NREL is a national laboratory of the U.S. Department of Energy, Office of Energy 

Efficiency & Renewable Energy, operated by the Alliance for Sustainable Energy, LLC.

Contract No. DE-AC36-08GO28308

Optimizing Geographic 

Allotment of Photovoltaic 

Capacity in a Distributed 

Generation Setting

Preprint

B. Urquhart

National Renewable Energy Laboratory and University 

of California–San Diego

M. Sengupta and J. Keller

National Renewable Energy Laboratory

To be presented at the 27th European Photovoltaic Solar Energy 

Conference and Exhibition

Frankfurt, Germany

September 24–28, 2012

Conference Paper

NREL/CP-5500-56539

September 2012
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1

OPTIMIZING GEOGRAPHIC ALLOTMENT OF PHOTOVOLTAIC CAPACITY IN A 

DISTRIBUTED GENERATION SETTING

Bryan Urquhart1,2, Manajit Sengupta,

1 and Jamie Keller1

1

National Renewable Energy Laboratory

15013 Denver West Parkway, Golden, CO 80401, USA

2

University of California at San Diego

9500 Gilman Drive, La Jolla, CA 92093, USA

ABSTRACT: A multi-objective optimization was performed to allocate 2 MW of photovoltaic (PV) among four candidate sites 

on the island of Lanai, Hawaii, such that energy was maximized and variability in the form of ramp rates was minimized. This 

resulted in an optimal solution set that provides a range of geographic allotment alternatives for the fixed PV capacity. Within 

the optimal set, a trade-off between energy produced and variability experienced was found, whereby a decrease in variability 

always necessitates a simultaneous decrease in energy. A design point within the optimal set was selected for study that

decreased extreme ramp rates by more than 50% while decreasing annual energy generation by only 3% more than the 

maximum generation allocation. To quantify the allotment mix selected, a metric was developed—called the “ramp ratio”—

that compares ramping magnitude when all capacity is allotted to a single location to the aggregate ramping magnitude in a 

distributed scenario. The ramp ratio quantifies simultaneously how much smoothing a distributed scenario would experience 

over single site allotment and how much a single site is being underutilized for its ability to reduce aggregate variability. This 

paper creates a framework for use by cities and municipal utilities to reduce variability impacts while planning for high 

penetration of PV on the distribution grid.

Keywords: distributed generation, optimization, geographic dispersion

1 INTRODUCTION

The variability of solar power provides integration 

challenges as a primary power source on the transmission 

system and poses potential power quality issues for 

distribution networks. A known level of fluctuation in solar 

power output will always exist because of natural variations in 

the solar position. Superimposed on the known fluctuations 

will exist additional variability in production from installed 

PV that depends primarily on weather conditions that

influence clouds and solar radiation at the location. It is well 

studied that aggregation of sites produces a smoother output 

of power on a per capacity basis [1–6]. These studies 

primarily address smoothing through geographic dispersion, 

and attempts have been made to mathematically model this 

phenomenon. On the other hand, there has been an apparent 

dearth of work on how to select the best geographic allocation 

of PV among candidate sites. This study examines the tradeoff between energy maximization and variability 

minimization while selecting and allocating PV generation 

among multiple candidate sites. In this paper, the geographic 

separation of the sites is small enough that we can assume that 

the problem we address is limited to the distribution grid. A 

study looking at minimizing the levelized cost of energy of a 

set of 12 PV generators using daily energy as input was 

performed by Roy [7]. Collins and Crowther [8] performed a 

multi-objective optimization that maximizes daily energy 

while minimizing generation shortfall using hourly input data 

and sites distributed across the state of Virginia. The latter is 

similar to this study in that it evaluated the trade-off between 

energy maximization and variability minimization; however,

the sites are far enough apart to be considered a transmission 

level problem. As previously mentioned, the scope of this 

study is restricted to geographic dispersion that can be 

assumed to not have the constraints imposed by balancing 

area coordination, line congestion, and other transmission 

issues.

