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Technical Report

NREL/TP-5D00-73207

February 2019

Highly Accurate Method for Real-Time 

Active Power Reserve Estimation for 

Utility-Scale Photovoltaic Power Plants

Vahan Gevorgian

National Renewable Energy Laboratory
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Office of Energy Efficiency & Renewable Energy

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Technical Report

NREL/TP-5D00-73207

February 2019

Highly Accurate Method for Real-Time 

Active Power Reserve Estimation for 

Utility-Scale Photovoltaic Power Plants

Vahan Gevorgian

National Renewable Energy Laboratory

Suggested Citation 

Gevorgian, Vahan. 2019. Highly Accurate Method for Real-Time Active Power Reserve 

Estimation for Utility-Scale PV Power Plants. Golden, CO: National Renewable Energy 

Laboratory. NREL/TP-5D00-73207. https://www.nrel.gov/docs/fy19osti/73207.pdf.
NOTICE

This work was authored by the National Renewable Energy Laboratory, operated by Alliance for Sustainable 

Energy, LLC, for the U.S. Department of Energy (DOE) under Contract No. DE-AC36-08GO28308. Funding 

provided by U.S. Department of Energy Office of Energy Efficiency and Renewable Energy Solar Energy 

Technologies Office. The views expressed herein do not necessarily represent the views of the DOE or the U.S. 

Government.

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Preface

This paper applies a robust technique for determining the available power from a curtailed 

utility-scale photovoltaic (PV) power plant. The proposed technique does not require deploying 

any additional equipment or sensors and is based only on the addition of new control logic to the 

existing power plant controller. Also, the proposed method is universally applicable to PV plants 

with any type of smart inverters and PV modules. Accurate determination of available power is 

important for using curtailed PV generation as a resource for various types of active power 

controls, such as spinning reserves and primary and secondary frequency control. For PV plants 

to be able to maintain the desired regulation range, the plant controller must be able to estimate 

the available aggregate peak power that all the plant’s inverters can produce at any point in time 

and ensure that the control error stays within the tolerance band at all times. In this paper, we 

explore a highly accurate control method that uses dedicated inverters within the plant as 

reference units and evaluates the available aggregate peak power for the whole plant under 

different cloud variability conditions.
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Acknowledgments

The author thanks Guohui Yuan and Anastasios Golnas of the U.S. Department of Energy’s 

Solar Energy Technologies Office for their continuous support of this project. Thank you also 

Mahesh Morjaria and other members of First Solar team.
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List of Acronyms

MPP Maximum power point

MPPE Maximum power point estimation

MPPT Maximum power point tracking

PV Photovoltaic
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Table of Contents

Introduction ................................................................................................................................................. 1

1 Proposed Method ................................................................................................................................. 4

2 Data ........................................................................................................................................................ 7

3 Statistical Analysis Methods............................................................................................................... 9

4 Placement of Reference Inverters..................................................................................................... 10

5 Correlation of Measured and Estimated MPPT Power for Entire PV Plant................................... 11

6 Maximum Power Estimation Error Statistics................................................................................... 17

7 Conclusions ........................................................................................................................................ 21

8 Future Plans........................................................................................................................................ 22

References................................................................................................................................................. 23
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List of Figures

Figure 1. Example of inaccurate maximum power estimation ..................................................................... 3

Figure 2. Example of maximum power estimation using a single reference inverter................................... 3

Figure 3. Large PV power plants under clear-sky conditions—only one reference MPP inverter is used... 5

Figure 4. Large PV power plant divided into control zones during cloud conditions .................................. 5

Figure 5. Snapshot of measured AC power output variability across the PV plant...................................... 7

Figure 6. Four variability cases used in the analysis..................................................................................... 8

Figure 7. Minimum curtailment level as a function of the number of reference inverters ........................... 8

Figure 8. Assigning different numbers of reference inverters within the array footprint. (Red dots 

represent sections of the array connected to reference MPPT inverters.).............................. 10

Figure 9. Calculating the error in estimated maximum power.................................................................... 10

Figure 10. Clear-sky case: a high level of accuracy in maximum power evaluation can be achieved 

even with a single reference inverter, and it can be further improved with four inverters. 

(Blue: measured total plant power when operating in MPPT mode; red: estimated total 

plant power for different numbers of reference inverters. The red cannot be distinguished 

from blue because of high accuracy of estimation.)............................................................... 11

Figure 12. Improving the accuracy of maximum power evaluation by using a larger number of 

reference inverters: moderate variability case. (Blue: measured total plant power when 

operating in MPPT mode; red: estimated total plant power for different numbers of 

reference inverters.) ............................................................................................................... 12

Figure 13. Improving the correlation between measured and evaluated maximum power by using a 

larger number of reference inverters: moderate variability case............................................ 13

Figure 14. Improving measured total plant power when operating in MPPT mode; red: estimated total 

plant power for different numbers of reference inverters.).................................................... 13

Figure 15. Improving the correlation between measured and evaluated maximum power by using a 

larger number of reference inverters: intense variability case ............................................... 14

