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Mismatch loss in photovoltaic systems
Available online at www.sciencedirect.com
ScienceDirect Solar Energy 105 (2014) 505–511 www.elsevier.com/locate/solener
Mismatch loss in photovoltaic systems Thomas S. Wurster ⇑, Markus B. Schubert University of Stuttgart, Institute for Photovoltaics, Pfaffenwaldring 47, 70569 Stuttgart, Germany Received 14 November 2013; received in revised form 10 April 2014; accepted 14 April 2014
Communicated by: Associate Editor Nicola Romeo
Abstract The effects of current mismatch and shading on the power output of single photovoltaic (PV) modules are well analyzed, but only few investigations address mismatch losses at a PV system level that also limit the annual energy yield. The simple question, what happens if PV strings with different numbers of modules are connected in parallel, has not yet been discussed in detail. In case of strings with unequal module count, the system builder must decide whether to use inverters with multiple maximum power point (MPP) trackers, module-power optimizers, or to shorten all strings for balancing the system. Our findings from this study open a new option. The numerical modeling of PV systems with strings of different length in parallel to several others which have an equal module count renders mismatch losses below 1% for most system configurations. For configurations where one string is one module shorter than the others, the mismatch losses fall below 0.5%. Therefore strings with unequal length may favorably connect to a cost-effective single-MPP inverter without causing significant energy yield losses. Moreover, typical thin film PV modules are less sensitive to mismatch than crystalline silicon based ones. Ó 2014 Elsevier Ltd. All rights reserved.
Keywords: Mismatch; Maximum power point tracking; MPPT; Partial shading; PV system
1. Introduction Most common photovoltaic (PV) modules comprise 60 or more solar cells. To maximize the energy yield it is crucial to match the parameters of the cells by binning during module fabrication. Since a standard PV module connects all solar cells within the module electrically in series, the cell current is the most important matching parameter (Bishop, 1988; Woyte et al., 2003). For small PV systems, consisting of just one PV string with a few PV modules connected in series, there is no need for the end user to consider parameter matching since this is done by the module manufacturer. For larger PV systems, however, where several PV strings are connected in parallel to increase the ⇑ Corresponding author. Tel.: +49 711 685 67246.
E-mail address: [email protected] (T.S. Wurster). http://dx.doi.org/10.1016/j.solener.2014.04.014 0038-092X/Ó 2014 Elsevier Ltd. All rights reserved.
system power, parameter matching becomes an issue. The parallel connection forces all strings to work at the same voltage leading to mismatch losses if a subset of strings would demand a different operating voltage than the others for reaching its maximum power point (MPP) (Woyte et al., 2003). The mismatch effect of the manufacturing tolerances was examined in detail by Chamberlin et al. Chamberlin et al. (1995), Spertino and Akilimali (2009) and there were numerous investigations on the effect of partial shading conditions (Quaschning et al., 1996; Rauschenbach, 1971; Bidram et al., 2012; Kjaer et al., 2005; Rogalla et al., 2010; Garcı´a et al., 2008). But sometimes there is simply not enough space on a roof to install the same number of PV modules in each string. The possibilities to avoid or reduce the mismatch of strings are manifold. Bidram et al. (2012) gave an over view of the various approaches. The main approaches to overcome mismatch
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losses are to either integrate a maximum power point tracker (MPPT) per PV string into the inverter (Kjaer et al., 2005), or to include power optimizers (Rogalla et al., 2010) in each PV module. While in theory both approaches decrease or cancel the mismatch losses, both raise system cost and reliability issues. Moreover, partial shading conditions were reported where single-MPPT inverters outperform multi-MPPT ones (Garcı´a et al., 2008). This contribution focuses on the mismatch losses caused by PV system configurations with unequal strings. The goal is to give an estimation of the losses and thereby to allow a system builder to decide whether an inverter with multiple MPP trackers, multiple inverters or power optimizers are necessary, or to which extent the mismatch losses can be neglected at all. Section 2 introduces the two numerical models and their computational flow implemented for quantifying the mismatch losses of the PV system configurations under investigation. Section 3 presents the results: Section 3.1 analyses one system configuration in detail to elucidate the origin of the mismatch losses, while Section 3.2 presents the calculated mismatch loss caused by varying mismatch conditions. The results show, that for a wide variety of system configurations the mismatch losses of unbalanced PV strings are within the measurement error of ±1% of standard test equipment, and thus can be neglected in practice. 2. Methodology Two models are used to quantify the mismatch losses originating from the operation of in parallel connected PV strings with different lengths, i.e. different numbers of identical modules per string. First, the ideal one-diode model (ODM) enables the estimation of the worst case mismatch losses because the gradient of its I/V characteristics close to the MPP is larger than for real PV modules (Section 2.2 and Fig. 5). Secondly, a numerical method computes PV string performance from measured I/V characteristics of a Suntechnics crystalline silicon (c-Si) module and a Schott amorphous silicon (a-Si) module (Table 2). The two models allow to compare the real-life mismatch losses with the worst-case ones deduced from the ODM. Both models calculate the I/V curves of PV strings Sn with a length n = 9–20 PV modules connected in series and use them to generate the I/V curves of various parallel connections. The comparison of individual string I/V curves and I/V curves of the parallel connections returns the mismatch losses. Both simulation models assume homogeneous in plane of array irradiance G = 1000 W/m2 and temperature T = 25 °C over the whole PV systems. The choice to neglect spatial and temporal variations of the local operating conditions simplifies and focuses our loss calculation, though it is an obvious restriction for comparing results with real-world performance data. Under this approximation, only one operating point serves to compare different
system configurations with special regard to our focus on the parallel connection of PV strings of different length. While high resolution data on the temporal variation of irradiance and temperature of PV systems are available (Zinsser et al., 2010), very little is known (Weigl et al., 2012) about spatial irradiance variations up to now. Therefore our approach excludes widely variable and mostly unknown extrinsic factors, like geometry, environment, partial shading and local weather, to provide a general understanding of the mismatch losses in parallel connections of PV strings with unequal length. 2.1. Ideal one-diode model The equation V I ¼ I SC I 0 exp 1 n1 V T nC
ð1Þ
