64543323; fax: +98 21 66406469. E-mail address: [email protected] (M.M. Ardehali). 0196-8904/$ - see front matter Ó 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.enconman.2010.07.051

linguistic rules for design of energy systems controllers can be accommodating to the need for ‘‘the translation of occupants sensation” with the least amount of fluctuation in thermal zone temperature. The implementation of AI-based control methodologies for lowering energy demand and consumption, while maintaining occupant thermal comfort, requires microprocessor technologies. The current direct digital control technologies have lead to substantial improvements in operation scheduling of heating ventilating, and air conditioning systems, however, the process of controlling the operation of these systems is still in need of improvements [3] and it has been reported that control algorithms programming are potential sources of energy inefficiencies [4]. To regulate energy input and achieve thermal zone temperature for maintaining occupant comfort, the most common controllers used in the industry utilize proportional (P), integral (I), and derivative (D) control algorithms. As the gain coefficients for the noted classical control algorithms affect the transient performance of energy systems, the procedures to determine them have been the subject of several studies [5–7]. There are two different approaches in determining the PID gain coefficients. In the first approach, the controller is identified as a static type and the gain coefficients of

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Nomenclature a Ai, Ao C c ce e f GE GCE GU hi, ho J K l M n N NB NS PB PS pm, pr q qx Qz t

allele of chromosome interior and exterior wall area, m2 specific heat, kJ/kg °C location of apex change of error error volumetric flow rate, m3/s error scaling factor error change scaling factor control signal scaling factor inside and outside convective heat transfer coefficients, W/m2 K cost function gain coefficient values of alleles 7, 8 and 9 values of alleles 4, 5 and 6 number of fuzzy membership functions values of alleles 10, 11 and 12 negative big negative small positive big positive small spacing parameter of membership functions and fuzzy rules values of alleles 13, 14 and 15 heat exchanger input energy or the controller energy, W thermal load, W time, min, s

the PID controller remain constant from the initial tuning through the entire operation period. In the second approach, the gain coefficients of PID controller are dynamically changed and re-calculated in each time step of operation period. Ardehali et al. [8] have shown that better results can be achieved, when fuzzy logic is used to determine the gain coefficients of PID (F-PID) controller for an energy system, as compared with its static-type counter part. In other studies, also, it has been shown that a F-PID controller performs better than a static-type PID controller [9,10]. Based on predicted-mean-vote comfort and illumination indices, as the reference inputs to the fuzzy-logic based controller and using Gaussian-type membership functions, Kolokotsa et al. [11] have utilized F-PID, F-PD and, adaptive F-PD control methodologies to reduce energy consumption of a system, for a heating application, by means of eliminating overshoots, undershoots, and fluctuations that contribute to excessive and wasted energy input. To control the heating valve along with other comfort-related mechanisms, in that study, the error value at the current time step, ej, and previous time step, ej1, are used as the reference inputs to the F-PI controller and, the change in error Dej = ej–ej1 is used as the reference input to the F-PD controller. In that study, the thermal zone air dynamics is taken into consideration, however, the dynamics of the thermal zone wall and heat exchanger, as the source for heating energy, are not accounted for. A self-tuning fuzzy controller is tested in the heating mode by regulating the steam flow rate to the steam coil by Huang and Nelson [12]. They showed that, it is possible to achieve excellent dynamic process control with less effort by using a self-tuning fuzzy controller. Tzafestas and Papanikolopoulos [13] have indicated that exploitation of fuzzy logic allows for using the human approach to propose rule-based solutions for designing fuzzy FPID controllers. Fuzzy reasoning is suitable for mathematical modeling of the human interpretation of how a controller should function, as denoted by Dounis et al. [14]. Fuzzy control strategies and

T V Z hs

q

temperature, °C volume, m3 zero characteristic angle density, kg/m3

Subscripts 0 initial 2 heat exchanger discharge, thermal zone inlet 3 thermal zone discharge a air ce change of error d derivative e error f final i integral j current discrete time m fuzzy membership function n number of fuzzy membership functions o output of fuzzy system p proportional r fuzzy rule base s seed point sp set point x heat exchanger w wall z zone

