Estimation of the largest expected photovoltaic power ramp rates

(1)

Applied Energy 278 (2020) 115636

Available online 5 August 2020

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

Estimation of the largest expected photovoltaic power ramp rates

Kari Lappalainen

^a^,^*

, Guang C. Wang

^b

, Jan Kleissl

^b

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

bUniversity of California, San Diego, Department of Mechanical and Aerospace Engineering, CA 92093-0411, United States

H I G H L I G H T S

•A simple method for the estimation of the largest expected PV power RRs is proposed.

•The largest power RRs are estimated from RRs in the PV system average irradiance.

•The proposed method showed superior performance compared to an existing method.

•The method enveloped the RR in the measured power over 99.99% of the time.

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

Power ramp rate estimation Power fluctuation

Photovoltaic power generation Partial shading

Energy storage sizing Irradiance transition

A B S T R A C T

Photovoltaic (PV) systems are prone to irradiance variation caused by cloud shadows leading to fluctuations in generated power. Since these fluctuations can be harmful to the operation of power grids, there is a need to restrict the largest PV power ramp rates (RR). This article proposes a method to estimate the largest expected PV power RRs. The only inputs of the method are the minimum PV system dimension and the measurements of point irradiance and cloud shadow velocity. Since cloud shadows cause the largest power RRs for well-designed large- scale PV power plants, the relation between the largest RRs in irradiance and power during partial cloud shading events was studied based on irradiance measurements. The largest RRs in PV power are estimated from RRs in the average irradiance across the PV system. The proposed method was validated using measured data of 57 days from two PV systems. It showed superior performance compared to an existing method enveloping the RR in the measured power over 99.99% of the time. The method can be used in design and component sizing of PV power plants.

1. Introduction

The power generated by photovoltaic (PV) power plants has a highly variable nature due to irradiance fluctuations, which are mainly caused by cloud shadows. As the share of grid-connected PV power production capacity increases, there is a growing potential for PV output power variability having negative effects on the reliability and power quality.

Some transmission system operators have specified limitations for power ramp rates (RR) of grid-connected generators [1]. For example, the Puerto Rico Electric Power Authority has set a 10% limit of rated power per minute for PV power RRs [2]. Yet variations in the power generated by PV plants can be many times larger than this limit. For example, up to 70% per minute power variations were measured at a 9.5 MWp PV plant [3].

Compliance with RR requirements has typically been achieved by

means of PV power output curtailment or energy storage systems (ESS) [4]. However, the application of power curtailment is limited to upward PV power ramps [5]. Aggregate PV module DC capacity is typically oversized in respect to the connecting inverter [6] such that the nominal DC power of a PV system is higher than the nominal AC power of the inverter connecting the system to the grid. Oversizing of a PV generator limits the output power to the inverter nominal power in the periods of high irradiance leading to energy losses. If the inverter is operating in power limiting mode, i.e., the maximum power of the PV generator is higher than the maximum DC power of the inverter, variations in DC power are not transmitted to AC power. The optimal DC/AC ratio, i.e., the ratio of the nominal PV DC power to the nominal inverter power, depends on many factors such as irradiance conditions and inverter characteristics [7]. Knowledge of the largest expected PV power RRs facilitates inverter and PV generator sizing.

* Corresponding author.

E-mail addresses: kari.lappalainen@tuni.fi (K. Lappalainen), g3wang@eng.ucsd.edu (G.C. Wang), jkleissl@ucsd.edu (J. Kleissl).

Contents lists available at ScienceDirect

Applied Energy

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

https://doi.org/10.1016/j.apenergy.2020.115636

Received 9 February 2020; Received in revised form 14 July 2020; Accepted 29 July 2020

(2)

The requirements for ESSs have been studied in several articles, for example in [8–10]. Martins et al. [11] presented a comparison of RR control methods for PV systems equipped with ESSs. Moreover, Cir´es et al. [12] compared levelized cost of energy of three RR control methods. The largest RRs determine the power capacity of the ESS needed to buffer all downward PV power ramps. Since increasing ESS power and energy ratings increases ESS costs, the sizing of the ESSs has to be balanced against benefits in RR mitigation. If the largest expected PV power RRs can be reliably estimated, the ESSs can be sized optimally and ramp rate mitigation costs can be decreased. Mitigation of all power variations by the ESSs requires knowledge of the largest expected PV power RRs to determine the needed ESS power and energy ratings.

Solar radiation variability has been studied comprehensively, e.g. in [13–15]. Also, PV output variations have been studied by several re- searchers: output power variations of various PV array configurations were studied in [16] and power variations of large-scale PV power plants were studied based on experimental measurements in [3,17] and based on simulations in [18]. The smoothing of PV power variation with increasing system size and by geographical dispersion have been studied thoroughly, e.g. in [19–21]. Moreover, output power variability of PV system fleets was studied in [22]. The relationship between the average irradiance and power output of PV plants was studied based on simulation in [23] and based on measurements in [24]. Moreover, it was studied in [17] how much RRs observed in the measurements of a single irradiance sensor differ from RRs observed from dispersed sensors and from power RRs of a 48 MW PV plant. However, only few methods for the estimation of PV power output variations suitable for planning of PV systems and sizing of their ESSs have been proposed. To be suitable for PV system planning, the approach should not be too complex nor require multiple measurements. Marcos et al. [25] presented a method to determine power variation using solely irradiance measurements at a single point and the PV system size. The method models a PV plant as a low pass filter. The drawback of the method is that the model needs to be adjusted for each PV system, which requires irradiance and power measurements. In [26], the method was applied to simulate power variation of a fleet of large-scale PV plants. Wang et al. [27] presented a method, which estimates the largest expected PV power RRs using the geometrical layout of the PV plant, point irradiance or PV power measurements and cloud shadow velocity.

In this article, a method for the estimation of the largest expected PV power RRs is proposed. The method is based on the idea of estimating the largest RRs in the power of a PV system from RRs in the average irradiance of the system. The average irradiance over the area of the PV system was calculated by averaging the measured point irradiance over a time interval defined by the measured cloud shadow velocities and the minimum system dimension. The only inputs of the method are the minimum system dimension and measurements of point irradiance and cloud shadow velocity. The advantages of the method are its simplicity and accuracy. The proposed method is not intended for power forecasting, but it is meant to be a tool for design and component sizing of PV power plants. The performance of the proposed method was compared with the method presented in [27]. While the proposed method has the same objective and is largely similar to [27], it advances the state-of-the- art over [27] in two aspects: a) [27] relies heavily on the quality and availability of cloud shadow velocity measurements. b) [27] does not take into account the effects of shadow edges and mismatch losses.

