AU2021101100A4

AU2021101100A4 - Novel Fractional order PID based robust controller designing tuned with hybridized GA-PSO algorithm for grid-connected photovoltaic system.

Info

Publication number: AU2021101100A4
Application number: AU2021101100A
Authority: AU
Inventors: Krishanu Chatterjee; Saibal Chatterjee; Shantanu Chatterjee; Abhijit Das; Akhtar Kalam; Rupam Maiti; Arghya Mukherjee; Gopal Shankar Mukherjee
Original assignee: Kalam Akhtar Dr; Chatterjee Saibal Dr; Chatterjee Shantanu Dr
Current assignee: Kalam Akhtar Dr; Chatterjee Saibal Dr; Chatterjee Shantanu Dr
Priority date: 2021-03-02
Filing date: 2021-03-02
Publication date: 2021-05-13
Anticipated expiration: 2029-03-02

Abstract

Novel Fractional order PID based robust controller designing tuned with hybridised GA-PSO algorithm for grid connected photovoltaic system. Abstract In this innovation a grid connected photovoltaic system is being presented where fractional order PID (FO-PID) controller is used as the current controller for the inner current loop of the PV system. The tuning of different parameters (Kr, KiKd, k, ) of FO-PID controller has been performed using fmincon, genetic algorithm (GA), particle swarm optimization (PSO) and hybrid GA-PSO algorithm. Comparison of different performance aspects using different optimization algorithms has been studied. The robustness of the system gain against sudden parametric variation will increase if the derivative of the phase with respect to frequency becomes zero, which signifies that slope of tangent of the phase curve of bode plot also becomes zero. Thereby the system will exhibit more or less constant peak overshoot for a specific range of rise time, which can be termed as iso-damping property in this innovation. In this invention the current controller of the system has been designed by using FOPID controller and it has been verified that flat phase curve is obtained when tuned by different optimization algorithm. Figures F-OPGRID VOLTAGE FITE GRIfd C D RLL1 Array Vc Ce VSI MPPT PWM7 BLOCK 9 Vde ab labc Eabc Cb dq -q "'d * V. v E, + LoM E, - Loi • FOPID FOPID5 VOLTAGE 'q. -0- CONTROL .Idrefq Eq Ed LOOP • d 6 - LL . .. . .... ... .... .... .. 0 3CURRENTCONTROLLOOP Figure 1

Description

Figures

RLL1 Array Vc Ce VSI

MPPT PWM7 BLOCK

9 Vde ab labc Eabc Cb dq -q F-OPGRID "'d * VOLTAGE FITE C D GRIfd V. v

E, + LoM E, - Loi

• FOPID FOPID5

VOLTAGE 'q. -0- CONTROL .Idrefq Eq Ed LOOP • d

6 - LL . .. . .... ... .... .... .. 0 3CURRENTCONTROLLOOP

Figure1

Novel Fractional order PID based robust controller designing tuned with hybridised GA-PSO algorithm for grid connected photovoltaic system.

Field of the invention The field of innovation is related to grid integration of photovoltaic systems through fractional order PI controller (FO-PID) to enhance stability under sudden disturbance by introducing phase flattening at crossover frequency. Moreover, along with the phase flattening property the proposed controller is tuned with three different optimization algorithms namely fmincon, GA, PSO, hybrid GA-PSO and different performance indices are compared to obtain the optimal solution.

Background of the invention In the present era growth population leads to increase in global energy consumption rate, as a result demand for energy is increasing. But conventional energy sources are limited and they cause environmental degradation. For these reasons use of renewable energy sources are rapidly developing. Out of all renewable energy sources solar energy is considered as economical and clean source of energy.

In order to convert the dc power generated by the photo voltaic array to an ac quantity which can be interfaceable with the utility grid, voltage source inverter is used in grid connected PV system. Thus the quality of power transmitted to the grid largely depends upon the quality of the current injected by the inverter to the grid. Numerous current control techniques have been developed in literature which can not only enhance inverter efficiency but also improves the dynamic performance, robustness and external disturbance rejection capability. All though all controllers perform the same task but controller wise their performance differs.

