Experimental study of the maximum power point characteristics of partially shaded photovoltaic strings

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Applied Energy 301 (2021) 117436

Available online 20 July 2021

(http://creativecommons.org/licenses/by-nc-nd/4.0/).

Experimental study of the maximum power point characteristics of partially shaded photovoltaic strings

Kari Lappalainen

^*

, Seppo Valkealahti

Tampere University, Electrical Engineering Unit, P.O. Box 692, FI-33101 Tampere, Finland

H I G H L I G H T S

•MPP characteristics of partially shaded PV strings were studied experimentally.

•The global MPP of photovoltaic strings was found to vary over a wide voltage range.

•It was found that variations in the global MPP voltage and power can be fast.

•Operation at the MPP closest to the nominal MPP voltage was found to have advantages.

A R T I C L E I N F O Keywords:

Photovoltaic power generation Maximum power point Power variation Partial shading

Maximum power point tracking Photovoltaic string

A B S T R A C T

Under non-uniform operating conditions, like partial shading, electrical characteristics of photovoltaic generators can have multiple maximum power points (MPP) and the voltage of the global MPP can fluctuate over a broad voltage range. Since the highly variable global MPP voltage poses challenges for MPP tracking, it would be advantageous to keep the inverter operation point all the time at voltages close to the nominal MPP voltage.

Earlier studies of MPP characteristics of photovoltaic generators have typically been simulation studies based on hypothetical shading patterns lacking knowledge of real operating conditions of photovoltaic cells. This article presents an experimental study of the MPP characteristics of partially shaded strings of 6 and 17 series-connected photovoltaic modules based on over 26000 measured current–voltage curves thus eliminating the shortcomings of earlier studies. Moreover, a scenario in which the MPP closest to the nominal MPP voltage is used the entire time as the operating point instead of the global MPP is studied for the first time. It was found that the global MPP of partially shaded photovoltaic strings varies over a broad voltage range and changes in its voltage and power can be extremely fast. The results show that the wide operating voltage range when the global MPP is followed can be significantly reduced by following the MPP closest to the nominal MPP voltage at a cost of negligible energy losses. The energy difference between the two MPPs was found to be insignificantly small from 0.03% to 0.35% of available energy.

1. Introduction

Photovoltaic (PV) power plants are constantly prone to changes in their operating conditions. The generated PV power varies highly due to fast irradiance fluctuations, which are mostly consequence of overpassing cloud shadows. With the increasing share of grid-connected PV power, fluctuations in PV output power have an emergent potential to negatively affect power quality and reliability. Some transmission system operators have already restricted the allowed power variations of grid-connected PV generators [1]. For instance, the Puerto Rico Electric Power Authority has set a 10 %/min limit of rated power for PV power

variations [2]. However, many times faster variations exist in the output power of PV generators than this limit. For example, up to 23 %/s power variations were measured for a 3.2 kWp PV string [3] and up to 70

%/min power variations were measured at a 9.5 MWp PV plant [4].

Output power variations of PV generators under overpassing cloud shadows have been studied comprehensively in [5,6] and a method for estimation of the largest expected PV power ramp rates was introduced in [7]. PV power plants comply with power variation requirements typically by means of energy storage systems or output power curtailment [8]. However, only upward power slopes can be restricted by power curtailment [9].

* Corresponding author.

E-mail addresses: kari.lappalainen@tuni.fi (K. Lappalainen), seppo.valkealahti@tuni.fi (S. Valkealahti).

Contents lists available at ScienceDirect

Applied Energy

journal homepage: www.elsevier.com/locate/apenergy

https://doi.org/10.1016/j.apenergy.2021.117436

Received 23 March 2021; Received in revised form 1 July 2021; Accepted 8 July 2021

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Fast irradiance fluctuations can negatively affect the operation of PV generators. Under homogeneous operating conditions, the non-linear electrical characteristics of PV generators have exactly one peak, i.e., maximum power point (MPP). However, under non-homogeneous operating conditions, the PV cells of a PV generator have divergent electrical characteristics shaping an electrical characteristic of the whole generator. In a series connection of PV cells, the same current flows through all the cells and the total voltage of the series connection is the sum of the individual cell voltages. The cell with the lowest short-circuit (SC) current limits the total current of the series connection. When the series connection current exceeds the SC current of the shaded cell, the shaded cell will be reverse-biased and act as a load. In that case, part of the power generated by the other cells is dissipated to heat in the shaded cell. That can lead to damaging of the shaded cell which is normally avoided by connecting bypass diodes in anti-parallel with a group of PV cells. Due to bypass diodes, the electrical characteristic of a PV system can have several MPPs. Under non-homogeneous operating conditions, a PV system can have at most as many MPPs as there are bypass diodes in the system. Only one of the MPPs represents true maximum power and is called the global MPP (GMPP) while the other MPPs are local MPPs (LMPP).

Existence of multiple MPPs can cause problems for MPP tracking (MPPT) and the voltage of the GMPP may fluctuate over a broad voltage range [3,10]. However, multiple MPPs and fast fluctuations in the GMPP voltage take place mainly only under considerable variation of irradiance over the PV generator. For well-designed PV power plants, that kind of a situation can only be caused by overpassing cloud shadows.