describes the I/V characteristics of an ideal one-diode model (ODM) of a crystalline PV module. Table 1 lists the simulation parameters and the resulting module parameters used for this study, namely the saturation current I0, ideality factor n1, thermal voltage VT, number of PV cells per module nC, the open circuit and MPP voltages VOC, Vmpp, short circuit and MPP currents Isc, Impp, and MPP power Pmpp. The chosen parameters reflect a state of the art crystalline silicon PV module. The ODM implements the basic function of a solar cell. On the one hand, matching its few parameters to the real characteristics of a PV module concludes in inaccurate results, especially for amorphous PV modules. On the other hand, the ideal behavior effects in the highest sensitivity to mismatch conditions, making it a perfect worst-case scenario. Fig. 1 shows a simplified flow chart of the simulation of the mismatch losses due to the different lengths of the strings. The simulation starts by loading the PV module parameters of Table 1. The string length variable n is set to the minimum string length n = nmin = 10. The simulation calculates the I/V curve of the string with a length of n PV modules. To obtain the string I/V curve from the module parameters of Table 1, the simulation uses Eq. (1) but multiplies the number of PV cells per module nC Table 1 One-diode model (ODM) parameters and the resulting PV module characteristics. Parameter I0 n1 VT nC VOC Vmpp Isc Impp Pmpp FF
Value
Unit 11
4.57 10 1.00 0.02569 60 39.7 34.8 8.50 8.10 283 83.3
A V V V A A W %
T.S. Wurster, M.B. Schubert / Solar Energy 105 (2014) 505–511 Table 2 Measured parameters of the SUNPower Suntechnics STM 200 FW (c-Si) and the Schott Solar ASIOPAK-30-SG (a-Si) PV modules. Parameter
Value
Denoted by Cell type VOC Vmpp Isc Impp Pmpp nC FF
c-Si Mono-Si 47.7 39.4 5.39 4.96 195 72 75.9
Unit a-Si Amorphous Si 50.1 37.7 0.90 0.75 28.5 2 30 66.4
V V A A W
507
following steps connect more long 11-module strings, until the 10-module string is connected to 39 of the 11-module strings. The next iteration step firstly connects k = 2 10module strings to one 11-module string, and then increments the number of long strings until 38 of the 11-module strings are connected to the two 10 module-strings. The subsequent routine calculates the mismatch losses LMM from the P/V curves according to the equation P max LMM ¼ 1 100 ð2Þ P par
%
by the number of modules per string n. A fixed voltage vector V = 0–1000 V with a resolution dV = 0.1 V for all strings Sn simplifies the parallel connection later. Subsequently, the variable n is incremented by one as long as n < nmax = 20 to complete this part of the simulation. The next step first sets n = nmin = 10 and the short string counter k = 1. The following subroutine calculates the P/V curves by summing up the current values of all connected strings for each discrete value of the voltage vector V and multiplying by V, in order to deduce the P/V from the I/ V curves. These routines execute for all possible parallel connections consisting of k short strings with n modules connected to 1 to (40 k) strings with n + 1 modules. Please note, that n denotes the module count within the k short strings. For example, if n = 10 and kmax = 40, the calculations starts by connecting k = 1 (short) strings of 10 modules length to the first long string of 11 modules. The
where Pmax is the sum of the power maxima of the individual strings and Ppar is the power maximum of these strings connected in parallel. If 39 of the n modules long strings were connected to one string of n + 1 module length, k becomes kmax, the simulation leaves the loop, increases the string length n by one, sets k = 1 and reenters the loop. If the maximum string length nmax is reached, the simulation completes by saving the results. 2.2. I/V curve fitting model Fig. 2 shows the three I/V curves utilized for the simulations. The ideal, analytical ODM yields the highest short circuit current and the steepest slope at MPP. The measured I/V data of the c-Si module exhibits a lower Isc and slope but a higher open circuit voltage. The data of a aSi module presents the lowest short circuit current and slope but the highest open circuit voltage. The second simulation model required for our loss investigations, applies a
Fig. 1. Flow chart of the mismatch simulation. After loading the PV module parameters and setting the string length n = nmin, a loop calculates all I/V curves of strings with a length of n = nmin to nmax. Then the number k of strings with less modules is set to one. The outer loop calls the inner loop for all considered length of strings n. The inner loop calls a subroutine that connects k short strings in parallel to 1 to (40 k) long strings, and afterwards increments the number of short strings during each cycle. After the mismatch losses were calculated for the maximum string length, the simulation saves results and ends.
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Fig. 2. I/V characteristics of the sample module according to the ODM, and the fit to the measured I/V curves of the real PV modules used in this study.
curve fitting algorithm to the measured I/V data of the c-Si and the a-Si modules. The curve fitting enables generating I/V curves of PV strings with an adjustable number of PV modules connected in series, and thereby using the same subroutines for the calculation of the mismatch losses as for the analytical ODM. In order to fit the measured data to a fit object, the “smoothing spline” function of Matlab is used. Fig. 3 shows the flow chart of this I/V curve fitting model which allows for investigating strings built from real measured module I/V characteristics. The program starts by loading the measured I/V data. Table 2 lists the key parameters of the c-Si and the a-Si modules. The next step
initializes the voltage vector V by generating nmaxnDP values from 0 V to nmaxVOC. The number of data points per PV module nDP results in a voltage step size of dV = nmaxVOC/nmaxnDP = VOC/nDP, where nmax is the maximum number of PV modules connected in series. After initializing the current vector I with nmaxnDP zeros and setting the string length n = nmin, the simulation initializes the strings voltage vector VS with nnDP data points from VS = 0 V to VS = VOC. The initialization with nnDP data points stretches the I/V curves from VOC to nVOC. For example, if nmax = 10,000, nDP = 1000 and n = 2, then the evaluation of the fit object returns an I/V curve with nDPn = 2000 data points spanning from VS(1) = 0 V to VS(2000) = VOC. Copying these values into the current vector I, the vector contains the I/V curve in its fields I(1) to I(2000) and I(2001) to I(10,000) is zero. Multiplying the current vector I with the original V, stretches the curve as the field V(2000) = 2VOC where VS(2000) = VOC and generates the P/V curve of the string with n = 2 modules connected in series. The loop continues until n = nmax, and the routine completes by saving the results. This procedure yields the same format of I/V and P/V data of the one-diode and the curve fitting models. Consequently, the very same method deduces the mismatch losses for both experimental and analytical input data. 3. Results Section 3.1 examines one PV system configuration in detail and thereby reveals the origin of the mismatch losses. It also compares the mismatch effect of a certain deviation from the MPP voltage for the ODM parameter set of Table 1 and the IV curve models parameter sets of Table 2.