Over the past few decades the controllers that had gained majority of the attention in the field of development are hysteresis regulators, linear PI regulators, PR regulators and predictive dead beat regulators. [ for example see, M. P. Kazmierkowski and L. Malesani, "Current control techniques for three-phase voltage-source PWM converters: a survey," in IEEE Transactions on Industrial Electronics, vol. 45, no. 5, pp. 691-703, Oct. 1998, doi: 10.1109/41.720325.]Some of the major advantages of hysteresis controllers are simple structure, rapid dynamic response and increased robustness. Instead of having certain advantage as mentioned above hysteresis current controller suffers from major drawbacks such as randomized switching frequency pattern, expensive filtration of the output waveform, increased stress on the switching devices and need of designing expensive filters in order to eliminate unwanted variable frequency noise which might cause premature and uncertain response in the grid voltage.

In order to increase the efficiency of tracking the reference signal along with increase the immunity of the system against uncertain parametric variations PI controller has gained prominence over other controllers in the field of photovoltaic applications. But increase in steady state and phase angle error when implemented in stationary reference frame owing to the limited gain of the controller at desired frequency makes it vulnerable.

One of the integral feedback compensator used in grid tied photo voltaic application is the Proportional Resonant (PR) controller. [For example see, S. Buso and P. Mattavelli, Digital Control in Power Electronics, ser. 978-1598291124. Denver: Morgan and Claypool Publishers, 2006.] Conventional Proportional-Integral (PI) or Proportional-Integral-Derivative (PID) controller introduces a pole with infinite gain at the origin and thus fails to minimise the steady state and phase angle error at the fundamental frequency unless and otherwise it is implemented in a synchronously rotating reference frame. On the contrary PR controller has a unique capability of injecting infinite gain at resonance frequency enabling it performance satisfactorily when exposed to AC signal in stationary reference frame. In grid connected photo voltaic application PI compensator outperforms PR controller when used in as a voltage controller for the D.C link capacitor voltage, while as PR controller introduces superior performance in terms of steady state error reduction, faithful tracking of sinusoidal reference and rejection of unwanted disturbance when used as a current controller in inner current control loop.

Model Predictive Control (MPC) is another controller topology which has been used in photo voltaic system (song et al.). This type of model predictive dead beat control technology proves to be advantageous in terms of faster dynamic response and precise tracking of the response signal. Presence of model based dead beat regulator makes it vulnerable towards sudden parametric changes which in turns result into increased inaccuracy and unprecedented delay.[For example see, R. Morfin Magafa, J. J. Rico-Melgoza, F. Omelas-Tellez and F. Vasca, "Complementarity Model of a Photovoltaic Power Electronic System with Model Predictive Control," in IEEE Transactions on Circuits and Systems I: Regular Papers, vol. 66, no. 11, pp. 4402-4414, Nov. 2019, doi: 10.1109/TCSI.2019.2923978.]

In recent past Fractional Order PI (FO-PI) or PID (FO-PID) controller has found its potential utility in different engineering applications owing to its satisfactory dynamic performance and increased control flexibility[ For example see, patent no: US 7,599,752 B2]. Though Fractional Order calculus is an old concept but its application in solving different engineering problems has gained attention over the last few decades. FOPID controller is generalised version of conventional integer order PID controller where the region of operation gets expanded from point to plane. Owing to its restriction in practical implementation limited research has been performed on the use of FO-PID controllers in power electronics application.

In this innovation fractional order PI controller is used for inner current control loop of the grid connected PV system. The tuning of the controller is performed by using different optimization algorithms and their performances are studied to find out which technique gives the best result.

Objectives of the invention The first objective of this patent is to perform a comparative study of various tuning methodologies of FO-PID controller for the current control loop of grid tied PV system in order to develop a closed loop control system for the same.

The second objective is to design a FO-PID controller tuned by different optimization algorithm for the current control loop of grid tied PV system to obtain phase flattening around gain crossover frequency.