The Maui Electric Company (MECO) owns and operates 

the island of Lanai’s electric power system. By comparison to 

mainland grids, the electric power system is small. The 

system energy is produced by a set of diesel generators 

located at the main power plant, of which one or two 

(depending on system load requirements) provide system 

frequency regulation using isochronous frequency control. 

Currently on the island there are two large distributed 

generation systems in addition to the main power plant. A 

large central station (1.2-MW) PV installation owned and 

named La Ola by Castle & Cooke is installed near the main 

diesel power plant that provides power to the grid through a 

purchase power agreement (PPA). Only recently has the La 

Ola PV system begun to generate power at full capacity 

because the PPA required that the system incorporate a 

battery energy storage system. This was to help mitigate 

ramping rates associated with the La Ola PV system. There is 

a combined heat and power (CHP) generator located near the 

end of the distribution circuit as well.

A team led by the U.S. Department of Energy’s National 

Renewable Energy Laboratory (NREL) and MECO are 

working with local developers on the island of Lanai to assess 

the economic and technical feasibility of increasing the 

contribution of renewable energy sources on the island, with a 

stated long-term goal of reaching 100% renewable energy as 

part of the Hawaii Clean Energy Initiative. For MECO, 

enabling the reliable installation (and determining the 

technical requirements associated with the reliable

installation) of additional renewable resources such as PV 

systems onto the electric power system is one way to assist in 

increasing the renewable energy penetration on Lanai.

The NREL team, with input from interested PV system 

installers, has been working to create and evaluate viable 

scenarios to assess the potential amount of PV systems that 

can be cost-effectively installed on Lanai. In support of this, 

NREL has installed equipment to measure the variability of 

the solar resource at several locations on the island. For more 

than one year, NREL has been capturing 3-sec timesynchronized solar radiation data at four locations on Lanai 

that represent some of the possible locations for additional PV 

systems. Figure 1 shows the locations of the 3-sec solar 

irradiance data collection points, as well as some of the 

existing generation and load centers. This data is available to 
2

help understand site resources and variability among sites;

however, appropriate system simulation models and 

assumptions are still needed.

This paper develops a design strategy for placement of PV 

generation by maximizing energy produced and minimizing 

variability and, more specifically, power ramp rates. To do 

this, the planned PV output was simulated from available 

solar measurements for one complete year from June 2010 to 

May 2011. A predetermined amount of PV (e.g., 2 MW) was 

geographically allocated among four locations with differing 

amounts at each site, and the impact on energy generation and 

variability was quantified. To formalize this process, a multiobjective optimization scheme was employed to generate 

Pareto-optimality (named for the Italian economist Vilfredo 

Pareto) curves using several variability criteria. Paretooptimality curves provide information on the trade-offs

between energy production and variability experienced. These 

curves are presented as a method for utilities to assess the 

trade-offs of maximizing energy production while minimizing 

system variability. The method presented provides a tool for 

selecting an acceptable level of ramping for an individual 

system.

2 METHODOLOGY

2.1 Experimental Setup

The island of Lanai is located at a latitude of 20.8o

N,

where the Walker and Hadley circulations generate the 

northeasterly trade winds. Trade wind cumulus clouds are 

dominant and cause large and frequent ramps in the global 

horizontal irradiance (GHI) because of the highly variable 

nature of that cloud type (Fig 2). GHI was measured at four 

prospective PV deployment locations (Fig. 1, Table I) on the 

island using Licor LI 200 silicon pyranometers capturing data 

at a rate of 3 sec. The La Ola station also had regular 

meteorological measurements of temperature and wind data. 

Data was collected from April 2010 to September 2011;

however, to ensure the summer half-year was not weighted 

more heavily in this study, only data from June 2010 to May 

2011 was used.

The Castle & Cook site is located in the center of the 

island, close to the mountains; the other sites are located in the 

southern portion of the island surrounded by much flatter 

terrain. Terrain-induced orographic lifting at the Castle & 

Cook resulted in higher levels of cloud formation and thus a 

higher frequency of large ramp events at that location. The 

Waste Water and Challenge LP sites were close to each other 

and thus exhibited stronger correlation than the other two 

sites. Hereinafter, the four sites are referred to by their 

designations indicated in both Fig. 2 and Table I.