Figure 16. Improving the accuracy of maximum power evaluation by using a larger number of 

reference inverters: extreme variability case. (Blue: measured total plant power when 

operating in MPPT mode; red: estimated total plant power for different numbers of 

reference inverters.) ............................................................................................................... 14

Figure 17. Improving the correlation between measured and evaluated maximum power by using a 

larger number of reference inverters: extreme variability case.............................................. 15

Figure 18. Improving accuracy for an extreme variability case. (Blue: measured total plant power 

when operating in MPPT mode; red: estimated total plant power for different numbers 

of reference inverters)............................................................................................................ 16

Figure 19. Reduction of control error with increasing numbers of reference inverters. (A: moderate 

variability case; B: extreme variability case.)........................................................................ 17

Figure 20. Mean error and standard deviation of error............................................................................... 19

Figure 21. Min and max estimation error ................................................................................................... 19

Figure 22. Skewness and kurtosis of error distribution............................................................................... 20

Figure 23. Correlation coefficients for all variability cases........................................................................ 20

Figure 24. Aerial photo of First Solar PV plant at the National Renewable Energy Laboratory’s 

National Wind Technology Center ........................................................................................ 21

Figure 25. Typical MW-scale PV inverter efficiencies .............................................................................. 22
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List of Tables

Table 1. Mean Peak Power Evaluation Error (% of Plant Rated Power).................................................... 18

Table 2. Standard Deviation of Peak Power Evaluation Error (% of Plant Rated Power).......................... 18

Table 3. Maximum Positive Peak Power Evaluation Error (% of Plant Rated Power) .............................. 18

Table 4. Maximum Negative Peak Power Evaluation Error (% of Plant Rated Power)............................. 18

Table 5. Skewness of Peak Power Evaluation Error Distribution (% of Plant Rated Power)..................... 19

Table 6. Kurtosis of Peak Power Evaluation Error Distribution (% of Plant Rated Power)....................... 19

Table 7. Correlation Coefficients Between Actual and Estimated Available Power for All 

Variability Cases.................................................................................................................... 20
1

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Introduction

All over the world, system operators and utilities are continually adapting their grid codes,

interconnection requirements, operational practices, and market mechanisms to make the 

integration of shares of fast-growing variable renewable generation both reliable and economic 

[1]. As power systems continue to evolve, the Federal Energy Regulatory Commission (FERC) 

noted that there is a growing need for a refined understanding of the services necessary to 

maintain a reliable and efficient system. In orders 755 and 784, FERC required improving the 

mechanisms by which frequency regulation services are procured and enabling compensation by 

fast-response resources such as energy storage. In addition, FERC recently issued a notice of 

proposed rulemaking to enable the aggregation of distributed storage and distributed generation 

[2]. The North American Electric Reliability Corporation’s (NERC) Integration of Variable 

Generation Task Force made several recommendations for requirements for variable generators 

(including solar) to provide their share of grid support, including active power control (APC) 

capabilities [3], [4]. Similar requirements for renewable energy plants have been introduced in 

Europe at both the transmission and distribution levels [5], [6]. In 2018, FERC Order 842 

amended the pro forma interconnection agreements to include certain operating requirements,

including maximum droop and dead band parameters, and sustained response provisions [7].

NERC’s BAL-003-1 standard on frequency response establishes target contingency protection 

criteria for each North American interconnection and individual balancing authorities (BAs)

within interconnections [8]. BAs are required to meet the minimum frequency response 

obligation (FRO), so the generating resources that are operated in a mode and range to meet the 

FRO need to have adequate headroom to respond to frequency transients and load-frequency 

control set points. Although establishing such headroom is not a problem for the conventional 

fleet, the varying nature of solar and wind generation makes it challenging to set and maintain 

adequate headroom by these varying resources.

In general, all system operators have processes and procedures in place to ensure grid reliability 

through monitoring market participant operation. For example, provisions of the California 

Independent System Operator (CAISO) tariff [9] set penalties for deviations from dispatch and 

regulation capacity for market participants that fail to perform in compliance. The permitted area 

of variation for performance requirements of resources used for various purposes is provided in 

the CAISO tariff [9]. The tolerance band is expressed in terms of energy (MWh) for generating 

units and imports from external dynamic system resources for each settlement interval and 

equals the greater of the absolute value calculated using either of the following methods: 

1. 5 MW divided by the number of settlement intervals per settlement period

2. 3% of the relevant generating unit’s maximum output (Pmax), as registered in the master 

file, divided by the number of settlement intervals per settlement period. 