supervisory techniques for control of temperature, humidity and CO2 concentration are compared with classical PID algorithm by Shepherd and Batty [15], and a reduction of the steady-state error is demonstrated. While fuzzy logic has been used as an effective tool in development of F-PID control algorithm, the performance of these controllers is limited to the extent of possible combinations of fuzzy systems characteristics such as, fuzzy rules, membership functions, and input and output scaling factors examined heuristically based on empirical knowledge and experimental trial and errors by the human designer [16]. However, it is hypothesized that the application of optimization techniques for determining fuzzy system characteristics results in optimal values of gain coefficients and a more desirable F-PID controller performance. Recently, the design of the fuzzy sets and the rule base has been automated by the use of artificial neural networks and genetic algorithms (GA) [16]. GAs have fast convergence to or near the global optimum as well as superior global searching capability in a space with complex searching surface and applicability to the search space where gradient information is unavailable. As compared with artificial neural networks, GAs are robust and comprehensive search techniques [16]. The use of GA for examining a much larger set of possible combinations of fuzzy system characteristics via different chromosomes that continually mutate and cross over, referred to as GAbased F-PID (GA–F-PID), results in optimal gain coefficients for FPID controllers. Based on the literature review, it is of interest to investigate the performance of the GA–F-PID controllers on the energy input and thermal zone temperature of a dynamic energy system where the time-dependent interactions of the major system components with each other and the exterior conditions are taken into consideration. The objective of this study is to examine the performance of PID, F-PID, and GA–F-PID controllers for enhancement of energy efficiency of a dynamic energy system. This study is organized as fol-

G. Jahedi, M.M. Ardehali / Energy Conversion and Management 52 (2011) 725–732

lows. In Section 2, the dynamic model for the energy system is presented. Section 3 discusses the different control algorithms used in this study. The performance evaluation of control methodologies and corresponding cost functions are discussed in Section 4. In Section 5, the simulation results for the examined control algorithms are given. Finally, Section 6 presents conclusions and recommendations. 2. Energy system model The differential equations describing the dynamic behavior of the energy system providing both heating and cooling for maintaining thermal zone temperature for occupant comfort, as shown in Fig. 1, are given by [8]

dT 2 ¼ qa fC a ðT o  T 2 Þ þ qx dt dT qa C a V z 3 ¼ qa fC a ðT 2  T 3 Þ þ hi Ai ðT w  T 3 Þ þ Q z dt dT qw C w V w w ¼ ho Ao ðT o  T w Þ  hi Ai ðT w  T 3 Þ dt

qx C x V x

ð1Þ ð2Þ ð3Þ

All parameters are defined in the nomenclature section. In the dynamic model of the system, Eq. (1) determines transient response of T2 according to input energy to the heat exchanger (qx). Temperature change in the thermal zone (T3) caused by various loads is determined by Eq. (2). Eq. (3) expresses the wall temperature changes (Tw) in relation to the temperature of inner and outer wall area. The thermo-physical properties for all materials are assumed constant. Further, it is assumed that uniform mixing exists and the thermal zone external and internal loads are constant. For simulation purposes, the heat exchanger provides both heating (positive) and cooling (negative), based on the control law that regulates energy input to the heat exchanger [8]. 3. Control algorithms 3.1. PID controller The PID controller algorithm is widely used for industrial automation tasks and thermal comfort heating and cooling applications where error, derivative of error, and integral of error are used in the calculation of control law [8]. 3.2. F-PID controller Due to the fact that fuzzy logic utilizes the simple qualitative expression of heuristic knowledge of the human experts, fuzzy controllers have been proposed for a variety of systems [8,11,13–