Effective MPPT is straightforward to implement during uniform operating conditions. However, during fast irradiance fluctuations and partial shading, conventional MPPT algorithms applying the hill-climbing method may get stuck at an LMPP instead of the GMPP [11]. In order to extract the highest possible output power under non-homogeneous operating conditions, several more complicated MPPT algorithms have been proposed [12]. For example, Zeˇcevi´c and Rolevski [13] recently presented a neural network approach to MPPT and Mirza et al. [14]

proposed slime mould optimisation and improved salp swarm optimi- zation MPPT algorithms. Most inverters have a certain allowed voltage range for proper operation and MPPT algorithms use a defined range of operating voltage to make sure that the GMPP is followed under changing operating conditions. Thus, the knowledge of the applicable operating range of the GMPP voltage of the installed PV generator is important for proper selection of the inverter and MPPT algorithm voltage ranges.

Although several novel MPPT algorithms have been proposed over the past few years for tracking the GMPP under the existence of multiple MPPs, the actual behaviour of the MPPs under varying operating conditions has received little attention. In [15], simplified equations were introduced to evaluate the voltage, current and power of multiple MPPs of partially shaded PV strings using only basic datasheet values. In [16], simplified equations were derived to estimate MPP characteristics of partially shaded PV arrays consisting of multiple parallel strings. Chen et al. [17] have proposed a method for modelling of PV modules at different ambient conditions, based on large datasets of measured current–voltage (I–U) curves. Actual MPP characteristics of PV generators have typically been studied by simulations based on fictitious irradiance values: MPP characteristics of separate PV modules were studied in [18];

I–U characteristics of a partially shaded PV module and a partially shaded series-connection of two PV modules were studied in [19];

voltage differences between LMPPs of a PV string of 3 modules were studied in [20]; GMPP voltage and power ranges of a PV string of 16 modules and of a series–parallel array of 16 modules were studied in [21,22], respectively; voltage differences between LMPPs of PV strings of up to 16 modules and GMPP voltage range of a PV string of 20 modules were studied in [23]; and number of MPPs of a string of 18 modules were studied in [24] under shading patterns of only two irradiance values. In all these simulation studies, hypothetical shading

patterns were used to study MPP characteristics of small PV arrays. MPP characteristics of PV arrays consisting of up to 1000 PV modules were studied in [25]. However, also this study was based on simulations and fictitious irradiance values. MPP currents and voltages of a string of 18 modules were studied by simulations in [26] under partial shading of only two fictitious irradiance values affecting the string. Moreover, the results were validated using irradiance measurements of a period of 7.5 h as an input for the simulation model. MPP characteristics of PV generators have been studied by simulations based on a comprehensive set of actual irradiance measurements in only few studies: ranges and fluctuation of GMPP voltage and the highest MPP voltage of various PV arrays of up to 500 modules were studied in [10]; and number of MPPs of various PV arrays during partial shading events by clouds were studied in [27]. In [21,23,26], the highest GMPP voltage of a string of series-connected PV modules was discovered to be less than 90% of the string nominal open-circuit (OC) voltage while the lowest GMPP voltage can be very low. In [10], these results were found to be valid for larger PV arrays composed of multiple parallel PV strings. Additionally, the fastest changes in the GMPP voltage were observed to be extremely fast:

abrupt voltage shifts of more than 75% relative to the nominal MPP voltage occurred when an LMPP at another voltage region than the GMPP became the GMPP [10]. In conclusion, earlier studies of this topical area have typically been simulation studies based on hypothetical shading patterns lacking knowledge of real operating conditions of PV cells.

Although simulations in few earlier studies [10,27] were based on irradiance measurements instead of fictitious irradiance values, they included several assumptions and simplifications. For example, operating temperature of PV modules was assumed constant. The assumptions and simplifications used in simulations will inevitably affect the obtained results, leading to uncertainty and reduced accuracy compared to actual measurements. MPP characteristics of PV generators have been studied based on I–U curve measurements only in [3], where measured I–U curves of 2 PV strings were analysed. However, analysis of the re- sults was quite brief. Moreover, a single example scenario based on a measured I–U curve of a string of 6 modules was presented in [24]. Thus, there is a need for a comprehensive experimental study of the MPP characteristics of partially shaded PV generators. The main reason why earlier studies of this topic have typically been based on simulations instead of actual measurements is that I–U curve measurements on PV string or array level are available only from few places. In this study, the shortcomings of earlier studies are eliminated by analysing an extensive set of measured I–U curves of 3 PV strings.

In this article, the MPP characteristics of partially shaded PV strings are analysed based on measured I–U curves. The experimental study is based on more than 26000 measured I–U curves of 3 PV strings located at Tampere, Finland, measured under partial shading. Moreover, for the first time, a scenario is studied in which the MPP closest to the nominal MPP voltage is used the entire time as the operating point instead of the GMPP. Since the highly variable GMPP voltage poses challenges for MPPT, it would be advantageous to keep the inverter operation point all the time at voltages close to the nominal MPP voltage. In this way, the operation of the PV power plant would be smoother, more straightforward and more predictable. The main novelty of this study is that, for the first time, the MPP characteristics of partially shaded PV strings are analysed comprehensively based on an extensive set of actual electrical measurements. Moreover, an equally comprehensive study on the optimal operating point of PV generators based on actual electrical measurements, as well as measured differences in the produced energy between the GMPP and the MPP closest to the nominal MPP voltage for PV strings, has not been presented earlier. The results of this study are particularly relevant for PV generator and system design as well as for MPPT algorithm development when striving for higher overall efficiency and power quality of PV power plants.