Fig. 3. Flow chart of the curve fitting model which links real-world I/V module data into our simulation study. After loading the PV module parameters, the voltage is initialized with nmax nDP spanning from 0 V to nmax VOC. The current vector I is initialized with nmax nDP zeros and n set to nmin. Then the simulation enters the loop to calculate the I/V curves. The loop first initializes the string voltage vector VS with n nDP values from 0 V to VOC then evaluates VS with the fit object to approximate the string current IS and the P/V characteristics. The result is an I/V curve which is stretched as compared to an evaluation with V. The next step increments the string length to n + 1. Once n = nmax is reached, the simulation concludes.
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The results justify the ODM as a worst case scenario. Section 3.2 gives a survey of all modeling results in their entirety, deducing the magnitude of mismatch losses for PV system configurations with n = 10–20 PV modules connected in series and m = 2–40 PV strings connected in parallel. 3.1. Origin of mismatch losses Fig. 4 shows the P/V characteristics of a PV system according to the ODM, for the sake of simplicity consisting of only m = 2 PV strings connected in parallel. String S1 is a series connection of n = 15 PV modules, while string S2 contains (n + 1)= 16 PV modules. The MPP voltage VMPP,# of the parallel connection P# is between the individual MPP voltages VMPP,S1 of PS1(V) and VMPP,S2 of PS2(V). The deviation of VMPP,# from VMPP,S1 and VMPP,S2 causes mismatch losses at both strings S1 and S2. The absolute power loss is Ploss = PS1(VMPP,#) + PS2 (VMPP,#) PS1(VMPP,S1) PS2(VMPP,S2) = 102 W. Using Eq. (2) this power corresponds to a relative mismatch loss P S1 ðV MPP;S1 Þ þ P S2 ðV MPP;S2 Þ Lfig4 ¼ 1 100 P S1 ðV MPP;# Þ þ P S2 ðV MPP;# Þ 4:247 kW þ 4:530 kW 100 ¼ 1:2%: ð3Þ ¼ 1 4:220 kW þ 4:455 kW
Fig. 5. More ideal I/V characteristics increase the sensitivity of a PV module to a voltage mismatch dV. The ODM renders the steepest slope followed by the c-Si module. Due to its low fill factor, the a-Si module responds less to a deviation from its MPP voltage especially for positive deviations.
The slope of the P/V curve close to the MPP controls the severity of a certain deviation from the MPP voltage. Fig. 5 presents the dependency of the mismatch loss L on the voltage mismatch dV of the ideal ODM and of the two curve fitting models. The curve of the ODM is completely within the other two curves and therefore results in the steepest gradient. For example, a voltage deviation
Fig. 6. Calculated mismatches loss L for PV system configurations with m strings in parallel. The single short strings consist of n = 10–20 modules in series connection, the m 1 in parallel connected long strings connect n + 1 modules in series. For systems with n > 11, the mismatch loss is as low as L < 0.5% depending on the number m of strings connected in parallel. The ODM, most sensitive to mismatch, shows the highest losses followed by the c-Si configurations. The a-Si configurations are significantly less prone to mismatch losses.
Fig. 4. Simulated power vs. voltage characteristics. Parallel connection of the strings S1 and S2 combines their characteristics PS1(V) and PS2(V) into the characteristics of the parallel connection P#(V). The global MPP voltage VMPP,# is between the MPP voltages VMPP,S1 and VMPP,S2 of the individual strings, thereby causing mismatch losses in S1 and S2.
of dV = + 2.5% Vmpp causes LODM(+2.5% Vmpp) = 0.93% mismatch losses in the ODM, while the c-Si module loses Lc-Si(+2.5% Vmpp) = 0.64% and a-Si module loses just La-Si(+2.5% Vmpp) = 0.32% of its power. Because the ODM is most sensitive to deviations from its MPP, it is well suited as a worst case reference. Also the ODM is very common for PV-system simulations.
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3.2. Magnitude of mismatch losses Fig. 6 presents the final results of the mismatch loss analyses. In order to assess the necessity or benefit of a string level MPPT, each of the simulated configurations incorporates m 1 identical strings of n + 1 modules length, plus one string with n modules, in parallel connection. The numerical simulation covers a wide variety of the PV module count n = 10–20 per string, and of the number m = 2–40 of strings connected in parallel. The nominal output power of the analyzed configurations ranges from P = 5 kWp to P = 230 kWp. Each line in Fig. 6 represents the series connection of a distinct number of n PV modules in the long strings, e.g. the top data set contains n + 1 = 11 PV modules in the long strings, and n = 10 modules in the additional shorter string. The dots mark the simulated configurations. The lines unite simulated configurations with a certain length n of their short string. The absolute value of the relative mismatch loss L decreases for m > 3 strings connected in parallel, independent of n. For all string lengths n P 10, configurations with LODM < 1% exist, but n P 11 is necessary to obtain LODM < 0.5%. The L = 0.5% level is important because most common energy and power meters are manufactured with a tolerance of ±0.5% to ±1%. Going into further detail, Zinßer et al. reported (Zinßer et al., 2008) that the limitations of the equipment to measure irradiance, temperature and DC power result in a total error of ±2.6 % for their sideby-side comparison of the outdoor performance of twelve different PV technologies. Such error margins are much larger than the mismatch losses of all system configurations deduced here. Consequently, a mismatch loss L < 1% is hard to detect with common measurement equipment. The effect of a single string with one module less, regardless whether caused by system design or by a fault, can only be detected by individual string monitoring, and only marginally reduces energy yield. Fig. 7 shows further mismatch simulations which in detail examine the case n = 15 of Fig. 6. This time, not the string length n is changed but the number of short strings k. While a higher initial number of short strings decreases the initial mismatch loss, the losses increase when further strings are connected in parallel. Despite the fact that the losses tend to further decrease once a certain number of strings are connected in parallel, majority of the configurations yield LODM P 1%. Consequently configurations of PV systems with more than one shorter string and a much higher number of longer strings should be avoided, while configurations with higher numbers of short strings than long strings yield lower mismatch losses. Fig. 8 presents a detailed analysis of the mismatch losses broken down to the string level. The curves show the losses of the long strings Ll, the short strings Ls and the total losses Ltot with k = 3 short strings connected to m k = 1–37 long strings. The initial losses of the long strings exceed
Fig. 7. Calculated relative mismatches loss LODM for PV system configurations with m strings in parallel. The k = 1 to 39 short strings consist of n = 15 modules, the long strings of n + 1 = 16 modules. The curves show that increasing the number k of short strings decreases the initial mismatch losses while the mismatch loss increase for higher numbers k of short strings. Configurations with just one long string connected to several short strings yield low mismatch losses in general.
