Evaluating the reliability of distributed photovoltaic energy system and storage against household blackout

0 Introduction

Large-scale grid outages are caused by several events,including extreme natural disasters,power system failures,vandalism,maloperations,and even terrorist attacks [1-5].According to the report released by the North American Electric Reliability Council,although sustained efforts were made to improve the grid reliability,there was no significant downward trend in the number of grid outages in the US from 1984 to 2006 [6].

Grid outages caused by extreme events have a low probability of occurrence but are harmful and accompanied by power supply cutoff to millions of households [7].Therefore,academics have been increasingly focusing on the concept of “resilience” ever since hurricane Sandy caused power cutoffs to more than 6.5 million households in the eastern US in 2012 [8-9].Resilience is the ability of a power system to mitigate losses after outages to restore normal power supply at the earliest opportunity [10].As an important approach to improving the grid resilience,combined photovoltaic-battery storage system (PV+BSS)has been increasingly installed in residential homes [11-12].When grid outages occur due to extreme events,the batteries can supply stored power generated using photovoltaic (PV)systems to households as opposed to no power supply by conventional generators,which can improve grid resilience and enable self-sufficiency for households.For example,as the blackouts caused by extreme weather conditions are common in the Borrego Springs Microgrid,San Diego,California,the PV+BSS was introduced to effectively ensure reliability [13].In [14],a multienergy transportation system was proposed to contribute toward grid resilience under extreme conditions.In [16],a multienergy microgrid power supply strategy,including wind turbines,PV systems,and battery storage was proposed to ensure that maximal critical load can be restored after an outage.By simulating the selfconsumption in Europe,in [17],the authors demonstrated that PVs or battery storage units contributed to the reliability of households; however,the implementation of full off-grid operation still needs to be explored.

Current research efforts are being focused on evaluating the value of battery storage,especially its reliability.In[15],power and energy capacity accessibilities were used to measure the influence of the PV+BSS on resilience; then,a location scheme for PV and battery storage was proposed.The authors in [18]note that for small-scale dwellings,investing in battery storage is profitable as it results in lower costs.In [19],a siting and sizing model was formulated on the condition that the loss-of-load penalty fee be used to quantify reliability.In [20],based on a user willingness model for payments,an optimal configuration method for battery energy storage systems was studied to improve the reliability of the distribution network under N-1 faults.In [21],individual user willingness on investing in the PV+BSS and reliability costs were examined.An evaluation system was developed in [22]to assess the economic benefits for households with distributed energy resource(DER) integration.Using the IEEE 14 and 118 bus systems,investing in DERs was shown to decrease home operating costs and increase generation flexibility.Based on reliability and economic value,an evaluation plan was proposed in [23]for household battery storage investments.However,these plans do not have detailed modeling for the probability of extreme events causing household blackouts and do not reflect the differences among different geographical regions with regard to battery storage investment.

To bridge these gaps,we propose a method for evaluating the reliability value of a PV+BSS by considering the probability of grid outages causing household blackouts.The contributions of this paper are summarized as follows:(1) A simple formula is derived to calculate an optimal strategy to provide household-level customers with a simple and straightforward expression for invested PV+BSS capacity.(2) We collected realistic datasets from 600 households of the US over the period of 2006 to 2015 and conducted case studies to demonstrate household-level customers’ willingness to invest in the PV+BSS against loss of load caused by blackouts.

The remainder of this paper is organized as follows.In Section II,we provide the data description.In Section III,the formulation of the optimal PV+BSS strategy is presented.In Section IV,we derive the analytical expression for the solution to PV+BSS installed capacity.Section V presents case studies based on realistic datasets,and Section VI presents some concluding remarks.

1 Data description

When power grid outages occur due to extreme events,the PV+BSS can help households achieve self-sufficiency by rapidly restoring power.In this study,to further measure the reliability value of the PV+BSS,we use the average outage time to calculate the probability of a grid outage based on data available from the National Electric Energy Testing Research and Applications Center (NEETRAC) in the US.

To help electric utilities solve the problems of efficiency and reliability in transmission and distribution,NEETRAC published reliability analysis results as “Baseline Project 17-048” [24],whose data on outages during 2006 to 2015 is available from IEEE.The annual customers’ reliability data collected by IEEE’s Distribution Reliability Work Group was also utilized herein.

Table1 Average outage durations in the US

Region Average outage duration (min)0 Span Sates or unknown 70.1 1 Northeast 74.5 2 Mid-Atlantic 115.3 3 Southeast 85.4 4 Midwest 65.1 5 Southwest 41.8 6 South 100.3 7 Northeast 127.9

The average annual outage duration is an important indicator for assessing grid reliability.To show the differences between location-based effects,we divided the US into eight regions; Table1 shows the average annual outage durations for these regions from 2006 to 2015,as derived from actual historical data.