Table I. Measurement site geographic position and designation

Site Designation Lat. [°N] Lon. [°W] Alt. [m]

Castle & Cook CC 20.81782 156.92107 464

Waste Water WW 20.74844 156.89694 110

Challenge LP CL 20.74102 156.90509 60

La Ola LO 20.76685 156.92291 381

2.2 Power Output Simulation

Power output was simulated using irradiance to DC power 

and DC to AC power algorithms that are part of the PVForm 

performance model (version 3.3) [9, 10]. This model has been 

widely used since its development, including in the latest 

version of PVWatts [11], and has been shown to have a 5% 

positive bias and RMS errors of less than 12% [12]. The 

model incorporates a temperature correction for irradiance to 

DC conversion as well as an efficiency reduction for low 

irradiance levels.

The transposition model used in PVForm, based on that of 

Perez et al. [13], to convert the horizontal irradiance 

measurements to plane-of-array (POA) irradiance was not 

used here. The goal of this study was to develop a 

methodology to optimize the benefits of geographic 

smoothing and did not aim to achieve high accuracy in the 

exact energy produced or ramp rates experienced. Both the 

energy output and ramp rates would increase if GHI was 

transposed to POA. Importantly, however, the trends would 

remain the same; that is the basis for the results presented 

here. An additional consideration is that the latitude was low; 

therefore, the tilt would be close to horizontal, so there was 

less error in making the horizontal simplification than would 

occur at higher latitudes.

Figure 2. GHI on June 20, 2011, showing typical large, cumulusinduced fluctuations on the island of Lanai, Hawaii.

It should be mentioned that direct performance model 

conversion of irradiance to power output is somewhat 

misleading. Large PV plants exhibit intra-plant smoothing 

similar to a low pass filter [14, 15]. Both Marcos et al. [14] 

and Lave et al. [16] developed Fourier and wavelet-based 

methods, respectively, to smooth modeled intra-plant power 

output so that it better matches observed output. Intra-plant 

smoothing was not considered here. The aggregate capacity of 

Figure 1. Map of the island of Lanai with four sensor locations 

indicated. The Lanai City and airport load centers are indicated, along 

with the La Ola PV system, the main island power plant, and a 

combined heat and power plant located near the coast.
3 

the four simulated sites was 2 MW, which is relatively small 

in total area, and it was assumed that direct use of a PV 

performance model does not impact the utility of the results 

presented. 

2.3 Multi-Objective Optimization: Maximizing Energy 

Generation While Minimizing Ramp Rate 

Using a predetermined installed capacity of PV to be 

allocated among the four candidate sites, an optimization was 

performed to determine which weighting (% of total) 

allocation at each site would provide the most energy while 

also having the lowest variability. To maximize only energy, 

one would simply place 100% capacity in the sunniest 

location. This would not provide the smoothest output in 

general, and if variability was a concern, utilities could take 

advantage of geographic dispersion and place some capacity 

in less sunny locations. This would result in a potential 

reduction in total energy produced but would also lead to 

reduction in the variability. To avoid defining a subjective 

cost penalty for each parameter being optimized, as is done in 

standard linear programming, the energy and variability 

criteria were treated as distinct objectives. This multiobjective optimization problem does not have a single unique 

solution, but instead has a range of solutions, known as the 

Pareto-optimal set, which describe a range of non-inferior 

alternatives for maximizing energy while minimizing 

variability. Arbitrarily selecting how to allocate capacity may 

reduce energy while not decreasing the variability 

significantly. For the Pareto-optimal set, the maximum energy 

output is achieved for a given level of variability. 

Figure 3 shows an example of the Pareto-optimal set for 

minimizing the 99th percentile of ramp rates while maximizing 

energy. Moving along this curve (also called the Paretooptimal frontier), one must trade energy generated for a 

reduction in variability, characterized here by ramp rates. At 

any point, in the Pareto-inferior region one can increase 

energy or decrease ramp rate without sacrificing the other, and 

corresponds to an inefficient selection of geographic 

allocation. Points along the curve are Pareto-efficient in that, 

to change one quantity, one must sacrifice the other. A system 

designer can then select the maximum level of variability a 

given electrical system can handle, and the corresponding 

point on the Pareto-optimal frontier will yield the geographic 

allocation for maximum energy production. 