To

Tsp

1

qx

Heat exchanger

Controller Fan 3 Exhaust air Qz

T2

2

727

15] including energy systems. The complete analysis of F-PID control of energy systems is given by Ardehali et al. [8]. For this study, triangular and trapezoidal membership functions are examined prior to using Gaussian membership functions, as the final choice, to map the input values to their appropriate membership values and, later, to determine the gain coefficients as the outputs. For the F-PID controller, the reference inputs used include the error ej = Tsp–T3,j and change in error Dej = (e3,j–e3,j1) where each input corresponds to five linguistic variables, hence, total of 25 rules are generated. The initial solution to the differential equations is used to determine the initial reference input values that are fed into the fuzzifier. The fuzzifier converts the real input variables into fuzzy input variables. Utilizing the fuzzy relation matrix, the fuzzy reasoning uses fuzzy input variables to obtain the fuzzy outputs. The defuzzifier then produces the real values of the controller gain coefficients so that a new control law for regulating qx can be determined and, as a result, Eqs. (1)–(3) can be solved for the next time step [8]. However, as it is discussed next, for the best performance of F-PID controllers, it is necessary that the respective fuzzy system characteristics are optimized. 3.3. GA–F-PID controller To determine the three optimal fuzzy systems required for specification of the three PID gain coefficients, GA is used in this study, as shown in Fig. 2. Because GA utilizes the concept of the survival of the fittest chromosome for GA–F-PID controller, several genetic operations, namely, reproduction, crossover, and mutation must be performed on the initially designed chromosome [17]. The chromosome is a string of data that contains information about each of the three fuzzy systems that must be developed for the GA–F-PID controller. In GA, the reproduction operation is used to generate a new population of children from an old population that is randomly chosen, based on the roulette wheel method. For crossover operation, two strings are chosen randomly and then divided into segments to form new strings. The new string replaces the old string if it results in lower value for the respective cost functions, as they are described later in Section 4, and it is said that the new string has a better fitness. The mutation operation is intended to avoid sub-optimal or near-optimal solution resulted from the crossover operation. To develop the chromosome structure, the parameters describing the characteristics of each fuzzy system, namely, number of membership functions, membership function spacing, rule base spacing, scaling factors, and rule base angle are encoded into a chromosome with 19 alleles [18,19]. The parametric values of the chromosome are listed in Table 1. The fuzzy system characteristics required for GA–F-PID are discussed next. 3.3.1. Number of membership functions The number of membership functions is an odd integer number limited between three and nine in this study (n = 3, 5, 7, 9). As shown in Table 1, the first, second and third alleles (a1, a2, a3) of the chromosome represent the number of membership functions for error input (ne), change of error input (nce) and output (no) of the fuzzy system, respectively. For illustration purposes, Fig. 3 shows the distribution of the number of membership functions, when ne = 5.

Wall T3 Sensor Thermal zone

Supply air

Tw To

hi

ho

Fig. 1. Heating and cooling energy system [7].

3.3.2. Membership function spacing The most common shape type for the membership functions are continuous triangular and gaussian functions. The gaussian functions are usually chosen when the rule base has few fuzzy rules, but when the number of fuzzy rules gets larger, gaussian type functions fire several rules that may contradict each other. However, triangular membership functions are simpler to use and the loca-

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Evaluation (Eq. 12)

Jsp , Jq x (Eqs. 10 and 11)

Evaluated Children Population Deleted members

Parents

Discard

Reproduction, crossover and mutation) New children

New chromosomes for fuzzy systems 1, 2 and 3 (ne,nce,no,pr,pm, θs,GE,GCE,GU)

+

e

Tsp

-

ce

GEp

d/dt

GCEp

Fuzzy system1 (Fuzzy sets1, Fuzzy rules1)

Kp

GUp

e GEd ce

d/dt

GCEd

e ce

GEi d/dt

GCEi

Fuzzy system2 (Fuzzy sets2, Fuzzy rules2)

Kd

GUd

PID Controller

qx

Energy System

T3

Eqs. (1)-(3) Fuzzy System3 (Fuzzy sets3, Fuzzy rules3)

Ki

GUi

Fig. 2. Block diagram for GA–F-PID controller.

Table 1 Parametric values of the variables encoded into the chromosome. Parameters

Values

a1 a2 a3 a4 a5 a6 a7 a8 a9 a10 a11 a12 a13 a14 a15 a16 a17

ne nce no Me Mce Mo le lce lo Ne Nce No qe qce qo hs GE

3, 5, 7 or 9 3, 5, 7 or 9 3, 5, 7 or 9 [0.1, 1] [0.1, 1] [0.1, 1] 1 or 1 1 or 1 1 or 1 [0.1, 1] [0.1, 1] [0.1, 1] 1 or 1 1 or 1 1 or 1 [0, 2p] 1 1 ; 21 ] [ 21

a18

GCE

a19

GU

[ 14 ; 14] [40, 50] for fuzzy system 1 [0.3, 0.6] for fuzzy system 2 [1000, 2000] for fuzzy system 3

NS

NB

1

PS

Z

PB

(a) pm=1 0.5

-1

Magnitude

Alleles

-0.8

-0.6

-0.2

NS

NB

1

-0.4

0

0.2

0.4

0.6

PB

PS

Z

1

0.8

(b) pm= 2 0.5 -1

-0.8

NB

1

-0.6

-0.4

-0.2

NS

0

0.2

0.4

Z

0.6

1

0.8

PS

PB

(c) pm= 0.5

0.5 -1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

Error Fig. 3. Effect of spacing parameter on membership functions of error input where ne = 5.

