Determining the optimum grid-connected photovoltaic inverter size

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Solar Energy 87 (2013) 96–116 www.elsevier.com/locate/solener

Determining the optimum grid-connected photovoltaic inverter size Song Chen ⇑, Peng Li, David Brady, Brad Lehman Department of Electrical and Computer Engineering, Northeastern University, 360 Huntington Ave., Boston, MA 02115, USA Received 24 January 2012; received in revised form 19 September 2012; accepted 20 September 2012 Available online 20 November 2012 Communicated by: Associate Editor Nicola Romeo

Abstract This paper discusses the practical factors that may influence the inverter sizing strategy. Effects of various factors are studied separately by isolating parameters in the simulations. These factors include irradiance and temperature conditions of the photovoltaic (PV) installation location, PV incentives, electricity rates, and inverter intrinsic parameters such as overload protection schemes and efficiency curves. Specifically, examples of nine different geographic locations in the US are simulated and discussed with realistic parameters to show that the optimum inverter size varies notably by location and context. Ó 2012 Elsevier Ltd. All rights reserved. Keywords: Optimum inverter sizing; Solar irradiance pattern; PV incentives

1. Introduction Conventionally, photovoltaic system inverters are sized based on the rated power of the PV panel installation. There are two typical methods for sizing the inverter: (1) most commonly the inverter is sized to approximately match the nominal PV array installation, i.e. a 10 kW rated (at STC) PV installation is sized with a 10 kW inverter, or (2) the inverter is downsized with the typical rule-of-thumb to take 70% of the nominal power PPV,nom of the PV panels (Burger and Ru¨ther, 2006; Macedo and Zilles, 2007; Luoma et al., 2012; Keller and Affolter, 1995; Islam et al., 2003). In this case, the same 10 kW PV installation is sized with approximately a 7 kW inverter. The rationale of downsizing the inverter is that irradiance levels in real installations only occasionally reach irradiance levels of the STC conditions

⇑ Corresponding author. Tel.: +1 6173734029.

E-mail addresses: [email protected] (S. Chen), [email protected] (P. Li), [email protected] (D. Brady), [email protected] (B. Lehman). 0038-092X/$ - see front matter Ó 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.solener.2012.09.012

(1000 W/m2 at Air Mass 1.5 and 25 °C cell temperature). Therefore, the additional investment on the inverter used to match the inverter size to the array’s “rated” power might not be recovered over the life-expectancy of the inverter. There is a tradeoff between maximizing energy yield and minimizing inverter cost. Only recently have there been analytic qualifications and conjectures as to what might be the optimum inverter size (Burger and Ru¨ther, 2006; Macedo and Zilles, 2007; Luoma et al., 2012; Keller and Affolter, 1995; Islam et al., 2003). For example, some researchers propose that undersized inverters cause considerable energy loss under high irradiance due to overload protection of inverter (Burger and Ru¨ther, 2006). In this case overload refers to over-irradiance, a condition in which the input power of inverters from the PV panel is more than their nominal rated allowable input power. This implies that clear sky irradiance locations might have noticeably different power yield compared to cloudy locations, even when they may experience the same average daily irradiance. Further, our preliminary work showed that the characteristics of the inverter protection scheme also strongly influence the decision on inverter sizing (Chen

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et al., 2010, 2011), i.e. whether the inverter has time delay before entering protection, thermal fold-back, or simply shuts down in over-current protection modes. This paper provides a more comprehensive view over the factors that influence the optimum inverter size for a given PV installation. There are multiple nonlinearly-dependent factors that influence inverter sizing recommendations, few of which have been previously discussed. Specifically, an algorithm to compute the proper Return On Investment (ROI) for an inverter is introduced. Previous literature did not consider important inter-relationships between government economic incentives, electricity rates, inverter protection strategies, PV operating temperature histograms, etc. The new ROI calculation examines all these influences for different inverter sizes in order to optimize the undersizing recommendation of the inverter that gives the highest cost savings. Fig. 1 generalizes the (highly nonlinear) effects various factors have on the optimum inverter size, with each factor having increasing parameter. Explicitly, the higher the irradiance level, PBI or electricity rates are, the larger inverter size is needed to achieve the economical optima. On the other hand, higher ambient temperature of a specific location, longer inverter protection delay, or higher inverter light-load efficiency will justify using a smaller inverter. Among these factors, irradiance level, PV panel working temperature and economic incentives are most influential to the optimum inverter size. Intuitively, one might expect that a large inverter is needed for a geographic location that has abundant sunlight or that is cool throughout the year. In fact, this is a conclusion of prior research in (Burger and Ru¨ther, 2006). However, our research shows that this is not precisely true: For most places, more sunlight exposure usually leads to hot climate: this high correlation between irradiance level and temperature results in the counteraction between these two factors. In this case, the exact optimum inverter size is not obvious, therefore requires quantity analysis. In fact, in some locations, the inverter sizing recommendation is

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only mildly influenced by either high irradiance or high operating temperatures because the two factors compete against each other (Las Vegas, Nevada when no government incentives are analyzed). The upfront incentives and rebates, granted by the government and/or utility companies, can either increase the optimum inverter size or decrease it, depending on the diverse policies for each location. These incentives can also be used to counter-act any influences of temperature or irradiance patterns on optimal inverter sizing. The most important factors that may influence the inverter sizing strategy are listed below, in three categories, each of which will be discussed in later sections (1) Meteorological factors – This includes irradiance patterns, wind speed and air temperature. Even for PV systems of the same size, the economically optimum inverter size can differ given different irradiance and air temperature patterns. (Wind effect is only discussed qualitatively and is a separate and emerging research topic (Jones and Underwood, 2000; Notton et al., 2005; Tina and Abate, 2008).) (2) Economic factors – A primary contribution of this research is to explain how economic factors influence inverter size recommendation. This topic has not previously been discussed in the literature, yet has dominant influence on inverter size recommendations. In this category, inverter pricing, electricity rate, and government incentives or feed-in tariffs influence optimum inverter sizing. Different types of incentives will lead to substantially different inverter sizing investment to optimize the Return On Investment (ROI). For example, in the USA, some states offer Performance Based Incentives (PBI) based on the actual power produced by the entire installation. Other states have incentives that are not performance based and are only based on cost or wattage of the installation. When there are substantial PBI incentives, the cost of a larger inverter can be justified. Otherwise, downsized inverters might be selected. Similarly, electricity rates reflect part of the cost of the energy losses due to undersizing of the inverter. However, the price of inverter counteracts the effects of PBI and electricity rate. Initial cost savings for undersizing an inverter tend to be linear ($/Watt) savings, while PBI and electricity rate production cost losses from undersizing follow an approximately parabolic increase with respect to energy lost due to underzing. So the intersection between the two cost curves will lead to the optimum inverter undersizing recommendation. The cost savings vs. inverter size curves will, therefore, look approximately parabolic, as discussed in later sections.

Fig. 1. Summary of effects on optimum inverter size when increasing the parameters of different influential factors. The effect of the upfront incentives and rebates is two-fold, depending on the policies. It will be discussed in Section 6.