The main novelty of this article is the description and experimental verification of a simple method for the estimation of the largest expected PV power RRs. The proposed method was validated and its performance was compared with an existing method using data from two PV systems located in San Diego, California. Moreover, for the first time, the relation between the largest RRs in PV power and irradiance was studied comprehensively. The relation was studied by simulations during almost 9000 shading transitions identified in irradiance data measured at Tampere, Finland. Those results are utilized to estimate the largest expected PV power RRs from RRs in irradiance. The estimation of the

largest expected PV power RRs is advantageous to PV, ESS, and inverter sizing.

The rest of the article is organized as follows. The simulation model and data used to study the relation between the largest RRs in PV power and irradiance are presented in Sections 2.1–2.3. In Section 2.4, the results related to the relation between the largest RRs in power and irradiance are presented and discussed. Section 3.1 introduces the data used to validate the proposed method. The proposed method is described in Section 3.2 and the procedure for its performance evaluation is introduced in Section 3.3. Section 3.4 introduces an existing method with which the proposed method is compared. Section 4.1 ex- amines the performance of the proposed method. Section 4.2 shows the comparison between the proposed method and the existing method. The results are further analyzed as a function of the power RR in Section 4.3 and the largest underestimation of measured power RR is examined in Section 4.4. The advantages and drawbacks of the proposed method are discussed in Section 5. Finally, the conclusions of the article are provided in Section 6.

2. Empirical relation between the largest ramp rates in irradiance and power

2.1. Electrical model for the PV modules

A PV submodule was used as a basic unit in the simulations. A PV submodule is a group of series-connected PV cells in a PV module pro- tected by a bypass diode. The electrical behavior of PV submodules was modeled by an experimentally verified MATLAB Simulink model [28], based upon the widely used one-diode model of a PV cell, which provides the following relationship between the current I and voltage U of a PV submodule:

I=Iph− Io

⎛

⎜⎝e^AkNsT/q^U+RsI − 1

⎞

⎟⎠− U+RsI Rsh

, (1)

where Iph is the light-generated current, Io the dark saturation current, Rs

the series resistance, A the ideality factor, T the operating temperature, and Rsh the shunt resistance of the PV submodule. The Boltzmann constant is represented by k, N_sis the number of PV cells in the submodule and q is the elementary charge. The bypass diodes were modeled using Eq. (1) by assuming Iph to be zero and Rsh infinite. Moreover, the bypass diodes were assumed to be at the same temperature as the submodules.

Electrical parameters of NAPS NP190GKg PV modules installed in the PV research plant of Tampere University were used in the simulations. The values of Rs and Rsh were obtained by fitting the characteristics of the simulation model to the measured characteristics of the PV modules by a method introduced in [29]. The fitting method is based on three points in the electrical characteristics of the PV module: the short- circuit (SC) current, the open-circuit (OC) voltage, and the maximum power point (MPP). It is stated in [29] that there is only one pair of values for Rs and Rsh for which the modelled MPP equals the one given by the manufacturer in standard test conditions (STC). This pair of R_s and Rsh can be solved from the following equation of MPP power derived from Eq. (1):

PMPP,STC=UMPP,STC

⎛

⎜⎝Iph,STC− Io,STC

⎛

⎜⎝e

UMPP,STC+RsIMPP,STC AkNsTSTC/q − 1

⎞

⎟⎠

− UMPP,STC+RsIMPP,STC

Rsh

⎞

⎟⎠, (2)

from which R_shcan be expressed as a function of R_sas

(3)

Rsh= UMPP,STC2+RsPMPP,STC

UMPP,STCIph,STC− UMPP,STCIo,STC

⎛

⎜⎝e

UMPP,STC+RsIMPP,STC AkNsTSTC/q − 1

⎞

⎟⎠− PMPP,STC

.

(3) More details of the simulation model and fitting procedure as well as the experimental verification of the model are presented in [28]. The MPP power, SC current, and OC voltage of the submodule under STC, given by the manufacturer, and the parameter values of the simulation model are compiled in Table 1. The ideality factor was selected to be 1.3, which is a typical value found in literature for silicon-based PV modules [30].

2.2. Irradiance transition model

Irradiance transitions caused by the edges of cloud shadows were modeled using the equation

G(t) = Gus− Gs

1+e^(t−^t⁰^)/b+Gs, (4)

where G is the irradiance, Gus and Gs are the irradiances corresponding to an unshaded and a fully shaded situation, respectively, and t is time [13]. Parameter b is related to the steepness of the irradiance change and its sign determines whether the change in irradiance is increasing or descending. Parameter t0 defines the midpoint of the irradiance change.

The shading strength (SS) of an irradiance transition, i.e., the attenua- tion of irradiance due to shading, is defined as

SS=Gus− Gs

Gus

. (5)

Using Eq. (4), irradiance transitions can be determined by four in- dependent variables: SS, b, apparent shadow edge speed ve, and apparent direction of movement α_eas presented in [31]. The apparent shadow edge velocity is the component of shadow velocity normal to the shadow edge. The duration of an irradiance transition was calculated as a product of b and the experimental regression coefficient of 7.67 [31].

The PV arrays simulated in this study consist of 6, 12, and 24 parallel strings of 23 series-connected PV modules. The PV strings of the arrays were located in straight lines from east to west without gaps between the modules and with a gap of 2.0 m between the strings. The PV modules were installed at a 45^◦tilt angle with respect to the horizon. The dimensions of the studied PV arrays, computed using the dimensions of the NP190GKg PV modules, are compiled in Table 2.

The electrical operation of the studied PV arrays was simulated based on the characteristics of the identified shadow edges moving across the arrays and three assumptions: 1) the cloud shadows were assumed to be wide enough to cover the whole PV array; 2) the edges of the cloud shadows were assumed to be linear across the PV array; and 3) the apparent velocities of the shadow edges were assumed to stay constant during each simulation period. These are reasonable approximations for the studied PV array sizes. A simulation period was the period when a

shadow edge was over the array, i.e., when the array was partially shaded due to the shadow edge. For example, for an increasing irradiance transition, this means that the period started when the first PV submodule of the array was less shaded than the others and ended when all the submodules where unshaded.

To make the proposed method simple and easily applicable, the STC irradiance (1000 W/m²) was used as Gus. Irradiances during the simulation period were calculated using Eq. (4) and the value of Gs was obtained using Eq. (5). In this way, actual irradiance measurements are not required for the simulations but only the values of SS, b, ve, and α_eare needed. The irradiance at the center of each submodule was used as the irradiance of the entire submodule for each time step of 0.1 s. To further simplify the proposed method, the effects of PV cell temperature were not taken into account, i.e., the operating temperature of the array was constant (at 25 ^◦C) during the simulations. The average irradiance over the array for each time step was calculated as the average of the irradiances at the centers of the submodules of the array. The experimental verification of the method to model the operation of PV modules during cloud shading transitions is presented in [32].