It should be noted that although we used ramp rates 

exclusively to quantify variability in this work, other 

metrics—such as ramp size, ramp frequency, or generation 

shortfall (if load information is available)—can be used and 

will be the subject of future study. Ramp rate rj

ሺt,∆tሻ at a time 

ݐ for an interval ∆t is defined as 

ݎ௝ሺݐ,∆ ݐሻ = ௣ೕሺ௧ା∆௧ሻି௣ೕሺ௧ሻ

∆௧ 	, (1) 

where pj

ሺtሻ is the power output at site j. The probability 

density of ramp rates over the one year studied (June 2010 to 

May 2011) for LO is given in Fig. 4 for different ∆ݐ. This 

shows the usual result that ramp rates are largest for the 

smallest ∆ݐ ;thus, for the optimization, only ramp rates at the 

3-sec data capture rate are used. Minimization of 3-sec ramp 

rates ensures that ramp rates for all other ∆t will always be 

smaller. No ramp rates were excluded in any computation 

step, even if very small, to ensure illumination of the true 

probability of occurrence of large ramp events in all statistical 

results. The cumulative distribution function indicated that 

99% of 3-sec ramp rates were below 4% of capacity, and 

99.9% of ramp rates were below 10% of capacity for the La 

Ola site. 

The multi-objective optimization employs a genetic 

algorithm using a tournament selection scheme described in 

[17]. The genetic algorithm tries many successive 

combinations of weights, pushing the Pareto frontier outward 

until no other variation of weight allocation can extend the 

curve farther into the optimal territory. During this process, 

the solution set in each successive iteration is combined 

(crossover) and “mutated,” and less fit alternatives are 

removed in an emulation of genetic evolution. The objective 

functions to be minimized are: 

 (2) , ௜ ݐ∆ሻ௜ݐሺ௝݌௝ݓ ∑ െ = ሻ࢝ሺ݂

݃௡ሺ࢝ሻ = ܲ௡൫หݓ௝ݎ௝ሺݐ௜,∆ ݐሻ ห൯ , (3) 

where ݓ ௝is the component of weight vector ࢝ corresponding 

to the geographic allocation for site j, and Pnሺ∙ሻ is the nth

percentile of the argument. Summation on j is implied for both 

equations (2) and (3). Equation (2) is the negative of total 

energy output by the aggregate system for configuration ࢝, 

and equation (3) is nth percentile of system aggregate ramp 

rate computed for the entire set of i ramp rates (i.e., ramp rates 

from the entire year data set). In equation (3), only the 

magnitude of the elements in the set are considered, denoted 

by the vertical bars. The constraints on the optimization are 

 , ሿ0,1ሾ௝ ∈ ݓ1, = ௝ݓ ௝∑

which requires that all weights sum to unity and that any 

given site can be weighted only from 0% to 100%. The 

notation used above contains a subtlety for computational 

purposes where both power and ramp rates used in equations 

(2) and (3) for individual sites are scaled so that each site is 

initially at 2 MW of capacity (8 MW total) so that when 

multiplied by the weights ࢝ the aggregate capacity is the 

desired 2 MW. This implementation and notation must be 

changed if intra-plant smoothing is considered. Power output 

and thus ramp rates are no longer independent of assigned 

capacity, and the power time series and thus ramp rates must 

be adjusted by means of a smoothing filter [14, 16]. 

