Parameter estimation of photovoltaic model via parallelparticle swarm optimization algorithmJieming Ma1,2,*,†, Ka Lok Man2, Sheng-Uei Guan2, T. O. Ting3 and Prudence W. H. Wong4
1School of Electronic and Information Engineering, Suzhou University of Science and Technology, No. 1 Ke Rui Road, Suzhou High-TechZone, Suzhou, Jiangsu Province, 215009, China2Department of Computer Science and Software Engineering (CSSE), Xi’an Jiaotong-Liverpool University, Science Building, No. 111Ren’ai Road, Suzhou Industrial Park, Suzhou, Jiangsu Province, 215123, China3Department of Electrical and Electronic Engineering (EEE), Xi’an Jiaotong-Liverpool University, Science Building, No. 111 Ren’ai Road,Suzhou Industrial Park, Suzhou, Jiangsu Province, 215123, China4Department of Computer Science, University of Liverpool, Ashton Building, Ashton Street, Liverpool, L69 3BX, UK
photovoltaic cells; modeling; parameter estimation; parallel algorithms; solar energy
Correspondence
*Jieming Ma, School of Electronic and Information Engineering, Suzhou University of Science and Technology, No. 1 Ke Rui Road,Suzhou High-Tech Zone, Suzhou, Jiangsu Province, 215009, China.†E-mail: [email protected]
Received 15 November 2014; Revised 21 March 2015; Accepted 6 May 2015
1. INTRODUCTION
The progress in photovoltaic (PV) research and develop-ment has been improving rapidly over the past years. Theprominent features of solar energy – clean, abundant, andrenewable – make the PV generation popular and promis-ing in various industrial applications [1].
Since the initial silicon PV cell was developed by usingthe single crystal, varieties of silicon materials have beenused to develop PV cells. For example, polycrystallineand amorphous silicon cells were designed to be lessenergy intensive. Thin silicon cells make a compromisebetween crystalline and amorphous cells and were reportedto achieve better efficiency and stability [2]. With numerousPV cells made of various semiconductor materials usingdifferent manufacturing processes, a general performanceestimation tool, known as PV electrical model, is crucialto predict the electrical characteristics of these cells beforeinstallation. Unfortunately, PV electrical model cannot be
directly used because of the lack of proper model parame-ters characterizing PV cells. The term parameter estima-tion refers to the process of using sample data toestimate parameters of the selected PV electrical model[3]. With the parameters obtained in such a way, the dif-ferences between simulated and experimental data can beminimized considerably.
In the literature [4], conventional parameter estimationmethods are classified into two categories:
i. analytical technique [5–8] represents model parame-ters mathematically by a series of equations;
ii. numerical technique [9–11] extracts parameters uti-lizing numerical methods to minimize the error ofthe applied model.
Feasible as they are, both of them have inevitable de-fects. The former method addresses the parameter estima-tion problem by analytical expressions in terms of the
INTERNATIONAL JOURNAL OF ENERGY RESEARCHInt. J. Energy Res. (2015)
Published online in Wiley Online Library (wileyonlinelibrary.com). DOI: 10.1002/er.3359
key points on the PV current–voltage (I–V) curve (e.g.,the maximum power point, short-circuit current Isc, andopen-circuit voltage Voc, etc.). Its errors can be significantand cannot be further improved if these fundamental ele-ments are incorrectly specified. Numerical parameter ex-traction is normally considered as an accurate approachin parameter estimation as all the measured data are usedin the calculation. It is axiomatic that its performance de-pends on the type of fitting algorithm, the cost function,and the initial values of parameters to be extracted [9].Moreover, many numerical algorithms can be computa-tionally expensive as the size of the required data is rela-tively large.