The third objective is to tune different parameters of FO-PID (Kp, Ki, K, X, ) usingfmincon algorithm, conventional Genetic algorithm (GA) algorithm, Particle swarm optimization (PSO)algorithm and hybridised GA-PSO optimization algorithm. A comparative study of different performance indices is also discussed in this innovation.

Summary of the invention The present innovation introduces a FOPID controller for the inner current control loop of the grid tied photovoltaic system. Introduction of the FOPID controller incorporates a flat region at the phase curve near the tangent frequency which in turn makes the system immune towards instantaneous impulsive disturbance. Furthermore, different soft computing techniques like GA, GA-PSO and hybrid GA-PSO has been used for optimal tuning of the FOPID controller and its performance has been analysed based on various integral performance errors like IAE, ITSE, etc. The results obtained in simulation clearly indicate that the FOPID tuned with hybridised GA-PSO algorithm gives superior performance in terms of time domain and frequency domain analysis.

Brief description of drawings Figure 1 illustrates the overall block diagram of the grid connected photovoltaic system with fractional order PI controller.

Figure 2 illustrates the step response of the system when tuned with fmincon optimization algorithm under varying loop gain.

Figure 3 illustrates the step response of the system when the controller is tuned with genetic algorithm (GA) under varying loop gain.

Figure 4 illustrates the step response of the system when the controller is tuned with particle swarm optimization (PSO) under varying loop gain.

Figure 5 illustrates the step response of the system when the controller is tuned with hybridised genetic algorithm particle swarm optimization (GA-PSO) under varying loop gain

Figure 6 illustrates the system's frequency response with different optimization techniques.

Figure 7 illustrates the optimal time domain performance index based tuning of FO-PID using fmincon algorithm.

Figure 8 illustrates the optimal time domain performance index based tuning of FO-PID using genetic algorithm (GA).

Figure 9 illustrates the optimal time domain performance index based tuning of FO-PID using particle swarm optimization (PSO) algorithm.

Figure 10 illustrates the optimal time domain performance index based tuning of FO-PID using hybridized genetic & particle swarm optimization (GA-PSO) algorithm.

Figure 11 illustrates the three phase inverter connected to the grid.

Detailed description of the invention Referring to figure 1, a novel FO-PID based control for grid integration of photovoltaic array is invented for efficient transmission and distribution of the electrical power.

As best shown in figure 1, two transformation blocks4 and helps to reduce the number of variables and assist the mathematical modelling of the system. These transformations are

. abc to up transformation, and *.Park transformation (or a to dq transformation).

The matrix for transforming the given abc parameters to up is given in equation (1)

1 1I - X - L1L x

x-l 2 2

Therefore, in up transformation a balanced three phase system is converted into a stationary two phase system. Park's transformation further transforms a stationary two phase system i.e. (up system) to a rotating two phase system (dq system). Therefore, it can be concluded that the resulted coordinate in up transformation is stationary whereas in the parks transformation the resulted coordinate is rotating in nature. The synchronous frame of a three phase system follows two stage transformation. Firstly, the 3 p output vector is transformed into a stationary up frame. Secondly, the up frame is transformed into

a rotating dq frame which rotates around the up frame with a constant angular frequency. Park's transformation is defined by equation (2) xdl FcosO sinO xal (2) Xq]I -sino cosO] X6

As shown best in figure 1, a voltage source inverter lis required to convert dc signal to ac signal for successful grid integration. Before feeding the ac signal to the grid it is passed through a filter 2 to cancel out the ambiguities. The analysis of the VSI and the filter together can be done by referring to figure 11.

Applying KVL in figure 11 equations (3)(4)(5) are obtained.