The probability of an extreme event,as mentioned above,is denoted by p,which can be obtained from the average outage time:

where ttotal=8760 h is the duration of one year.

Based on (1),we obtained the probability of household blackouts from 2006 to 2015 from the eight regions.These results are visually depicted in Fig.1,and color bar indicates the magnitude of p.

Fig.1 Probability of household blackouts from 2006 to 2015

2 Model formulation

In this study,we assume that the individual household system comprises an electrical load,a PV,and battery storage [25].These household systems are considered to be unique to each household and are therefore unrelated so that individual systems can be analyzed separately after blackouts caused by extreme events.

Considering the time interval as ΔT=1 h,the entire day is divided into two time periods,namely daytime(6:00-19:00) and nighttime (19:00-24:00,0:00-6:00),with Tn and Td representing the daytime and nighttime periods,respectively.

2.1 Household

First,the three parts of the household system are individually modeled.According to historical data,the electrical load in typical household can be counted in hours and denoted as l(t).Then,the accumulated loads during the daytime (Ld) and nighttime (Ln) are

After installing PV systems,households can not only achieve self-sufficiency but also send the excess power to the grid.Based on historical data,an output per hour of 1 kW can be achieved for a PV battery,which is denoted by pPV(t).Therefore,the total daytime output of a PV battery with capacity Kp is

To reduce the complexity of the model,we simplify the battery storage part; Kb represents the energy capacity of household battery storage as the depth of discharge,which is denoted as μb.Ignoring the dynamic charge and discharge process,the actual usable capacity of storage is given as μbKb.To provide more convenient investment guidance to households,an idealized battery storage model that only considers the stored energy capacity was selected.Empirical parameters are used to quantify and correct the discrepancies between the idealized and realistic models,which consider the power limit and charging/discharging process of the battery storage operation in case studies.

2.2 Investment problem

In the PV+BSS,the decision variables for individual household investors are Kp and Kb.Considering the economic value,reliability value,and facility cost of the PV+BSS,the investment worth is evaluated.Therefore,the optimal investment can be obtained by solving

where CE represents the economic value; CR represents the reliability value; CF represents the facility cost,and I represents the revenue of a household relying on the PV+BSS after blackouts.

Households can generate profits by using or selling electricity from the PV+BSS,which is the embodiment of the economic value of the scheme.CE is approximately proportional to Kp and Kb as

where αp and αb are the annual economic values of the PV+BSS per unit capacity.

The facility cost is also proportional to the capacity.In addition,a one-time fixed cost that includes site and labor is also taken into account.To generate good profits after installation,it is necessary to discount the one-time fixed cost; then,the facility cost is expressed as

where γp and γb are the one-time fixed costs for the PV+BSS;βp and βb are the investment costs of the PV+BSS per unit capacity.

Fig.2 shows the reliability value realization mechanism of the PV+BSS after blackouts.The left-side figure shows the loss of household load represented by ΔL(0,0) in the absence of the PV+BSS.The loss of household load includes the accumulated loads during the daytime and nighttime:

Fig.2 Reliability value realization mechanism

In the right-side panel in Fig.2,the reliability value is reflected in the following two cases:

(1) During the daytime,only PV provides electricity to the household,while the loss of load is

The extra solar energy generated is stored in the battery once Ld is small or μpKp is large.Taking the capacity limitation of the battery storage into account,the stored energy Ls of the battery during Td is expressed as

(2) During the nighttime,the battery discharges its daytime stored energy to supply the nighttime load.Thus,the lost load in this period can be expressed correspondingly as

Therefore,the reliability value of the PV+BSS with capacity according to (Kp, Kb) can be measured by the amount of avoided load loss.

where p is the probability mentioned in (1); πR is the unit value of load loss of a household.

From (4),(5),(6),and (11),the final expression of the individual household revenue for an installed PV+BSS can be easily derived as:

In the next section,we discuss the maximum value of this expression for each case from a mathematical point of view.

3 Model analysis

It is obvious from (12) that I is a binary linear function of (Kp,Kb),where the domains of Kp and Kb are both greater than zero.Since I is piecewise linear,it cannot be derived at its extreme points.Therefore,this section discusses the function in a piecewise manner to determine the optimal investment strategy (K* p,K* b) and maximum revenue Imax.

To highlight the effect of the reliability value,we assume that lp=αp-βp＜0 and lb=αb-βb＜0; additionally,let lp=αp-βp+pπRμp and lb=αb-βb+pπRμb,whose positive and negative cases will be discussed.Therefore,under different values of Kp and Kb,the values of I are as shown in Table2.