The rest of this article is organised as follows. The used measurement data is introduced in Section 2.1 and the procedure for identification of

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partial shading events is introduced in Section 2.2. Section 3.1 examines the behaviour of the GMPP of PV strings. After that, the scenario in which the MPP closest to the nominal MPP voltage is used the entire time as the operating point instead of the GMPP is investigated in Sec- tion 3.2. The results and their significance are further discussed in Section 4. Finally, Section 5 provides the conclusions of the article.

2. Methods and data 2.1. Measurement data

Measured I–U curves of 3 PV strings of the PV power research plant of Tampere University [28] were utilised to study the MPP characteristics of partially shaded PV strings. I–U curves measured during seven partly cloudy days in summers 2019 and 2020 were analysed for each string.

Measurement period of each day was from 9:00 to 18:00 (UTC +2). The layout of the studied PV strings is shown in Fig. 1. The studied strings were String 1 and String 2, consisting of 17 series-connected PV modules, as well as String 4, which is composed of 6 series-connected PV modules. The studied PV strings are composed of NAPS NP190GK modules. Each module consists of 3 submodules of 18 polycrystalline silicon solar cells. Each submodule is protected by an anti-parallel- connected bypass diode. The nominal standard test conditions (STC) current, voltage and power (P) values of the modules are compiled in Table 1. Details of the studied strings are compiled in Table 2. Seven PV modules of String 1, six modules of String 2 and two modules of String 4 are equipped with irradiance and temperature measurements with a sampling frequency of 10 Hz. Photodiode-based SP Lite2 pyranometers are mounted at the same 45^◦tilt angle and 157^◦azimuth angle from north to east as the PV modules to measure the irradiance incident on the modules. The back-sheet temperature of the modules was measured by Pt100 temperature sensors.

An I–U curve was measured once per second during the measurement period, using an I–U curve tracer utilizing the electronic load method.

The I–U curve was traced by loading the PV system with a dynamic resistance that can alter the output current of the system. Parallel connected IGBTs act as an electronic variable load. The IGBTs are gate controlled with a ramp signal for opening and closing channels of the transistors. The measurement sweep direction is from OC to SC. The current was measured by Tektronix TCP312A current probe with Tek- tronix TCPA300 current probe amplifier (accuracy ±1%) and the voltage by LeCroy AP031 differential voltage probe (attenuation accuracy ±2%, output offset ≤5 mV). Each measured I–U curve consists of 4000 measurement points.

The measured I–U curves were pre-processed by the following procedure. First, the points with identical measured voltage value were replaced with a single point by averaging their measured current values.

After that, clearly abnormal measurement points were removed. A point was removed if its power was lower or higher than the power of the previous and next measurement point with a difference of more than 1.3 times the mean power difference between adjacent measurement points in the vicinity of the point (previous and next nine measurement points).

Finally, the measured current and voltage were smoothed separately using smooth.m function in MATLAB. Fig. 2 shows an example of measured P–U curves of String 1 illustrating the used pre-processing procedure.

2.2. Identification of partial shading events

Only the time when the studied PV strings were under non-uniform irradiance conditions is of interest in this study. Partial shading events during which the irradiance difference between the ends of the string was momentarily at least 200 W/m²were identified from the measurement data. Partial shading event analysis started when the difference between the measurement values of irradiance sensors at the ends of the string (see Fig. 1) exceeded 50 W/m²and ended when the irradiance difference no longer exceeded 50 W/m².

The total number of identified partial shading events for String 1 was 830 and their total duration was 125 min and 27 s. 1210 partial shading events were identified for String 2 with a total duration of 283 min and 16 s and 211 partial shading events for String 4 with a total duration of 25 min and 47 s. The average duration of the identified partial shading events was 9.1, 14.0 and 7.3 s for Strings 1, 2 and 4, respectively. As Fig. 1. Partial layout scheme of the PV power research plant of Tampere

University indicating the locations of PV modules and strings, the locations of irradiance and temperature sensors, and the heights (H) of surrounding build structures.

Table 1

Nominal STC values of NAPS NP190GK PV modules.

Parameter Value

PMPP, STC 190 W

IMPP, STC 7.36 A

UMPP, STC 25.8 V

ISC, STC 8.0 A

UOC, STC 33.0 V

Table 2

Details of the studied PV strings. The nominal STC values were calculated using the nominal STC values of NAPS NP190GK PV modules presented in Table 1.

String Number of

modules PMPP, STC

(W) UMPP, STC

(V) UOC, STC

(V) Length

(m)

1 17 3230 439 561 28.8

2 17 3230 439 561 40.4

4 6 1140 155 198 8.9

Fig. 2.Example of an original and pre-processed measured P–U curve of String 1 in the vicinity of the GMPP.

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expected, the number and duration of the identified partial shading events increased with increasing physical length of the PV string. The daily numbers and durations of the identified partial shading events are presented in the Appendix. In total, more than 26000 I–U curves measured during the partial shading events were analysed.

3. Experimental results

In this section, the MPP characteristics of partially shaded PV strings are studied based on more than 26000 I–U curves measured during the identified partial shading events. The behaviour of the GMPP voltage and power is investigated in Section 3.1. In Section 3.2, a scenario is studied for the first time in which the MPP closest to the nominal MPP voltage is used the entire time as the operating point instead of the GMPP.