2.3. Data from Tampere University

Five months (May–September 2013) of irradiance data of the PV research plant of Tampere University was utilized to study the relation between ramp rates in irradiance and power. The data was measured by irradiance sensors S2, S5, and S6 (Fig. 1) with a sampling frequency of 10 Hz. The sensors were photodiode-based SP Lite2 pyranometers (Kipp&Zonen) installed at an azimuth angle of 157^◦from north to east and a tilt angle of 45^◦. The location and details of the used sensor triplet are presented in Fig. 1. More details of the measurement system are available in [33].

In total 9097 shadow edges, consisting of 4438 increasing and 4659 decreasing irradiance transitions, were identified in the measurement data using the method offered in [13]. The criterion for irradiance transition was at least a 40% SS. The SS and b of the identified transitions were determined by curve fitting Eq. (4) to the irradiance data and minimizing the root-mean-square deviation between the curve fit and the irradiance data. The curve fitting procedure has been described in more detail in [13]. Thereafter, the apparent velocities, i.e., the apparent speeds and directions of movement, of the shadow edges were

Table 1

Parameter values of the simulation model for the PV submodules and bypass diodes.

Parameter Value

PMPP, STC 63.3 W

ISC, STC 8.02 A

UOC, STC 11.0 V

A 1.30

Rs 0.110 Ω

Rsh 62.6 Ω

Abypass 1.50

Io, bypass 3.20 μA

Rs, bypass 20.0 mΩ

Table 2

Numbers of modules and dimensions of the studied virtual PV systems.

Number of modules (parallel ×

series) Dimensions

(m) Diagonal

(m) Area

(m²)

6 ×23 14.2 ×33.9 36.8 481

12 ×23 30.4 ×33.9 45.5 1030

24 ×23 62.7 ×33.9 71.3 2128

Fig. 1. Partial layout scheme of the PV research plant of Tampere University.

The used sensor triplet is indicated with the black triangle.

(4)

determined from the time lags of the irradiance transitions between the three irradiance sensors by the method given in [34]. To enable reliable determination of apparent velocities, the irradiance sensors were located close to each other at the same elevation and forming close to a right triangle.

2.4. Relation between the largest ramp rates in irradiance and power An example relation between the largest RRs in the average irradiance and power of a PV array during a cloud shading event is illustrated in Fig. 2 for the 6 ×23 array. The maximum rate of change (or RR) in the power of the array is larger than in the average irradiance over the array.

The main reason why the largest RR in power is larger than the one in irradiance is mismatch losses. Mismatch losses are defined as the difference between the sum of power of individual PV submodules of a PV array operating separately at their maximum power point, and the actually generated maximum power of the whole PV array. Since the output power plus the mismatch losses in Fig. 2 are only slightly smaller than the normalized irradiance, the difference between the normalized irradiance and the normalized power is mainly due to mismatch losses.

The remaining difference between the irradiance and power +mismatch losses curves results from the fact that the relative output power of a PV submodule decreases faster than the relative irradiance received by the submodule when that irradiance is lower than the STC irradiance [16].

In this example, the largest change in power during 1 s was 2.6% while in irradiance it was 0.24%. The large difference results from the steps on the power curve. These steps exist also on the irradiance curve but, because of mismatch losses, they are larger in the power curve. A step exist when the transition zone moves over a group of PV submodules. A cloud shading event during which there is a large difference between the largest RRs in irradiance and power was selected for this example. Thus, the example does not represent typical cloud shading events.

Next, the ratio between the largest RRs in power and irradiance during the identified shading transitions was examined. This ratio RRP, max/RR_{G, max}is called the RR ratio from here on out. The RR ratio is larger than one for every shading transition. Since the measurement data from UCSD was measured with 2 s intervals, the largest RRs were computed from the largest observed changes in irradiance and power occurring in 2 s. The average and maximum RR ratios for the studied PV arrays during the identified shading transitions are presented in Table 3.

The average RR ratios were around 1.1 decreasing with increasing array size. The largest observed RR ratios were from 5.4 to 6.1 increasing with increasing array size.

The distributions of the RR ratio for the studied PV arrays during the identified shading transitions are presented in Fig. 3. For most of the shadow edges, the largest power RR was less than 10% larger than the

largest RR in irradiance and the number of shadow edges in each bin decreased with increasing RR ratio. 42% of the shadow edges caused RR ratios larger than 1.1 for the 6 ×23 array. This share decreased with the increasing array size being 22% for the 24 ×23 array.

Fig. 4 presents a scatter plot between the largest irradiance RR and the RR ratio for the 6 × 23 PV array during the identified shading transitions. RR ratios larger than the 99th percentile are marked in blue dots. The RR ratio decreased with increasing maximum irradiance RR.

RR ratios larger than 3.0 were observed only during shading transitions during which the largest RR in irradiance was less than 0.3 %/s.

The largest RR ratios were caused by shading transitions with short duration and slow apparent speed. The average duration and apparent speed of the transitions causing RR ratios larger than the 99th percentile were 6.8 s and 3.2 m/s while these values for all the identified shading transitions were 15.1 s and 8.5 m/s. Consequently, the average length of the transitions causing RR ratios larger than the 99th percentile, calculated as the product of the apparent speed and the duration of the transition, was much smaller (18.3 m) than that of all the identified transitions (113 m).

The curve fitted to the largest RR ratios in Fig. 4 follows the equation RRP,max

RRG,max

=c1RRG,maxc2, (6)

where c1 and c2 are fit constants. Using Eq. (6) the maximum RR in power can be estimated from the maximum RR in irradiance as RRP,max=c1RRG,max(1+c2). (7)

The values of c1 and c2 for the studied PV arrays during the identified shading transitions are presented in Table 4.

3. Maximum power ramp rate model 3.1. Data from University of California, San Diego

The proposed method was validated using data from two PV systems located on the EBU2 building (32^◦52^′53.1^′′N, 117^◦13^′59.2^′′W) at the University of California, San Diego (UCSD). The layouts of the PV systems are illustrated in Fig. 5. The PV modules were installed at a 20^◦tilt angle with respect to the horizon and a 225^◦azimuth angle from north.

PV system A (marked in red in Fig. 5) consists of a total of 210 PV

Fig. 2. Average irradiance, output power, and mismatch losses for the 6 ×23 PV array during the movement of an identified shadow edge over the array.

Irradiance is normalized by 1000 W/m²and power and mismatch losses are normalized by the nominal power of the array.

Table 3

RR ratios for the studied PV arrays during the identified shading transitions.

PV array Average RR ratio (–) Maximum RR ratio (–)

6 ×23 1.13 5.41

12 ×23 1.11 5.84

24 ×23 1.08 6.14

Fig. 3.Distributions of the ratio between the largest RR in power and average irradiance for the studied PV arrays during the identified shading transitions.