More recently bio-inspired metaheuristic algorithms,such as genetic algorithm [12], particle swarm optimiza-tion [13,14], bacterial foraging algorithm [15], patternsearch (PS) [16], simulated annealing (SA) [17], differen-tial evolution [18,19], and cuckoo search [20] have beenproposed to determine the values of PV-model parameters.Albeit accurate, most of these methods apply multipleagents or particles in random search and do not facilitatea meaningful improvement in computational efficiency.Today’s programming environments, such as Open Com-puting Language (OpenCL), are more multifaceted, andthey enable an algorithm to be executed in a wide rangeof central processing units (CPUs), digital signal proces-sors, field programmable gate arrays, and graphic process-ing units (GPUs) [21]. These programming environmentsor application programming interfaces exploit the capabil-ities of computing devices using the languages that only re-quire the highest-level descriptions of parallel processmanagement [22].
With the aim of distributing the workload of a parame-ter estimation algorithm appropriately to computing de-vices in parallel mode, this paper presents a form ofcomputation in which the PSO-based parameter estimationalgorithm is carried out simultaneously. It is desirable thatthe parallel particle swarm optimization (PPSO) outper-forms the sequential particle swarm optimization (SPSO)in two aspects:
i. the computational speed tends to be faster than theSPSO with the same amount of work load;
ii. more computational units can be utilized in optimi-zation, and thus, it is scalable.
The accuracy and computational efficiency of the pro-posed method are evaluated by identifying the parametersof the two most widely applicable PV electrical models.The remainder of this paper is organized as follows. Thenext section briefly illustrates PV electrical models. Thisis followed by the problem formulation in Section 3.Section 4 elaborates on the SPSO, followed by the imple-mentation of the proposed parallel method, the PPSO. Ex-tensive simulation is run on CPUs and GPUs, and theobtained results are discussed in Section 5. Finally,Section 6 derives conclusions with some proposed insightsfor future work.
2. MATHEMATICAL MODELING OFPHOTOVOLTAIC DEVICES
2.1. Single-diode model
As briefly discussed in the introduction, PV system de-signers are usually interested in modeling PV devices asmapping of the electrical characteristics is of significanceto the understanding, optimization, and development ofPV power harvesting systems.
The elementary PV device is a PV cell, which is basi-cally a semiconductor diode that generates a reverse cur-rent when its p–n junction is exposed to light. Thisreverse current is termed as photocurrent Iph. In darkness,the PV cell behaves like a diode, and thus, its dark I–Vcharacteristics are usually mathematically expressed byShockley diode equation [23]:
ID1 ¼ Io1 eVDA1Vt � 1
� �; (1)
where VD represents the electrical potential difference be-tween the two ends of the diode, Io1 denotes the reversesaturation current, and A1 is the diode ideality factor. Vt
is known as thermal voltage, and its value can be esti-mated as a function of temperature T, namely, Vt = kT/q,where k and q represent the Boltzmann constant(1.380650 × 10� 23 J/K) and the electron charge(1.602176 × 10� 19 C), respectively. Assume that thesuperposition principle holds, the full I–V characteristicis simply the sum of the dark and illuminated I–V charac-teristics:
I ¼ Iph � Io1 eV
A1 Vt � 1� �
: (2)
In the literature [24] [25], 2 is also the mathematicalexpression of an ideal PV model, in which the Iph ismodeled as a current source. As reported by the authorsin [23] and in [26], the output current I is dependent onthe resistances of p and n bodies, the contact resistanceof the n layer with the top metal grid, the resistance ofthe grid, and the contact resistances of the metal basewith the p semiconductor layer, as well as the leakagecurrent of the p–n junction. These losses are roughly rep-resented by series resistance Rs and shunt resistance Rp in thesingle-diode model, whose circuitry diagram is shown inFigure 1(a). The corresponding equivalent circuit equationis expressed in 3 [27]:
I ¼ Iph � Io1 eVþIRsA1Vt � 1
� �� V þ IRs
Rp: (3)
2.2. Double-diode model
The dark characteristics of PV cells have been intensivelystudied by many researchers. A simple approach of im-proving the single-diode model is to model the junction
recombination, which can be achieved by adding a seconddiode in parallel with the first one [28]. Figure 1(b) showsthe circuitry diagram of the double-diode model, and itsmathematical model equation is as follows:
I ¼ Iph � Io1 eVþIRsA1 Vt � 1
� �� Io2 e
VþIRsA2 Vt � 1
� �� V þ IRs
Rp;
(4)
where Io1 and Io2 are the reverse saturation currents of thefirst and second diode, respectively. Similarly, the seconddiode’s ideality constant is denoted by A2.