(3) Inverter intrinsic characteristics – The intrinsic parameters of the candidate inverter selected to be installed with a PV system may also have impacts

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on the final sizing strategy. Two of the crucial parameters are the protection scheme of the inverter and its efficiency curve along the load-to-nominal ratio. To date, the implied assumption behind inverter protection loss calculation has been that the inverters enter downscaled protection mode as soon as over-irradiance events occur, thereby causing immediate power loss. Usually, this is achieved by having the inverter force the PV array to operate at a higher voltage than its maximum power point voltage, thereby reducing PV current and clamping to a safe power level. However, there might be a time delay before inverters protect themselves from over-current due to high irradiance. During such time delay, which is determined by the protection scheme of the inverter, the inverter can still handle full input power without derating. This is because thermal time constants for inverter internal electronics are long (slow), and the inverter may still operate at safe temperatures for a short time period (Meysenc et al., 2005). Thus, over-irradiance conditions may not lead to lost energy if they do not last long. Further, previous research often assumed PV inverters operated at constant efficiency, independent of the load. This is unrealistic, particularly in low power or scaled-back overpower protection schemes. Our approach is to take realistic inverter efficiency curves (from independent test lab results (International Energy Agency, 2002; Ginn et al., 1997; Go Solar California, 2011) and protection delays and then include them into the cost-optimization analysis. Although the inverter size recommendation for highest ROI depends on all the above factors, it is helpful to investigate the influence of each parameter while all other parameters are fixed. Therefore, the outline of the paper is as follows: First, the basic overall methodology to compute the optimal inverter size for highest ROI is explained in Section 2. The baseline example case of a 10 kW PV installation is introduced, along with fundamental cost analysis and PV performance equations. Section 3 discusses the irradiance level effects and ambient temperature effects on Return On Investment (ROI) on the baseline example with all other parameters fixed. Section 4 explains the influences of economic factors, such as government incentives, electricity rates and inverter costs, once again, while varying only one parameter at a time. Section 5 isolates the influence of inverter protection schemes and electrical efficiency. Finally, Section 6 allows all these parameters to interact for optimization of ROI for the same PV installation at nine different geographic locations. The interesting result is that several of the installations in the USA (Eugene, Oregon; Las Vegas, Nevada; Lanai, Hawaii) has significantly different inverter size recommendation. This result is due to the interaction of all the parameters, discussed in Sections 3–5, which vary based on geographic location.

2. Problem statement and methodology for solution 2.1. Problem statement Inverters take a considerable portion of PV system capital cost. A common tradeoff in inverter sizing is the balance between energy yield and inverter investment. Conventional practices are to (i) let nominal input power of inverter to approximately match the nominal output power of PV array: Pinv,nom  PPV,nom; (ii) undersize inverter to around 70% of PV array size: Pinv,nom = 70%PPV,nom (Burger and Ru¨ther, 2006; Keller and Affolter, 1995). Both practices are empirical and might not be sizing the inverter optimally with respect to cost and return analysis. If PPV,nom is rated under Standard Test Condition (STC)1 with irradiance GSTC = 1000 W/m2, inverter size can be described as a nominal irradiance level GTh or the ratio between Pinv,nom and PPV,nom, denoted R (Macedo and Zilles, 2007; Luoma et al., 2012): P inv;nom GTh ¼ ¼R P PV;nom GSTC

ð1Þ

where Pinv,nom is DC input rated power of inverter; PPV,nom is rated PV installed power; R is the inverter downsize coefficient. When 0 < R < 1, the inverter is undersized. For R > 1 the inverter is being oversized and will have higher power rating that the PV installed power. For example, for a STC-rated 10 kW PV system, with approach (i), GTh = 1000 W/m2, the inverter downsize coefficient is R = 1, and the inverter is rated at Pinv,nom  10 kW; while with approach (ii), GTh = 700 W/m2, the inverter downsize coefficient is R = 0.7, and the inverter is rated at Pinv,nom  7 kW. Problem Statement: Given a PV installation with rated power PPV,nom and inverter input power Pinv,nom = R  PPV,nom, where R is defined in (1). Select the optimal value of R that yields maximum 10 year cost savings compared to an inverter that is sized with R = 1. 2.2. Simulation methodology and algorithm As previously discussed in Section 1, the solution to the problem statement is a trade-off between cost savings in downsizing an inverter and earnings lost by the reduction of energy produced and/or from government incentives. The process for computing total cost Ctotal for a given inverter size is broken into five steps, illustrated by Fig. 2, which has been simulated in MATLAB. Referring to Fig. 2, time-stamped meteorological data (horizontal irradiance (W/m2) and ambient temperature (°K or °C) for an entire year is imported into the MATLAB program. In this paper, the resolution of the data is sampled each 1-min. From these data arrays, the output 1 STC: Standard Test Condition of PV modules: irradiance = 1000 W/ m2, cell temperature = 25 °C, air mass = 1.5.

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Fig. 2. General steps for computing total cost for a given inverter size Pinv,act.

power of the PV arrays is derived. This output PV power is used in Step 2 as the input power of the inverter. Relying upon user-specified inverter efficiency curves, protection delays and downsize coefficient (R), the inverter output at each instant of time is calculated. This can be integrated in Step 3 to calculate the annual energy yield of the inverter. Once the energy yield of the inverter is calculated, its capital gain can be computed in Step 4, provided that the performance based incentives (PBI) and local electricity rates (ER) are known. Similarly, the upfront costs of the installation can be calculated. For simplicity, we assume that the annual capital gain is the same each year. So the total (10 year) cost can be calculated in Step 5 by subtracting the capital gain from the upfront costs. The MATLAB program can be swept for different values of inverter downsize coefficient R to determine the highest ROI. Similarly, it is possible to compare cost savings for each R < 1 to the baseline condition of when the inverter rated power is set equal to the PV rated power, i.e. R = 1. In this algorithm, we are optimizing inverter size and keeping the PV array size fixed. Each Step in the algorithm is now explained in more detail, in the sub-sections below. 2.2.1. Preliminary step: meteorological data import The meteorological data sets are composed of irradiance and ambient temperature data which were both acquired by experimental 1-min sampling at each geographic location. The 1-min data are used because they preserve most

of the details of fast changing irradiance while maintaining a reasonable data size and computation load (Burger and Ru¨ther, 2006; Islam et al., 2003; Wide´n et al., 2010). They are time-stamped so that the ambient temperature at each irradiance condition is known. This is required to correlate the effects of temperature to output power of a PV panel. The irradiance and temperature data are taken from two sources: (1) the weather stations run by National Renewable Energy Laboratory’s (NREL) Measurement and Instrumentation Data Center (MIDC); and (2) the Solar Radiation Monitoring Laboratory of University of Oregon. A total of nine different geographic locations are inspected and compared in this paper. They are: Eugene, Oregon (University of Oregon – Solar Radiation Monitoring Laboratory, 2009); Las Vegas, Nevada (University of Nevada – Las Vegas, 2009); Lanai, Hawaii (Solar Resource & Meteorological Assessment Project (SOLRMAP) at La Ola Lanai, 2010); Oahu, Hawaii (Solar Resource & Meteorological Assessment Project (SOLRMAP) at Kalaeloa Oahu, 2011); Lowry Range, Colorado (Colorado State Land Board, 2011); Oak Ridge, Tennessee (Oak Ridge National Laboratory, 2011); Sacramento, California (Sacramento Municipal Utility District, 2011); Phoenix, Arizona (Solar Resource & Meteorological Assessment Project (SOLRMAP) Southwest Solar Technologies, 2011); Prewitt, New Mexico (Solar Resource & Meteorological Assessment Project (SOLRMAP) Escalante Tri-state, 2011). Each of the locations has distinct meteorological pattern in terms of irradiance and temperature. The

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effects will be discussed in detail in later sections. Our approach utilized horizontal irradiance data, and therefore the PV panels are assumed horizontal tilt. However, as long as the irradiance data is for the tilt angle of the panel, the methodology below remains valid. 2.2.2. PV panel power output (Step 1) The ideal PV panel output power is calculated by assuming the maximum power point tracking (MPPT) power is proportional to the irradiance level (Jiang et al., 2012; Huusari and Suntio, 2012): P PV ðtÞ ¼