3.1. Behaviour of the global maximum power point

Fig. 3 presents the distributions of the GMPP voltages for the studied PV strings during the identified partial shading events. The peaks of the distributions are around 90% of the nominal MPP voltages. The GMPP voltage was most of the time below the nominal value since the operating temperature of the PV modules was typically higher than the STC temperature of 25 ^◦C. The average temperatures of Strings 1, 2 and 4 were 37.5, 45.2 and 41.0 ^◦C, respectively, during the partial shading events. These temperature differences explain part of the differences between the GMMP voltage distributions in Fig. 3. The string temperature was calculated as the average of the measured string module temperatures. The share of the time when the GMPP voltage was higher than the nominal value was 17.0%, 4.4% and 3.3% for Strings 1, 2 and 4, respectively.

The measured GMPP voltage ranges are compiled in Table 3 for the studied PV strings during the partial shading events. The highest measured GMPP voltages were below 90% of the nominal OC voltage in line with the earlier simulation studies [21,23,26]. Moreover, the voltage ranges are largely in accord with the simulation results presented in [10] and the experimental results presented in [3]. String 4 has much higher minimum GMPP voltage, and thus narrower GMPP voltage range, than Strings 1 and 2 since String 4 is physically shorter due to which the differences in operating conditions between the PV modules are smaller. This is in line with [10], where it was found that the length of the strings has an essential impact on the lower limit of the GMPP voltage range of a PV array, whereas the electrical array configuration and the number of parallel-connected strings have only slight effects on the GMPP voltage range.

The distributions of the GMPP powers are shown in Fig. 4 for the studied PV strings during the identified partial shading events. As expected, the weight of the distribution shifts to lower power values as the string length increases, because larger irradiance differences can occur in longer PV strings, increasing the presence of MPPs at low voltages and powers. For all the studied strings, the GMPP power was almost all the Fig. 3. Distributions of the measured GMPP voltages for the studied PV strings

during the partial shading events. The voltages are with respect to the nominal MPP voltages at STC.

Table 3

Measured ranges of the GMPP voltage for the studied PV strings during the partial shading events.

String Minimum voltage with respect to UMPP, STC (%)

Maximum voltage with respect to UMPP, STC (%)

Minimum voltage with respect to UOC, STC (%)

Maximum voltage with respect to UOC, STC (%)

1 37.9 113.7 29.6 88.9

2 31.0 109.0 24.2 85.2

4 81.3 106.3 63.5 83.1

Fig. 4. Distributions of the measured GMPP powers for the studied PV strings during the partial shading events. The powers are with respect to the nominal MPP powers at STC.

Fig. 5.Scatter plots between the GMPP power and voltage for Strings 1 and 4 during the partial shading events (a) and uniform irradiance conditions (b). The values are with respect to the nominal MPP values at STC.

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time smaller than the nominal MPP power. This is reasonable since the irradiance incident on the PV modules varies during the day and the highest theoretical clear sky irradiance during a year in the Tampere region is just over 900 W/m²for PV modules installed at an optimal angle. Further, as stated earlier, the operating temperature of the modules was typically higher than the STC temperature decreasing also the power compared to operation at STC. The lowest measured GMPP power was 10.8%, 8.3% and 12.8% relative to the nominal MPP power, and the highest was 113.0%, 114.5% and 123.0% for Strings 1, 2 and 4, respectively. Thus, GMPP power higher than the nominal MPP power was measured for all the studied strings. The reason for the high power values is the cloud enhancement phenomenon [29]: because of photons scattering off clouds near the direct path of sunbeams, irradiance in partly cloudy conditions can be higher than under clear sky.

Fig. 5 (a) presents scatter plots between the GMPP power and voltage for Strings 1 and 4 during the identified partial shading events. For String 1, the range of the GMPP voltage tapers as the GMPP power increases. GMPP voltages from 38% to 112% occurred for String 1 with Fig. 6. Relative cumulative frequencies of the rate of change in the GMPP voltage for the studied PV strings during the partial shading events. The values are with respect to the nominal MPP voltages at STC.

Table 4

Rates of change in the GMPP voltage and power for the studied PV strings during the partial shading events with respect to the nominal MPP values at STC.

String Average rate of change in voltage (%/s)

Maximum change of voltage during one second (%)

Average rate of change in power (%/s)

Maximum change of power during one second (%)

1 2.2 62.7 4.9 37.2

2 2.7 65.4 3.3 43.8

4 1.5 6.5 7.5 32.5

Fig. 7. Relative cumulative frequencies of the rate of change in the GMPP power for the studied PV strings during the partial shading events. The values are with respect to the nominal MPP powers at STC.

Fig. 8. (a) Six consecutive power–voltage curves of String 1 measured during an identified partial shading event on May 20, 2020. (b) The irradiances measured by three irradiance sensors in the string during the period in (a).

Vertical green lines in (b) indicate the moments when the curves in (a) were measured.

Fig. 9.(a) Four consecutive power–voltage curves of String 4 measured during an identified partial shading event on May 29, 2020. (b) The irradiances measured by three irradiance sensors in the string during the period in (a).

Vertical green lines in (b) indicate the moments when the curves in (a) were measured.

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Table 5

Shares of time (%) during the partial shading events when the studied PV strings had two or more MPPs and when the MPP closest to the nominal MPP voltage was not the GMPP.

String 2 or more MPPs The MPP closest to nominal was not the GMPP

1 34.9 5.7

2 58.4 10.6

4 18.9 0.5

Table 6

Measured voltage ranges of the MPP closest to the nominal MPP voltage for the studied PV strings during the partial shading events.