(5)

modules and is connected to the grid by five SMA Sunny Boy 7000US inverters. PV system B (marked in blue in Fig. 5) is a bit smaller than PV system A and consists of 182 PV modules. It is connected to two SMA Sunny Boy 5000US and three SMA Sunny Boy 7000US inverters. The AC powers of the PV systems were measured by the inverters with a sampling frequency of 0.5 Hz. The details of the PV systems are compiled in Table 5.

The proposed method utilizes the measurements of local irradiance and cloud motion vectors (CMV) to estimate the largest expected PV power ramp rates of the PV systems. A global horizontal irradiance (GHI) sensor and a cloud speed sensor (CSS) are installed on the same rooftop as PV systems A and B (Fig. 5). The GHI values were measured with a sampling frequency of 0.5 Hz. The CSS consists of an array of nine phototransistors. The operation and features of the CSS are described in

detail in [35]. The procedure to calculate cloud shadow velocity v from the detected apparent shadow edge velocities ve is presented in Ap- pendix A Since the CSS requires cloud edge passages to determine CMV, its data acquisition rate is irregular. The produced shadow movement direction α is defined relative to north. Because cloud shadows cause the largest PV power RRs for the studied PV systems, only partially cloudy days are of interest. Thus, fully clear days or overcast days with only small RRs in the measured irradiance were excluded from the study.

Data from 57 partially cloudy days with synchronous GHI, cloud speed, and power measurements were used to validate the proposed method.

3.2. Model description

The power RR estimate was calculated from the largest RR in the average irradiance of the PV system utilizing the simulation results presented in Section 2.4, specifically Eq. (7). Since the dimensions of the PV systems of UCSD EBU2 (Table 5) are close to the dimensions of the 6

×23 PV array (Table 2), the simulation results of the 6 ×23 PV array in Section 2.4 were utilized in the estimation of the power RRs of the actual PV systems of EBU2.

The DC capacities of the UCSD PV systems are oversized in respect to the nominal AC powers of their connecting inverters (see Table 5). The oversized PV DC capacity is “clipped” by the inverter as visible in the daily power production profile of the system. Oversizing causes the largest RRs in the AC power to be smaller than the largest RRs in the DC power of the PV system. While we do not model the clipping, the modeled DC power ramps are a conservative assumption for the worst AC ramp rates.

The only measurements needed for the estimation of power RRs of a PV system are irradiance and cloud shadow speed from a single point near the PV system. First, the average irradiance of the PV system is estimated from measured point irradiance. The average irradiance over the land area of the PV system is calculated by averaging the measured irradiance over a time interval centered around the present time and defined by the ratio of the PV system shorter dimension and the measured cloud shadow velocity. The cloud shadow velocity is calculated from the detected apparent shadow edge velocities as presented in Appendix A. Since the purpose of the method is to estimate the largest expected PV power ramp rates, the minimum dimension of the PV system is used in the averaging of the irradiance as a conservative estimate.

The minimum dimensions of the rectangles marked in Fig. 5 are used for UCSD EBU2. Assuming that the irradiance on a tilted surface changes with approximately the same RR as GHI, GHI measurements are substituted for actual plane of module irradiance measurements.

Due to the infrequent CSS measurements, the largest RR in the average irradiance in a 10 min period around present time is used as the RR in irradiance from which the power RR estimate is calculated using Eq. (7) and the values of Table 4. Small power ramps even exist in clear conditions and are not necessarily tied to cloud speeds or mismatch losses. Such ramps are difficult to model yet immaterial for PV system planning. Therefore we prevented these small ramps from causing ramp violations by setting RRP, max ≥0.2 %/s even if the measured point irradiance was constant. The computational procedure of the proposed method is illustrated in Fig. 6.

Fig. 4. Scatter plot between the largest RR in average irradiance and the ratio between the largest RR in power and average irradiance for the 6 ×23 PV array during the identified shading transitions and curve fit to the largest values.

Table 4

Parameter values of the proposed power function of Eq. (7) for the studied PV arrays.

PV array c1 (–) c2 (–)

6 ×23 3.323 − 0.2481

12 ×23 3.167 − 0.2301

24 ×23 3.022 − 0.2117

Fig. 5. Aerial view of PV systems A (red rectangle) and B (blue rectangle) installed on the rooftop of UCSD EBU2. The locations of the CSS and GHI sensor are indicated with a green and yellow circle, respectively. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

Table 5

Numbers of modules, nominal powers, and dimensions of the PV systems installed on the rooftop of UCSD EBU2.

PV system Number of PV modules

Nominal DC power (kW)

Nominal AC power (kW)

Dimensions

(m) Diagonal

(m)

A 210 43.1 35.0 15.0 ×34.2 37.3

B 182 37.3 31.0 33.0 ×14.4 36.0

(6)

3.3. Performance evaluation

The performance of the proposed method is evaluated following [27]. First, the compliance indicator σ is calculated as the ratio of the measured power RR to the estimated power RR

σ(t) =RRmeasured(t)

RRestimated(t). (8)

The measured RR complies with the estimate if σ≤1. The performance of the proposed method can be evaluated by considering the compliance in evaluation time windows with certain lengths. The largest σ in each non-overlapping evaluation window is denoted by μ. The noncompliance rate ε is defined as

ε= (

1− Ncpl

Nw

)

, (9)

where Ncpl is the number of compliance events, i.e., evaluation windows for which μ≤1, and Nw is the total number of evaluation windows. The overestimation of the RR estimate is evaluated by the degree of overestimation δ expressed as

δ= (∑_N_cpl

j=1

(1− μ_j⁾ Ncpl

)

. (10)

The summation term in Eq. (10) is not a continuous series, but it includes only the compliance events.

The selection of the length of evaluation time windows is arbitrary and the selection affects the values of ε and δ. As can be concluded from Eqs. (9) and (10) short window lengths will lead to smaller ε but larger δ. Thus, we present the results of ε and δ as a function of evaluation time window starting from window length of 2 s, which is the resolution of the data. The upper limit of the studied evaluation time windows was selected to be 30 min since this time scale is of interest to transmission system operators.

In addition, the validation of the proposed method includes the

difference between the estimated and measured power RRs with respect to the nominal AC power of the PV system. Positive differences indicate overestimation and negative differences indicate underestimation.