Fig. 8. Mismatch losses for k = 3 and k = 13 short strings broken down into the losses in the long strings Ll, losses in the short strings Ls and the resulting total losses Ltot. The short strings consist of n = 15 modules, the long strings are one module longer. While the loss Ls of the short strings increases when m increases, the loss Ll of the long strings decreases. In both cases the long strings are the majority and therefore Ll dominates the total loss.
the initial losses of the short strings, i.e. Ls < Ll. Connecting more long strings in parallel, increases the losses of the short strings Ls but also decreases the losses of the long strings Ll. At m = 8 Ll and Ls intersect, and Ls > Ll for m > 8. Because the long strings become the majority they dominate the total losses Ltot when m further increases. Fig. 9 finally plays through the reduction in string length of the short string. Reducing the string length by more than one module results in a severe increase of the mismatch loss L. The six curves show a reduced string length of n = 15 to n = 10 modules compared to the long string with 16 modules connected in series. Configurations with a n 2 string length and shorter can cause the complete loss of the power of the short strings by forcing them into their open circuit
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‘mismatched by design’ is more prone to mismatch losses caused by additional extrinsic influences like shading. On the other hand, less electronic components imply less risk of failure and enhance longevity, thus encouraging the most simple system design that provides similar annual energy yield. Further investigations will compare the results of this general study with outdoor performance data under real environmental conditions. Acknowledgements
Fig. 9. One short string connected in parallel to m 1 long strings. The length of the short string is n = 15–10 modules. The reduction of the length of the short string increases the mismatch loss dramatically. For strings with three modules less than the long strings there are system configurations where optimum voltage Vmpp,# of the parallel connection exceeds the open circuit voltage VOC of the short string, causing a complete loss of its power.
condition. Therefore PV system configurations with strings reduced in length by more than one module are not recommended. 4. Conclusion Whereas many investigations on mismatch losses in PV arrays exist, only few studies address mismatch losses at the system level. This contribution investigates the power losses arising from string length mismatch in PV generators with 10–20 modules per string and up to 40 strings connected in parallel, corresponding to a power range of P = 5 kWp to P = 230 kWp. Moreover, the ratio of short and long strings is varied, and the effect of real IV characteristics of monocrystalline and amorphous silicon modules vs. the worst case one-diode model IV data are evaluated. In general, there are numerous obstacles hindering such system level mismatch analyses, e.g. a wide variety of PV system configurations, arising from cell and module manufacturing as well as from wiring variations. Local and temporal irradiance and weather fluctuations make it difficult to compare the performance of PV systems, even if installed next to each other. By blanking out these ‘reality issues’, the modeling of a wide variety of PV system configurations reveals that the relative mismatch loss is limited to L < 1% for most common PV installations with parallel strings, if only one of the strings counts one module less than the others. Furthermore the simulations show that thin film technologies with lower efficiency, like amorphous silicon, benefit from their lower fill factor as deviations from their MPP cause lower mismatch losses as compared to high performance crystalline silicon technologies. These findings underline that most PV systems are very robust by design. This robustness questions the requirement of expensive multiple-MPPT inverters or power optimizers. Important to note, however, such a system
The authors gratefully acknowledge instructive discuss ions with R. Merz, M. Groll, C. Buchholz, and M. Seehuber of REFUsol (Metzingen, Germany). We thank J.H. Werner for encouraging and supporting this work, P. Lill and J. Kistner for careful proofreading of the manuscript. References Bidram, A., Davoudi, A., Balog, R.S., 2012. Control and circuit techniques to mitigate partial shading effects in photovoltaic arrays. IEEE J. Photovolt. 2 (4), 532–546. Bishop, J.W., 1988. Computer-simulation of the effects of electrical mismatches in photovoltaic cell interconnection circuits. Solar Cells 25, 73–89. Chamberlin, C.E., Lehman, P., Zoellick, J., Pauletto, G., 1995. Effects of mismatch losses in photovoltaic arrays. Sol. Energy 54 (3), 165–171. Garcı´a, M., Maruri, J.M., Marroyo, L., Lorenzo, E., Pe´rez, M., 2008. Partial shadowing, MPPT performance and inverter configurations: observations at tracking PV plants. Prog. Photovolt.: Res. Appl. 16 (6), 529–536. Kjaer, S., Pedersen, J., Blaabjerg, F., 2005. A review of single-phase gridconnected inverters for photovoltaic modules. IEEE Trans. Ind. Appl. 41 (5), 1292–1306. Mathworks.com [Internet]. Natick: Matlab programming environment. Quaschning, V., 1996. Simulation der Abschattungsverluste bei solarelektrischen Systemen. Zugl.: Berlin, Techn. Univ., Diss., 1996, 1st ed. Ko¨ster, Berlin. Rauschenbach, H., 1971. Electrical output of shadowed solar arrays. IEEE Trans. Electron Dev. ED-18, 483–490. Rogalla, S., Burger, B., Goeldi, B., Schmidt, H., 2010. Light and shadow—when is MPP-tracking at the module level worthwhile. In: Proc. 25th Europ. Photovolt. Solar Energy Conf., Valencia, Spain, pp. 3932–3936. Spertino, F., Akilimali, J.S., 2009. Are manufacturing IV mismatch and reverse currents key factors in large photovoltaic arrays? IEEE Trans. Ind. Electron. 56 (11), 4520. ¨ chsner, P., Weigl, T., Nagl, L., Weizenbeck, J., Zehner, M., Augel, M., O Giesler, B., Becker, G., Mayer, O., Betts, T.R., Gottschalg, R., 2012. Modelling and validation of spatial irradiance characteristics for localised irradiance fluctuations and enhancements. In: Proc. 27th Europ. Photovolt. Solar Energy Conf., Frankfurt, Germany, pp. 3801– 3804. Woyte, A., Nijs, J., Belmans, R., 2003. Partial shadowing of photovoltaic arrays with different system configurations: literature review and field test results. Sol. Energy 74 (3), 217–233. Zinßer, B., Makrides, G., Schubert, M.B., Georghiou, G., Werner, J.H., 2008. Temperature and intensity dependence of twelve photovoltaic technologies. In: Proc. 23rd Europ. Photovolt. Solar Energy Conf., Valencia, Spain, pp. 3249–3252. Zinsser, B., Makrides, G., Schubert, M.B., Georghiou, G.E., Werner, J.H., 2010. Rating of annual energy yield more sensitive to reference power than module technology. In: Proc. 35th IEEE Photovoltaic Specialists Conference, pp. 1095–1099.