String Minimum voltage with respect to UMPP, STC (%)

Maximum voltage with respect to UMPP, STC (%)

Minimum voltage with respect to UOC, STC (%)

Maximum voltage with respect to UOC, STC (%)

1 79.6 111.8 62.2 87.4

2 79.6 111.2 62.3 86.9

4 81.3 104.4 63.5 81.6

Table 7

Rates of change in the voltage and power of the MPP closest to the nominal MPP voltage for the studied PV strings during the partial shading events with respect to the nominal MPP voltages at STC.

String Average rate of change in voltage (%/s)

Maximum change of voltage during one second (%)

Average rate of change in power (%/s)

Maximum change of power during one second (%)

1 1.7 17.6 4.9 42.4

2 1.3 23.4 3.3 55.1

4 1.5 7.3 7.4 32.5

Fig. 10. Relative cumulative frequencies of the power difference between the GMPP and the MPP closest to the nominal MPP voltage for the studied PV strings when there were two or more MPPs. The power difference is presented with respect to the nominal MPP power at STC.

Table 8

Differences in the voltage and power between the GMPP and the MPP closest to the nominal MPP voltage for the studied PV strings during the time of having two or more MPPs. The differences in voltage and power are with respect to the nominal MPP values at STC.

String Average difference in voltage (%)

Maximum difference in voltage (%)

Average difference in power (%)

Maximum difference in power (%)

1 2.5 64.9 0.3 22.6

2 4.7 69.7 0.9 53.0

4 0.2 11.6 0.1 15.1

Fig. 11.Example of the behaviour of the power and voltage of the GMPP and the MPP closest to the nominal MPP voltage (CMPP) for String 1 on May 20, 2020. The powers and voltages are with respect to the nominal MPP values at STC.

Table 9

Differences in the produced energy between the GMPP and the MPP closest to the nominal MPP voltage for the studied PV strings with respect to the energy produced when operating at the GMPP (%) during the time of having two or more MPPs, during the total duration of the partial shading events and during the entire measurement period.

String More than 1 MPP Partial shading events Entire measurement period

1 0.66 0.20 0.20

2 1.99 1.00 0.35

4 0.14 0.02 0.03

Table A1

Daily numbers and durations of the identified partial shading events for String 1.

Day Number of identified partial

shading events Duration of partial shading events

June 16, 2019 115 22 min 48 s

June 19, 2019 178 13 min 21 s

May 12, 2020 111 17 min 39 s

May 20, 2020 178 28 min 2 s

August 14,

2020 73 15 min 16 s

August 15,

2020 92 11 min 34 s

August 16,

2020 83 16 min 47 s

Table A2

Day Number of identified partial shading

events Duration of partial shading

events June 20,

2019 125 29 min 5 s

June 22,

2019 412 44 min 45 s

June 19,

2020 138 40 min 59 s

June 20,

2020 139 33 min 10 s

June 21,

2020 230 62 min 40 s

June 22,

2020 74 36 min 49 s

June 23,

2020 92 35 min 48 s

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GMPP powers lower than 30%, whereas the GMPP voltage was between 83% and 103% when the GMPP power was higher than the nominal MPP power. For the sake of comparison, corresponding scatter plots are presented in Fig. 5 (b) during uniform irradiance conditions. There is a considerable difference in the GMPP voltage range of String 1 between the partial shading events and uniform irradiance conditions. During uniform irradiance conditions, almost all the measured GMPP voltages were between 80% and 100% and GMPP powers were concentrated significantly closer to nominal MPP power. Whereas for String 4, the GMMP voltages are limited to a much smaller voltage range between 80% and 100% of the nominal MPP value due to the much shorter length of the string. Also, the difference in the GMPP voltage range between partial shading and uniform irradiance conditions is quite slight. Irra- diance conditions of String 1were considered uniform when all the irradiance measurements of the string were within 10% of the highest measured value. For String 4, the corresponding limit was 5%.

Fig. 6 shows the cumulative frequencies of the rate of change in the GMPP voltage for the studied PV strings during the identified partial shading events. The figure reveals a large difference between the strings.

Fast changes in the GMPP voltage were much more common for Strings 1 and 2 than for the shortest String 4. The GMPP voltage of Strings 1, 2 and 4 changed faster than 5 %/s for 6.4%, 8.2% and 1.7% of the total duration of the partial shading events, respectively. The largest change in the GMPP voltage in one second was over 60% for Strings 1 and 2 while it was only 6.5% for String 4. It is obvious that the extremely fast variations existing in the GMPP voltage of Strings 1 and 2 cause problems for MPPT leading to energy losses. The average and maximum changes of the GMPP voltage during one second are compiled in Table 4.

The average rate of change in the GMPP voltage was from 1.5 to 2.7 %/s for the studied strings being the higher the longer is the string.

The cumulative frequencies of the rate of change in the GMPP power are presented in Fig. 7 for the studied PV strings during the identified partial shading events. As expected, average changes in the GMPP power were the faster the shorter was the string since changes in average irradiance over a land area are the faster the smaller is the area. How- ever, the largest observed change of power between consecutive measurements, i.e., within one second, was the larger the longer was the string. The average and maximum changes of the GMPP power during one second are compiled in Table 4. The measured average rates of change in the GMPP power were from 3.3 to 7.5 %/s being in line with the simulation results obtained for corresponding PV array sizes in [30].