This CAISO tariff and similar requirements from other system operators imply that accurate realtime estimations of available maximum power from the curtailed photovoltaic (PV) plant are 

important for avoiding excessive penalty payments if utility-scale PV plants become market 

participants for energy and various reliability services related to active power controls. 
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A typical modern utility-scale PV power plant is a complex system of large PV arrays and 

multiple power electronic inverters, and it can contribute to mitigating the impacts on grid 

stability and reliability through sophisticated, automatic “grid-friendly” controls. To provide 

active power reserves (or a headroom margin) for up-regulation that can be automatically 

dispatched as needed, the PV plant needs to operate below its maximum power point (MPP);

however, evaluating that MPP in curtailed mode is not a trivial task, especially for large PV 

power plants during various types of variable conditions caused by clouds. One paper [10]

proposed an experimentally validated maximum power point estimation (MPPE) method that

operates in real time using irradiance and cell temperature measurements to ensure that sufficient 

reserve power is available. Another paper [11] proposed an advanced real-time MPPE algorithm 

by applying curve fitting on voltage and current measurements obtained during inverter 

operation. Some previously proposed MPPE methods used offline prediction and employed

regression analysis or neural networks [12], [13]. These methods seem to be accurate but might 

require excessive processing power. Others proposed methods for real-time calculation [14]–[16]

by making assumptions that reduce the accuracy of the PV model or, in some cases, require 

knowledge that is not typically available on PV module data sheets[14], [17]. Another important 

limitation of MPPE methods is that modifications are needed based on inverter types and 

topologies. For example, in single-stage inverters (no DC/DC conversion), the power reserve 

capability can be achieved by inverter control modifications [18], [19]. In two-stage systems 

(inverter and DC/DC converter), however, the DC/DC converter control needs to be modified 

instead [20]. This makes the use of maximum power estimation for curtailed PV systems 

challenging and highly dependent on inverter make and topology, types of modules used in PV 

plants, and accurate knowledge of the inverter and PV module parameters. 

The variability of solar PV output in the regulation reserve time frame among various arrays 

within a large-scale (~50-MW) solar PV plant in the southwestern United States is discussed in 

[21]. Although the distributions of changes in aggregate power output throughout all timescales 

considered were clustered around a strong peak at zero, the distributions at all timescales 

exhibited significant instances of higher magnitude ramps in the tails of the histograms. The 

results achieved in [21] were very important because the method presented in this paper was also 

tested using the data from the same PV power plant.

In 2015, a demonstration project was conducted in the U.S. territory of Puerto Rico using a 20-

MW grid-connected PV power plant [22]. This plant was controlled to provide different types of 

reliability services to the island’s grid including various types of active power controls. Testing 

on this plant provided real data on levels of uncertainty that can be introduced by traditional 

MPPE methods based on irradiance and temperature measurements as well as inverter I–V 

characteristics. One example of such uncertainty is shown in Figure 1 during operation when the 

PV plant was responding to an automatic generation control (AGC) signal sent from the system 

operator. The AGC system assumed that there was still some available headroom for upregulation because its evaluation was based on the available plant power value that was 

communicated by the power plant controller (PPC). The calculated available power is overly 

optimistic, however, and inverters are not able to produce as much power because they are 

already operating at the maximum peak power point. Note that the mismatch at lower 

curtailment level shown in Figure 1 is because of the minimum curtailment limit set by the plant 

owner for economic reasons.
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Figure 1. Example of inaccurate maximum power estimation

Figure 2. Example of maximum power estimation using a single reference inverter

A different method for estimating the maximum power was used during the demonstration 

testing of a 300-MW PV power plant in California [1]. In this case, a single 4-MVA inverter was 

taken from the AGC scheme and was set to operate at the power level determined by its

maximum power point tracking (MPPT) algorithm. The measured AC power of this inverter was 

used as an indicator of available power for the other 79 inverters (80 inverters total), so the plant 

was able to operate with fixed 30 MW of headroom (Figure 2). This method also has inherent 

uncertainties because it assumes uniform solar resource conditions across the whole 300-MW 

plant. Fortunately, cloud conditions were favorable for this method to be acceptable because 

there was a clear sky above the plant during most of the testing day. 

The method proposed in this paper is also based on using dedicated reference inverters within a 

curtailed PV power plant for estimating the maximum available power; however, it is based on

the use of multiple reference inverters to achieve high levels of real-time maximum power 

estimation under conditions of extreme variability.
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1 Proposed Method

For utility-scale PV power plants to be able to maintain the desired regulation range or spinning 

reserve levels, the plant controller must be able to estimate the available aggregate peak power 

that all the plant’s inverters can produce at any point in time. The available power is normally 

estimated by an algorithm that considers solar irradiation, the characteristics and temperatures of

PV modules I–V , inverter efficiencies, etc.; however, this method has many uncertainties, 

depends on the availability of accurate system models, and does not account for other factors 

such as panel soiling because of dust. The proposed method can determine the available peak 

power of the PV plant and maintain desired reserves with high levels of accuracy without the use 

of external sensors or devices. The existing plant hardware and controls can perform this task 

after the addition of the new optimized control algorithm in power plant controller software.

Under clear-sky conditions, a single PV inverter can be used as a reference for the whole plant to 

determine the available power at any point in time (Figure 3). Under variable-cloud-cover 

conditions, however, a single-inverter method will not be accurate enough for large PV power 

plants. Instead, we propose a concept that can accurately allocate reserves for PV power plants 

by: 

• Creating dynamic virtual control zones in the PV plant

• Determining which PV inverter is to serve as a reference by operating at MPP

• Determining the optimal dispatch interval for the reference MPP inverters based on the

rate of change of the power in each zone (i.e., indicator of cloud movement) 

• Determining the optimized combinations of the curtailment set points for participating 

inverters in each zone for the maximum aggregate inverter efficiency (or minimum 

electric losses in the plant) for every control interval.