(1) KpKb=0

Owing to the discontinuity of the sign function sgn(∙),the expression is discussed when either Kp or Kb is equal to 0.According to the value of I in the first row and the first column of Table2,it can be deduced that since lb＜0,I decreases as Kb increases from 0,and I1=I(0,0)=0 is the only maximum value.When Kp increases from 0,the final slope is lp＜0; if l’ b＜0,I1 is the maximum value of I; and if l’ b＞0,then I2=I(ld/μp,0) is the maximum value.

(2) KpKb≠0

At this time,I assumes values from the rest of Table2,which can be regarded as three planes in the rectangular coordinate system in space (Kp,Kb,I) intersecting at the point ((Ld+Ln)/μp,Ln/μb,I3),where I3=I((Ld+Ln)/μp,Ln/μb).Obviously,the maximum value is at most I3.

Combining the above two discussions,there are three possible situations for the optimal strategy,and the corresponding maximum values of I are I1,I2,and I3.

Table2 Value of I varies with Kp and Kb

I Kp 0 (0,Ld/μp) (Ld/μp,(Ld+Ln)/μp) ((Ld+Ln)/μp, Kp,max)Kb 0 0 l’ pKp-γp lpKp-γp+pπRLd(0,Ln/μb) lbKbγb l’ pKpγp+lbKb-γb lpKp-γp+l’ bKbγb+pπRLd(Ln/μb,Kb,max)lpKp-γp+lbKbγb+pπR(Ld+Ln)

4 Case studies

Based on the actual historical load data of 600 households and empirical data of PV outputs,the optimal investment strategy calculation is presented herein for each household,along with the corresponding annual reliability value and revenue,assuming that the PV+BSS is already installed.The data and parameter which needed in the investment strategy of PV+BSS capacity are detailed in V-A,and we will compare the spatiotemporal investment strategies in V-B and V-C.

4.1 Data and parameters

Figs.3 and 4 show the load data of 600 households and for 1 kW PV output,respectively.The load data are presented in the form of cumulative distribution functions(CDF),as in [26],for the daytime and nighttime periods.Similarly,the PV output data of an entire year is computed as in [27]and presented in the form of the CDF,from which it is clearly observed that the PV output first increases and then decreases during the daytime.

From Fig.3,we see that the load in the daytime is much larger than that in the nighttime,which indicates the significance of the PV and battery storage time-sharing operation.

Fig.3 also reveals the laws of Ld and Ln,while Fig.4 demonstrates the distribution of μp,which are all necessary parameters for further calculations,according to (13).

In addition,μb=1 to fully utilize the battery storage capacity and πR=11.96 $/kWh [28]to emphasize the value of the load loss,with the other relative parameters set as in Table3 [29],[30].All parameters are presented by day according to the payback period,with the one-time fixed cost discounted.Once an optimal investment strategy (K* p,K* b) is determined,the daily and annual CR and Imax can be calculated.

Fig.3 CDFs of the loads of 600 households during daytime and nighttime

Fig.4 One kilowatt PV output in one year

Table3 Relative parameters of the PV+BSS

Economic value Facility cost One-time fixed cost PV αp=0.1226 $/kW βp=0.1323 $/kW γp=0.1090 $Battery αb=0.1350 $/kWh βb=0.1360 $/kWh γb=0.1557 $

4.2 Data-driven correction

Herein,a benchmark model is proposed to calibrate the simple calculation model above and to compensate for the process of battery storage charge/discharge models and power limits.The benchmark model of the PV+BSS’s reliability-based investment strategy is given as

Subject to:

where l(t) is the load used by the household consumer at time t,and lN(t) is the net load.pPV(t) is the power produced by the solar panel at time t,pmax is the maximum battery storage power,and pcha(t) and pdis(t) are the battery charge and discharge power values at time t,respectively.e(t) is the energy stored in the battery at time t,CBSS is the capacity of the battery,and SOCmin and SOCmax are the minimum and maximum state of charge (SOC) of the battery,respectively,with η being the efficiency of the storage battery.Equations(17) to (19) show the various limitations of the solar panel and battery,and (20) depicts the battery charging and discharging process.

We obtained results using the benchmark and theoretical models to compare the two and to obtain a data-driven correction.The deviation between the actual and theoretical results can be described as a quadratic function:

where D is the deviation by which the data drives the phase to correct the theoretical reliability value model.

4.3 Regional differences

Herein,we present the calculation of the optimal capacity of the PV and battery for the eight regions according to (13).The optimal investment strategies of the PV+BSS systems are shown in shown in Fig.5,Fig.6,and Table4.It is worth noting that we only study the regional differences,so the average probability from 2006 to 2015 for each region is used for the calculations.