3.4. An existing method for power ramp rate estimation

The performance of the proposed method was compared with an existing method proposed by Wang et al. [27] which is referred to as the existing method or W2019. In that method, the largest power RR is estimated from the movement of an irradiance field, consisting of an unshaded and a shaded area, over the PV system. The RR estimate is calculated utilizing shadow velocity, irradiance (or power), and the PV system dimension as

RRP,max= ±(

Lv|cosα| +Wv|sinα| − v²Δt|sinαcosα|)Pcs|ktmax− ktmin|

LW ,

(11) where L and W are the dimensions of the PV system, α is the difference between the azimuth angles of the measured shadow movement direction and the PV system, Δt is time interval (2 s), Pcs is clear sky PV power, and kt is clear sky index [27]. Wang et al. used the power produced on the most recent clear day as Pcs and the largest and smallest kt from recent history (30 min) as ktmax and ktmin. ktmax is not necessarily the kt in clear sky condition, but the largest value in the 30 min interval, which can be higher (cloud enhancement) or lower (overcast) than the clear sky value. Either measured irradiance or PV power can be used as P_cs. The RR estimate was calculated only if CMVs were available in the 30 min time window.

W2019 is based on the following five assumptions: 1) The unshaded and shaded area are larger than the PV system; 2) The unshaded and shaded area are steady and homogeneous during the time interval; 3) The unshaded and shaded area are separated by a line, i.e., the effects of shadow edges were not taken into account; 4) The power of the PV system is directly proportional to the average irradiance of the system, i.

e., mismatch losses were not taken into account; 5) The effects of Fig. 6. Procedure for computing power RR estimate.

(7)

inverter and PV cell temperature were not taken into account.

In the model proposed in this paper, the aforementioned assumptions 1), 2), and 5) were used when studying the relation between RRs in irradiance and power, i.e., solving the parameter values of Eq. (7).

However, the effects of shadow edges and mismatch losses were taken into account when solving the parameter values, i.e., the assumptions 3) and 4) were not used. Shadow edges are taken into account as the actual irradiance time series is used to calculate the average irradiance over the PV system. In the proposed method, the minimum dimension of the PV system was used in the averaging of the irradiance, whereas the actual angle between the PV system and cloud shadow movement direction was taken into account in W2019. Thus, the proposed method needs only the minimum dimension of the PV system, whereas W2019 needs both dimensions of the rectangular PV system.

4. Results

4.1. Performance of the proposed method

In Fig. 7, ε for the proposed method over the set of 57 days is presented as a function of evaluation time window. As expected, ε increased with increasing length of the evaluation time window. The measured power RRs comply well with the estimates of the method: ε for 30 min evaluation time window is 5.0% for PV system A and 2.0% for system B.

The noncompliance rate is clearly smaller for PV system B than for system A with all evaluation time window lengths.

Fig. 8 presents δ for the proposed method over 57 days as a function of the evaluation time window. Since δ is calculated as the average of the smallest overestimations of compliance events, δ decreased with increasing length of the evaluation time window. At most of the studied window lengths, δ was smaller for PV system B than for system A. The results of Figs. 7 and 8 indicate that the proposed method works better for system B than for system A. Only at evaluation time windows shorter than 2 min, δ was slightly smaller for PV system A than for PV system B.

The cumulative frequencies of the daily ε for the 2 min evaluation time window are presented in Fig. 9. The measured RRs complied with the estimates of the proposed method most or all of the time on the studied days. For 73.7% and 91.2% of the days ε_{2 min}for systems A and B, respectively, was smaller than 0.5%. During 18 (system A) and 36 (system B) days ε was 0%, i.e., the measured RR complied with the estimate all the time. There were only 4 and 2 days for systems A and B, respectively, for which ε_{2 min}was larger than 1.0%.

Fig. 10 presents the cumulative frequencies of the daily δ for 2 min evaluation time for PV systems A and B. The daily δ_{2 min}varied greatly:

the smallest values were 53% and 51% for systems A and B, respectively, and the largest value was around 79% for both systems. For 54.4% and 56.1% of the days the δ_{2 min}for systems A and B, respectively, was smaller than 70%.

Based on Figs. 7–10 there seems to be a clear difference in the performance of the proposed method between PV systems A and B. This difference is larger in ε than in δ. The proposed method fails to envelop the measured RR of system A more often than the measured RR of system B (Figs. 7 and 9). However, the degree of overestimation is typically smaller for system B than for system A (Fig. 8). To further illustrate the differences in the performance of the proposed method between PV systems A and B, the relative cumulative time distributions of the difference between the estimates and the RRs in the measured powers of the systems are presented in Fig. 11. The two curves in Fig. 11(a) are almost on top of each other meaning that there are no considerable differences between the PV systems. The share of time when the Fig. 7. Noncompliance rate ε over 57 days as a function of the evaluation time

window for the UCSD PV systems.

Fig. 8.Degree of overestimation δ over 57 days as a function of the evaluation time window for the UCSD PV systems.

Fig. 9.Relative cumulative frequency of daily noncompliance rate ε _{for 2 min} evaluation time window for the UCSD PV systems.

Fig. 10. Relative cumulative frequency of daily degree of overestimation δ for 2 min evaluation time window for the UCSD PV systems.

(8)

difference was between 0 and 5 %/s was about 71.5% for both PV systems. In these situations, the proposed method enveloped the measured power RR with an overestimation of less than 5 %/s. The largest overestimation was 45.4 %/s for both PV systems. The share of time when the difference was negative, i.e., the method failed to envelop the measured power RR, was 0.012% and 0.006% for systems A and B, respectively. Large relative difference in these shares exposes why there is such a large difference in ε between the methods (Fig. 7). Fig. 11(b) further illustrates the frequency and degree of underestimation for systems A and B. Underestimation during a single time step leads to noncompliance of the corresponding evaluation time window. Thus, there can be large difference in ε between the systems although the differences between the systems in Fig. 11 are small. The largest underestimation was 4.7 and 6.1 %/s for systems A and B, respectively.

4.2. Comparison with an existing method

The performance of the proposed method was compared with W2019. The same dataset of 57 days as in Section 4.1 was used for the performance comparison between the methods. The performance was compared for both PV systems of UCSD EBU2 only for the periods for which the estimates could be calculated by W2019.

Fig. 12 presents ε for the methods as a function of the evaluation time window. Both methods failed to envelop the measured power RR of PV system A more often than that of system B. The noncompliance rate of the proposed method was smaller than for W2019; in other words the proposed method outperformed. The noncompliance rate of the proposed method for PV system B improved substantially.

The comparison of δ between the methods is presented as a function of the evaluation time window in Fig. 13. The proposed method had smaller δ for all evaluation time windows. The relative difference in δ between the methods was the smallest at 2 s window length and increased with increasing window length. W2019 shows only minor differences of δ between the PV systems. As for the proposed method, δ was smaller for system A than for system B.