GðtÞ P PV;nom GSTC

ð2Þ

where G(t) is the irradiance level at time t. For example, when the irradiance level is 600 W/m2, the output power of the 10 kW PV array is 6 kW. However, the increase of PV cells working temperature may lead to a drop of their energy conversion efficiency. Typically the PV panels are rated with temperature coefficients of short-circuit current (ISC) and open-circuit voltage (VOC) of the panel (King et al., 1997). These specifications are usually subtle on determining the actual efficiency drop when temperature rises. Research has proposed models to analyze the thermal effects on overall cell efficiency (Dubey et al., 2009; Tiwari et al., 2011). This paper adopts these models as a tool to evaluate the performance of the PV panels and its effects on the inverter sizing strategy. The solar irradiance incident to the panel will raise the solar cells’ working temperature above ambient temperature. In most cases, this working temperature is higher than the nominal reference temperature at which the efficiency of the panel is claimed. This leads to a drop of cell efficiency, virtually decreasing the power output of the PV panel than its nominal output, even with the same standard irradiance level of GSTC = 1000 W/m2. Therefore, the PV output power needs to be adjusted including the temperature degradation effects. The adjusted PV output power is then given by: P PV ðtÞ ¼

gc GðtÞ P PV;nom gref GSTC

ð3Þ

where gc, the solar cell’s temperature dependent efficiency, is given by: gc ¼ gref ½1  bref ðT c  T ref Þ þ cref log10 G

ð4Þ

where gref is the nominal efficiency at reference temperature Tref (usually Tref is 25 °C according to STC) and at irradiance of GSTC = 1000 W/m2. bref is the temperature coefficient of the PV module (typical value is 0.004, in K1 or °C1) (Notton et al., 2005; Skoplaki and Palyvos, 2009). For the ideal case that ignores temperature dependency, Tc = Tref and cref = 0. Then (3) reduces to (2). Two types of construction layouts of the PV panel are often considered for crystalline PV cell, opaque (glass to Tedlar) and semi-transparent (glass to glass), each of which has its own relationship between cell temperature and

ambient temperature given material and irradiance. We assume the PV panels have the opaque structure in our simulations. The relationship between cell temperature and ambient temperature is given by (Dubey et al., 2009; Tiwari et al., 2011; Andrews et al., 2012; Dekker et al., 2012; Siddiqui et al., 2012; Tsai and Tsai, 2012) Tc  Ta ¼

½sg fac bc þ ð1  bc ÞaT  gc bc gGðtÞ U LM

ð5Þ

where sg is the transmittance of the glass; bc is packing factor of the collector; ac is absorptivity of the collector; aT is transmittance of Tedlar; ULM is overall heat transfer coefficient, blacken surface to ambient (W/m2 K); gm is gc sg bc; gc is temperature dependent PV electrical efficiency; G(t) is the solar irradiance at the moment (W/m2). Generally, parameters sg, bc, ac, and aT are intrinsic parameters of the PV panel. Some of these constants are specified by the manufacturer; while others are constants based on the materials of construction and are not specified. ULM, the term for the overall heat transfer coefficient, is a combination of heat loss coefficients of Ub, Utc,a and UTc,f, which are heat transfer coefficients through the panel back, front glass cover, and flowing air, respectively. Since the wind effect is not included in the simulations in this paper, ULM is the addition of Ub and Utc,a, which is specified as of the order of 10 W/m2 K in (Dubey et al., 2009) for typical PV panels. Table 2 lists the typical values of the aforementioned parameters found in a typical PV panel (Dubey et al., 2009). These typical values are used as parameters when simulations are run with thermal effects. In general, the wind speed is negatively correlated to the cell temperature Tc, and positively correlated to cell efficiency gc. In other words, given the irradiance level and ambient temperature at a certain moment, the stronger the wind, the more efficient PV cells are. This may affect the inverter sizing strategy in a way that in a windy area, the temperature degradation of the PV panel’s output may be counteracted, and a larger inverter may be chosen compared to when there is no wind. However, this is left for future research, particularly since wind speeds are intermittent, their cooling effects depend on angle and azimuth of incident light, and this effect is still an emerging research field (Jones and Underwood, 2000; Notton et al., 2005; Tina and Abate, 2008; Andrews et al., 2012; Siddiqui et al., 2012; Tsai and Tsai, 2012). Finally, (3)–(5) can be used to compute the output power of the PV panels. First, collector temperature, Tc, for each time instant is calculated using (5) utilizing the time stamped data samples for irradiance G(tm) and ambient temperature Ta. With Tc known (3) and (4) are used to calculate the output power PPV (tm) for each 1-min irradiance data sample. 2.2.3. Inverter output power (Step 2) Define Pinv,gen as the actual inverter output power generated at any given time. This generated power is

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Table 1 Typical values of the parameters in thermal analysis for the PV panel. Parameter

Value

Parameter

Value

Parameter

Value

Parameter

Value

gref Tref bref

12% 25 °C 0.004 °C1

cref sg bc

0.12 0.95 0.83

ac aT Ub

0.9 0.95 3.3 W/m2 K

Utc,a

7.5 W/m2 K

Table 2 Benchmark condition of the analysis for Sections 3–5. PV system size (PPV,nom) Electricity rates (ER) Inverter unit price (qinv)b PV costb NRE cost (NRE)b Time period of simulation a b

10 kW $0.2/W $0.65/W $2.4/W $1.4/W 10 years

Inverter efficiency Inverter protection delay (TT) Performance-based incentives (PBI) Federal tax incentives (sfed) State rebate/incentives Cell temperature (TC) Thermal effects of PV panels

Constant 95% No delay Nonea 30% of system cost Nonea 25 °C Not considered

Incentives depend on state; the baseline case assumes no state incentives or rebates. Price data of PV panels, inverters, and NRE are sourced from (International Energy Agency, 2002; Solarbuzz, xxxx; International Energy Agency, 2007; Devlin, 2008; Aboudi, 2011).

influenced by: (i) the output power generated from the PV panels, PPV; (ii) the nominal power of the inverter Pinv,nom = R  PPV,nom, which represents the maximum input power the inverter can handle from the PV array. In the event of over-irradiance, (PPV > Pinv,nom), the inverter either shuts down (Pinv,gen = 0) or limits the output to its nominal power (Pinv,gen = Pinv,nom); (iii) inverter protection delay TT and (iv) inverter efficiency curve. Factors (iii) and (iv) are nonlinear and explained in detail in Section 5. Essentially, though, the goal of Step 2 is to compute the inverter efficiency that is defined in a special way as being ginv = Pinv,gen/PPV,nom. This represents the ratio of the instantaneous inverter generated power divided by the maximum possible DC power of the PV installation for the given sampled data instant of time. This ratio includes the nonlinearities associated with shut-downs or power limiting of the inverter. 2.2.4. Energy yield (Step 3) For each minute tm, Pinv,gen(tm) is assumed to be unchanged throughout the minute, and the energy generated in this minute is Pinv,gen(tm)/60 (W h). The annual energy yield of the whole system is calculated through summation of the time series of inverter output power Pinv,gen (t) over all the data points for a year: Eyield;annual ¼

1yr X P inv;gen ðtm Þ m¼1

60

ðW hÞ

ð6Þ

2.2.5. Cost analysis (Steps 4 and 5) The determination of optimum inverter size is based on the PV-system-level cost analysis, which includes component cost, incentives, irradiance data, etc. The total upfront cost Cupfront when purchasing and installing the PV system can be expressed as by:

C upfront ¼ C PV þ C inv þ NRE  V

ð7Þ

where CPV is the cost of purchasing PV panels, Cinv is the cost of the inverter, V is the combination of all incentives and rebates, and NRE is non-recurring expenses that includes the costs for necessary balance-of-system (BOS) and installation labors. The total upfront incentive V can be broken down to three different types: (i) incentives based on percentage of total system cost (CPV + Cinv + NRE); (ii) incentives based on PV panels’ nominal DC power output (PPV,nom) and (iii) incentives based on system AC power output, which is the nominal AC power output of the inverter (Pinv,act or R  PPV,nom). Then the V term in (7) can be written as: V ¼ T fed þ T state þ Rdc þ Rac ¼ C upfront ðsfed þ sstate Þ þ cdc P PV;nom þ cac RP PV;nom

ð8Þ

where sfed and sstate are the percentage rates granted by US federal government and state governments of type (i), and cdc and cac are rates for type (ii) and (iii) incentives in $/W. Specific geographic locations may have different incentive policies; some terms in (8) may be zero in that certain types of incentives are not granted for that location. Different types of incentives will be explained further in detail in Section 4. The “effectiveness” of an undersized inverter is analyzed by calculating “Total savings” of a certain sizing scheme compared to the baseline condition. In the baseline condition, the system size, i.e. the nominal output power of the PV panels under STC, is 10 kW, which is typical for residential or small businesses in the USA. The inverter is chosen based on Pinv,nom = PPV,nom = 10 kW, i.e. GTh = GSTC = 1000 W/m2 or R = 1. With the unit cost of $0.65/ W (Solarbuzz, 2011), the baseline inverter cost would be $6500.

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The cost analysis considers inverter cost, inverter efficiency, and cost for energy loss, due to excess irradiance, at given performance-based incentive PBI level. For any inverter size Pinv,nom = Pinv,act = R  PPV,nom, the total savings Stotal can be expressed as S total ¼ C total;10kW  C total;act

ð9Þ

where Ctotal,10kW is the total cost (equipment prices less incentives and earnings from energy yield Iyield,act, as shown by (10) and (11)) when using a 10 kW inverter; Ctotal,act is the total cost when using an undersized inverter of which the nominal power is Pinv,act. The Total Savings in (9) will never exceed the total savings from upfront costs. That is, for an inverter with downsize coefficient R < 1, the cost savings for the inverter is estimated at $0.65/W  (1  R)  10 kW, assuming the baseline cost of the inverter is $0.65/W for the 10 kW rated PV system. Notice that as R decreases, the upfront cost savings increase linearly. This, though, is counteracted by the earnings loss from the downsized inverter. In particular, the annual earnings from power generation are calculated by assuming that the inverters work under downscaled protection mode limiting their maximum power:

1yr X P inv;gen ðtm Þ

60  1000 m¼1

 ðER þ PBIÞðkW hÞ

ð10Þ

ER is the electricity rate at the geographic location, and PBI denotes the Performance-based incentives. For example typical values for the US cost of electricity may be ER = $0.16/kW h and PBI = $0.20/kW h. Essentially, for these typical values, customer is being paid $0.36/kW h for each kW h produced by the inverter output. So, the cumulative amount of energy lost by undersizing the inverter must be less than the savings gained by lower cost of the inverter for there to be cost savings associated with the downsizing. These issues will be discussed in Section 4. Then the total energy earnings in the assumed lifetime of the inverter are given as: I yield;act ¼ I yield;annual  Inverter lifetime

ð11Þ

The inverter lifetime in this paper is assumed to be 10 years, which is the normal warranty period for a standard inverter. Therefore, the total cost with given inverter size Pinv,act is given by C total;act ¼ C upfront;act  I yield;act

S total ¼ DC upfront  DI yield ¼ ðC upfront;10kW  C upfront;act Þ  ðI yield;10kW  I yield;act Þ

ð13Þ

The first term in (13), DCupfront, is the upfront cost savings when choosing a smaller inverter; while the second term, DIyield, is the cost of additional energy lost with a small inverter that cancels out part of the upfront cost savings throughout the inverter’s service life. In particular, when constant K = K(PPV,nom, sfederal, sstate, qinv, cac) is determined, DCupfront is a negative-sloped linear function of R with zero crossing when R = 1 DC upfront ¼ ð1  RÞP PV;nom ½ð1  sfederal  sstate Þqinv þ cac  ¼ K  P PV;nom ð1  RÞ

ð14Þ

where qinv is the unit cost of inverter in $/W. On the other hand, the DIyield term is a non-linear, distribution dependent function of data sets of ambient temperature Ta (t), and irradiance G(t). 2.3. Baseline conditions

I yield;annual ¼ Eyield;annual  ðER þ PBIÞ ¼

difference of the energy yield earnings between two inverter sizing schemes.

ð12Þ

where Cupfront,act is the upfront cost calculated from (7). By computing Ctotal,act for all values of R, 0 < R < 1, it is possible to find the optimal size of the inverter that gives the maximum ROI. Inserting (12) into (9), the total savings can be written alternatively as the difference of upfront costs minus the

A set of baseline conditions are listed in Table 2. In Sections 3–5, a single parameter is varied, while all other parameters are fixed to default values in Table 2. For example, in Section 4.2(a), effects of performance-based incentives are being discussed, then the simulation calculates the optimum inverter size and total savings for values of PBI from $0/kW h to $0.6/kW h while leaving other conditions as Table 1 stated. Inverter price plays a determinant part in deciding the optimum inverter size. It is a unit cost per watt of the inverter nominal power (qinv). In (14), qinv affects the parameter K and changes the slop of the linear DCupfront curve. In this paper, the inverter price is assumed to be $0.65/W in the simulations and analysis (Solarbuzz, 2011). Note that this is a simplified assumption for proof of concepts. In reality, the inverter price is a more complex parameter to be considered: (a) Unit price is not constant at all power levels – even for the products from similar manufacturer in the same location, the unit price of inverter changes with the power level. As an example, the unit price for a 10 kW Fronius IG-Plus 10.0–1 inverter is listed around $0.63/W, while the 5 kW Fronius IG-Plus 5.0–1, which is from the same product line and manufacturer, costs $0.78/W. The effect of this unleveled pricing may imply that larger inverters may possibly be justified for they cost less per watt. However, it should be noted that the change of unit price among different power levels is neither linear nor continuous.

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(b) Unit price is not constant at the same power level – the prices of specific inverters may be higher if: they are made by different manufacturers, have different reliability, or have more functions, even compared to other inverters of the same power level. This is a case-dependent fluctuation upon a baseline average price. For simplicity, this paper assumes that the inverter’s unit price remains constant of $0.65/W within the power range of discussions, as the latest survey suggests (Solarbuzz, 2011). After all the factors are isolated and analyzed, in Section 6, all parameters are allowed to vary together in order to obtain the proper optimum inverter size. 3. Meteorological factors This section isolates effects of meteorological factors, such as irradiance patterns and temperature while holding all other factors constant. Two baseline test cases are utilized to illustrate the influence from Eugene, Oregon (USA) and from Las Vegas, Nevada (USA). These two cases were selected because of their meteorological contrasts. Eugene is temperate and cloudy, while Las Vegas is hot and sunny. 3.1. Irradiance patterns The sizing strategy of PV inverters will be influenced by the irradiance patterns of specific locations. Fig. 3 shows the experimentally measured distribution histogram of horizontal total irradiance throughout year 2009 for two different locations, Eugene, OR (University of Oregon – Solar Radiation Monitoring Laboratory, 2009) and Las