tions of their apexes must be specified. Due to their limited overlapping, triangular membership functions provide smooth transitions on the control surface [16]. In this study, membership functions have triangular form and a relation used to determine the locations of triangular apexes along the universe of discourse is as follows

2

cm;k

3pm;k i 5 4 ¼   nj 1 2

i ¼ 1; 2; . . . ;

n1 and k ¼ e; ce; o 2

ð4Þ

729

G. Jahedi, M.M. Ardehali / Energy Conversion and Management 52 (2011) 725–732

where cm,k specifies the locations of the apexes and n is the number of membership functions as noted above in Section 3.3.1. The value of the pm,k is determined by

pm;k ¼ ðMk Þ ;

k ¼ e; ce; o

ð5Þ

where Mk is limited in [0.1, 1] and specified in alleles a4, a5 and a6. The apex of the central membership function is always fixed on zero and apexes of the remaining membership functions are symmetric on the left and right part of the universe of discourse. Parameter lk takes 1 or 1 and causes the membership functions for error input, change of error input and output of the fuzzy system to concentrate near the zero or on the extremes, respectively, and it is encoded into the alleles a7, a8 and a9. The effect of pm on the locations of apexes along the universe of discourse is demonstrated in Fig. 3. 3.3.3. Rule base spacing In this study, the concept of phase plane of error and change of error is used to derive fuzzy rule base for the fuzzy system [18,19]. The phase plane of error and change of error is segmented into the regions with same consequents and, seed and grid points in the phase plane, as shown in Fig. 3 by stars and circles, respectively, are used to derive the rule base. To do this, it is required to determine the coordination of these points in the phase plane. The coordination of grid points in the phase plane are determined by calculating the apexes of the membership functions for the error and change of error by following relation, which has a similar form as Eq. (4)

2

cr;k

3pr;k i 5 4  ¼  ; nk 1 2

i ¼ 1; 2; . . . ;

n1 and k : e; ce; o 2

ð6Þ

where cr,k gives the locations of the apexes, n is the number of membership functions, and pr,k, as the parameter for the rule base spacing, is determined by qr;k

pr;k ¼ ðNk Þ

k ¼ e; ce; o

;

ð7Þ

where the value of Nk, similar to Mk, is limited between [0.1, 1], and is encoded into the alleles a10, a11 and a12. Parameter qk takes 1 or 1 that causes the membership functions to concentrate near the zero or on the extremes, respectively. The values for qk are encoded into the alleles a13, a14 and a15. The coordination of grid points are then determined by the location of the centers of error and change of error membership functions in the phase plane. Using the spacing parameter for rule base, the coordination of seed points are determined by the following relations:

"

i xs ¼ n1

#pr;k

2

ys ¼ xs tanðhs Þ

;

i ¼ 1; 2; . . . ;

n1 and k : e; ce; o 2

ð8Þ ð9Þ

where hs as the characteristic angle, is limited between [0, 2p] encoded into the allele a16. Having the coordination of seed and grid points, the consequent of the fuzzy rule for every grid point is specified by finding the nearest seed point to the considered grid point. Fig. 4 shows the positioning of seed and grid points in the phase plane, in which hs = 45°, pr,e = 0.9, pr,ce = 0.9, and pr,o = 1. The derived rule base from the Fig. 4 is shown in Table 2. 3.3.4. Scaling factors In this study, all universes of discourse are limited between 1 and 1, and scaling factors are used to give appropriate values to the input and output variables of the fuzzy system. As shown in Fig. 2, scaling factors for error input, change of error input and output of the fuzzy system are represented by GE, GCE and GU, respectively. Ranges of variation for these parameters are defined by real values

Segmentation line

NB

Change of error

lk

NB

NS NS Z

Z

PS

PS PB

PB NB

NS

PS

Z

PB

Error Fig. 4. Seed and grid points in phase plane of error and change of error used to derive the rule base.