-V+ RLa + L di" +E =0 dt a

-V + RLb + L di +EhO (4) dt

-V,+ RLc + L dC + E =0 (5) dt Where, L is the filter and grid inductance per phase RL is the inductor and grid resistance per phase Va, Vb and Vc are the per phase output voltage of the inverter. Ea, Eb and Ec are the grid voltages of the three- phase grid. ia, ib and ic are the three-phase current injected to the grid Conversion of the equations (3)(4)(5) into up domain gives equation (6).

did] RL __1 dt L [d d d diq RL Q L Vq-Eqj dt _ Li From equation (6) using parks transformation equation (7) can be obtained did] RL dt L d d d di RL L V -E _dtiL Li Splitting the matrix (7) into d and q axis components, di~ RL 1 (8) dt L L(V -EI) The equation (8), which is basically a d-axis equation, contains q-axis component also. The aim is to design a model which is perfectly decoupled. Hence, the target model should be in the form of did RL • 1 (9) =_-_1+--uI dt L L Where, ud is defined as, Ud -Vd-Ed LiO (10)

That means to generate the required ud, the inverter output voltage V should satisfy, Vd -Ud + Ed+ i, (11) Similarly, the KVL equation for the q-axis current is expressed in equation (12), diq RL 1 (12) _=- 1i +-uq dt L L Where, Uq=V-Eq+Laid (13)

As shown best in figure 1, the PLL block 10is responsible for the minimization of the phase and frequency errors by the help of negative feedback

Performing Laplace transform in equation (9) and equation (12), equation (14) and equation (15) can be obtained respectively

S RL (14) L L RL 1q()+ U S (15) .' S L L Rearranging equation (14) and equation (15), equation (16) & equation (17) can be obtained.

id(s)=ud(s)x RL!R (16)

iq(s)=uq(s)x RL S +j (17)

From equation 16 and 17 it can easily be observed that the open loop transfer function is a first order system. As there is a specific time delay caused due to the switching of the inverter switches by the PWM pulses so, the overall open loop transfer function of the current control loop comes out to be first order time delay system. The delay is assumed to be equal to one switching period T. the switching frequency is taken here to be 1kHz so the T is equal to 0.001.

The open-loop transfer function of the overall system with the fractional order PI (FO-PI) controller can be represented as gain in equation (18). G~s~~() ()K( 1Kl ( K (18) G (s)= G(s)G,(s)=K, + Tsa +1

( Where Gopen (s)= Open loop transfer function of the system.

Gc(s)= Transfer function of the FO-PI controller.

Gp(s)= Transfer function of the first order plus time delay based plant.

For the controller design the parameters L, a and T are already known, but the parameters Kp, Ki and X needs to be estimated. The transfer function of the FO-PI controller can be expressed as

Ge(s)=K,1+K K K = K,1I+ =I. K,1+ Kp-(19) co(9

Since j=e2

2 Then e =cos I 2 ) +jsin(' 2 )

Therefore, Ge(s)=K 1+ Kco- (20) Cos (/IY + jsin (I2 Then the phase of the open-loop system at gain cross-over frequency can be represented by equation (21)

Arg Gope,(j )= tan-' ( - tan-' Lco (21) 1+ Kgo;," Cos I'2X

Where, X =1+ To cos(a 2 ) and Y = T sin(ac 2 ).

On applying the design constraint as given in equation (30), equation (22) can be derived.

p sin (22)

a 1 tan-'-(2)y2 ) - ta tan- 1 Y-w -Lo =-rc +#0 1+Kof cos (

I (2

,Kpc-' sin '

Or, (A2 'l=tan tan-'± + L +# (23) 1+Kp-DAcos

Then, Ki and X can be related as

-A f(24) -K Af-D2 w-tcos +wsin I (2 )I (2)

Where, D2 = tan tan-'Y +)L+#M

Open-loop gain using FOPI controller at the gain cross-over frequency can be represented as given in equation (25).