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Vegas, NV (University of Nevada – Las Vegas, 2009). Each occurrence represents a minute time interval of data sampled at the irradiance level. The horizontal axis represents the irradiance level; and each bar in the histogram describes the number of minutes at which the irradiance falls into a 50 W/m2 span. Hence, Fig. 3 depicts the distribution of the minute data at different irradiance levels, i.e., the irradiance patterns of both geographic locations. Further, the average irradiance throughout the year is 324 W/m2 in Eugene, and 463 W/m2 in Las Vegas. The effects of irradiance patterns on the PV inverter sizing strategies are extracted by running the simulation with the parameters of Table 1 on the irradiance data G(t) for Eugene and Las Vegas. Fig. 4 shows the total savings curves with different inverter sizes. All factors (temperature, economic incentives) are fixed and only the downsize coefficient, R, is varied. In Fig. 4, the y-axis is cost savings compared to a 10 kW inverter and the x-axis represents downsize coefficient, R, in percentage. As R varies, the savings curves have an approximately parabolic shape with maximum savings at an optimal R. There will always be a $0 savings at R = 1 (Pinv,nom = PPV,nom = 10 kW). As R decreases, the DCupfront dominates the Iyield term in (13). But as the inverter is downsized more (R further decreases), the DIyield term eventually grows and becomes greater than DCupfront, which leads to the second $0 savings x-axis crossing. The results show that the optimum inverter size would be substantially affected by irradiance patterns. In Eugene, the irradiance distribution is weighted heavily to instances of low irradiance, representing a low irradiance climate in this region; while in Las Vegas, the distribution is weighted more to instances of higher irradiance around 800–1000 W/m2. The results of cost analysis turn out that the optimum point inverter size is different

Fig. 3. Distribution of total horizontal irradiance data (data for Eugene, OR, and Las Vegas, NV, 2009).

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clearly, R = 0.7 is not viable for every geographic location. That is, if a 7 kW inverter were used for the 10 kW PV system in Las Vegas, the cost for energy lost would overwhelm the upfront savings on inverter cost and cause $1700 of financial loss due to the use of an undersized inverter. To summarize this effect, in cloudy areas (Eugene) that have the irradiance distributed heavily around the low irradiance region, the strategy of undersizing the PV inverters can be considered. The undersizing strategy may not be necessary in sunny areas. However, this calculation does not take into account the temperature effects of the PV panels, which will be discussed below.

3.2. Temperature

Fig. 4. Optimum inverter size for a 10 kW system at Eugene, OR and Las Vegas, NV. Based on year 2009 data. Account for 10 year period (The bottom figure is the same figure as the top one with different axes scales).

in Las Vegas than in Eugene (optimum R  0.84 for Eugene, versus R  0.94 for Las Vegas), even for the same PV system with the same rate of incentives. The results can be better explained using the cost analysis from Section 2.2.5. For illustration purpose only, assume R = 0.7. From Table 2 baseline conditions, sstate = 0%, cac = 0, and qinv = $0.65/W, therefore, for both Eugene and Las Vegas, the parameter K in linear cost savings term DCupfront equals to (100%–30%-0%)  $0.65/W + 0 = $0.455/W. For Eugene, the cost for energy loss from the simulation is $633 for 10 years, resulting in total savings of $95 if the inverter nominal power is 7 kW (R = 0.7). For Las Vegas, the cost for energy loss DIyield would be $2430, resulting in total savings of $-1702. The negative total savings indicates that the cost for energy loss due to a small inverter exceeds the upfront cost savings. At Las Vegas, it would be less economical to choose a 7 kW inverter than to use a 10 kW inverter for the 10 kW PV system. Fig. 4 also shows that the total savings would also be negative for Eugene when R < 0.68. Furthermore, conventional choice of R = 0.7 would lose money in Las Vegas –

The efficiencies of PV panels specified by manufacturers are tested under a certain set of standardized conditions. In most cases, the Standard Test Condition (STC) is used to show the reference performance of a certain PV panel. STC is defined as (1) Solar irradiance: 1000 W/m2; (2) Cell temperature: 25 °C or 77 °F and (3) Air Mass (AM): 1.5. The nominal DC output power and efficiency from the manufacturer’s datasheet are measured under this Standard Test Condition. While the actual power output is proportional to the irradiance level, it is inversely proportional to the cell temperature. Since the cell temperature is also correlated to irradiance, the DC output of the PV panel is then changed nonlinearly with irradiance. In other words, higher irradiance may be helpful in energizing the PV cell to produce more power, but it also increases cell temperature, which reduces cell efficiency. This insight is particularly important when considering the temperature effects in different geographic locations that have distinct irradiance and ambient temperatures. The relationship between temperature and cell efficiency was discussed in Section 2.2.2. Fig. 5 shows the differences this thermal effect of the PV panels might make in Eugene and Las Vegas. In both locations, the optimum inverter sizes are smaller, comparing to Fig. 3. The difference in optimum inverter sizes are R = 0.66 for Eugene and R = 0.68 Las Vegas, compared to R = 0.84 and R = 0.94 in Fig. 3, respectively. The corresponding maximum savings are $1900 versus $520 in Eugene, and $1850 versus $120 in Las Vegas. The results indicate that for a geographic location that is sunny and hot, such as Las Vegas, the additional irradiance level may be counteracted by the degradation of the PV panels due to a higher cell temperature. In both locations, including the temperature effects of solar cell efficiency leads to a smaller inverter size recommendation. In particular, in Las Vegas, the temperature effects render more substantial change in both optimum inverter size (from 94% to 68%) and total savings (from $120 to $1850) than in Eugene. The left column of Fig. 5 shows histograms of ambient temperature in Eugene and Las Vegas. It shows that in Las Vegas, in nearly half of the time, the tempera-

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Fig. 5. The effects of ambient temperature on optimum inverter in Eugene and Las Vegas, considering thermal degradation effects of the PV panels.

ture is above 25 °C, whereas in Eugene, the ambient temperature is below 25 °C for most of time. Despite the fact that the average irradiance in Las Vegas is more than 25% greater than Eugene, the inverter size recommendations are similar (68% compared to 66% downsize). This gives different conclusions than (Burger and Ru¨ther, 2006; Keller and Affolter, 1995; van der Borg and Burgers, 2003). Further, the results indicate that the maximum savings substantially increase when temperature effects are considered. 4. Economic factors This section isolates the economic factors and discusses their effects separately in each subsection. Economic factors include incentives, electricity rates, and inverter price. In each subsection, we vary one parameter while fixing others to the baseline condition. 4.1. Government incentives In the USA, PV incentives, granted by both the Federal and State governments, might have considerable effects on inverter sizing. There are generally three categories of PV incentives: (a) tax credits; (b) system purchase rebates and (c) performance-based incentives (PBI) (US Department of Energy, 2011; Yamamoto, 2012). PV incentive programs in the USA are summarized in Table 3, which

includes different state incentives that depend on which state the PV system is installed. Tax credits are classified as income tax credits, sales tax exemptions, or property tax exemptions. This paper reviews only the effect of income tax credits, which is most common. Note that although state income tax credit varies by state, the United States allows 30% federal tax credit for the expenditure of a PV system with no upper limit (US Department of Energy, 2011). In most states, the income tax incentives are based on total system cost, including PV panels, inverters, balance of system, and labor. Inverter sizing strategy might have to consider these policies in a specific location. Larger and more expensive inverters permit more energy yield and sometimes more tax credits. Such effect may be less important when the system size is large and the incentive cap is met. Most of the state rebates are based on rated DC power of PV arrays, which implies that with the same installed PV system size, a smaller inverter can reduce overall system expenses and permit the same amount of state rebates. The type (a) tax credits and type (b) rebates are mostly based on system installation costs and rated power, so these incentives only affect the upfront cost savings DCupfront of (14). Moreover, only the incentive types that are related to inverter size or cost have effects on the cost savings. In some cases, the design of incentive policy may have significant influence of inverter sizing strategy. Take Las Vegas for example. As the analysis results in Section