ScienceDirect Solar Energy 105 (2014) 505–511 www.elsevier.com/locate/solener
Mismatch loss in photovoltaic systems Thomas S. Wurster ⇑, Markus B. Schubert University of Stuttgart, Institute for Photovoltaics, Pfaffenwaldring 47, 70569 Stuttgart, Germany Received 14 November 2013; received in revised form 10 April 2014; accepted 14 April 2014
Communicated by: Associate Editor Nicola Romeo
Abstract The effects of current mismatch and shading on the power output of single photovoltaic (PV) modules are well analyzed, but only few investigations address mismatch losses at a PV system level that also limit the annual energy yield. The simple question, what happens if PV strings with different numbers of modules are connected in parallel, has not yet been discussed in detail. In case of strings with unequal module count, the system builder must decide whether to use inverters with multiple maximum power point (MPP) trackers, module-power optimizers, or to shorten all strings for balancing the system. Our findings from this study open a new option. The numerical modeling of PV systems with strings of different length in parallel to several others which have an equal module count renders mismatch losses below 1% for most system configurations. For configurations where one string is one module shorter than the others, the mismatch losses fall below 0.5%. Therefore strings with unequal length may favorably connect to a cost-effective single-MPP inverter without causing significant energy yield losses. Moreover, typical thin film PV modules are less sensitive to mismatch than crystalline silicon based ones. Ó 2014 Elsevier Ltd. All rights reserved.
Keywords: Mismatch; Maximum power point tracking; MPPT; Partial shading; PV system
1. Introduction Most common photovoltaic (PV) modules comprise 60 or more solar cells. To maximize the energy yield it is crucial to match the parameters of the cells by binning during module fabrication. Since a standard PV module connects all solar cells within the module electrically in series, the cell current is the most important matching parameter (Bishop, 1988; Woyte et al., 2003). For small PV systems, consisting of just one PV string with a few PV modules connected in series, there is no need for the end user to consider parameter matching since this is done by the module manufacturer. For larger PV systems, however, where several PV strings are connected in parallel to increase the ⇑ Corresponding author. Tel.: +49 711 685 67246.
E-mail address: [email protected] (T.S. Wurster). http://dx.doi.org/10.1016/j.solener.2014.04.014 0038-092X/Ó 2014 Elsevier Ltd. All rights reserved.
system power, parameter matching becomes an issue. The parallel connection forces all strings to work at the same voltage leading to mismatch losses if a subset of strings would demand a different operating voltage than the others for reaching its maximum power point (MPP) (Woyte et al., 2003). The mismatch effect of the manufacturing tolerances was examined in detail by Chamberlin et al. Chamberlin et al. (1995), Spertino and Akilimali (2009) and there were numerous investigations on the effect of partial shading conditions (Quaschning et al., 1996; Rauschenbach, 1971; Bidram et al., 2012; Kjaer et al., 2005; Rogalla et al., 2010; Garcı´a et al., 2008). But sometimes there is simply not enough space on a roof to install the same number of PV modules in each string. The possibilities to avoid or reduce the mismatch of strings are manifold. Bidram et al. (2012) gave an over view of the various approaches. The main approaches to overcome mismatch
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losses are to either integrate a maximum power point tracker (MPPT) per PV string into the inverter (Kjaer et al., 2005), or to include power optimizers (Rogalla et al., 2010) in each PV module. While in theory both approaches decrease or cancel the mismatch losses, both raise system cost and reliability issues. Moreover, partial shading conditions were reported where single-MPPT inverters outperform multi-MPPT ones (Garcı´a et al., 2008). This contribution focuses on the mismatch losses caused by PV system configurations with unequal strings. The goal is to give an estimation of the losses and thereby to allow a system builder to decide whether an inverter with multiple MPP trackers, multiple inverters or power optimizers are necessary, or to which extent the mismatch losses can be neglected at all. Section 2 introduces the two numerical models and their computational flow implemented for quantifying the mismatch losses of the PV system configurations under investigation. Section 3 presents the results: Section 3.1 analyses one system configuration in detail to elucidate the origin of the mismatch losses, while Section 3.2 presents the calculated mismatch loss caused by varying mismatch conditions. The results show, that for a wide variety of system configurations the mismatch losses of unbalanced PV strings are within the measurement error of ±1% of standard test equipment, and thus can be neglected in practice. 2. Methodology Two models are used to quantify the mismatch losses originating from the operation of in parallel connected PV strings with different lengths, i.e. different numbers of identical modules per string. First, the ideal one-diode model (ODM) enables the estimation of the worst case mismatch losses because the gradient of its I/V characteristics close to the MPP is larger than for real PV modules (Section 2.2 and Fig. 5). Secondly, a numerical method computes PV string performance from measured I/V characteristics of a Suntechnics crystalline silicon (c-Si) module and a Schott amorphous silicon (a-Si) module (Table 2). The two models allow to compare the real-life mismatch losses with the worst-case ones deduced from the ODM. Both models calculate the I/V curves of PV strings Sn with a length n = 9–20 PV modules connected in series and use them to generate the I/V curves of various parallel connections. The comparison of individual string I/V curves and I/V curves of the parallel connections returns the mismatch losses. Both simulation models assume homogeneous in plane of array irradiance G = 1000 W/m2 and temperature T = 25 °C over the whole PV systems. The choice to neglect spatial and temporal variations of the local operating conditions simplifies and focuses our loss calculation, though it is an obvious restriction for comparing results with real-world performance data. Under this approximation, only one operating point serves to compare different