Table 2 Derived rule base. ce

e

NB NS Z PS PB

NB

NS

Z

PS

PB

NB NB NS NS Z

NB NS NS Z PS

NS NS Z PS PS

NS Z PS PS PB

Z PS PS PB PB

of corresponding variables and scaling factors are used for each variable to give the appropriate range, and their values are encoded into a17, a18 and a19 alleles. 4. Performance evaluation In this study, the performance of each control methodology is evaluated based on energy and comfort cost functions. The energy cost function is given by [8]

J qx ¼

Z

tf

ðqx Þ2 dt

ð10Þ

t0

where the quadratic form of energy required by the heat exchanger, either positive for heating or negative for cooling, is integrated over the entire time period of evaluation. Because the deviations from a setpoint temperature are the main cause for the occupant discomfort in a thermal zone and, doubling the deviation more than doubles the discomfort [20], the penalty for deviation from a desired temperature setpoint is computed based on the following comfort cost function.

J sp ¼

Z

tf

ðT sp  T 3 Þ2 dt

ð11Þ

t0

While the performance of all three controllers, namely, PID, FPID, and GA–F-PID are compared based on cost functions given by Eqs. (10) and (11), to apply GA to F-PID for optimizing the fuzzy system characteristics, a fitness function for assessment of the fitness of all chromosomes is needed. As it may vary depending on the problem type, the fitness function for assessing the maximum fitness of chromosomes for reproduction, during the optimization of fuzzy system characteristics of GA–F-PID, is chosen as the inverse of the sum of both cost functions for energy and comfort, given by

Fitness function ¼

1 J sp þ J qx

ð12Þ

G. Jahedi, M.M. Ardehali / Energy Conversion and Management 52 (2011) 725–732

Note that because GA is usually used for maximization, the fitness function given in the inverse form results in a minimum value for the sum of both costs that must be attained by the fittest chromosome having the maximum fitness. Through iterative search, the GA seeks for generations that evolve towards an optimal design of a F-PID controller. In this study, GA–F-PID controller is examined in two different forms, namely, global form and local form. For the global form, all possible combinations of fuzzy system characteristics in the search domain are explored by GA for finding the fittest chromosome for all discrete time intervals during the entire operation period. For the local form, however, GA is used in each discrete time interval to find the fittest chromosome for implementation.

4.5 4 3.5 3

Fitness

730

2.5 2 1.5 1

Maximum fitness

0.5

Mean fitness

5. Simulation results

0

K p 2 ½K p;min ; K p;max  ) K p 2 ½40; 50

ð13Þ

K i 2 ½K i;min ; K i;max  ) K i 2 ½0:3; 0:6

ð14Þ

K d 2 ½K d;min ; K d;max  ) K d 2 ½1000; 2000

ð15Þ

Based on the output ranges for the gain coefficients given by Eqs. (13)–(15), the range of reference input values for ej and Dej are determined as [8]

ej 2 ½ej;min ; ej;max  ) ej 2 ½21; 21

ð16Þ

Dej 2 ½Dej;min ; Dej;max  ) Dej 2 ½4; 4

ð17Þ

For the design of GA–F-PID controller, one chromosome with 19 alleles, having parametric values listed in Table 1, is used for every fuzzy system. Fig. 5 illustrates the convergence trend of GA for maximum and mean fitness during the successive generations and search process for GA–F-PID controller. The performances of the PID, F-PID, global form GA–F-PID and local form GA–F-PID algorithms are simulated and, the results from evaluation of the costs functions are compared in Table 4. It is observed that, due to the

Table 3 Parametric values for thermo-physical properties of the energy system [8].

qa Vx

qx Cx Vx Vz hi Ai

qw Cw Vw ho Ao

3

1.14 kg/m 1 kJ/kg °C 6544 kg/m3 0.12 kJ/kg °C 0.09 m3 28.3 m3 9.4 W/m2 9 m2 271.4 kg/m3 13.6 kJ/kg °C 1.35 m3 34.1 W/m2 9 m2

0

10

20

30

40

50

60

70

80

90

100

Generation Fig. 5. Convergence trend of GA in finding best fitness for GA–F-PID controller.

Table 4 Energy and comfort costs comparisons for PID, F-PID, and global and local form GA–FPID control algorithms. Controller

J qx

PID F-PID GA–F-PID (global form) GA–F-PID (local form)

Jsp

3.57 2.73 1.74 1.15

Enhancement with respect to PID (%)