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Table 3 USA PV incentive programs in 2011 (US Department of Energy, 2011), depending on state. Incentive type

Number of states

Range of amount

Range of caps

Tax credits State rebates Performance-based incentives (PBI) USA Federal tax credit

26 33 33

15–50% of system cost or PV wattage $0.75–$3/W nominal DC power $0.0025–$1.5/kWhr 30% of system cost

$1000–$25,000 $2000–$500,000 N/A No limit

6.2 shows, with the government rebate calculated by the PV system’s AC output power, (cac = $1.7/WAC), although undersizing the inverter by 1 W saves inverter’s capital cost by $0.65, the customer is losing government rebate by $1.7. In total, that is a $1.05-loss per reduced watt of inverter nominal power. Therefore, the optimum inverter size for Las Vegas would be R = 1, regardless of other factors, and the total savings would be zero. 4.2. Energy cost In this paper, energy cost is the price the customer has to pay for the energy losses due to using a small inverter. The energy loss is priced with two parameters: (1) the performance-based incentives (PBI) rewarded by the government or the utility companies based on actual electric energy generation of the whole PV system and (2) the electricity rates the customer has to pay to buy the energy lost due to reduced inverter size. Both parameters quantize the cost of the energy wasted when a smaller inverter is chosen and the irradiance level is high. The effects of these two parameters are explained below. 4.2.1. Performance-based incentives In some states a portion of incentives are granted based on performance of the system, where a rebate is given for each kW h generated by the inverter ($/kW h). Generally this type of incentive is high compared to local electricity rates. It might notably affect the overall optimum inverter size as shown in Table 4 and Fig. 6. Larger inverters might be chosen to produce more energy in order maximize PBI instead of choosing a smaller inverter to save on initial costs. The analysis for PBI effects assume all parameters except for PBI are held constant with values from Table 2. Table 4 and Fig. 6 summarize cost savings with PV incentives for Eugene (left) and Las Vegas (right). The optimum inverter size increases as PBI increases. In fact, inverter selection optimized for no PBI might lose money when Table 4 Optimum inverter size and total savings with different Performance-Based Incentives for a 10 kW PV installation in Eugene, OR in 10 years. PBI ($/kW h)

Optimum inverter size (R)

Maximum savings ($)

0 0.1 0.3 0.6

0.70 0.77 0.83 0.86

1900 1500 1200 1000

PBI incentive is added. For example, the 94% downsize inverter recommendation for Las Vegas will actually lose money when PBI is greater than $0.3/kW h (once again, though, this does not consider temperature effects on the output power). In fact, the optimum inverter size would be larger than R = 1 when PBI is greater then $0.3/kW h. So a system owner in Las Vegas was possible to take advantage of the high PBI and oversized his/her inverter in order to extract more energy. 4.2.2. Electricity rates The effects of local electricity rates are similar to that of the PBI, in that when the inverter protects itself by limiting the power output, there is energy loss from the PV input to inverter output. Losing this part of energy costs the user not only the potential gain of PBI, but also the electricity rate if the user needs to draw such amount of energy from the grid to compensate usage needs. The analysis for PBI effects assume all parameters except for ER are held constant with values from Table 2. Figs. 7 and 8 show the trends of maximum savings and optimum inverter size with respect to the change of electricity rates. The PBI in both figures are zero. In both figures, the trends of the maximum savings and optimum size are identical to the PBI cases discussed last section: as ER increases, the optimum inverter size increases. 5. Inverter intrinsic parameters In this section, we discuss how inverter intrinsic parameters may affect the inverter sizing strategy. Inverter overload protection schemes, inverter protection delay, and inverter efficiency curve are being discussed. 5.1. Inverter protection scheme 5.1.1. Measures of inverter overload protection To protect the inverter from overload, there are two major schemes: (1) fold-back: the inverter converts only the rated power by forcing the PV system to work at a non-maximum-power point of the PV system; and (2) shut-down: inverter shuts down when overloading occurs and turns back on when load is within limit. Fig. 9 illustrates the need for inverter overload protection. In foldback protection, the lost power is illustrated in the grey shaded areas of the curve. In this case, the inverter clamps to its rated power, even though the PV panel may wish to produce higher than Pinv,nom. On the other hand, for shut-

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Fig. 6. Optimum inverter size for a 10 kW system at Eugene, OR (left column) and Las Vegas, NV (right column), based on year 2009 data. Savings are for 10 year period.

Fig. 7. Optimum inverter size and total savings with different electricity rates – data of Eugene, without considering thermal effects and assumed constant inverter efficiency (95%).

down inverter protection, the generated power in the inverter becomes zero in any time instant the curve in Fig. 9 is shaded grey. Therefore, the fold-back scheme outperforms the shut-down scheme in terms of total savings, because in

fold-back protection mode, inverters are converting rated power rather than nothing during overload conditions, i.e. over-irradiance conditions in this paper, reducing energy lost.

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Fig. 8. Optimum inverter size and total savings with different electricity rates (ER) – data of Las Vegas, without considering thermal effects and assumed constant inverter efficiency (95%).

Fig. 9. Normalized output power with protection delay.

5.1.2. Inverter protection delay An inverter’s protection delay time is commonly designed by the inverter industry. During this delay time, some energy loss due to high irradiance can be recycled. This section discusses the effect of TT in inverter sizing. TT in this paper is defined as the protection time delay when input power of inverter is 150% of its nominal power. For example, if TT = 2 min and R = 70%, when the irradiance is around 1050 W/m2, the inverter would ideally be able to handle the over-irradiance without entering protection mode for 2 min. Irradiance higher or lower than 1.5GTh decreases or increases the actual time delay proportionally. This, of course, leads to the two different efficiency curves of an inverter shown in Fig. 10: (1) the inverter efficiency for the brief time period less than TT before the inverter protection activates and (2) the inverter efficiency when over-irradiance lasts longer than TT and power scale-down or inverter shut-down has been executed.

Past analysis had assumed the inverter immediately enters protection mode and that every watt exceeding GTh was wasted since the inverter would limit the power output of the PV array at GTh (Burger and Ru¨ther, 2006; Keller and Affolter, 1995). In this paper, we consider that the inverter can operate in over-irradiance for a brief time period, as described above, at reduce efficiency as in Fig. 10. This means the inverter does not waste energy due to entering protection mode in for short over-irradiance events. Fig. 9 shows the comparison between cases with and without the protection time delay TT = (t2  t1) mins. Dark areas depict the extra power rescued by this time delay, while the lightly shaded areas represent the wasted power. This is a zoomed graph and the time span is 250 min. From t1 to t2, although the irradiance is beyond nominal level, the inverter can still output the extra power for TT minutes. The inverter downscales the PV output

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Fig. 10. Typical inverter efficiency curve as a function of load (irradiance) (Go Solar California, 2011).

power to match the nominal inverter power from t2 to t3, during which the extra irradiance is wasted. Similar procedure happens during other over-irradiance events. Spikes that last less than TT minutes are not interrupted by the inverter.

Fig. 11 shows the total savings with different TT value (TT = 0, 2, and 5 min), while, all other parameters are fixed to the baseline case. In 10 years’ life span, up to $400 more total savings are realized with TT = 5 min compared to TT = 0 min in Lanai, HI, and $100 more in Las Vegas.

Fig. 11. Total inverter cost savings vs inverter size as % of system size for different TT of Las Vegas, NV (left column) and Lanai, HI (right column) (No performance-based incentives).

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Fig. 12. Efficiency curves of different simulation cases.