2.3. Photovoltaic module model
In a large PV generation system, PV modules are used asbasic components rather than PV cells because the outputpower of a PV cell is limited at high voltage levels.Researchers develop the PV module model so as to pre-dict the I–V characteristics before modeling the wholesystem.
Because the PV module is a packaged, connected as-sembly of Ns PV cells, its output voltage and resistanceare scaled in accordance with the following rules:
V ′ ¼ Ns�V ; I′ ¼ I;
R′s ¼ Ns�Rs; R′p ¼ Ns�Rp;(5)
where V′ and R′s here represent the voltage and series re-sistance of the PV module, respectively.
For the convenience of description, the models discussedin this paper are abbreviated as follows: (i) SDC, single-diode cell; (ii) DDC, double-diode cell; (iii) SDM, single-diode module; and (iv) DDM, double-diode module.
3. PROBLEM FORMULATION
Based on an optimization algorithm, the parameter estima-tion method minimizes the differences between calculatedcurrent and measured data by adjusting PV parameters[20]. After importing several parameters, the parameterestimation algorithm starts evaluating possible solutionsby using the objective function with the measured I–Vdata. In general, the objective function is formulated bythe root mean square (RMS) error frms, which serves toaggregate absolute differences into a single measure ofpredictive power. If the number of experimental data isdenoted by N, the RMS error can be mathematicallydescribed by the following equation:
f rms ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1N
XNd¼1
f D V^; I^;X� �� �2
vuut ; (6)
where V^ and I^denote the measured voltage and current, re-spectively, fD(V^, I^, X) is the objective function for the dth
data, and X is a vector representing the model parameters.Take the SDC for an example, f D V^; I^;X�
is a homoge-neous form of 3, namely,
f DðV^; I^;XÞ ¼ Iph � Io1 eVþ I RsA1 Vt � 1
�� V^þ I^Rs
Rp� I^;
�(7)
where X is a vector involving the model parameters Iph, Io1,A1, Rs, and Rp.
4. PARAMETER ESTIMATIONALGORITHM
4.1. Sequential particle swarm optimization
By mimicking the swarm behavior of fishes and birds,Kennedy and Eberhart [29] developed a nature-inspiredmetaheuristic algorithm in 1995. This derivative-freemethod is particularly suited for continuous variable prob-lems and has been successfully applied to many engineer-ing optimization problems. In [30], Kennedy et al.implemented the algorithm in a procedural C-program.We call it SPSO in this paper.
The basic idea behind the SPSO is to search a space byadjusting the trajectories of particles, which represent pos-sible solutions of the objective function. The pseudocodedepicting the SPSO is shown in Algorithm 1. Assume thatthe swarm size is P and the problem dimension is D. Theith (i= 1, 2,…,P) particle in jth (j= 1, 2,…,D) dimensionis denoted by xi,j. Similarly, the ith velocity in jth dimensionis vi,j.
The SPSO firstly initializes the algorithm parameters(e.g., inertia weight, learning parameters, etc.) as wellas the velocity and position of each particle. In aniteration t (t = 1, 2,… tmax), the fitness of particles areevaluated individually by its objective function. When
(a)
(b)
Figure 1. Circuitry diagram of photovoltaic models (a) single-di-ode model (b) double-diode model.
a particle i arrives a location that is better than any posi-tions it arrived, it records the new position as local bestposition pbesti. In a swarm of particles, there are P localbest positions. Among them, the one with the best solu-tion is termed as global best position gbest in the litera-ture. Kennedy and Eberhart proposed that themovements of particles are mainly attracted toward thepbesti and gbest, and the new position of a particle initeration t + 1 can be mathematically expressed in the fol-lowing manner:
xtþ1i;j ¼ xti;j þ vtþ1
i;j ; (8)
where vtþ1i;j is the velocity, expressed as
vtþ1i;j ¼ wvti;j þ α∈1 xti;j � gbestt
� �þ β∈2 xti;j � pbestti
� �:
(9)
In 9, the notations α and β are the learning parameters.Typically, α≈ β≈ 2. The two random vectors ∈ 1 and ∈ 2
are in the range between 0 and 1. The inertia weight w isused to balance global and local search abilities. It can betaken either as a constant from 0.5 to 0.9 for simplicity ora linear function in terms of iteration t. In this paper, thevalue of w is defined as
wt ¼ wmax � wmax � wminð Þ t
tmax; (10)
where wmax and wmin represent the maximum and mini-mum of the w, respectively.