The idea for such a method is shown in Figure 4. The plant controller allocates virtual dynamic 

control zones consisting of two or more inverters depending on the cloud conditions over the 

plant. Then, a single inverter in each virtual group is operated at MPP and is used as a reference 

for determining the maximum available power for the zone so that appropriate curtailment set 

points can be sent to all participating inverters within the zone. After a certain time interval, the 

process is repeated, ensuring accurate reserve allocation by the whole plant and avoiding 

excessive curtailments. 
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Figure 3. Large PV power plants under clear-sky conditions—only one reference MPP inverter is 

used

Figure 4. Large PV power plant divided into control zones during cloud conditions

In this report, we use the following abbreviations:

• NNtttttttttt: total number of inverters in the PV power plant

• NNzz zzzzzz: number of control zones (the same as the number of reference inverters NNrr rr )

• NNii ii = NNtttttttttt

NNzz zzzzzz

: number of inverters in each control zone

• PPtttttttttt.mm mm: total power production of the plant when all inverters are operating at MPP

• PPmm mm,ii: power production of zone ii when all inverters in the zone are operating at MPP

• PPmmmmmm ,ii: power production by a single inverter operating at MPP in zone ii.

The estimated maximum available power from the plant using instantaneous power produced by 

single MPPT inverters in each zone (blue inverters in Figure 4) is determined as:

PPtttttttttt.mm mm = ∑ NNii ii ∙ PPmmmmmm ,ii

NNzz zzzzzz

ii=1 (1)
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Plant curtailment set point as a percentage of estimated maximum available power from the plant

is determined as:

PPtttttttttt = (1 − ∆PP) ∙ PPtttttttttt.mm mm (2)

where ∆PP (in per units) is a curtailment set point. For example, ∆PP = 0.1 means that the plant is 

expected to operate with 10% active power reserve margin or at 90% of PPtttttttttt.

Power set points to all individual inverters in zone ii that are participating in the curtailment 

scheme (orange inverters in Figure 4) can then be calculated:

PPii ii,ii = PPmmmmmm ,ii ∙

NNiiiiii(1−∆PP)−1

NNiiiiii−1 (3)

Therefore, the power production of zone ii can be calculated as:

PPzz zzzz,ii = PPmmmmmm ,ii + (NNii ii − 1) ∙ PPiiiiii,ii (4)

And the total power production of the plant operating with the curtailment set point ∆PP is:

PP = ∑ PPzz zzzz,ii

NNzzzz zzzz

ii=1 = [NNiiiiii(1 − ∆PP) − 1] ∙ ∑ PPmmmmmm ,ii

NNzzzz zzzz

ii=1 (5)
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2 Data

In this work, we examined the applicability of the proposed method using solar PV output power 

data from different arrays in a single utility-scale (~50-MW) PV plant in the western United 

States. The plant consists of 96 individual inverters, each rated at 0.5 MW. We used 1-s power 

data from each individual inverter collected from the plant during a period of several months,

allowing us to analyze the accuracy of the proposed method under different resource variability 

scenarios. 

A depiction of the geographic layout of the individual 0.5-MW PV arrays is shown in Figure 5. 

The figure also shows a snapshot of the variability of the measured AC electric power among 

different sections of the plant. After analyzing many days of measured data, we decided to use 

plant data sets from four different days characterizing four different variability scenarios: (1) 

clear sky, (2) moderate variability, (3) intense variability, and (4) extreme variability. Onesecond time series for each selected variability case is shown in Figure 6A. For comparison, we 

show the 1-s rate of change of the total plant power (Figure 6B) and the frequency distribution of 

1-s power changes (Figure 6C). The extreme variability case shown in Figure 6C had the largest 

1-s changes in plant power output during the whole period of observation. The maximum 

positive and negative 1-s changes in power are +0.6 MW and -0.42 MW, respectively (or 

+1.25% and -0.87% of the plant’s rated power). One observation from Figure 6C is that the

distribution of the 1-s power changes for the extreme variability case is nonsymmetric, and the 

distribution has a longer positive tail. This can be explained by the different up- and down-ramp 

limit settings in the inverters. When selecting these data sets, care was taken to make sure that

the variability in the output was caused by changing solar irradiance conditions only, not trip-off

events.

Figure 5. Snapshot of measured AC power output variability across the PV plant
8

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Figure 6. Four variability cases used in the analysis

Note that the proposed method is expected to provide an accurate estimation of available peak 

power only for curtailment levels that are above a certain minimum curtailment threshold. This 

threshold is determined by the number of inverters from equations (1)–(5) as follows. For 

example, for the case of NNrr rr = 48, the are 48 inverters operating at MPPT, leaving the 

remaining 48 inverters available for curtailment. Therefore, the plant can be curtailed only down 

to 50% of available power for NNrr rr = 48. The calculated minimum curtailment for different 

numbers of reference inverters is shown in Figure 7. For smaller numbers of reference inverters, 

the plant can be curtailed to much lower levels, as shown in Figure 7. 