system configurations with special regard to our focus on the parallel connection of PV strings of different length. While high resolution data on the temporal variation of irradiance and temperature of PV systems are available (Zinsser et al., 2010), very little is known (Weigl et al., 2012) about spatial irradiance variations up to now. Therefore our approach excludes widely variable and mostly unknown extrinsic factors, like geometry, environment, partial shading and local weather, to provide a general understanding of the mismatch losses in parallel connections of PV strings with unequal length. 2.1. Ideal one-diode model The equation V I ¼ I SC I 0 exp 1 n1 V T nC
ð1Þ
describes the I/V characteristics of an ideal one-diode model (ODM) of a crystalline PV module. Table 1 lists the simulation parameters and the resulting module parameters used for this study, namely the saturation current I0, ideality factor n1, thermal voltage VT, number of PV cells per module nC, the open circuit and MPP voltages VOC, Vmpp, short circuit and MPP currents Isc, Impp, and MPP power Pmpp. The chosen parameters reflect a state of the art crystalline silicon PV module. The ODM implements the basic function of a solar cell. On the one hand, matching its few parameters to the real characteristics of a PV module concludes in inaccurate results, especially for amorphous PV modules. On the other hand, the ideal behavior effects in the highest sensitivity to mismatch conditions, making it a perfect worst-case scenario. Fig. 1 shows a simplified flow chart of the simulation of the mismatch losses due to the different lengths of the strings. The simulation starts by loading the PV module parameters of Table 1. The string length variable n is set to the minimum string length n = nmin = 10. The simulation calculates the I/V curve of the string with a length of n PV modules. To obtain the string I/V curve from the module parameters of Table 1, the simulation uses Eq. (1) but multiplies the number of PV cells per module nC Table 1 One-diode model (ODM) parameters and the resulting PV module characteristics. Parameter I0 n1 VT nC VOC Vmpp Isc Impp Pmpp FF
Value
Unit 11
4.57 10 1.00 0.02569 60 39.7 34.8 8.50 8.10 283 83.3
A V V V A A W %
T.S. Wurster, M.B. Schubert / Solar Energy 105 (2014) 505–511 Table 2 Measured parameters of the SUNPower Suntechnics STM 200 FW (c-Si) and the Schott Solar ASIOPAK-30-SG (a-Si) PV modules. Parameter
Value
Denoted by Cell type VOC Vmpp Isc Impp Pmpp nC FF
c-Si Mono-Si 47.7 39.4 5.39 4.96 195 72 75.9
Unit a-Si Amorphous Si 50.1 37.7 0.90 0.75 28.5 2 30 66.4
V V A A W
507
following steps connect more long 11-module strings, until the 10-module string is connected to 39 of the 11-module strings. The next iteration step firstly connects k = 2 10module strings to one 11-module string, and then increments the number of long strings until 38 of the 11-module strings are connected to the two 10 module-strings. The subsequent routine calculates the mismatch losses LMM from the P/V curves according to the equation P max LMM ¼ 1 100 ð2Þ P par
%
by the number of modules per string n. A fixed voltage vector V = 0–1000 V with a resolution dV = 0.1 V for all strings Sn simplifies the parallel connection later. Subsequently, the variable n is incremented by one as long as n < nmax = 20 to complete this part of the simulation. The next step first sets n = nmin = 10 and the short string counter k = 1. The following subroutine calculates the P/V curves by summing up the current values of all connected strings for each discrete value of the voltage vector V and multiplying by V, in order to deduce the P/V from the I/ V curves. These routines execute for all possible parallel connections consisting of k short strings with n modules connected to 1 to (40 k) strings with n + 1 modules. Please note, that n denotes the module count within the k short strings. For example, if n = 10 and kmax = 40, the calculations starts by connecting k = 1 (short) strings of 10 modules length to the first long string of 11 modules. The
where Pmax is the sum of the power maxima of the individual strings and Ppar is the power maximum of these strings connected in parallel. If 39 of the n modules long strings were connected to one string of n + 1 module length, k becomes kmax, the simulation leaves the loop, increases the string length n by one, sets k = 1 and reenters the loop. If the maximum string length nmax is reached, the simulation completes by saving the results. 2.2. I/V curve fitting model Fig. 2 shows the three I/V curves utilized for the simulations. The ideal, analytical ODM yields the highest short circuit current and the steepest slope at MPP. The measured I/V data of the c-Si module exhibits a lower Isc and slope but a higher open circuit voltage. The data of a aSi module presents the lowest short circuit current and slope but the highest open circuit voltage. The second simulation model required for our loss investigations, applies a
Fig. 1. Flow chart of the mismatch simulation. After loading the PV module parameters and setting the string length n = nmin, a loop calculates all I/V curves of strings with a length of n = nmin to nmax. Then the number k of strings with less modules is set to one. The outer loop calls the inner loop for all considered length of strings n. The inner loop calls a subroutine that connects k short strings in parallel to 1 to (40 k) long strings, and afterwards increments the number of short strings during each cycle. After the mismatch losses were calculated for the maximum string length, the simulation saves results and ends.