Normally, lower and upper boundaries are set to ensurethe particles are within the predetermined range. If the ve-locity or position of a particle exceeds the upper bound, itwill be reset to the maximum and vice versa. The algorithmwill then continue to evaluate the fitness, and a new itera-tion starts. The SPSO will not stop searching for better so-lutions until it meets the stopping criterion.
4.2. Implementation of parallel particleswarm optimization
The workload behaviors can be generally classified intotwo types: data intensive and control intensive. In fact,there is no best architecture that runs optimally on alltypes of workloads. According to [21], control-intensiveapplications tend to run faster on super-scalar CPUs,where significant computing efforts have been devotedto branch prediction mechanisms, while data-intensiveapplications tend to run fast on vector architectures,where the same operation is applied to multiple data itemsconcurrently.
The structure of SPSO has a mix of the workload char-acteristics. Consider the fitness evaluation function. In pro-cedural C-program, the RMS errors are computed particleby particle in a for loop. In order to parallelize this func-tion, we choose to generate a separate execution instanceto perform fitness evaluation for each particle. Figure 2 de-picts the concurrent process. With the measured I–V data,the RMS errors can be calculated concurrently in a kernel,which actually is a piece of code executing tasks on amulti-core processor. The fitness evaluation process for aparticle is independent of any other particle, and therebypossesses significant data level parallelism. On the otherhand, the function updating the swarm’s velocities and
Figure 2. Parallel computing framework utilizing a swarm ofparticles.
positions, especially the process checking whether thevalues exceed the predefined bounds, can be assigned tothe category of control-intensive applications because it in-volves explicit flow-control constructs such as if-then-else.From these considerations, it is desirable that a program-ming framework with the capability of execution across awide range of device types so that the workload can beexecuted most efficiently on a specific style of hardwarearchitecture. The OpenCL, managed by the nonprofittechnology consortium Khronos Group, is such a heteroge-neous programming framework that supports a wide rangeof levels in parallelism and efficiently maps to a variety ofcomputing devices [21]. A host and a device-side languageare both defined in the OpenCL. The former offers a man-agement layer that supports efficient plumbing of compli-cated concurrent programs, while the latter maps theheavy work load into a wide range of memory systems.
In our implementation, the main program was writtenin OpenCL code. Application programming interfaceswere used to configure a context which allows commandsand data passing to the device. Figure 3 represents thewhole algorithmic flow of the proposed PPSO. After ini-tialization, velocities and positions of particles aretransfered from the host to the device. In the OpenCL ker-nel function, we choose to decompose fitness evaluationsto perform the evaluations concurrently on a multi-processor device. Global synchronization or barrier func-tion is applied to ensure that all of the fitness evaluationsare completed before they are transfered back to the host.The local best and global best positions are aided to deter-mine the new velocities and positions of particles. Thealgorithm will then return to the parallel process by evalu-ations through the objective function until the stoppingcriterion is satisfied.
Figure 3. Flow chart of the parallel particle swarm optimization algorithm (a) the main program (b) parallel evaluations of root meansquare (RMS) errors.
The proposed PPSO was implemented in OpenCL, andsimulations were performed under Microsoft’s Windows 764-bit operating system. Its algorithm parameters were setas the learning factors c1 = c2 = 2, the maximum inertia factorwmax = 0.9, and the minimum inertia factor wmin = 0.4. Withthe aim of conducting a comprehensive evaluation, bothsingle-diode model and double-diode model were appliedin parameter estimation. The experimental I–V data of a57mm diameter commercial silicon PV cell (R.T.C. France)and a PVmodule (Photowatt-PWP 201) comprising 36 poly-crystalline silicon PV cells are considered as test examples inthis paper. Their values were obtained under the controlledconditions from an automated measuring system with aCBM8096 microcomputer as demonstrated in [10]. It is as-sumed that all the silicon cells in a PV module are identicaland work under the same temperature (R.T.C. France PV cellat 33°C and Photowatt-PWP 201 PV module at 45°C).