2.26 1.88 1.03 0.85

J qx

Jsp

– 23.5 51.2 67.8

– 16.8 54.4 62.4

use of GA for optimizing F-PID controller, global GA–F-PID and local GA–F-PID control methodologies, in comparison with F-PID controller, result in enhanced energy efficiency by lowering energy costs by 36.2%, and 57.8%, respectively. Similarly, comfort costs are lowered by 45.2%, and 54.7%, respectively. Also, the performances of global GA–F-PID and local GA–F-PID control methodologies are found more desirable as compared with that of PID, where the energy costs are lowered by 51.2% and 67.8%, respectively, and the comfort costs are decreased by 54.4%, and 62.4%, respectively. Figs. 6 and 7 show the transient responses for T3 and qx. As compared to those of global form GA–F-PID, F-PID, and PID and controllers, it is observed that there is no overshoot in the case of local 36

GA-F-PID (local form) GA-F-PID (global form) F-PID PID

34

Thermal zone temperature, T3 (°C)

For the energy system shown in Fig. 1, the transient and steady state response is simulated for the operation period of 900 s and the sampling time is 10 s as suggested by Nesler and Stocker [7]. The set point temperature for the thermal zone air is 24 °C. As the initial conditions for solving Eqs. (1)–(3) for the first time step (j = 1), the initial temperatures for the thermal zone wall, heat exchanger and thermal zone air are 35 °C. Table 3 shows the parametric values for the thermo-physical properties of the energy system described by Eqs. (1)–(3). The initial tuning of the PID controller is accomplished based on the quarter wave damping criteria suggested by Zigler–Nicholes [5], and the gain coefficients are determined as Kp = 44.1, Ki = 0.42 and Kd = 1146. For the F-PID controller, the range of output values for gain coefficients are determined as [8]

32 30 28 26

24 22 20

0

100

200

300

400

500

600

700

800

900

Time(s) Fig. 6. Comparison of transient thermal zone temperature response for PID, F-PID, and global and local form GA–F-PID controllers.

731

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×10

4

×10

1.5

0

-2

-4

-6

GA-F-PID (local form) GA-F-PID (global form) F-PID PID

-8

-10

1.4

1.3

1.2

1 0

100

200

300

400

500

600

700

800

900

GA-F-PID (Local form) GA-F-PID (Global form) F-PID PID

1.1

0

Fig. 7. Comparison of transient heat exchanger input energy response for PID, FPID, and global and local form GA–F-PID controllers.

44

43

42

40

GA-F-PID (local form) GA-F-PID (global form) F-PID PID 0

100

200

300

400

500

600

700

800

200

300

400

500

600

700

800

900

Fig. 10. Variation of Kd for PID, F-PID, and global and local form GA–F-PID controllers.

GA–F-PID controller. It is also shown that the optimum fuzzy systems characteristics in both global and local form GA–F-PID controllers result in achieving setpoint temperature and reaching steady state in fewer time steps, which turns out as the reason for the superior performance, as compared to other control algorithms examined in this study. The application of optimization in each time step results in the best performance by the local form GA–FPID, where no overshoots and undershoots in both heat exchanger and thermal zone temperature responses, thereby yielding a more energy efficient operation of the system. Figs. 8–10 show the variation of PID gain coefficients due to implemented algorithms during the process time. While the gain coefficients remain constant for PID controller, the time-varying selection of gain coefficients are determined as the key factor influencing time responses for heat exchanger energy input and thermal zone temperature for F-PID and global and local form GA–F-PID controllers.

45

41

100

Time(s)

Time(s)

Proportional gain coefficient, (Kp)

3

2

Derivative gain coefficient, (Kd)

Heat exchanger input energy, qx (W)

4

900

6. Conclusion and final comments

Time(s) Fig. 8. Variation of Kp for PID, F-PID, and global and local form GA–F-PID controllers.

Integral gain coefficient, (Ki)

0.55

GA-F-PID (local form) GA-F-PID (global form) F-PID PID 0.5

This study demonstrates the effects of various control algorithms on energy efficiency and the level of comfort related to maintaining setpoint temperature in a thermal zone. It is concluded that the application of GA for optimum design of fuzzy controller characteristics results in optimum performance, both for entire time interval and every discrete time interval. It is also shown that local GA–F-PID led to better controlling performance with no overshoot and undershoot and minimum settling time than global GA–F-PID. The local GA–F-PID and global GA–F-PID controllers enhance energy demand and comfort conditions during transient operation periods. As a result, it is anticipated that the utilization of local GA–F-PID and global GA–F-PID result in lower equipment initial and operating costs, as the equipment capacity and number of fluctuations are minimized. For future work, experimental validation of the study is recommended.

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References

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Time(s) Fig. 9. Variation of Ki for PID, F-PID, and global and local form GA–F-PID controllers.

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