Figure 3. The Pareto-optimal set for maximizing energy while 

minimizing the 99th percentile of ramp rates. The curve is shown to 

divide two regions: an unattainable set of energy, ramp rate 

combinations; and an inferior set where individually either energy can 

be increased or ramp rates decreased without compromising the other. 
4

3 RESULTS

To assess the trade-off between ramp rates and energy 

production, multiple ramp rate percentiles were used as 

minimization criteria. Figure 5 (a) shows the 97th, 98th, 99th, 

and 99.5th percentile Pareto-optimal curves, along with an 

interpolated color swath to indicate the spread in site 

weighting. The darker color indicates that the standard 

deviation of the four weight values was low. The upper bound 

of 8.26 MWh was the maximum production possible for a 2-

MW plant during the one year studied; thus, the allocation 

was 100% at the CL site, with corresponding high standard 

deviation of the four weights. The near verticality of the 97th

percentile showed that by changing the weights, not much 

improvement was made in reducing the ramp rates. At higher 

percentiles, such as 99.5, significant gains could be seen by 

adjusting the weights, and ramp rates dropped by nearly 40% 

for a 3% reduction in energy generated.

Taking a more expanded view of the high end of ramp 

rate space, Fig. 5 (b) shows ramp rate percentiles up to the 

99.999th percentile. The space between this percentile and the 

100th percentile is spanned by only 43 data points. More than 

one year of data is needed to provide statistical context for 

percentiles beyond the “five nines” level, and this was the 

maximum percentile considered here. At 99.999th percentile, a 

more than 50% reduction in the magnitude of extreme ramps 

can be achieved with only a 3% drop in energy produced 

when site allocations are properly selected. In Fig. 5 (b), the 

color shading indicates that for lower ramp rates, geographic 

allocation was more evenly distributed. As shown, for each 

energy level, the site weightings appeared somewhat 

independent of ramp rate percentile. This was a positive result 

because it implied that geographic dispersion lowers ramp 

rates at every level. The multi-objective optimization was run 

to simultaneously minimize two ramp rate percentiles (e.g., 

99th and 99.99th percentile) while maximizing energy, and the 

result indicated that the geographic allocation used to reduce 

ramp rates in one percentile was beneficial to the other. This 

means that optimal solutions at low ramp rates have similar 

site allocations, a fact not clear from looking at the standard 

deviation of weights alone. The three-dimensional plot is not 

shown because perspective effects make it difficult to 

interpret on the written page. The site weighting and ramp 

rates for a selection of points in Fig. 5 (b) are given in Table II 

as a function of energy and ramp rate percentile.

A system designer is bound by economic constraints and 

cannot simply minimize ramp rate while disregarding a 

potentially significant drop in energy output. The exact 

requirements are of course system dependent, but here an 

example case was selected where energy was decreased from 

only the 8.26-MWh maximum to an 8-MWh design point 

(OPT). The site weights for this case are given in Table III, 

and the resulting ramp rate percentiles are given in Table IV. 

For comparison, Table IV also includes the ramp rates at 

different percentiles when all 2 MW was allocated to each site 

separately. As expected, the ramp rates of OPT were smaller 

at all percentiles, but the important result is that energy was 

not significantly reduced from the maximum achievable level. 

In fact, it was significantly higher than could be achieved with 

100% allocation at either CC or LO. Comparing OPT to the 

equal allocation case (EQ) shows that one can reduce ramps 

over OPT, but energy dropped by nearly 9% above the 

maximum and 6% above OPT. In the limit of zero correlation 

of ramp rates between sites, the EQ scenario would be the best 

geographic allocation for ramp rate minimization. Because 

there was a small degree of correlation, even at short time 

scales, a multi-objective optimization minimizing the 99th and 

99.99th percentile and ignoring energy output yielded the 

lowest overall ramp rates (MIN, Table IV). The MIN results 

marginally improved both ramp rate reduction and energy 

production over EQ.

Figure 4. Probability distribution of ramp rates for several time 

steps at the La Ola site. Ramp rates are shown as a percentage of 

total power output capacity. Small or negligible ramp rates were 

included; thus, the curves indicate true probability of events.

Figure 5. Pareto-optimal sets for (a) the 97th, 98th, 99th, and 99.5th

percentile and (b) the 99th, 99.5th, 99.9th, 99.99th, and 99.999th

percentile. The standard deviation of the geographic allocation (as a 

fraction of the total), is shown as a color swath where darker color 

indicates low deviation and lighter color indicates a single site that has 

a majority allocation.
5 

Table II. Site allocation weight values on the Pareto-optimality curves for minimization of the 99th, 99.9th and 99.99th percentile of ramp rates, shown as a 

function of aggregate daily energy produced. Aggregate system ramp rates are also shown. 