Extensive simulation results and statistical analysis arepresented in the subsequent sections. Section 5.1 studiesthe parameter estimation capability by evaluating the evo-lution performance and distribution of fitness values forthe proposed PPSO method. Besides RMS error, the meanabsolute error ē is used to evaluate how close the simulatedcurrent values I are to the measured data Î. It is mathemat-ically expressed as
e ¼ 1N
XNd¼1
Id � I^dj: (11)
In Section 5.2, we demonstrate how the PPSO methodoutperforms its sequential version in terms of computa-tional speed. Speedup is used to qualify the ratio of se-quential execution time to parallel execution time:
S ¼ Ts
Tp; (12)
where Ts is the execution time of sequential algorithm onthe host processor and Tp is the execution time of parallelalgorithm on multi-core devices.
5.1. Parameter estimation capability
Table I shows the estimated parameters for different PVelectrical models obtained from the best of 30 runs of theproposed PPSO method, in which the swarm size andmaximum iteration number are set to 2048 and 80,000,respectively.
In order to make a comprehensive comparison, the pa-rameters estimated by the other methods, such as leastsquare optimization [10], pattern search (PS) [16], andsimulated annealing (SA) algorithms [17], are also listed
Table I. Estimated parameters of different PV electrical models using various methods.
in Table I. Among these test results, the ē obtained by thePPSO with DDC achieves the lowest value, recording6.6415E-4, which is 51.85% lower than the SA with thesame PV electrical model, and is approximately 3% lowerthan the PPSO with the SDC. It is observed that ē can bereduced if we apply the DDC instead of SDC. However,the complicated model DDM does not always give accu-rate simulation results in parameter estimation. In the sim-ulation results for the PV module, the accuracy cannot beimproved by using the DDM.
Figure 4 shows the qualitative representation of theaverage evolution performance of the PPSO method fordifferent electrical PV models. The fitness value, namely,the RMS error, is averaged over 30 runs of the applied
parameter estimation methods. In general, the average fitnessof PPSO drops dramatically in the convergence traces, espe-cially before the first 2000 iterations. The average fitness ofthe SDM reaches the lowest value after 10,000 iterationsby using the PPSO with 2048 particles as seen in the plots.Whichever model we use, the algorithm with a larger swarmsize tends to be faster in terms of convergence speed.
Based on the previously discussed analysis, the PPSOshows its consistent performance of extracting the parame-ters from the experimental data with a high accuracy.Figure 5 further demonstrates the distribution of the fitnessvalues obtained from the PPSO method after 20,000 gene-rations. The swarm size is respectively set at 64, 256, 512,1024, and 2048. It is observed that the medium values tend
Figure 5. Box plots depicting distribution of fitness values obtained by the PPSO with relevant PV models (a) single-diode cell,(b) double-diode cell, (c) single-diode module, and (d) double-diode module.
Figure 6. The execution time and average fitness of sequential and parallel parameter estimation for photovoltaic models on Intel i7-4770 k central processing unit (a) sequential particle swarm optimization (SPSO) with single-diode cell (SDC), (b) SPSO withdouble-diode cell (DDC), (c) SPSO with single-diode module (SDM), (d) SPSO with double-diode module (DDM), (e) parallel particle
swarm optimization (PPSO) with SDC, (f) PPSO with DDC, (g) PPSO with SDM, and (h) PPSO with DDM.
to decrease with the increasing of the particle number,which agrees well with the simulation results in Figure 4.The trend of the decrease implies that the PPSO with alarge swarm size has a higher possibility of achieving goodfitness value without changing the iteration number. In thissense, the PPSO can improve the accuracy in a unit time ona specified device.