K.KP 1+Kp " cos +K +o- sin (25) G2(jGcO)I = X2 +Y 2 According to the second constraint (29)

+ KwI sin 12 (26) K.Kj 1+KpaI- cos Z; 2 2

X2+Y (27) Or,1 )=K 1+K~w- cos 2 +2 ~2±K~w- sin~j~

According to the fourth constraint (31)

KPw 1 sin A (28) 2F -E2 = 0 )

2 A + 2KW cos2 + K.2

Where, E2= aT<-X1sin 2 YeOS +L X2y (si(f 2 ) Yo- ff)±

Based on the equations the values of Kp, Ki and can be analysed that maximizes the value of co - &, This optimization problem is solved using fmincon optimization algorithm, Genetic Algorithm (GA), Particle Swarm Optimization (PSO) algorithm and hybrid GA-PSO algorithm. Different algorithm techniques are used to obtain the optimum result. The fractional order controller tuning is tuned based on two ways which are as follows.

1) Frequency domain specifications

2) Time domain based optimal control tuning

In order to perform frequency domain analysis special attention should be given to the following parameters.

a. Omission of the steady state error for closed loop transfer function can be achieved using FOPID controller. b. The gain crossover frequency can be expressed as follows G(joc) = C(jao)P(jo) = 1 (29) c. The phase margin cDm can be expressed as Arg [G(jco)] = Arg [C(jo) P(jo)] -;Tc+# (30) d. To ensure robustness against gain variation the phase of the open loop transfer function across the gain crossover frequency must be constant. Thereby the derivative of the phase of the OLTF w.r.t. frequency becomes zero which is represented in equation (29).

d(Arg[G(j.)]) =0 (31)

e. In order to reject noise at higher frequency the magnitude of the closed loop transfer function of the overall system must be a very small quantity. So, for a specified frequency the magnitude of the closed loop transfer function must be less than some specified gain which is specified in equation (32) where A is the specified gain at specified frequency.

T( jw)|= C(j)P(j) 1+C(jcO)P(jo) <AdB;Vw>wradIs (32) 1+C(jw)P(jw) dB f. In order to eliminate unwanted disturbance at the output and achieve superior reference tracking the sensitivity function must have a small magnitude at lower frequency range. This criteria is expressed in mathematical form as shown in equation (33) where B specified gain of the sensitivity function.

S( jo) = IBdB;IVo ! ,rad / s (33) 1+C(jw)P(jw ) dB The control signal of the FOPID controller for a control loop error in time domain is expressed as

u(t)=KJe(t)+IK .D-[e(t)]+KdD"[e(t)] (34)

Where D denotes the integral action of fractional order and D" denotes the differential operator of fractional order. The error minimization criteria are customized by proper selection of a time domain integral performance index in order to have an improved control action i.e.

a. Integral of Absolute Error (IAE) (35) JAE = |e,(t)|dt 0

b. Integral of Square Error (ISE): (36) ISE = ej(t)dt 0

c. Integral of Time multiplied Absolute Error (ITAE): (37) ITAE f t e (t)|dt 0

d. Integral of Time multiplied Square Error (ITSE): (38) ITSE = terr(t)dt 0

e. Integral of Squared Time multiplied by Error, all to be Squared (ISTES): 2 (39) ISTES f[t2e, (t) ]dt 0 f. Integral of Squared Time multiplied by Square Error (ISTSE):

(40) ISTSE = t2er2(t)dt 0

The multiplication of the time factor with power > 1 in the performance indices expressed in equation (36),(38),(39) and (40) decreases the probability of oscillation in the time domain response curve and helps to minimize the settling time of the closed loop transfer function of the system. Similarly, high value of the powers in the error term assists in significant reduction in the probability of high overshoot. It is not possible to evaluate an integral practically upto infinity. Therefore, the upper limit should be chosen sufficiently high so that the transients occurring in the system decays out satisfactorily within that time interval.

The different algorithms used for optimization are Particle Swarm Optimization, genetic algorithm, hybrid GA-PSO etc. Firstly, the particle swarm optimization PSO algorithm is based on the social behaviour of birds and fishes. It was originally developed by Kennedy and Eberhart in 1995. The mathematical equations used in PSO for the searching process are:

The position updating equation is given as follows.

xi (t+1)--xi(t)+vi (t+1) (41) The velocity updating equation is given as follows:

vi(t+1)=vi(t)+c(pi-xi(t))RI+c 2 (g-xi(t))R2 (42)

Where pi is termed as "personal best" of the particle, while g is the "global best". ci(cognitive coefficient) and c 2(social coefficient)are real valued acceleration constant, usually lying in the range of 0 to 4 (0 5ci,c 2 < 4). On the other hand, the two random numbers which are generated from a uniform distribution in [0, 1] are R1 and R2 .The different procedures for performing the PSO algorithm is represented in the form of a flow chart as shown best in figure 12.