The relative cumulative time distributions of the difference between the estimated and measured power RRs of the PV systems for both methods are presented in Fig. 14. Again, the curves for PV systems A and B for the proposed method were almost identical. In contrast, the existing method provided a smaller difference for PV system B than for system A. The share of time when the proposed method enveloped the measured power RR with an overestimation of less than 5 %/s was 46%

for both PV systems. The corresponding share for W2019 was 24% for PV system A and 25% for system B.

The differences in the performance of the two methods are further illustrated in Tables 6 and 7 where the RR overestimation and underestimation are compared. Both methods succeeded to envelop the RRs in the measured power over 99.9% of the time. The average overestimation for the proposed method was 9.2 and 9.3 %/s for PV systems A and B, respectively. For W2019, these average values were 14.6 and 13.6 %/s.

The largest overestimations were 45 %/s for the proposed method and about 50 %/s for W2019.

The share of time when the RR in the measured power was larger than the estimate of the proposed method was only 0.015% for PV system A and 0.005% for system B. Underestimation of the RRs in measured power was more common for W2019 at 0.041% and 0.051%

of the time for PV systems A and B, respectively. However, the average underestimation for W2019 was smaller than for the proposed method.

The total numbers of cases when the proposed method failed to envelop the power RR of PV systems A and B during the set of 57 days were 90 and 30, respectively. For W2019, the corresponding numbers of failures were 251 and 317. The largest underestimation for the proposed method was 4.7 and 6.1 %/s for systems A and B, respectively. As for W2019, the largest underestimation of the power RR of PV system A was much larger than for the proposed method, 12.9 %/s. However, the largest Fig. 11. (a) Relative cumulative frequency of the difference between the esti-

mated and measured power RRs over 57 days. (b) Blow-up of the figure at negative differences.

Fig. 12.Noncompliance rate ε for the proposed method and W2019 as a function of the evaluation time window.

Fig. 13.Degree of overestimation δ for the proposed method and W2019 as a function of the evaluation time window.

(9)

underestimation of the power RR of system B for W2019 was only 3.8

%/s.

The RRs in the measured powers of PV systems A and B and the estimates of the methods on one day (March 20, 2018) are presented in Fig. 15. There are discontinuities in the existing method around 11:30 and 16:30 since CMV measurements were not available. By contrast, the proposed method provided the RR estimates for all times when measured irradiance was available. The accuracy of W2019 relies heavily on the quality and availability of CMV measurements while the proposed method does not equally depend on CMV measurements but produces RR estimate even when CMV measurements are not available.

During periods when CMV measurements were not available, the proposed method enveloped the RRs in the measured power at a cost of moderate overestimation.

Fig. 15 illustrates how the overestimation of W2019 is typically much larger than that of the proposed method. The estimates of the proposed method followed the measured RRs better than W2019,

especially during periods when there were only small variations in the measured power, e.g. around 9:50, 11:45, 14:30, and 14:50. In these cases, the overestimation of W2019 was up to 16 kW/s. The RRs in the measured power complied with the estimate of the proposed method throughout the day whereas W2019 failed to envelop the RR of both PV systems at 9:00 and 9:12. Moreover, W2019 underestimated the RR of system A for several minutes around 10:05 and 10:30 and the RR of system B two times around 16:50.

In conclusion, the proposed method showed superior performance compared to W2019. The proposed method failed to envelop the measured RRs in power less frequently than W2019, especially for PV system B. Moreover, the overestimation of the proposed method was typically much smaller than that of W2019.

The comparison of Figs. 7 and 12 reveals that the noncompliance rates of the proposed method for the limited dataset used in the comparison did not differ much from those calculated for the whole original dataset. The overestimation of the measured power RRs was somewhat smaller for the original dataset (Fig. 8) than for the limited dataset (Fig. 13). The reason for the difference is that the original dataset included more periods with low RRs in measured power, i.e., fully clear sky or overcast conditions. The number of data points was 817,852 for the limited dataset and 1,205,934 for the original dataset. The largest overestimations and underestimations occurred during intense partly cloudy conditions and were included in both of the datasets.

4.3. Results as a function of the power ramp rate

Next, it was studied how the proposed method succeeded to envelop different power RRs. These results are relevant from the point of view of PV system planning. Fig. 16 presents ε for the 2 s evaluation time window over the set of 57 days as a function of the power RR limit. For both PV systems, ε increased with increasing RR limit meaning that the method succeeded to envelop almost all small power RRs, and the power ramps exceeding the estimate are typically very steep. Still, the proposed method succeeded to envelop over 98.5% of the extreme ramp rates with absolute value larger than 10 %/s. Those RRs were very rare existing less than 0.1% of the time.

In Fig. 17, δ for the 2 s evaluation time window is presented as a function of the power RR limit. The degree of overestimation decreased with increasing RR limit. The proposed method typically overestimates gentle power ramps more than steep ramps. This behavior is desirable as for PV system planning applications, it is essential that the method successfully envelops most of the largest PV power RRs with a relatively small degree of overestimation.

Fig. 18 presents a scatter plot between the underestimation of the proposed method and the measured power RR. The largest underestimation (6.1 %/s for system B) took place when the absolute value of the RR in the measured power was nearly 30 %/s. However, all the other underestimations larger than 4 %/s existed when the absolute value of the measured power RR was smaller than 8 %/s. In 50% of the underestimation cases, the absolute value of the RR in the measured power was less than 1.5 %/s. It can be concluded that most of the underestimations were small and existed with gentle power ramps, which are not of concern for PV planning applications.

The results of Figs. 16–18 are in line with the earlier finding that the proposed method works better for PV system B than for system A.

Especially, the proposed method failed to envelop the power RR of system A more often than for system B. One reason for this is that the PV systems are not rectangular. The minimum dimensions of the rectangular land areas of the outermost points of the PV systems (see Fig. 5) were used as the minimum dimensions of the systems in irradiance averaging. PV system A has more empty space within the rectangle than system B. Moreover, the minimum dimension for most of system A is shorter than the minimum dimension of the rectangle. Thus, the average irradiance over the actual area of PV system A may change faster than was estimated. Whereas for PV system B, the minimum dimension of the Fig. 14.(a) Relative cumulative frequencies of the difference between the

estimated and measured power RRs for the proposed method and W2019 (b) Blow-up of the figure at negative differences.

Table 6

Overestimation for the proposed method and W2019 over 57 days.

Method PV system Share of time (%) Average (%/s) Maximum (%/s)

Proposed A 99.985 9.22 45.4

Proposed B 99.995 9.27 45.4

W2019 A 99.959 14.63 50.5

W2019 B 99.949 13.58 49.7

Table 7

Underestimation for the proposed method and W2019 over 57 days.

Method PV system Share of time (%) Average (%/s) Maximum (%/s)

Proposed A 0.015 1.08 4.68

Proposed B 0.005 0.96 6.13

W2019 A 0.041 0.50 12.92

W2019 B 0.051 0.27 3.79

(10)

rectangle equals the minimum dimension of the PV system. In addition, the GHI sensor is located further from system A than from system B.