From another perspective, the PPSO executes particleevolution processes concurrently with the applied comput-ing device, and in such a way, the efficiency of parameterestimation can be improved. The speedup, as well as theparallel efficiency of the proposed PPSO method, will bediscussed in the subsequent subsection.
5.2. Speedup and parallel efficiency
In the implementation of PPSO, we follow a hybrid ap-proach whereby the fitness is evaluated in the kernel andthe other processes (e.g., position updating and velocityupdating) are performed in the host device. The resultsaveraged over 30 trials of the PPSO with 20,000 iterations.To show how much the parallel processing speeds up thefitness evaluation function, the execution time on the hostand the device are denoted by bars with light and darkcolors separately. A comparison of the total execution timeand fitness values, both measured in the proposed PPSO-based parameter estimation and its sequential counterpart forPV electrical models, is made in Figure 6. In Figure 6(a)–(d),
we observe that the execution time on the fitness evaluationmakes up much larger percentage than that on the otherfunctions in sequential processing. Except for the fitnessevaluation, the codes of the PPSO and SPSO are exactlythe same, and therefore, their execution time on the hostis similar. As seen in Figure 6(e)–(h), the computation timetakes in fitness valuation function can be significantly re-duced by the PPSO. In addition, the total execution timefor the DDC is longer than the one for SDC. This happensbecause the computational complexity of the double-diodemodel is higher than that of the single-diode model.
To further evaluate the parallel performance of the pro-posed PPSO algorithm, We evaluate the speedup of PPSOon a number of multi-core computing devices, whichincludes Intel Core i3-3220 CPU (2 cores, 2 threads,3.3 GHz), Intel Core i5-3470 CPU (4 cores, 4 threads,3.2 GHz), Intel Core i7-4770K CPU (4 cores, 8 threads,3.5GHz), NVIDIAGeForce GTX 760 GPU (1152 ComputeUnified Device Architecture (CUDA) cores, 980MHz),NVIDIA GeForce GT 620 GPU (96 CUDA cores,700MHz), and AMD Radeon R9 200 GPU (2048 streamprocessors, 1150MHz).
Table II lists the average speedup of the PPSO with dif-ferent swarm size on these devices. From the simulation re-sults, some conclusions can be drawn. In most tests, thespeedup of the PPSO is above 1. In other words, the execu-tion time of the PPSO is normally shorter than that of theSPSO. Moreover, the parallel program with larger swarm
Table II. PPSO’s speedup on heterogeneous computing platforms.
size tends to perform at a faster speed. The exception madeby the PPSO with a swarm size of 64 particles. Its speed iseven lower than the corresponding sequential version onGT 620 and GTX 760, which implies the overheads on datacommunication and kernel scheduling are more significanton the two GPUs. With more applied particles, the speedupappears to be a larger ratio. This is because the speedup onthe applied multi-core devices over the host processor islarge enough to compensate for the initial data transfer cost.From Figure 6(a) and (e), we can conclude that the PPSOcan achieve better fitness values if taking the same amountof execution time as the SPSO. Similar trends are observ-able in the speedup for the DDC, SDM, and DDM. Amongthese tests, the parallel program with Intel i7-4770 k andAMD R9 200 series exhibits the minimum execution time,recording an average speed up ratio from 3.4426 to 4.6062for a swarm size set of 2048 particles.
6. CONCLUSION
In this work, a parallel computing paradigm has beenshown to speed up the parameter estimation process for var-ious PV models. The proposed PPSO implemented inOpenCL can be executed in a wide range of multi-corecomputing devices. Fitness evaluations were performedconcurrently on multi-processor devices, and the simulationresults showed that the PPSO with 2048 particles is capableof accelerating the computational speed by at least 64% onthe relevant computing devices. The PPSO records low cal-culation errors and shows improvement in terms of compu-tational speed. Hence, it is evident that the PPSO possessesexceptional capability in the parameter estimation. Becausebranching is difficult for all the computing devices, espe-cially GPUs, the process of updating positions and veloci-ties of particles in the PPSO has not been parallelized.This shall be investigated in our future work. Also, we in-tend to explore more aggressive multi-core computerswhich are equipped with more powerful GPU capabilities.
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