The amount of improvement is case and location dependent: Las Vegas is a sunny area, where the irradiance is at a comparatively constant high level during daytime. The improvement made by a protection delay would be less substantial. On the other hand, the weather in Lanai is not as sunny as Las Vegas. A simple standard deviation calculation of the irradiance data from these two geographic locations shows that the irradiance in Lanai is changing more substantially over time than in Las Vegas. (rLanai = 361.3 W/m2 and rVegas = 309.1 W/m2). The reason is, in sunny areas, over-irradiance events usually last for a longer time without fluctuation. The result is the inverter does not have time to cool down and would be in protection mode in most over-irradiance minutes. In cloudy areas, many over-irradiance events only last for a few minutes thereafter the irradiance drops below GTh for a while. This kind of fluctuation permits the inverter to cool down; so another protection delay TT can be realized in the next over-irradiance event.

5.1.3. Inverter cooling system The facts revealed (b) imply that it might be desirable to extend the inverter’s protection delay time at a reasonable cost. Development of inverters’ cooling schemes would be the direction of achieving this goal. Usually a fan is not a preferred option due to its reliability issues. However, a fan can be added to a convection cooling inverter to provide additional resistivity to overloading. A proposed control scheme might be turning on the fan only when over-irradiance event is observed and for TT minutes at maximum. For example, in Eugene, OR, over-irradiance events below 5 min count only 10575 min, which is translated into fan duty ratio of less than 2% in a year. Typically the cooling fan’s lifetime is considered to be 3 years with 100% duty ratio. With such a low duty ratio in this application (2%), the fan should serve the 10-year expected lifespan of the inverter. In this context, the reliability would not be compromised while the cooling is improved under overload conditions. If the

Fig. 13. The effect of the inverter efficiency curves.

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Table 5 Parameters for Eugene, OR simulations. PV system size (PPV,nom) Electricity rates Inverter cost Thermal effects of PV panels State rebate/incentives

10 kW $0.085/kW h $0.65/W Considered $1.75/W DC up to $20,000

fan does fail, the inverter controller can reset the protection scheme to TT = 0 with no harm to the system. 5.2. Inverter efficiency curve Inverter efficiency is another factor that affects the optimum size. Different sized inverters under the same irradiance would have different efficiency. Inverter efficiency ginv usually varies with inverter load (Notton et al., 2010). Fig. 10 presents the typical inverter efficiency (average efficiency of several commercial PV inverters, as reported in (International Energy Agency, 2002; Ginn et al., 1997; Go Solar California, 2011; Vignola et al., 2008). Also, its efficiency during TT is assumed to be the same as the 100%-load efficiency. Generally, there are three operating regions in a typical inverter efficiency curve: 5.2.1. No load or load is too light (G < Gstart) When there is no load or the load is very light, or the input voltage is too low, the inverter is not turned on. This is the case when the irradiance is too low or at night. In this region, the inverter efficiency would be zero. Typically, inverters require 10% of nominal power to turn on and convert energy. 5.2.2. Normal operation (Gstart 6 G 6 GTh) When the load is 10–100% of its nominal power, the inverter is operating in the normal operation region, where most models maintain near constant efficiencies.

Inverter efficiency Inverter protection delay (TT) Performance-based incentives (PBI) Federal tax incentives by utility, plus $3/W DC tax credit, up to $6000

SMA SB7000US No delay $0.468/kW h 30% Of total system cost

When irradiance is below GTh, i.e., the output DC power from PV panels does not exceed the nominal power of the inverter (G 6 GTh, PPV 6 Pinv,nom), the inverter’s conversion efficiency can be written as: P AC;inv ginv ¼ ð15Þ P PV where PAC,inv is output AC power of the inverter and PPV is the PV power output. 5.2.3. Overload (G > GTh) When the irradiance is too high, the input DC power of the inverter would be too much, pushing the inverter to operate in the overload region. In this region, the inverter may enter the protection mode, in which it shuts down completely (zero efficiency) or converts only its nominal power with full-load efficiency and discards excessive power. While in the case that G > GTh and t > TT, the inverter’s efficiency curve, shown in Fig. 11, decreases after the irradiance G is above GTh (700 W/m2 in the case when R = 0.7). Then we can define maximum rated efficiency of the inverter as: grated ¼

P AC;nom P inv;nom

ð16Þ

where PAC,nom is the generated AC output power of the inverter when its DC input is Pinv,nom = R  PPV,nom. Then for over-irradiance event PPV = Pinv,nom + DPdc, where PPV is given by (3). Then the efficiency can be written as

Fig. 14. Optimum inverter size for Eugene, OR, under realistic conditions.

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Table 6 Parameters for Las Vegas, NV simulations. PV system size (PPV,nom) Electricity rates Inverter cost Thermal effects of PV panels State rebate/ incentives

gover-rated 

10 kW

Inverter efficiency

$0.125/kW h

Inverter protection delay (TT) $0.65/W Performance-based incentives (PBI) Considered Federal tax incentives $1.7/W AC, up to $85,000

P ac;nom grated  P inv;nom þ DP dc 1 þ P DP dc inv;nom

SMA SB7000US No delay $0.098/kW h 30% Of total system cost

ð17Þ

DPdc denotes the extra power that the PV array could produce if there were no power limits on the inverter, always assumed to operate at its maximum power point. This occurs at conditions higher than GTh. As the irradiance goes higher, DPdc increases, causing the efficiency under overirradiance (gover-rated) to decrease, as shown in Fig. 10. If the protection delay is considered, during this delay (t 6 TT), the inverter would still convert all the fed-in power with the efficiency that is assumed to be the same as the full-load power. Fig. 12 shows inverter efficiency curves for three different inverters: (1) ideal 95% constant efficiency, with no minimum turn-on power requirement; (2) SMA SB7000US PV inverter and (3) Sysgration SG-4000 PV inverter (Go Solar California, 2011). In Fig. 12, all efficiency curves are shown with the fold-back inverter overload protection which applies Eq. (17) when the load is above 100% of inverter nominal power. When the inverter is overloaded, the efficiency is assumed to be the same as when it is 100% loaded.

Fig. 13 shows the effect of taking into account the inverters efficiency curve. Fig. 13 shows three different cases: constant inverter efficiency of 95% with no minimum turn-on power, efficiency curve of the SMA’s model SB7000US, and efficiency curve of another product, Sysgration’s model SG-4000 (Go Solar California, 2011). Both efficiency curves were tested by testing laboratories and enlisted by the California Energy Commission with a uniform protocol (Go Solar California, 2011) and are typical efficiency curves for grid-connected inverters. The optimum inverter sizes in the three cases are 84%, 81% and 80%; and the corresponding total savings are $523, $804, and $893, respectively. The effect of the inverter’s efficiency originates mostly from the light load conditions. In other words, one of the major differences of the three efficiency curves shown in Fig. 13 is when the load ratio is under 50% of the inverter’s nominal power. The difference is even more drastic when the load is extremely light, which is 0–20%. When the input power is under 10%, most inverters do not turn on at all, resulting in zero efficiency. Also, in light load (10–20%) when the inverter is on, the efficiency is low. The light-load performance of the inverter could occur in PV applications near sunrise, sunset, and during cloudy conditions. If a smaller inverter is used, some of these light-load events may be eliminated, causing the inverter to operate more often near its nominal power efficiency. This is more justified in some specific geographic locations where there are more low-irradiance events and less high-irradiance events. 6. Summary examples for different geographic locations: allowing all parameters to vary In previous sections, we isolated and discussed each of the factors that may influence the optimum inverter size, by fixing other factors to the baseline condition. In reality,

Fig. 15. Optimum inverter size for Las Vegas, NV, under realistic conditions.