The following example illustrates the behaviour of the P–U curve and the GMPP of a PV string during fast changes in irradiance causing partially shading. Six consecutive P–U curves of String 1 are shown in Fig. 8 (a) measured during a partial shading event with extremely high rates of change in the GMPP voltage and power. The largest change in the GMPP voltage of String 1 during one second, 62.7% with respect to the nominal MPP voltage, was measured on May 20, 2020. That voltage shift occurs between the second and third lowest curve in Fig. 8 (a). The GMPP voltage changed from 166 V to 441 V within one second and the decrease in power was 398 W. During the previous second, the change in

the GMPP voltage was 146 V while the GMPP power decreased 779 W.

During the second before that, the changes in the GMPP voltage and power were 110 V and 778 W, respectively. Thus, the GMPP power decreased 1557 W, or 48% with respect to the nominal MPP power, in two seconds, while the decrease of the GMPP voltage was 255 V, or 58%

with respect to the nominal MPP voltage. During the partial shading event, the GMPP voltage was most of the time close to the nominal MPP voltage while the GMPP power decreased from 2994 to 451 W. This example demonstrates why the GMPP power distributions in Fig. 4 are flatter than the GMPP voltage distributions in Fig. 3.

The irradiances measured at the ends and middle of the string during the partial shading event are presented in Fig. 8 (b). During the period of Fig. 8, a dark cloud shadow covered the string starting from the PV modules on the right-hand side of the string (see Fig. 1). Before the irradiance transition, the middle part of the string was under cloud enhancement. At the beginning of the partial shading event (the up- permost curve in Fig. 8 (a)), the PV string was unshaded and the P–U curve of the string had smooth shape with only one MPP. On the other hand, the third lowest curve shows an example of the typical shape of the P–U curve of a partially shaded PV system with multiple MPPs. The power is quite even over a wide voltage range from 100 to almost 500 V.

Under that kind of conditions, large and fast changes in the GMPP voltage may occur.

The next example illustrates how the P–U curve and the GMPP of a short PV string behave during fast changes in irradiance. Fig. 9 (a) shows four consecutive P–U curves measured during a partial shading event of String 4 with the largest change of the GMPP power in one second. It occurred between the second and third curve from the top when the GMPP power decreased 370 W, or 32.5% of the nominal MPP power.

The GMPP power decreased 66% of the nominal MPP power from 1132 to 375 W during the partial shading event while the GMPP voltage was quite even varying between 143 and 156 V. Fig. 9 (b) shows the irradiances measured at the ends of the string during the same partial shading event as the dark cloud shadow moved on top of the string.

The shape of the P–U curves in Fig. 9 (a) illustrates how the irradiance differences between the modules of short PV strings are quite small and how there is typically only one MPP during even fast irradiance transitions due to moving clouds. This results from the fact that cloud shadows produce gentle irradiance transitions, resulting in only minor irradiance differences between adjacent modules. The average length of irradiance transitions caused by cloud shadows at Tampere region is around 150 m [31], i.e., much longer than the length of the studied PV strings. Since String 4, consisting of only six modules, is much shorter than String 1, its voltage range of nearly constant power under partial shading cannot be as wide as in the case of String 1 (see Fig. 8 (a)). This explains why the fastest changes in the GMPP voltage are much faster for Strings 1 and 2 than for String 4 as shown in Fig. 6 and Table 4.

3.2. Operation of PV strings at the MPP closest to the nominal MPP voltage instead of the global MPP

The shares of time when the studied PV strings had two or more MPPs and when the MPP closest to the nominal MPP voltage was not the GMPP during the identified partial shading events are compiled in Table 5. The strings had two or more MPPs from 19% to 58% of the total duration of the partial shading events and the share increased with increasing physical length of the string. Correspondingly, also the share of time when the MPP closest to the nominal MPP voltage was not the GMPP increased as the physical length of the string increased. For the physically longest String 2, the time share was over 10%, while for the shortest String 4 it was only half a percent.

The measured voltage ranges of the MPP closest to the nominal MPP voltage during the identified partial shading events are compiled in Table 6 for the studied PV strings. For all the studied strings, the highest measured voltages of the MPP closest to the nominal MPP voltage were close to the highest measured GMPP voltages. For String 4, the lowest Table A3

Day Number of identified partial

shading events Duration of partial shading events

May 28, 2020 12 2 min 7 s

May 29, 2020 49 7 min 32 s

May 30, 2020 28 2 min 2 s

August 7,

2020 77 7 min 55 s

August 10,

2020 12 2 min 8 s

August 11,

2020 22 1 min 58 s

August 12,

2020 11 2 min 5 s

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measured voltage of both the MPPs was the same. However, for Strings 1 and 2, the MPP closest to the nominal MPP voltage had a much higher minimum voltage than the GMPP, i.e., the MPP closest to the nominal MPP voltage had considerably narrower voltage range than the GMPP.

The average and maximum rates of changes in the voltage and power of the MPP closest to the nominal MPP voltage are compiled in Table 7.

For Strings 1 and 2, the average and maximum rates of change in the voltage of the MPP closest to the nominal MPP voltage were significantly lower than those of the GMPP. This result indicates that following of the MPP closest to the nominal MPP voltage would not pose as hard challenges for MPPT as following of the GMPP. Thus, following of the MPP closest to the nominal MPP voltage could reduce energy losses caused by fallible MPPT. However, for String 4, the average rate of change of both the MPP voltages was the same and the measured maximum voltage change was only a bit larger for the MPP closest to the nominal MPP voltage than for the GMPP. For all the studied strings, the average rate of change in the power of the MPP closest to the nominal MPP voltage was very close to that of the GMPP. For Strings 1 and 2, the maximum observed change of power during one second was larger for the MPP closest to the nominal MPP voltage than for the GMPP. For String 4, the maximum measured change in the power was equal for both of the MPPs.