99th pctile 99.9th pctile 99.99th pctile 

Energy 

[MWh/day] 

Weights 

[%] 

Ramp 

Rate 

[kW/s] 

Weights 

[%] 

Ramp 

Rate 

[kW/s] 

Weights 

[%] 

Ramp Rate

[kW/s] 

CC WW CL LO CC WW CL LO CC WW CL LO 

7.55 ‒ ‒ ‒ ‒ ‒ ‒ ‒ ‒ ‒ ‒ 22 23 29 26 93 

7.60 ‒ ‒ ‒ ‒ ‒ 21 27 30 22 64 20 24 32 24 96 

7.70 17 26 36 21 35 20 28 34 18 67 18 25 37 20 103 

7.80 13 27 43 17 36 16 27 42 15 73 14 29 41 16 112 

7.90 11 27 49 13 38 21 34 41 4 77 12 32 44 12 121 

8.00 8 23 60 9 42 16 33 50 1 86 8 34 50 8 135 

8.10 5 23 67 5 46 8 29 60 3 98 5 35 55 5 146 

8.20 2 19 78 1 52 3 15 81 1 128 2 32 65 1 168 

8.26 0 0 100 0 66 0 0 100 0 158 0 0 100 0 253 

Legend: Castle Cook (CC), Waste Water (WW), Challenge LP (CL), La Ola (LO) 

To quantify the level of geographic smoothing of ramp 

rates, a new metric termed the “ramp ratio” was introduced. 

The ramp ratio RRj

n for the j

th site at the nth percentile is 

defined as 

RRj

n= Pn൫หrj

ሺti,∆tሻห൯

Pn൫หwjrj

ሺti,∆tሻห൯ 	, (4) 

where summation over j is implied by repeated indices. 

Effectively, ramp ratio compares ramping magnitude when all 

capacity is allotted to a single location to the aggregate 

ramping magnitude in a distributed scenario with allocation 

weights wj. For a time period with no variability, such as on a 

clear day, and assuming all sites are proximal enough that the 

solar resource is nearly the same, the ramp rates of an 

individual site are a result of changes in solar geometry. In 

this case, the ramp ratio had a value close to one. This 

corresponded to a high correlation between sites because they 

increased and decreased in unison with the movement of the 

sun. When there was variability because of the presence of 

clouds, the correlation between sites began to decrease at 

short time scales. This resulted in compensating ramps when 

the plants were distributed, which ultimately resulted in 

smoothing. This characteristic showed up as ramp ratios that 

were greater than one, indicating that the total system output 

was smoother on a per capacity basis than the ݆

th constituent 

site. The value of the ramp ratio quantified the level of 

increased geographic smoothing, e.g. RRj

99=2 indicates that 

the aggregate system output was two times smoother than site 

j at the 99th percentile of ramp rates. It may so happen that the 

ramp ratio was less than one. This would imply that a 

particular site was actually less variable than the aggregate. In 

this case, the weights allotted to the sites were obviously not 

optimal, with greater weights having been allotted to sites 

with high variability at the expense of low variability sites. In 

such a scenario, reallocation of capacity would be beneficial. 

Several examples of the ramp ratio RRj

99 are given in 

Figure 6 for different allocation scenarios across different 

months. The EQ case (Fig. 6 (a)) shows that smoothing was 

not uniform for all months and tended to be higher during the 

summer. Figure 6 (b) shows the OPT case. The island of 

Lanai currently has 1.2 MW of PV installed at the La Ola site, 

so it is of practical interest to look into the scenario where the 

remainder of the 2 MW is distributed equally among other 

sites. Figure 6 (c) shows the Lanai case study scenario (LAN, 

Table III). Overall, in comparison to Fig. 6 (b) and Fig. 6 (c), 

EQ had the highest and most consistent RRj

99, which was to 

be expected for nearly uncorrelated signals. During the winter 

half-year, WW and CL, the sites near each other, had lower 

RRj

99 in EQ and thus were less variable than CC and LO. This 

is one reason why the OPT case weighted WW and CL more 

heavily than the other sites. Figure 6 (b) reflects this increase 

in weighting because the ramp ratios of WW and CL were 

more uniform across the year. These two sites also produced 

more energy on average and therefore had a higher weighting 

Table III. Site geographic allocation percentage for seven scenarios: 