1. Initialize the particle position and velocity for each particle. 2. Estimate the fitness value of each particle. 3. The particles fitness value is analogized with thepBest for each individual particle. If the latest fitness value of the particle is better than thepBest, then thepBestgets replaced with the latest value and the position of the particle after kth iteration is set as p 4. The particle with the best fitness value is selected and set asgBest. The position of the particle corresponding to the best fitness value is set as pk 5. The equations for the estimation of the particle's position and velocity in particle swarm optimization technique is given as follows xk= xk + v (43)

v r+cr p -) (44)

6. Process is repeated until optimum result is obtained.

Secondly, genetic algorithms (GAs) are adaptive meta-heuristic search algorithms based on natural selection. This was first invented by John Holland in the 1960s. The different steps involved in GA are as follows.

1. Initialize a randomly generated population consisting of n number of chromosome.

2. Estimate the fitness value of each individual present in the population. 3. Repeat the following steps until and otherwise a new population is created. 4. Create a new population by following the new steps: * [selection] Pick two parent chromosome of better fitness value from the population. Better the fitness value more is the chance of getting selected. * [crossover]Cross over the selected parents to form the new infant with the help of the crossover frequency. In the absence of the crossover, the infant is a precise copy of the parents. * [Mutation]Mutate the infants further at each locus to produce more infants. * [Accepting] Add the new infants to the new population 5. The process is repeated until the targeted n number of infants are created 6. The present population is replaced with the latest population. 7. The process is repeated until global optimum is reached. The different steps involved in hybrid GA-PSO are as follows.

1. Initial position xos of the ith particles randomly taken x,- (xminx) from a uniform distribution in the range[xmin,X-]. xmin: design variables' lower bound Xma: design variables' upper bound 2. Velocity vector is used to update the current position Pbest of each particle in the swarm based on the memory gained by each particle. 3. Update of current position of each particle gbestis done from the knowledge of the swarm as a whole. 4. Position and velocity are updated after each iteration x4 =4 + vk (45)

Where,

v = wv + Cr p- + c2r (46)

5. Some particles of new generation are selected and then GA is applied to them separately.

GA,,, = GAn max- (GA m -- GAnun, mi) (47) PSOmax iter

Where, PSO: - current PSO iteration PSOmax iter: - maximum number of generation in PSO 6. New population is generated by replacing the points in current population with better points via genetic algorithm. 7. After that population size and maximum iterations of GA changes w.r.t iterations of PSO as follows.

GsGAmnPS + GAmas-GAmin PS) (48) PSOmitr

GA iter = GA itcr+ PSOi f (GAmaxiter GAminit,) (49) PSOm -it

8. Process is repeated until global optimum is reached. The controller is tuned with fiincon, GA, PSO and hybridised GA-PSO respectively. The different Parameter initialization for different types of algorithm are best shown in the following table

SL. Genetic Algorithm(GA) Particle swarm optimization Hybrid GA-PSO No. (PSO) 1. Generation = 1000 Swarm size = 50 GAminps= 10 2. Crossover: Two-point C1 = 1.5 GAminIT = 10 crossover 3. Population size = 20 C 2 = 1.5 GAnumiMax =20

4. Crossover Probability = Moment of inertia = 5 GAnumMin= 1 0.7 5. Mutation Probability = 0.5 Iterations = 30 B = 15 6. Iterations = 30 y= 10

As shown best in figure 2 the time domain step response of the closed loop system is depicted with 200% variation in the loop gain when tuned with fmincon. This variation results in large increase in the maximum overshoot accompanied by absence of the iso damping property. The comparative study of the time domain performance indices of the FOPI controller tuned with fmincon is tabulated in table 1.