The results in Tables 6 and 7 show that the MPP closest to the nominal MPP voltage has some advantages compared to the GMPP. In the case of Strings 1 and 2, the MPP closest to the nominal MPP voltage had a clearly narrower voltage range and much lower rates of voltage change than the GMPP. On the other hand, for the shortest String 4, only minor differences between the MPPs were observed in the voltage ranges and in the rates of change of the voltage and power. But how big the differences between the MPPs are in voltage and power and how much energy is lost if the PV system does not work in GMPP but in the MPP closest to the nominal MPP voltage? In the following study, only the time when there were multiple MPPs was considered.

The relative cumulative frequencies of the power difference between the GMPP and the MPP closest to the nominal MPP voltage are presented in Fig. 10 for the studied PV strings. For String 4, as presented in Table 5, the MPP closest to the nominal MPP voltage was almost the entire time the GMPP. Moreover, when the MPP closest to nominal was not the GMPP, the power difference between the MPPs was typically very small.

This is seen also in Fig. 10 as minimal power differences between the two MPPs. For Strings 1 and 2, much larger power differences existed between the MPPs up to 20% and 50% with respect to the nominal MPP power, respectively. The power differences between the MPPs increased considerably with increasing PV string length. For 1.6%, 6.7% and 0.3%

of the time when there were more than one MPP, the power difference between MPPs was more than 5% in Strings 1, 2 and 4, respectively.

The average and maximum differences in the voltage and power between the GMPP and the MPP closest to the nominal MPP voltage are compiled in Table 8 for the studied PV strings. The differences in the voltage and power between the MPPs increased with increasing physical length of the PV string. The average difference in voltage was from 0.2%

to 4.7%. The largest measured voltage difference between the MPPs was over 60% for Strings 1 and 2 and less than 12% for String 4. The average difference in the power between the MPPs was from 0.1% to 0.9%

indicating that operation at the MPP closest to the nominal MPP voltage instead of the GMPP will have only minor effect on energy production.

Fig. 11 shows an example illustrating the behaviour of the power and voltage of the GMPP and the MPP closest to the nominal MPP voltage under rapidly varying operating conditions. Although the powers of the MPPs varied greatly during the period under review, the differences in voltage and power between the MPPs mostly remained small. However, momentary voltage and power differences existed between the MPPs. A large difference in voltage between the MPPs existed around 10:05:25 during a fast irradiance transition. The same partial shading event is discussed in the example of Fig. 8. During the partial shading, the GMPP voltage dropped momentarily below 170 V (40%) while the voltage of

the MPP closest to the nominal MPP voltage remained in proximity to the nominal MPP voltage the entire time.

Based on the results shown above, it is evident that certain advantages can be achieved by operating PV systems constantly at the MPP closest to the nominal MPP voltage instead of the GMPP. Keeping the inverter operating point at high voltages close to the nominal MPP voltage would make the PV system operation more predictable and straightforward. Next it is studied how much energy would be lost by operating at the MPP closest to the nominal MPP voltage instead of the GMPP. The differences in the produced energy between the GMPP and the MPP closest to the nominal MPP voltage are presented in Table 9 for the studied PV strings. The differences in the produced energy between the MPPs were small in all strings but increased strongly with increasing string length. The relative energy lost due to operating at the MPP closest to the nominal MPP voltage instead of the GMPP was from 0.1%

to 2.0% during the time of having multiple MPP and only from 0.02% to 1.0% during the identified partial shading events. It is worth noting that only the time when the studied PV strings were partially shaded was considered, so the energy losses are completely negligible.

To further illustrate the overall impact of operating at the MPP closest to the nominal MPP voltage instead of the GMPP, the energy difference was calculated also for the entire measurement period of 63 h.

During the entire measurement period, only from 0.03% to 0.35% of available energy was lost due to operating at the MPP closest to the nominal MPP voltage instead of the GMPP. Since the daily measurement period was from 9:00 to 18:00, these values represent the total amount of energy lost due to operating at the MPP closest to the nominal MPP voltage instead of the GMPP. The results show that certain advantages can be achieved by operating constantly at the MPP closest to the nominal MPP voltage instead of the GMPP at a cost of marginal energy losses. The main reason why the relative energy difference between the MPPs was equal (String 1) or even higher (String 4) during the entire measurement period than during the identified partial shading events is that only partial shading events during which the irradiance difference between the ends of the string was momentarily at least 200 W/m²were identified and considered. In practise, the operating conditions of PV strings are never absolutely uniform, as there are always small differences in irradiance and temperature between the PV cells of the strings.

4. Discussion

The presented comparison between the studied strings is not totally impartial since the operating conditions of the studied strings differ somewhat from each other. Although the strings are installed close by each other, they are differently shaded by the neighbouring built structures. Strings 1 and 4 are installed very close to each other and differences in shadings they experience by built structures can be considered small. Moreover, different days were considered for each string. It would have been ideal to measure all the studied strings simultaneously. Unfortunately, that kind of measurements were not available. However, all the measurements were performed on May, June and August and most of the measurement days were close to each other.