CC, WW, CL, and LO allocate all capacity at a single site; OPT uses 

weighting for the 8-MWh design point; EQ equally weights each site; 

MIN weighting mix shows results from minimization of the 99th and 

99.99th percentile ramp rates; and LAN is a Lanai case study. 

Site Geographic Allocation [%] 

Site CC WW CL LO 

CC 100 0 0 0 

WW 0 100 0 0 

CL 0 0 100 0 

LO 0 0 0 100 

OPT 9 30 53 8 

EQ 25 25 25 25 

MIN 22 26 26 26 

LAN 13 13 13 61 

Legend: Castle Cook (CC), Waste Water (WW), Challenge LP (CL), 

La Ola (LO), Optimum Allocation (OPT), Equal Allocation (EQ), 

Minimum Ramp Allocation (MIN), Lanai case study (LAN) 

Table IV. Energy production and ramp rates at the 99th, 99.9th, 

99.99th, and 99.999th percentile for seven weighting scenarios: CC, 

WW, CL, and LO show results for single site allocation at each 

respective site; OPT shows results for the 8-MWh design point; the 

EQ case uses 25% weight at each site; MIN uses a site mix to 

minimize ramp rates irrespective of energy generation; and LAN is a 

Lanai case study scenario.

Site Energy 

[MWh/day] 

Ramp Rate 

[kW/s] 

99th pct 99.9th

pct 

99.99th

pct 

99.999th

pct 

CC 6.93 86 198 314 401 

WW 8.19 71 166 266 362 

CL 8.27 66 158 253 339 

LO 6.72 78 177 276 358 

OPT 8.00 41 89 140 189 

EQ 7.53 35 65 94 127 

MIN 7.55 35 64 93 126 

LAN 7.13 49 111 173 225 

Legend: Castle Cook (CC), Waste Water (WW), Challenge LP (CL), 

La Ola (LO), Optimum Allocation (OPT), Equal Allocation (EQ), 

Minimum Ramp Allocation (MIN) , Lanai case study (LAN) 
6

in the optimization. The energy generation and ramp rates of 

the LAN scenario were well within the Pareto-inferior region 

(Table IV). Fig. 6 (c) shows the ramp ratio was lower than 

OPT or EQ, and in fact dipped below one in November. The 

ramp ratios below one means these sites were underutilized in 

this scenario for their ability to decrease ramping during the 

winter.

The previous discussion for ramp ratio used the 99th

percentile as the baseline for comparison of monthly and 

inter-site variability. Use of a different percentile, specifically 

higher percentiles, provided a different quantification for 

understanding variability. Extremely large ramp rates—

caused by the passage of an individual cumulus cloud with the 

direct solar beam along with greater than average cloud 

enhancement just beyond the leading or trailing edge—are 

infrequent and typically affect only a single site. These 

extreme ramp events at one site will generally not have any 

correlation to other locations. Aggregation of sites under these 

conditions will therefore invariably result in smoothing, which 

is reflected by higher ramp ratios. The larger the ramp event 

(and thus higher percentile), the larger the inter-site decorrelation; therefore, the ramp ratio tends to increase as 

percentile considered increases.

The probability distributions (PDFs) of large ramp events 

are shown in Fig. 7. Reviewing the design point case OPT, the 

ramp rates were reduced significantly over all four single site 

allocation scenarios. The EQ case had the lowest extreme 

ramp rates by a significant margin. The LAN case reduced the 

variability over the single site cases, but ramping was larger 

than OPT by almost an order of magnitude toward the upper 

tail of the distribution, and it generated more than 10% less 

energy per annum. The juxtaposition of the OPT case to that 

of LAN underscores the benefit of designing a system for 

optimal use of the available solar resource while taking 

maximum advantage of its spatial de-correlation at the time 

scale considered.