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Fig. 2. I/V characteristics of the sample module according to the ODM, and the fit to the measured I/V curves of the real PV modules used in this study.
curve fitting algorithm to the measured I/V data of the c-Si and the a-Si modules. The curve fitting enables generating I/V curves of PV strings with an adjustable number of PV modules connected in series, and thereby using the same subroutines for the calculation of the mismatch losses as for the analytical ODM. In order to fit the measured data to a fit object, the “smoothing spline” function of Matlab is used. Fig. 3 shows the flow chart of this I/V curve fitting model which allows for investigating strings built from real measured module I/V characteristics. The program starts by loading the measured I/V data. Table 2 lists the key parameters of the c-Si and the a-Si modules. The next step
initializes the voltage vector V by generating nmaxnDP values from 0 V to nmaxVOC. The number of data points per PV module nDP results in a voltage step size of dV = nmaxVOC/nmaxnDP = VOC/nDP, where nmax is the maximum number of PV modules connected in series. After initializing the current vector I with nmaxnDP zeros and setting the string length n = nmin, the simulation initializes the strings voltage vector VS with nnDP data points from VS = 0 V to VS = VOC. The initialization with nnDP data points stretches the I/V curves from VOC to nVOC. For example, if nmax = 10,000, nDP = 1000 and n = 2, then the evaluation of the fit object returns an I/V curve with nDPn = 2000 data points spanning from VS(1) = 0 V to VS(2000) = VOC. Copying these values into the current vector I, the vector contains the I/V curve in its fields I(1) to I(2000) and I(2001) to I(10,000) is zero. Multiplying the current vector I with the original V, stretches the curve as the field V(2000) = 2VOC where VS(2000) = VOC and generates the P/V curve of the string with n = 2 modules connected in series. The loop continues until n = nmax, and the routine completes by saving the results. This procedure yields the same format of I/V and P/V data of the one-diode and the curve fitting models. Consequently, the very same method deduces the mismatch losses for both experimental and analytical input data. 3. Results Section 3.1 examines one PV system configuration in detail and thereby reveals the origin of the mismatch losses. It also compares the mismatch effect of a certain deviation from the MPP voltage for the ODM parameter set of Table 1 and the IV curve models parameter sets of Table 2.
Fig. 3. Flow chart of the curve fitting model which links real-world I/V module data into our simulation study. After loading the PV module parameters, the voltage is initialized with nmax nDP spanning from 0 V to nmax VOC. The current vector I is initialized with nmax nDP zeros and n set to nmin. Then the simulation enters the loop to calculate the I/V curves. The loop first initializes the string voltage vector VS with n nDP values from 0 V to VOC then evaluates VS with the fit object to approximate the string current IS and the P/V characteristics. The result is an I/V curve which is stretched as compared to an evaluation with V. The next step increments the string length to n + 1. Once n = nmax is reached, the simulation concludes.
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The results justify the ODM as a worst case scenario. Section 3.2 gives a survey of all modeling results in their entirety, deducing the magnitude of mismatch losses for PV system configurations with n = 10–20 PV modules connected in series and m = 2–40 PV strings connected in parallel. 3.1. Origin of mismatch losses Fig. 4 shows the P/V characteristics of a PV system according to the ODM, for the sake of simplicity consisting of only m = 2 PV strings connected in parallel. String S1 is a series connection of n = 15 PV modules, while string S2 contains (n + 1)= 16 PV modules. The MPP voltage VMPP,# of the parallel connection P# is between the individual MPP voltages VMPP,S1 of PS1(V) and VMPP,S2 of PS2(V). The deviation of VMPP,# from VMPP,S1 and VMPP,S2 causes mismatch losses at both strings S1 and S2. The absolute power loss is Ploss = PS1(VMPP,#) + PS2 (VMPP,#) PS1(VMPP,S1) PS2(VMPP,S2) = 102 W. Using Eq. (2) this power corresponds to a relative mismatch loss P S1 ðV MPP;S1 Þ þ P S2 ðV MPP;S2 Þ Lfig4 ¼ 1 100 P S1 ðV MPP;# Þ þ P S2 ðV MPP;# Þ 4:247 kW þ 4:530 kW 100 ¼ 1:2%: ð3Þ ¼ 1 4:220 kW þ 4:455 kW
Fig. 5. More ideal I/V characteristics increase the sensitivity of a PV module to a voltage mismatch dV. The ODM renders the steepest slope followed by the c-Si module. Due to its low fill factor, the a-Si module responds less to a deviation from its MPP voltage especially for positive deviations.
The slope of the P/V curve close to the MPP controls the severity of a certain deviation from the MPP voltage. Fig. 5 presents the dependency of the mismatch loss L on the voltage mismatch dV of the ideal ODM and of the two curve fitting models. The curve of the ODM is completely within the other two curves and therefore results in the steepest gradient. For example, a voltage deviation
Fig. 6. Calculated mismatches loss L for PV system configurations with m strings in parallel. The single short strings consist of n = 10–20 modules in series connection, the m 1 in parallel connected long strings connect n + 1 modules in series. For systems with n > 11, the mismatch loss is as low as L < 0.5% depending on the number m of strings connected in parallel. The ODM, most sensitive to mismatch, shows the highest losses followed by the c-Si configurations. The a-Si configurations are significantly less prone to mismatch losses.
Fig. 4. Simulated power vs. voltage characteristics. Parallel connection of the strings S1 and S2 combines their characteristics PS1(V) and PS2(V) into the characteristics of the parallel connection P#(V). The global MPP voltage VMPP,# is between the MPP voltages VMPP,S1 and VMPP,S2 of the individual strings, thereby causing mismatch losses in S1 and S2.
of dV = + 2.5% Vmpp causes LODM(+2.5% Vmpp) = 0.93% mismatch losses in the ODM, while the c-Si module loses Lc-Si(+2.5% Vmpp) = 0.64% and a-Si module loses just La-Si(+2.5% Vmpp) = 0.32% of its power. Because the ODM is most sensitive to deviations from its MPP, it is well suited as a worst case reference. Also the ODM is very common for PV-system simulations.