Figure 7. Minimum curtailment level as a function of the number of reference inverters
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3 Statistical Analysis Methods

Some of the statistical background that was used in the 1-s maximum power evaluation analysis 

is described in this section using [23], [24], and [25]. We used a histogram as a graphical method 

to display the shape of the distribution for control errors for maximum evaluated power data. The 

class intervals, or bin widths, were determined using the square-root method, which calculates 

the number of bins as a square root of the number of data points in the sample [26] (NNbb =

�NNdd ). Only daytime production hours (6 a.m.– 7 p.m.) were used in the analysis, totaling 

NNdd = 46,800 (equal to the number of seconds from 6 a.m.–7 p.m.). The bin widths were then 

calculated directly from the maximum and minimum values in the data. For data sets with a large 

number of points, a histogram essentially resembles continuous frequency distributions because 

of the large number of bins.

In addition to the histogram analysis, standard deviation, and mean, we applied two more

statistical measures to the characterization of the error distributions: skewness and kurtosis. 

Skewness is a measure of symmetry (or, more precisely, the lack of symmetry) for a given data 

set, and it is defined as:

γγ = ∑ ( ii−μμ) NN 3 ii=1

(NN−1)σσ3 (6)

where γγ is the skewness, μμ is the mean, σσ is the standard deviation, and NNis the number of data 

points in the set. The skewness for a normal distribution is zero, and any symmetric data set 

should have a skewness near zero.

Kurtosis is the degree of peakedness of a distribution. It is a measure of the magnitude of the 

peak of the distribution or, conversely, how fat-tailed the distribution is. It is defined as:

KK = ∑ ( ii−μμ) NN 4 ii=1

(NN−1)σσ4 − 3 (7)

In this definition of kurtosis, data sets with a higher value of kurtosis tend to have a distinct peak 

near the mean and decline rather rapidly.

Correlation coefficients are used in the statistics to measure the strength of the relationship 

between two data sets. In this analysis, we used the Spearman’s rank correlation coefficient, 

commonly used to measure the degree of association between two variables:

RR = 1 − 6 ∑ ddii

2

( 2−1) (8)

where RR is the Spearman’s rank correlation coefficient, xx and yy are elements of data sets, d is the 

difference between the ranks of the corresponding variables, and is the number of observations.

It is general practice in statistical analysis of acquired time series data to remove the outliers 

(data points that lie an abnormal distance from other values in a sample). This is usually 

accomplished by removing values that are greater than a certain percentile value (for example, 

the 99th percentile). In this analysis, we did not perform any outlier screening in the ramp rate 

data. The reason for this is that extreme ramp rate events that lie in the tails of the observed 

distribution are of significant interest in the context of this work and will have an impact on the

accuracy of the proposed maximum power evaluation method.
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4 Placement of Reference Inverters

The total plant 1-s production data and data from 96 individual inverters for four variability cases 

were used to evaluate the ability of the proposed method to accurately predict the maximum 

available power from the entire plant for different numbers of reference inverters. The following 

numbers of control zones with a single reference inverter in each zone were used in the analysis

of the number of reference inverters: NN = 1, 2, 3, 4, 6, 8, 12, 24, 48. A simple algorithm was 

developed to select the locations of the reference inverters using an equidistant approach. The 

selected locations of the reference inverters for each case are shown in Figure 8, where the red 

dots represent single 0.5-MW arrays with their inverters operating in MPPT mode (reference 

inverters). For example, for NN = 1, there is only one reference inverter and the algorithm choose

it to be in the center of the plant. For NN = 2, the are two reference inverters placed in two 

sections of the plant, etc. For NN = 48, there are 48 reference inverters out of the total 96 

inverters. In this case, the algorithm breaks the whole plant into pairs of individual inverters 

where one inverter operates in MPPT reference mode.

Figure 8. Assigning different numbers of reference inverters within the array footprint. (Red dots 

represent sections of the array connected to reference MPPT inverters.)

Figure 9. Calculating the error in estimated maximum power

The estimated maximum power of the plant can be calculated using Eq. (1) for any number of 

reference MPPT inverters. Then, the estimation error can be evaluated as shown in Figure 9.
11

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5 Correlation of Measured and Estimated MPPT 

Power for Entire PV Plant

Correlation measures the linear dependence between two variables, and correlation values fall 

within the range from -1 to 1. A value of 1 indicates that one variable is a positive linear function 

of the other, a value of -1 means that one variable is a negative linear function of the other, and 0 

indicates a lack of correlation entirely. In this section, we use the Spearman’s rank correlation 

coefficient (Eq. 8) to measure the linear correlation between measured 1-s total plant power 

operating in MPPT mode with estimated maximum available peak power from the plant 

operating in curtailed mode at different numbers of reference inverters with locations shown in 

Figure 8. The Spearman’s rank correlation method does not carry any assumptions about the 

frequency distribution of the data sets, allowing us to determine whether two ranked variables 

covary (i.e., as one variable increases or decreases, the other variable tends to follow). Further 

statistical analysis of the evaluated power will be performed in Section 7.