Kp Ki k %Mr tr ts Ess FOD

IAE 8.072501 6.011203 1.512702 7.931 0.23202 2.493 0.005402 0.884301

ISE 1.506303 1.546401 0.574101 34.71 0.22604 2.011 0.005703 0.879203

ITAE 1.395201 9.513902 1.432903 27.00 0.41001 2.472 0.005601 0.932102

ITSE 0.577102 1.914004 0.982501 29.00 0.23203 1.817 0.005107 1.067106

ISTES 1.556104 1.01901 0.912606 30.73 0.44305 3.86 0.005405 1.432505

ISTSE 3.501601 5.513903 1.065402 11.51 0.32202 3.585 0.005302 1.274602 Table comparativestudy of different performance indices usingfmincon

As shown best in figure 3 and figure 4 the time domain step response of the closed loop system varying from 100% to 300% is depicted when the controller is tuned with GA and PSO respectively. This variation results in little increase in the maximum overshoot accompanied by the reduction in the rise time of the system. The comparative study of the time domain integral performance indices of the FOPI controller when tuned with GA and PSO algorithm is tabulated in table 2 and 3 respectively.

K, Ki k %Mr tr ts Ess FOD

IAE 3.278301 1.770012 1.12323 4.71 1.8501 6.173 0.005704 0.843401

ISE 0.660706 3.603 0.94332 34.203 0.33106 2.841 0.005906 1.142906

ITAE 2.080405 0.913904 1.059103 5.10 0.22404 4.472 0.005803 1.59804

ITSE 3.914103 3.913906 1.31704 6.162 0.53602 2.984 0.006102 1.941908

ISTES 0.359101 0.913907 0.863202 12.80 1.1201 3.742 0.005604 1.048302

ISTSE 2.640305 3.000109 1.021706 5.823 2.0206 7.091 0.005501 0.905501 Table 2comparativestudy ofdifferent performance indices using genetic algorithm

Kp Ki k %Mr tr ts Ess FOD

IAE 2.160102 1.019401 0.153302 2.51 1.7205 4.0301 0.005602 0.878405

ISE 2.89604 5.513906 0.652104 25.32 0.33903 2.0702 0.005405 0.800203

ITAE 5.993501 1.988902 1.053205 6.534 0.69604 1.9405 0.005501 0.502405

ITSE 3.435906 6.496906 0.998201 19.91 0.64802 4.0904 0.005206 1.395404

ISTES 4.282302 9.513904 1.06508 24.30 0.48601 3.5401 0.005102 1.281601

ISTSE 1.651101 1.913903 0.536206 6.532 0.69603 1.9403 0.005501 1.092207 Table 3comparative study of different performanceindices usingparticle swarm optimization algorithm

As shown best in figure 5 the time domain step response of the FOPI controller tuned with hybrid GA-PSO is depicted with a gain variation of nearly 500%. This variation resulted into no increase in the maximum overshoot accompanied by the decrease of the rise time. Thus the system satisfactorily exhibits iso-damping property when tuned with hybridised GA-PSO optimization algorithm. The comparative study of the time domain integral performance indices of the FOPI controller when tuned with hybrid GA-PSO algorithm is tabulated in table 4.

K, Ki k %MP tr ts Ess FOD

IAE 5.985101 0.913902 1.028503 1.752 0.81203 1.251 0.00583 0.17583

ISE 0.231703 0.989903 0.215405 7.926 1.8804 5.683 0.00542 1.45154

ITAE 7.919504 5.513905 1.192507 1.174 0.28702 0.344 0.00574 0.03052

ITSE 4.903206 9.513904 1.334508 3.019 0.68406 4.672 0.00556 1.48897

ISTES 0.584705 2.011202 0.729103 12.94 0.19809 1.116 0.00578 0.41946

ISTSE 1.692802 1.019391 0.862001 1.245 1.9502 3.034 0.00524 0.40854 Table 4comparativestudy of different performance indices using hybrid genetic algorithmparticleswarm optimization