4 CONCLUSIONS

It is well known that geographic dispersion provides a 

smoother overall output on a per capacity basis than 

centralized generation, but the optimal allocation to generate 

the most energy output while minimizing the variability has 

not been widely studied. A multi-objective optimization 

scheme was employed to investigate if variability can be 

significantly minimized while maintaining high levels of 

energy production. It was shown that an optimal set of 

alternative geographic allocation exists to maximize energy 

production and minimize variability. With proper geographic 

allotment of generation, ramp rates were reduced by 50%;

whereas energy was reduced by only 3% over the maximum 

production case. A useful result for system designers is that 

the exact level of variability can be tuned to an acceptable 

level, with the understanding that if the solution is optimal, 

then energy generation must be compromised as indicated by 

the shape of the Pareto-optimal set.

In addition to showing that there exists a clear energyvariability trade-off, a new metric was introduced, termed the 

ramp ratio, which can be used to assess multisite geographic 

allocations and their effect on variability. This ratio showed

simultaneously how much smoothing the aggregate system 

experiences over any individual site, and also if any particular 

site has significantly less variability per unit capacity than the 

other sites in the mix. It is important to note that the ramp 

ratio was developed to assess optimality of capacity allocation 

among multiple sites; therefore, looking at a single site’s ramp 

ratio in isolation does not provide a complete set of 

information. The ramp ratio can be extended into a curve in 

the temporal dimension by segmenting the input data set (as 

done in this work) or into the magnitude dimension by 

looking at different percentiles of variability. 

The methods applied here will be extended to other 

variability criteria beyond ramp rates in future work. The 

metrics of energy and ramp rate are directly derived from 

resource data, but for power systems application at the 

distribution level, voltage fluctuations and cost will be the key 

metrics optimized in the kind of analysis presented here. 

Additional future work will involve implementing this multiobjective optimization with a feeder simulation where 

Figure 6. Monthly ramp ratios at the 99th percentile for (a) the equal allocation scenario EQ, (b) the 8-MWh design point scenario OPT, and 

(c) a Lanai case study LAN reflecting currently installed capacity at La Ola.

Figure 7. Probability distribution of ramp rates over the 99th

percentile. The sign of the ramp was not considered. The distribution 

was scaled such that the probability of all ramps over the 99th

percentile summed to unity. Allocating all capacity to four sites 

separately is shown (thin lines), along with three multisite allocation 

cases (thick lines).
7

economic benefit is maximized and voltage variations are 

minimized. In this study, no restriction was placed on the 

quanta of generation installed at any given site, and no site 

was given a minimum threshold to be considered active. For a 

realistic feeder simulation, generation siting will be practically 

limited by space constraints and by operations and 

maintenance constraints. These elements need to be included 

so their impact on the Pareto-optimal set of alternatives can be 

quantified.

This paper in effect creates a framework that can be used 

by cities and municipal utilities that would like to see a high 

penetration of PV and would be interested in reducing the 

variability impacts on the distribution grid. The development 

that energy and variability have a clear trade-off, and that this 

trade-off can be optimized for unique scenarios, offers an 

opportunity to communities and utilities to leverage this to 

their advantage. The methods presented can be incorporated 

into power system design and simulation tools so that 

designers of future PV systems can assess geographical 

allocations that provide the maximum benefit. This work 

offers only a glimpse of what can be achieved in maximizing 

the utility of the solar resource through optimized geographic 

allotment.

It is important to note that such studies are valuable before 

deployment of PV; however, to conduct such studies, there is 

a need to collect high-resolution time-synchronized solar 

radiation data sets for a reasonably extended period of time. 

Such deployments are relatively inexpensive but can provide a 

high level of benefit when planning high level penetration of 

PV on the distribution grid.

ACKNOWLEDGEMENT

This work was supported by the U.S. Department of 

Energy under Contract No. DE-AC36-08-GO28308 with the 

National Renewable Energy Laboratory.

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