First, we analyze the ability of the proposed method to estimate the available peak plant power 

for a clear-sky case (as defined in Figure 6). In this case, even with one reference inverter, there 

is a very strong correlation between measured and estimated 1-s peak power data, as shown in 

Figure 10 (upper graphs). This observation is consistent with results of the test conducted on a

300-MW PV plant as described in [1]. With increasing numbers of reference inverters, the 

correlation becomes even stronger. For example, with only four reference inverters, the 

correlation is basically ideal as shown in Figure 10 (lower graphs).

Figure 10. Clear-sky case: a high level of accuracy in maximum power evaluation can be achieved

even with a single reference inverter, and it can be further improved with four inverters. (Blue:

measured total plant power when operating in MPPT mode; red: estimated total plant power for 

different numbers of reference inverters. The red cannot be distinguished from blue because of

high accuracy of estimation.)
12

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Figure 11. Maximum power estimation error for different reference inverter locations (clear-sky 

case)

Changing the location of reference inverter in clear-sky case has very small impact on mean 

estimation error (blue dots) as shown in Figure 11 for two consecutive clear-sky days.

These observations suggest that even with a low number of reference inverters, the proposed 

method can track the true available power with a high degree of correlation on a second-bysecond basis for clear-sky conditions. This picture changes drastically for cases with variability 

caused by cloud movements, as further described using results shown in Figure 12–Figure 18 for 

three different variability conditions.

Figure 12. Improving the accuracy of maximum power evaluation by using a larger number of 

reference inverters: moderate variability case. (Blue: measured total plant power when operating 

in MPPT mode; red: estimated total plant power for different numbers of reference inverters.)

1

2

3 4 5

7 6

8 9

10

Moving reference 

inverter location

-4

-3

-2

-1

0

1

2

3

4

5

6

7

0 1 2 3 4 5 6 7 8 9 10 11

% of Rated Plant Power

LOCATION

Clear sky - day 1

Mean Max error+ Max error -

-4

-3

-2

-1

0

1

2

3

4

5

6

7

0 1 2 3 4 5 6 7 8 9 10 11

% of Rated Plant Power

LOCATION

Clear sky - day 2

Mean Max error+ Max error -
13

This report is available at no cost from the National Renewable Energy Laboratory (NREL) at www.nrel.gov/publications.

Figure 13. Improving the correlation between measured and evaluated maximum power by using a

larger number of reference inverters: moderate variability case

Figure 14. Improving measured total plant power when operating in MPPT mode; red: estimated 

total plant power for different numbers of reference inverters.)

N=1 N=2 N=3

N=4 N=6 N=8

N=12 N=24 N=48
14

This report is available at no cost from the National Renewable Energy Laboratory (NREL) at www.nrel.gov/publications.

Figure 15. Improving the correlation between measured and evaluated maximum power by using a

larger number of reference inverters: intense variability case

Figure 16. Improving the accuracy of maximum power evaluation by using a larger number of 

reference inverters: extreme variability case. (Blue: measured total plant power when operating in 

MPPT mode; red: estimated total plant power for different numbers of reference inverters.)

N=1 N=2 N=3

N=4 N=6 N=8

N=12 N=24 N=48

N=1

N=2 N=3

N=4 N=6 N=8

N=12 N=24 N=48
15

This report is available at no cost from the National Renewable Energy Laboratory (NREL) at www.nrel.gov/publications.

Figure 17. Improving the correlation between measured and evaluated maximum power by using a

larger number of reference inverters: extreme variability case

Calculations performed for three variability cases (moderate, intense, and extreme) demonstrated 

continuous improvements in the accuracy of estimated maximum PV plant power. This can be 

observed from both 1-s time series graphs and 1-s correlation X-Y charts for the moderate 

variability case (Figure 12 and Figure 13), intense variability case (Figure 14 and Figure 15), and 

extreme variability case (Figure 16 and Figure 17), respectively. For all three cases, the best 

correlation is achieved with the largest number of reference MPPT inverters. This is explained 

by the varying nature of solar irradiance across the plant footprint, causing the diversity in the 

output power level between individual inverters. Therefore, having a larger number of dedicated 

reference MPPT inverters scattered all over the PV power plant would help achieve better 

accuracy in peak power estimation. Figure 18 demonstrates this fact at a higher resolution for the 

extreme variability case. This 10-min snapshot shows how the increasing numbers of reference 

inverters help improve the maximum power evaluation accuracy under varying conditions. 

N=1 N=2 N=3

N=4 N=6 N=8

N=12 N=24 N=48
16

This report is available at no cost from the National Renewable Energy Laboratory (NREL) at www.nrel.gov/publications.

Figure 18. Improving accuracy for an extreme variability case. (Blue: measured total plant power 

when operating in MPPT mode; red: estimated total plant power for different numbers of reference 

inverters) 
17

This report is available at no cost from the National Renewable Energy Laboratory (NREL) at www.nrel.gov/publications.