Figure 6 shows the phase and magnitude plot of the system w.r.t. frequency (i.e. bode diagram) using different optimization techniques. It can be easily observed from the figure that the phase plot shows that flat phase region around the flat phase region around the tangent frequency when tuned with genetic algorithm, particle swarm optimization and hybrid GA-PSO algorithm whereas the flat phase region is very minimal when tuned with fmincon algorithm. Flat phase region along the tangent frequency ensures the robustness of the system against impulsive disturbance. The more the extent of flat phase region around the tangent frequency higher will be the system immunity against certain disturbances. It can also be concluded from figure 6 that the extent of the flat phase region is maximum when corresponding system is tuned with hybrid GA-PSO algorithm. Thereby exhibiting iso-damping property.

From figure 7-10 it can be observed that ITAE error criteria among all the performance index for tuning of FOPID controller results into minimal overshoot and faster dynamic response. It can also be concluded from these figures that IAE performance when tuned with hybridized GA-PSO algorithm with a peak overshoot of 1.75% along with settling time (ts) of 1.25 sec. It can also be inferred from these figures that ISE performance ISE performance index results into lower overshoot of 7.92% when tuned with hybrid GA-PSO algorithm but the settling time gets increased. Furthermore, it can also be concluded that ISTES performance index when tuned with hybrid GA PSO optimization algorithm results into optimal performance in terms of reaching the steady state with minimal peak overshoot. ISTES performance index results into superior performance with a peak overshoot of 12.8% when tuned with GA based optimization algorithm but the settling time gets increased by 3.74 sec while it gets decreased by 1.11 sec when tuned with hybrid GA-PSO algorithm.

Claims

Novel Fractional order PID based robust controller designing tuned with hybridised GA-PSO algorithm for grid connected photovoltaic system.

We claim, 1. A FO-PID controller is used in the inner current control loop in the grid tied photovoltaic system.
2. The use of FO-PID controller as described in claim 1 introduces phase flattening at corner frequency which enhances immunity of the system towards instantaneous disturbance.
3. Introduction of the FO-PID controller in the current control loop not only increases robustness but also provides extra degree of freedom.
4. The control parameters of the FO-PID controller i.e. K, K 1, KD, X and g are tuned using fmincon, Genetic algorithm(GA), Particle swarm optimization(PSO) and hybrid GA-PSO algorithms.
5. Different performance parameters like IAE, ISE, ITAE, ITSE, ISTES, ISTSE has been analysed for different algorithms as mentioned in claim 4.
6. The optimum result in the photovoltaic system is obtained through GA-PSO algorithm where increase in scalar gain up to a level of 500% reduces the rise time while maintaining the same overshoot.

Figures

8 1 2 FILTER GRID RL L PV RL L Array Cdc VSI Vdc RL L 2021101100

MPPT PWM 7 BLOCK mabc 9 ab Iabc Vdc  Eabc c dq md mq

Vdc* Vd

Ed  Liq Eq  Lid + + + +

ud uq

FOPID FOPID 5 ab ab 4 c c dq + + - dq Iqref = 0 VOLTAGE - CONTROL Idref Iq Ed LOOP Id  Eq

PLL 1 6 0 3 CURRENT CONTROL LOOP

Figure 1

Figure 3 Figure 2

Figure 5 Figure 4

Figure
7 Figure 6

Figure 9 Figure 8

Figure 10

L Ea Va RL ia

L PWM Vb RL ib Eb INVERTER GROUND L Ec Vc RL ic

Figure 11

Start

Set particle position and velocity

Estimate the fitness value of each particle

Set the local best fitness and local best position 2021101100

Set global best fitness

Update velocities and position of each particle.

Estimate the fitness value of each particle.

NO Latest fitness < local best fitness YES Put local best fitness = latest fitness

Latest fitness < global best fitness YES Put global best fitness = latest fitness

NO Terminating criterion met?

YE
S End Figure 12