4.4. Largest underestimation

The RR in the measured power of PV system B and the estimate of the proposed method on January 9, 2018 are presented in Fig. 19. The largest underestimation over 57 days occurred on that day around 14:00 when a downward ramp exceeded the estimate of the proposed method.

At that moment, the measured power RR was − 9.05 kW/s, i.e., − 29.2

%/s. It was the only moment when the underestimation exceeded 5 %/s.

The other failures to envelop the power RR on that day occurred around 11:50 (an upward and a downward ramp) and around 12:20 (a downward ramp). On that day, ε_{2 min}was 1.02% and δ2 min 56.3%. The daily ε₂

min was the second highest on the set of 57 days while the highest value was 1.48%. There was only one other day when the proposed method failed to envelop the daily largest RR of PV system B. Thus, Fig. 19 provides an example of a day when the performance of the proposed system was at its worst.

Fig. 15.Comparison between the measured power ramp rates on March 20, 2018 and the estimates of the proposed method and W2019.

Fig. 16.Noncompliance rate ε for 2 s evaluation time window over 57 days as a function of the power ramp rate limit (absolute value).

Fig. 17. Degree of overestimation δ for 2 s evaluation time window over 57 days as a function of the power ramp rate limit (absolute value).

Fig. 18.Scatter plot between the underestimation of the proposed method and the absolute value of the measured power ramp rate over 57 days.

(11)

5. Discussion

An obvious advantage of the proposed method is the low number of required measurements. The method needs only the measurements of point irradiance and cloud shadow velocity. The same measurements are sufficient also for the simulations by which the parameter values of Eq.

(7) are solved. If an instrument to measure cloud shadow velocity, like the CSS, is not available, cloud shadow velocity can be determined from the measurements of three irradiance sensors. On the other hand, one shortcoming of the proposed method is that other weather conditions than irradiance, like temperature, were not taken into account. How- ever, temperature variations over a few seconds are small and with typical temperature coefficients of 0.05%/K, they do not have a large effect on the output power variations.

The proposed method is not geographically restricted, but it can be applied universally. As demonstrated in this article, the simulations to obtain the parameter values by which the maximum RR in power is estimated can be executed using irradiance measurements from a different location than the PV system. The irradiance measurements in the simulations were from Northern Europe while the PV systems whose power RRs were estimated were located in California. In the light of the assumptions used in this study, it does not matter where the irradiance data for the simulations is measured. Moreover, actual irradiance measurements are not compulsory for the simulations since the STC irradiance was used as Gus. Thus, only the values of SS, b, ve, and α_eare needed for the simulations. Measured distributions of those variables are presented for example in [31]. When the proposed method is used to estimate the largest expected PV power RRs of a PV system of a certain size, there is no need to run the simulations with as many shadow edges as in this study, but only with the shadow edges causing the largest RR ratios, i.e., large mismatch losses. Slowly moving dark shadows with steep edges were found to cause the largest mismatch losses [36].

The RR ratio is obtained using an experimentally verified electrical simulation model of a PV module containing certain simplifications and assumptions. The PV submodules were modeled by the widely used one- diode model of a PV cell, which is a simplification of the more realistic and accurate two-diode model, providing the right trade-off between accuracy and complexity for the presented analysis. The results of the simulations could slightly change if different PV modules were used as a reference for the simulation model. However, the basic behavior would not change because the electrical characteristics of crystalline silicon PV modules are essentially identical. Thus, the obtained results are valid for all PV systems consisting of crystalline silicon PV modules as also demonstrated in this article. The proposed method is suitable for all kinds of PV systems by replacing the electrical simulation model (Eq.

(1)) with a simulation model adjusted to the specific PV technology. It was found in [16] that the electrical connections between the modules of a PV array have only minor effects on the output power variation of PV systems. This aspect was also demonstrated in this article since the electrical and physical configurations of the PV arrays studied by

simulations may differ from those of the PV systems of UCSD EBU2.

One source of inaccuracy of the proposed method is that the average irradiance of a PV system is calculated from an irradiance measurement of a single sensor. There might be situations when the PV system is fully or partly shaded by a cloud shadow that does not cover the sensor, resulting in an underestimation of the power RRs. On the other hand, a cloud shadow might move over the sensor without covering the PV system entirely, resulting in an overestimation of the power RRs. Such sources of inaccuracy can be minimized by placing the irradiance sensor near the center of the PV system.

The assumptions of shadow edges used for the simulations may have led to a conservative estimate of mismatch losses and the relation between the largest RRs in irradiance and power. For example, only one cloud shadow was assumed to be over the PV system. However, real PV systems can be shaded simultaneously by multiple cloud shadows that might lead to faster changes in the average irradiance and power of the systems. However, multiple cloud shadows are unlikely for the small PV systems that were analyzed in this study. In addition, irradiance differences may exist within the unshaded and shaded area, i.e., before and after an irradiance transition caused by a cloud shadow edge, increasing mismatch losses. Moreover, the mathematical model of Eq. (4) is a fit to many irradiance transitions, but may smoothen out fluctuations during actual irradiance transitions and reduce the largest RRs in irradiance compared to actual irradiance transitions. Thus, larger RRs can be expected to occur in the average irradiance and power output of real PV systems and real mismatch losses, which are the main reason for the difference between RRs in irradiance and power, might momentarily be larger than in the simulations.

The fact that the largest expected RRs in the AC power of the PV systems were estimated based on the relation between RRs in the average irradiance and DC power requires further discussion. The DC/

AC ratio of the PV system has naturally an effect on the largest RRs in the AC power. The larger the DC/AC ratio the more the power production profile of system is clipped. The more the clear sky AC power is clipped the smaller are the power ramps when a cloud shadow moves over the PV system. The DC/AC ratios of the PV systems of UCSD EBU2 were similar at 1.23 and 1.20 for systems A and B, respectively. Moreover, maximum power point tracking and variations in inverter efficiency have an effect on the RRs in the AC power.

6. Conclusions

This article proposed a method to estimate the largest expected PV power RRs. The proposed method estimates the largest RRs in the power of a PV system from RRs in the average irradiance of the system. The average irradiance over the land area of the PV system was calculated by averaging the measured point irradiance over time defined by the measured cloud shadow velocities and the minimum system dimension.

The proposed method has the advantage that it requires only a small number of measurements. The only inputs of the proposed method are Fig. 19.Comparison between the measured power ramp rates of PV system B on January 9, 2018 and the estimate of the proposed method.

(12)

the minimum system dimension and the measurements of point irradiance and cloud shadow velocity.

The relation between the largest RRs in PV power and irradiance was studied comprehensively for the first time. The relation was studied by simulations during almost 9000 shading transitions identified in measured irradiance data.