6 Maximum Power Estimation Error Statistics

Figure 19A and Figure 18B show the frequency distribution of error in estimating the maximum 

available PV plant power for different numbers of reference inverters in the moderate and 

extreme variability cases, respectively. These distributions were calculated for a large number of 

bins and are shown in logarithmic scale as a visual representation of the difference in the

estimation error between the number of inverters and the large range of values. These 

distribution shapes are concentrated in the center, with large visible tales for fewer numbers of 

reference inverters. The distribution tales drop significantly with increasing numbers of reference 

inverters, achieving essentially a no-tail distribution when N = 24 or 48 for both the moderate 

and extreme variability cases.

Figure 19. Reduction of control error with increasing numbers of reference inverters. (A: moderate 

variability case; B: extreme variability case.)

Table 1–Table 4 list the average, standard deviation, and min/max available plant power 

estimation errors, respectively, for all four variability cases as a percentage of plant rated power. 

The average (or mean) values and standard deviation of the error distributions are changing with 

the number of reference inverters, reaching insignificant levels with larger numbers of reference 

inverters. Similarly, the largest 1-s positive and negative peak power estimation errors reduce

significantly with increased numbers of reference inverters. For example, in the extreme 

variability case, the maximum positive evaluation error was reduced from 50.5% (N=1) to 7.9% 

(N=48) of rated plant power, and the maximum negative evaluation error was reduced from -

55.8% (N=1) to -9.2% (N=48) of rated plant power.

A.

B.

Increasing N Increasing N

Increasing N Increasing N
18

This report is available at no cost from the National Renewable Energy Laboratory (NREL) at www.nrel.gov/publications.

Skewness is a measure of symmetry. Table 5 shows that the distribution of errors is not exactly 

symmetric, with larger numbers of positive or negative tails, depending on the variability case 

and number of reference inverters. This means that particular periods of data collection 

sometimes produced larger numbers of positive 1-s errors than negative errors, and vice versa, 

depending on the variability conditions and number of reference inverters used in the estimation 

process.

Kurtosis is the measure of peakedness of distribution. Based on the kurtosis values in Table 6, it 

appears that the distribution of peak power estimation errors is relatively more peaked for higher 

variability cases. More-peaked distribution means that the higher frequency of smaller errors is 

concentrated around the central axes.

Table 1. Mean Peak Power Evaluation Error 

(% of Plant Rated Power)

Nref Clear Sky Moderate

Variability

Intense

Variability

Extreme

Variability

1 -1.572 -1.243 -1.411 -0.656

2 -1.325 -0.925 -1.242 -0.547

3 -1.034 -0.785 -1.084 -0.31

4 -0.984 -0.686 -1.039 -0.254

6 -0.703 -0.531 -0.894 -0.216

8 -0.643 -0.457 -0.652 -0.152

12 -0.39 -0.383 -0.443 -0.088

24 -0.129 -0.058 -0.281 -0.038

48 -0.035 -0.037 0.062 0.014

Table 2. Standard Deviation of Peak Power 

Evaluation Error (% of Plant Rated Power)

Nref Clear Sky Moderate

Variability

Intense

Variability

Extreme

Variability

1 0.939 6.437 8.180 6.275

2 0.735 6.190 7.525 5.934

3 0.546 5.605 6.877 5.267

4 0.527 5.296 6.417 5.081

6 0.361 4.601 5.668 4.656

8 0.326 3.831 5.317 3.976

12 0.192 2.472 3.019 3.481

24 0.091 1.817 2.386 2.106

48 0.070 0.977 1.408 1.156

Table 3. Maximum Positive Peak Power 

Evaluation Error (% of Plant Rated Power)

Nref Clear Sky Moderate

Variability

Intense

Variability

Extreme

Variability

1 3.329 58.729 45.800 50.566

2 2.091 54.389 43.859 47.678

3 1.213 55.146 42.003 45.254

4 1.183 45.527 41.808 45.408

6 0.564 34.479 39.957 39.582

8 0.494 28.121 33.959 28.893

12 0.315 15.270 19.935 22.891

24 0.2468 14.744 16.666 12.391

48 0.325 9.155 8.966 7.958

Table 4. Maximum Negative Peak Power 

Evaluation Error (% of Plant Rated Power)

Nref Clear Sky Moderate

Variability

Intense

Variability

Extreme

Variability

1 -3.635 -44.829 -52.92 -55.844

2 -3.210 -44.682 -45.702 -37.143

3 -2.289 -44.145 -46.761 -31.521

4 -2.177 -44.526 -38.979 -31.880

6 -1.654 -43.433 -28.235 -31.312

8 -1.461 -35.303 -27.046 -23.73

12 -0.916 -24.506 -17.528 -24.624

24 -0.482 -14.248 -12.806 -14.987

48 -0.256 -6.779 -7.777 -9.265
19

This report is available at no cost from the National Renewable Energy Laboratory (NREL) at www.nrel.gov/publications.