Since the study was defined to consider partly cloudy days, the use of the measurements of consecutive days was restricted. All the measurement days are presented in the Appendix. The purpose of this works was to study MPP characteristics of partially shaded PV strings. The study was not defined to study only partial shading events resulting from move- ment of clouds, although moving clouds are the main reason of partial shading for well-designed PV power plants. Thus, small differences in operating conditions and shading causes between the studied strings are not important relative to the scope of the study.

The procedure used to identify partial shading events affects the obtained results. The used procedure is based on the irradiance difference between the ends of the PV strings. Thus, some partial shading events, where only the middle part of the string was shaded, might have not been identified. The probability of these situations increases with

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increasing physical length of the string. However, these situations are rare for the studied strings as the typical size of cloud shadows is much larger than the length of the strings [31]. For Strings 1 and 2, irradiance and temperature measurements from several PV modules were available while for String 4 the operating conditions were measured only from the ends of the string. We opted to identify partial shading events based on the irradiance difference between the ends of the string so that uniform procedure was used for all the strings. More sophisticated procedure for identification of partial shading events could have been used if operating condition measurements for all the PV modules had been available.

Moreover, the selected limits for identification of partial shading events affect the obtained results. However, the used limits ensured that only the time of the studied strings being partially shaded was included in the study.

An obvious source of inaccuracy of the results is the sampling frequency of measurement data. The analysed I–U curves were measured with a 1 Hz sampling frequency. It has been stated in [28,32] that a sampling frequency of the order of 10 Hz is sufficient for identifying the fastest phenomena related to PV power generation. The use of a higher sampling frequency would affect mainly the fastest measured rates of change in voltage and power. Its effect to the voltage distributions of MPPs and energy differences between the MPPs would be minimal.

Moreover, the geographical location of the studied strings affected the obtained results. Thus, the exact values represent only high-latitude locations. However, the conclusions drawn from the results and the general observations of the studied phenomena are not geographically restricted but can be applied universally.

The results show that certain advantages can be achieved by operating constantly at the MPP closest to the nominal MPP voltage instead of the GMPP at a cost of only marginal energy losses. This is an important finding from PV generator and system design point of view. In Section 3.2, the energy differences between the MPPs were calculated assuming that the strings would operate all the time at the MPP, i.e., MPPT was assumed to be ideal. However, in practise, MPPT algorithms cannot follow the MPP perfectly during fast irradiance transitions. The results show that the MPP closest to the nominal MPP voltage has a clearly narrower voltage range and much slower voltage variations than the GMPP. Thus, MPPT meets more challenges when trying to follow the GMPP than the MPP closest to the nominal MPP voltage. Thus, the energy difference between the MPPs would be somewhat smaller in practise than presented in this study. In some cases, following of the MPP closest to the nominal MPP voltage would even yield more energy than following of the GMPP. Another discrepancy between this study and the real operation of PV systems is that PV power capacity is typically oversized with respect to the connecting inverter such that the generator nominal DC power is higher than the inverter nominal AC power [33]. Oversizing of PV power capacity limits the PV output power to the inverter nominal power during the periods of high irradiance. If the inverter operates in power limiting mode, i.e., the maximum power of the generator is higher than the inverter nominal power, the operating point of the inverter is moved to higher voltages than the GMPP voltage to decrease the current and power of the inverter. Thus, operating in power limiting mode results in losses of available energy production.

Taking the oversizing of the PV generator into account and studying of the effects of the oversizing on the studied phenomena were out of the scope of this study but are interesting topics for future work increasing the importance and practical value of the results.

5. Conclusions

In this article, MPP characteristics of partially shaded photovoltaic strings were studied based on measured I–U curves. The experimental study was based on more than 26000 I–U curves of 3 photovoltaic strings measured during partial shading events. The global MPP voltage of partially shaded photovoltaic strings was found to vary over a broad voltage range: the global MPP voltage of the longer strings of 17 series-

connected photovoltaic modules varied from below 40% to around 110%, with respect to the nominal MPP voltage, while the global MPP voltage of the shorter string of 6 modules varied from 81% to 106%. So, the global MPP voltage range clearly increased with increasing string length. It was found also that the variations in the global MPP voltage and power can be fast. The average rate of change in the global MPP voltage was from 1.5 to 2.7 %/s during the partial shading events being the larger the longer is the string. The largest measured changes in the global MPP voltage and power during one second were over 60% and about 40% with respect to the nominal MPP values, respectively.

Moreover, a scenario in which the MPP closest to the nominal MPP voltage is used the entire time as the operating point instead of the global MPP was studied for the first time. Operation at the MPP closest to the nominal MPP voltage was found to have advantages compared to operation at the global MPP. In the case of the longer strings, the MPP closest to the nominal MPP voltage has a clearly narrower voltage range and much lower rates of voltage change than the global MPP. Moreover, the energy difference between the MPPs was found to be insignificantly small from 0.03% to 0.35% of available energy. Incompleteness of practical MPP tracking was not taken into account in this study meaning that the energy difference between the MPPs would be even smaller in practise than presented in this study. The results show that it would make sense to follow the MPP closest to the nominal MPP voltage instead of the global MPP, as by that way the wide voltage range of the global MPP can be significantly reduced at a cost of negligible energy losses compared to operating at the global MPP.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgments

The authors acknowledge the financial support from Business Finland and KAUTE Foundation for the research reported in this article.

Appendix A. Daily numbers and durations of the identified partial shading events

The daily numbers and durations of the identified partial shading events for Strings 1, 2 and 4 are compiled in Tables A1, A2 and A3, respectively.

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