The proposed method was experimentally validated and its performance was compared with an existing method using data of 57 days from two PV systems with nominal AC powers of 35 and 31 kW. The proposed method enveloped the measured power RRs more frequently than the existing method. Moreover, the RR overestimation of the proposed method was typically much smaller than that of the existing method. The proposed method enveloped the RR in the measured power over 99.99% of the time. The method enveloped most of the largest measured PV power RRs with a relatively small degree of overestimation. That is essential for PV system planning applications. The method failed to envelop the daily largest RR of PV systems A and B only four and two times, respectively. The proposed method can be applied universally as a tool for design and component sizing of PV power plants.

CRediT authorship contribution statement

Kari Lappalainen: Conceptualization, Methodology, Software, Formal analysis, Writing - original draft. Guang C. Wang: Software, Data curation. Jan Kleissl: Conceptualization, Writing - review &

editing.

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 Alfred Kordelin Foundation funded the research visit of K.

Lappalainen to UCSD during which the study presented in this article was carried out.

Appendix A. Calculation of cloud shadow velocity from detected apparent shadow edge velocities

The original CSS algorithm presented in [35] uses the assumption of linear cloud shadow edges and detects the apparent shadow edge velocity ve. This component of shadow velocity perpendicular to the shadow edge underestimates the actual shadow speed v. However, the largest expected PV power RRs determined in this study are a function of v. Thus, the cloud shadow velocity v was calculated from the apparent shadow edge velocity using a weighted non-linear regression of v and α to the NCMV CMVs collected in a certain time period (30 min) using

wivⁱ_e=vcos(

αⁱ_e− α⁾, (A1)

where wi is the weighting factor calculated based on the time difference between the timestamp of the i^thCMV ti and the present time to as wi=⃒

⃒tf− to

⃒⃒− |ti− to| +1, ∀i∈ [1,NCMV], (A2)

where tf is the timestamp furthest from to in the time period. As the result of the weighting, the CMV closest to present has the largest weight and the CMV with the timestamp furthest from the present time has the smallest weight, i.e., unity.

If the NCMV CMVs collected in the time period show only small variation of α_e(less than 20^◦), the shadow movement direction is almost perpendicular to the shadow edge and the regression of Eq. (A.1) is not needed. In these cases, as well as in cases where the number of CMVs in the time period is too small for reliable regression (less than 9), the CMVs are decomposed into north–south and east–west directions, and the median value of each is then used to recompose one median filtered CMV.

References

[1] Cabrera-Tobar A, Bullich-Massagué E, Aragüés-Peñalba M, Gomis-Bellmunt O.

Review of advanced grid requirements for the integration of large scale photovoltaic power plants in the transmission system. Renew Sustain Energy Rev 2016;62:971–87.

[2] Gevorgian V, Booth S. Review of PREPA technical requirements for interconnecting wind and solar generation. NREL/TP-5D00-57089. Golden, CO, USA: National Renewable Energy Laboratory; 2013.

[3] Marcos J, Marroyo L, Lorenzo E, Alvira D, Izco E. Power output fluctuations in large scale PV plants:one year observations with one second resolution and a derived analytic model. Prog Photovolt Res Appl 2011;19:218–27.

[4] Shivashankar S, Mekhilef S, Mokhlis H, Karimi M. Mitigating methods of power fluctuation of photovoltaic (PV) sources – A review. Renew Sustain Energy Rev 2016;59:1170–84.

[5] Chen X, Du Y, Wen H, Jiang L, Xiao W. Forecasting-based power ramp-rate control strategies for utility-scale PV systems. IEEE Trans Ind Electron 2019;66:1862–71.

[6] Wang HX, Munoz-García MA, Moreda GP, Alonso-García MC. Optimum inverter sizing of grid-connected photovoltaic systems based on energetic and economic considerations. Renew Energy 2018;118:709–17.

[7] Peippo K, Lund PD. Optimal sizing of solar array and inverter in grid-connected photovoltaic systems. Sol Energy Mater Sol Cells 1994;32:95–114.

[8] Kakimoto N, Satoh H, Takayama S, Nakamura K. Ramp-rate control of photovoltaic generator with electric double-layer capacitor. IEEE Trans Energy Convers 2009;

24:465–73.

[9] Marcos J, Stork¨el O, Marroyo L, García M, Lorenzo E. Storage requirements for PV power ramp-rate control. Sol Energy 2014;99:28–35.

[10] Makibar A, Narvarte L, Lorenzo E. On the relation between battery size and PV power ramp rate limitation. Sol Energy 2017;142:182–93.

[11] Martins J, Spataru S, Sera D, Stroe D-I, Lashab A. Comparative study of ramp-rate control algorithms for PV with energy storage systems. Energies 2019;12. https://

doi.org/10.3390/en12071342.

[12] Cir´es E, Marcos J, de la Parra I, García M, Marroyo L. The potential of forecasting in reducing the LCOE in PV plants under ramp-rate restrictions. Energy 2019:188.

https://doi.org/10.1016/j.energy.2019.116053.

[13] Lappalainen K, Valkealahti S. Recognition and modelling of irradiance transitions caused by moving clouds. Sol Energy 2015;112:55–67.

[14] Tomson T, Tamm G. Short-term variability of solar radiation. Sol Energy 2006;80:

600–6.

[15] Lave M, Kleissl J. Solar variability of four sites across the state of Colorado. Renew Energy 2010;35:2867–73.

[16] Lappalainen K, Valkealahti S. Output power variation of different PV array configurations during irradiance transitions caused by moving clouds. Appl Energy 2017;190:902–10.

[17] Van Haaren R, Morjaria M, Fthenakis V. Empirical assessment of short-term variability from utility-scale solar PV plants. Prog Photovolt Res Appl 2014;22:

548–59.

[18] Chen X, Du Y, Lim E, Wen H, Yan K, Kirtley J. Power ramp-rates of utility-scale PV systems under passing clouds: Module-level emulation with cloud shadow modeling. Appl Energy 2020;268.. https://doi.org/10.1016/j.

apenergy.2020.114980.

[19] Lappalainen K, Valkealahti S. Smoothing of Output Power Variation with Increasing PV Array Size. In: Proceedings of 2018 IEEE innovative smart grid technologies - Asia Conference; 2018. p. 190–5. https://doi.org/10.1109/ISGT- Asia.2018.8467900.

[20] Marcos J, Marroyo L, Lorenzo E, García M. Smoothing of PV power fluctuations by geographical dispersion. Prog Photovolt Res Appl 2012;20:226–37.

[21] Lave M, Kleissl J, Arias-Castro E. High-frequency irradiance fluctuations and geographic smoothing. Sol Energy 2012;86:2190–9.