S. Chen et al. / Solar Energy 87 (2013) 96–116 Table 7 Parameters for Lanai, HI simulations. PV system size (PPV,nom) Electricity rates Inverter cost Thermal effects of PV panels State rebate/ incentives

10 kW $0.255/kW h

Inverter efficiency

Inverter protection delay (TT) $0.65/W Performance-based incentives (PBI) Considered Federal tax incentives 35% Tax credit of system total cost, up

SMA SB7000US No delay $0.218/kW h 30% Of total system cost to $500,000

for specific geographic locations, most of the parameters are known, such as historical meteorological pattern, electricity rates, and incentive/rebate policy. In this section, three examples are analyzed with realistic parameters for the specific geographic locations. Initially, three locations are explained in detail: (1) Eugene, OR, a cloudy city which has moderate irradiance and cool weather. (2) Las Vegas, NV, a sunny city which has high availability of solar irradiance but is very hot at the mean time. (3) Lanai, HI, a place at the middle of the Pacific Ocean that is sunny for most of the time but suffers from fast changing irradiance due to the frequent passage of scattered clouds. Later, 9 geographic locations in the United States summarized to prove the generalization of the proposed method. 6.1. Eugene, Oregon (OR) Table 5 summarizes the parameters for Eugene. These parameters are realistic local data of Eugene, taken from sources (Solarbuzz, 2011; US Department of Energy, 2011; US Energy Information Administration, 2011). In Eugene, both the utility company and the state of Oregon grant rebates based on the DC power output of the PV

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system. The PV customer would receive $1.75/W DC up to $20,000 from utility company PG&E, and $3/W DC from state government up to $6000. Since in this example, the size of PV system is fixed at 10 kW. The optimum inverter size is 7.1 kW. One might estimate a $6/W cost for the entire PV system (Solarbuzz, 2011), which leads to total cost of the PV system to be $42,600 before rebate. The total state rebate and tax incentive for this installation would be $17,500 + $6,000 = $23,500. Additionally, the United States federal government grants tax incentives at 30% of total system cost, which is $12,800 in this case. Therefore, the initial cost of the 10 kW PV system is $6300. Fig. 14 shows the optimum inverter size for Eugene, OR. The model parameters are listed in Table 5. Compared to Fig. 2 where baseline parameters are used, Fig. 14 implies that by choosing the optimum inverter size (71% of PV rated system size), total savings of $2300 is permitted over 10 years. This represents >30% of initial investment after incentives. 6.2. Las Vegas, Nevada (NV) Table 6 summarizes the realistic parameters for Las Vegas, from sources (Solarbuzz, 2011; US Department of Energy, 2011; US Energy Information Administration, 2011). The State of Nevada’s rebate program is based on the AC wattage output of the whole PV system, which means the amount of rebate is calculated with the size of inverter. This rebate program encourages PV customers to choose large inverters to match the DC output from the PV panels, with an overwhelming $1.7 W compared to inverter cost of $0.65/W. However, as the simulations in Fig. 15 show, the inverter size should not be larger than PV panels’ DC output capacity. As a result, in Las Vegas, the optimum inverter size is 100% of the PV rated system size with zero total savings. One can receive $1.7/

Fig. 16. Optimum inverter size for Lanai, HI, under realistic conditions.

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Table 8 Parameters and results for nine geographic locations.

Electricity rates ($/kW h) State rebate/incentives* PBI ($/kWhr) Optimum inverter size (% of system size) Maximum savings ($) PV system size (PPV_nom) PV panel thermal effects US Federal tax incentives Inverter efficiency curve Inverter cost Inverter protection delay

Eugene

Las Vegas

Lanai

Lowry range

Oak ridge

Sacramento

Oahu

Phoenix

Prewitt

0.085 $23.5ka max 0.468 71%

0.125 $1.7/WAC 0.098 100%

0.255 35%b 0.218 76%

0.122 $1/WDC 0.0945 72%

0.089 N/A 0.27 69%

0.158 $0.25/WAC $2.25/WDC 0.15 71%

0.255 35%b 0.218 74%

0.095 $1.5/WDC 10%b N/A 67%

0.085 6%b 0.127 74%

2337 0 10 kW Considered 30% of total system cost SMA SB7000US $0.65/W No delay (TT = 0)

1005

1456

2181

1004

1042

1213

1149

* State rebates and incentives have various forms: (1) $ per nominal AC watt, (2) $ per nominal DC watt, (3) % of total system cost, and their combinations. a State rebates for Eugene are $1.75/WDC up to $20,000 plus $3/WDC of state tax credit up to $6,000. b Percent symbol means rebates are calculated in percentage of total system cost.

W  10,000 W = $17,000 of state rebates with this inverter size selection. Therefore, state rebate policy can be used to counteract factors that may lead to inverter downsizing, such as the high operating temperatures in Las Vegas. Previously, Fig. 5 indicated that the high ambient temperatures in Las Vegas are favorable to smaller wattage inverters. Because state rebates are based on AC power and not DC power, this trend is counteracted. 6.3. Lanai, Hawaii (HI) Using the parameters in Table 7, Fig. 16 shows the analysis result for Lanai, HI. Lanai has the highest electricity rates among the three locations, at $0.255/kW h. The State of Hawaii grants 35% of tax credit based on the total cost of the system, including PV panel cost, inverter cost, and other balance-of-system costs. Lanai has higher irradiance than Eugene; its rebate program also returns part of the inverter investment. These cause larger optimum inverter size (76%) than in Eugene (71%), and permitting less total savings ($1000 versus $2300 in Eugene). 6.4. Other geographic locations Using the same methodology, data from additional geographic locations are analyzed and the results are presented in Table 8. A total of nine locations scattered around the United States are listed. For all the cases, the PV system size is assumed to be 10 kW; inverter cost to be $0.65/W; inverter efficiency curve to follow the typical model of SMA SB7000US; inverter overload protection scheme to be fold-back with no time delay; and the US federal tax incentive to be 30% of total system cost with no upper limit. 7. Conclusions/discussions There are multiple factors that influence the inverter sizing strategy. Three types of factors are discussed: meteoro-

logical, economic, and inverter characteristics. This paper isolates and explains each factor on inverter Return-OnInvestment. The factors, however, are interdependent on each other, as the examples in Section 6 illustrate. Additionally, the inverter overload protection delay, a characteristic of inverter that is often ignored in PV inverter sizing, is also analyzed. The protection delay allows recycling some of the energy losses that may be caused by the use of smaller inverters. The summary examples in Section 6 reveal some interesting results with respect to the incentive policies. From the policy makers’ point of view, incentives are the measures to encourage the expansion of solar PV energy output. This means that the government wants as much power being generated as possible from the PV systems. The goals of upfront rebates are to reduce the PV customers’ initial cost for the system and to expand the viability for more residents and businesses, whereas the performance-based incentives are designed to reward the power generating performance of the PV systems. Las Vegas grants the upfront rebate based on inverter size (AC output), permitting the government to get the most of power from the PV systems. In the meantime, with the same rebate per watt, the Nevada government’s expense of paying the rebate is kept at the same level as, or even less than the Oregon government, who grants upfront rebate based on PV array size (DC output). The Nevada rebate will lead to an inverter optimum ROI when a full size inverter is used. The Oregon rebate encourages a downsized inverter, which will of course lead to lower energy output. Therefore, governments may be better served by implementing AC power rebates. On the other hand, from the customers’ point of view, the first priority is often personal financial benefit and cost reduction. Several types of incentives, either upfront or performance-based, help the customers to reduce system cost and shorten the systems’ pay-back periods. To achieve the financial optima, PV customers may take advantage of

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