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3.2. Magnitude of mismatch losses Fig. 6 presents the final results of the mismatch loss analyses. In order to assess the necessity or benefit of a string level MPPT, each of the simulated configurations incorporates m 1 identical strings of n + 1 modules length, plus one string with n modules, in parallel connection. The numerical simulation covers a wide variety of the PV module count n = 10–20 per string, and of the number m = 2–40 of strings connected in parallel. The nominal output power of the analyzed configurations ranges from P = 5 kWp to P = 230 kWp. Each line in Fig. 6 represents the series connection of a distinct number of n PV modules in the long strings, e.g. the top data set contains n + 1 = 11 PV modules in the long strings, and n = 10 modules in the additional shorter string. The dots mark the simulated configurations. The lines unite simulated configurations with a certain length n of their short string. The absolute value of the relative mismatch loss L decreases for m > 3 strings connected in parallel, independent of n. For all string lengths n P 10, configurations with LODM < 1% exist, but n P 11 is necessary to obtain LODM < 0.5%. The L = 0.5% level is important because most common energy and power meters are manufactured with a tolerance of ±0.5% to ±1%. Going into further detail, Zinßer et al. reported (Zinßer et al., 2008) that the limitations of the equipment to measure irradiance, temperature and DC power result in a total error of ±2.6 % for their sideby-side comparison of the outdoor performance of twelve different PV technologies. Such error margins are much larger than the mismatch losses of all system configurations deduced here. Consequently, a mismatch loss L < 1% is hard to detect with common measurement equipment. The effect of a single string with one module less, regardless whether caused by system design or by a fault, can only be detected by individual string monitoring, and only marginally reduces energy yield. Fig. 7 shows further mismatch simulations which in detail examine the case n = 15 of Fig. 6. This time, not the string length n is changed but the number of short strings k. While a higher initial number of short strings decreases the initial mismatch loss, the losses increase when further strings are connected in parallel. Despite the fact that the losses tend to further decrease once a certain number of strings are connected in parallel, majority of the configurations yield LODM P 1%. Consequently configurations of PV systems with more than one shorter string and a much higher number of longer strings should be avoided, while configurations with higher numbers of short strings than long strings yield lower mismatch losses. Fig. 8 presents a detailed analysis of the mismatch losses broken down to the string level. The curves show the losses of the long strings Ll, the short strings Ls and the total losses Ltot with k = 3 short strings connected to m k = 1–37 long strings. The initial losses of the long strings exceed
Fig. 7. Calculated relative mismatches loss LODM for PV system configurations with m strings in parallel. The k = 1 to 39 short strings consist of n = 15 modules, the long strings of n + 1 = 16 modules. The curves show that increasing the number k of short strings decreases the initial mismatch losses while the mismatch loss increase for higher numbers k of short strings. Configurations with just one long string connected to several short strings yield low mismatch losses in general.
Fig. 8. Mismatch losses for k = 3 and k = 13 short strings broken down into the losses in the long strings Ll, losses in the short strings Ls and the resulting total losses Ltot. The short strings consist of n = 15 modules, the long strings are one module longer. While the loss Ls of the short strings increases when m increases, the loss Ll of the long strings decreases. In both cases the long strings are the majority and therefore Ll dominates the total loss.
the initial losses of the short strings, i.e. Ls < Ll. Connecting more long strings in parallel, increases the losses of the short strings Ls but also decreases the losses of the long strings Ll. At m = 8 Ll and Ls intersect, and Ls > Ll for m > 8. Because the long strings become the majority they dominate the total losses Ltot when m further increases. Fig. 9 finally plays through the reduction in string length of the short string. Reducing the string length by more than one module results in a severe increase of the mismatch loss L. The six curves show a reduced string length of n = 15 to n = 10 modules compared to the long string with 16 modules connected in series. Configurations with a n 2 string length and shorter can cause the complete loss of the power of the short strings by forcing them into their open circuit
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‘mismatched by design’ is more prone to mismatch losses caused by additional extrinsic influences like shading. On the other hand, less electronic components imply less risk of failure and enhance longevity, thus encouraging the most simple system design that provides similar annual energy yield. Further investigations will compare the results of this general study with outdoor performance data under real environmental conditions. Acknowledgements
Fig. 9. One short string connected in parallel to m 1 long strings. The length of the short string is n = 15–10 modules. The reduction of the length of the short string increases the mismatch loss dramatically. For strings with three modules less than the long strings there are system configurations where optimum voltage Vmpp,# of the parallel connection exceeds the open circuit voltage VOC of the short string, causing a complete loss of its power.
condition. Therefore PV system configurations with strings reduced in length by more than one module are not recommended. 4. Conclusion Whereas many investigations on mismatch losses in PV arrays exist, only few studies address mismatch losses at the system level. This contribution investigates the power losses arising from string length mismatch in PV generators with 10–20 modules per string and up to 40 strings connected in parallel, corresponding to a power range of P = 5 kWp to P = 230 kWp. Moreover, the ratio of short and long strings is varied, and the effect of real IV characteristics of monocrystalline and amorphous silicon modules vs. the worst case one-diode model IV data are evaluated. In general, there are numerous obstacles hindering such system level mismatch analyses, e.g. a wide variety of PV system configurations, arising from cell and module manufacturing as well as from wiring variations. Local and temporal irradiance and weather fluctuations make it difficult to compare the performance of PV systems, even if installed next to each other. By blanking out these ‘reality issues’, the modeling of a wide variety of PV system configurations reveals that the relative mismatch loss is limited to L < 1% for most common PV installations with parallel strings, if only one of the strings counts one module less than the others. Furthermore the simulations show that thin film technologies with lower efficiency, like amorphous silicon, benefit from their lower fill factor as deviations from their MPP cause lower mismatch losses as compared to high performance crystalline silicon technologies. These findings underline that most PV systems are very robust by design. This robustness questions the requirement of expensive multiple-MPPT inverters or power optimizers. Important to note, however, such a system
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