Table 5. Skewness of Peak Power Evaluation 

Error Distribution (% of Plant Rated Power)

Nref Clear Sky Moderate

Variability

Intense

Variability

Extreme

Variability

1 0.2495 2.085 -0.235 0.513

2 0.2185 1.755 -0.0213 0.562

3 0.2836 1.562 -0.212 0.555

4 0.2937 0.978 -0.005 0.740

6 0.305 -0.330 0.189 0.547

8 0.358 -0.376 0.136 0.241

12 0.373 -1.275 -0.182 -0.453

24 0.0244 1.0313 0.399 -0.644

48 1.264 0.377 0.571 -0.366

Table 6. Kurtosis of Peak Power Evaluation 

Error Distribution (% of Plant Rated Power)

Nref Clear Sky Moderate

Variability

Intense

Variability

Extreme

Variability

1 2.236 23.844 11.249 15.053

2 2.101 21.602 11.089 15.133

3 2.236 25.288 11.582 15.344

4 2.135 22.589 11.835 16.554

6 2.312 23.201 11.853 15.017

8 2.329 20.951 10.753 13.372

12 2.521 21.197 11.095 14.589

24 3.271 18.951 11.266 12.994

48 5.207 19.70 10.823 15.165

For a better visual perception of the data shown in Table 1–Table 6, we show the same data in 

graphical form to demonstrate the impact of increased numbers of reference inverters on 

improving the accuracy of the estimation statistics of the available power. Figure 20A and Figure

19B show reduction trends for mean estimation error and error standard deviation with increased 

numbers of reference inverters, respectively. Similarly, Figure 21A, Figure 20B, Figure 22A, and

Figure 21B show improvements on min/max error and distribution skewness and kurtosis,

respectively.

Figure 20. Mean error and standard deviation of error

Figure 21. Min and max estimation error
20

This report is available at no cost from the National Renewable Energy Laboratory (NREL) at www.nrel.gov/publications.

Figure 22. Skewness and kurtosis of error distribution

In addition, Table 7 shows how correlation coefficients between actual and estimated 1-s 

available power improve significantly with increasing numbers of reference inverters for all four 

variability conditions. The same data are consolidated and shown in Figure 23 as well. 

Table 7. Correlation Coefficients Between Actual and Estimated Available Power for All Variability 

Cases

Nref Clear Sky Moderate

Variability

Intense

Variability

Extreme

Variability

1 0.995 0.979 0.947 0.947

2 0.998 0.983 0.955 0.947

3 0.999 0.987 0.963 0.956

4 0.999 0.988 0.968 0.959

6 0.999 0.990 0.974 0.967

8 0.999 0.992 0.976 0.977

12 0.999 0.994 0.992 0.990

24 1.000 0.998 0.995 0.995

48 1.000 0.999 0.998 0.998

Figure 23. Correlation coefficients for all variability cases 
21

This report is available at no cost from the National Renewable Energy Laboratory (NREL) at www.nrel.gov/publications.

7 Conclusions 

In this report, we examined the peak power estimation method for curtailed PV power plants 

based on using dedicated reference inverters within a plant. The proposed technique does not 

require deploying any additional equipment or sensors and is based only on the addition of new 

control logic to the existing power plant controller. Based on our calculations using measured 1-s 

power production data from the entire PV power plant and individual inverters, the method has 

the potential to produce highly accurate real-time estimates of available aggregate peak power 

that all the plant’s inverters can produce at any point in time and ensure that the control error 

stays within the desired tolerance bands at all times.

To complete the validation of the proposed PV plant peak power evaluation method, First Solar 

and the National Renewable Energy Laboratory are planning to conduct demonstration testing on 

a large utility-scale PV power plant installed at the National Wind Technology Center under a

cooperative research-and-development collaboration. An aerial photo of the plant is shown in 

Figure 24. This PV plant is rated for 430 KWac and uses six PV string inverters. The architecture 

of the PPC for this PV plant allows for easy control modifications, so the proposed concept can 

be validated with real hardware under real resource variability conditions. The National Wind 

Technology Center test site offers unique testing conditions for this experiment because of the

extreme solar resource variability conditions present at the site.

Figure 24. Aerial photo of First Solar PV plant at the National Renewable Energy Laboratory’s 

National Wind Technology Center
22

This report is available at no cost from the National Renewable Energy Laboratory (NREL) at www.nrel.gov/publications.

8 Future Plans

Areas of future research include the development of optimized control strategies to achieve the

minimum peak power estimation error under all types of operational scenarios for PV plants. 

Several optimization objectives must be resolved by the reserve allocation controller, which will 

dispatch reference inverters every predetermined time interval, ∆T (every 10 s, for example):

1. Determine the optimal number of control zones, NNzz zzzzzz, for the given variability 

conditions. (This could be determined from the forecast as well.)

2. Determine the optimal location of a reference inverter in each zone.

3. Determine the optimal dispatch interval, ∆T, for the reference MPPT inverters. (This will 

probably be longer for clear-sky days and shorter for highly variable days.)

4. Determine the optimized combinations of curtailment set points for participating inverters 

in each zone for maximum aggregate inverter efficiency (or minimum electric losses in 

the plant) for every ∆T interval. (Typical inverter efficiencies are shown in Figure 25.)

5. Use the short-term energy storage to correct estimation errors.

Figure 25. Typical MW-scale PV inverter efficiencies
23

This report is available at no cost from the National Renewable Energy Laboratory (NREL) at www.nrel.gov/publications.

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