Table4 Optimal investment strategy for the PV+BSS

Region Optimal storage capacity/kW Optimal solar capacity/kW 0 16.08 11.85 1 16.09 11.86 2 16.37 12.75 3 16.14 12.23 4 16.09 11.46 5 15.30 10.06 6 16.14 12.68 7 16.62 12.91

Fig.5 Optimal investment strategy for the BSS in different regions

Fig.6 Optimal investment strategy for the PV in different regions

Based on the optimal investment strategies for the BSS and PV separately,the annual reliability value and total revenue of the PV+BSS are shown in Table5.From Fig.7,it can be seen that the reliability value of each region is positively correlated with the probability of blackouts.Region 7 has the highest median value of CR,which reaches$251.17 with the highest probability of outages,where the local households are the most suitable candidates for PV+BSS investment.Under the optimal investment strategy,the median value of Imax is $115.48,which indicates that half the residents who have the PV+BSS can benefit by more than $115.48 each year.However,Region 5 has the lowest median value of CR,which is $81.64 even under the optimal investment strategy.Correspondingly,the median values of (K* p,K* b) are 10.06 kW and 15.30 kWh,respectively,and the median value of Imax is $26.04.

Table5 Reliability value of and total revenue from the PV+BSS

Region Reliability value/$ Total revenue/$0 137.03 55.10 1 145.89 59.87 2 226.78 102.58 3 166.60 70.93 4 126.82 49.71 5 81.64 26.04 6 195.81 86.37 7 251.17 115.48

Fig.7 Reliability values of PV+BSS for different regions

4.4 Annual differences

The probability of a grid outage causing household blackouts (p) varies over the years,which implies that household investors require different investment strategies.Therefore,in addition to the regional differences,the temporal differences should be studied.Figs.8 and 9 show the optimal PV+BSS capacities in the eight regions of the US from 2006 to 2015.For each curve in the diagram,one end of the line is connected to the year and the other end is connected to the region,indicating the corresponding relationships between the areas and years; the thickness of the line indicates the investment strategy.According to Figs.8 and 9,based on the differences in probabilities of blackout,households from the eight regions are shown to require distinct strategies for PV+BSS installations.In 2010,because of the lengthy outage duration,the investment strategy in Region 6 was maximum,with the investments of the PV and BSS noted as 17.12 kW and 13.27 kWh,respectively.Region 5 has 0 optimal capacities for the PV and battery in 2008 and 2012,which indicates that it would not be worth investing in the PV+BSS for these two years.

Fig.8 Investment strategy for BSS in different regions and years

Fig.9 Investment strategy for PV in different regions and years

Under the optimal investment decision (K* p,K* b),CR and Imax for a region are not identical over the years,as shown in Fig.10.In Fig.11,for longer annual outage durations,the reliability values for investment in the PV+BSS are larger.For example,Region 4 has the largest outage duration in 2007 such that CR and Imax are maximum,with median values of $260.08 and $120.18,respectively.The shortest blackouts occurred in 2013,where CR and Imax reduced to $48.82 and $9.77,respectively.Region 5 is not a worthy investment in 2008 and 2012 because the reliability value is 0 owing to the shorter annual outage durations.

Fig.10 Median annual reliability values and total revenues of the PV+BSS for different regions and years

Fig.11 Relationship between median reliable value of the PV+BSS and annual outage duration

In the above analysis,it is worth noting that the regions with a higher probability of blackouts would benefit more through the proposed strategy,which shows a positive correlation trend.This is because the reliability value of PV+BSS is measured by the amount of avoided load loss,so the longer the average outage time,the greater the benefits the household users will get from it.On the other hand,a smaller scale investment strategy could be useful elsewhere,this is due to even if the power outage time is relatively small in some years,it is still profitable most of the time.Therefore,household users are willing to implement the proposed method and model in the system to meet the selfsufficiency in power failure.

5 Conclusions

In this study,the reliability value evaluation model of a distributed PV+BSS was developed by considering the probability of grid outages.Case studies were performed based on 10-year historical load data from eight regions in the US; the results of these evaluations demonstrate the following:

(1) The proposed model can comprehensively consider the economic and reliability values of the PV+BSS,which are useful as guidance for residents planning on installing the PV+BSS in extreme weather regions.

(2) Regional differences can be observed from the reliability value and investment strategy results of the PV+BSS.The reliability values are particularly high in regions where the annual outage durations are long.Among the eight regions in the US,Region 7 has the longest annual outage duration,with the highest reliability value of $251.17 for the given period.

(3) The simulations also show differences in the annual results.The longest annual outage durations were recorded in 2006,2007,and 2010 for Region 7,where the reliability values of the PV+BSS were a maximum of $183.68,$170.31,and $89.86,respectively.

In our future work,two issues deserve further study:i) Consider the sharing scheme for battery storage between households.ii) A comprehension model needs to be studied which considers detailed physical process of battery storage such as charge,discharge and loss.

Acknowledgements

This work was supported by National Natural Science Foundation of China (Project 51907064),in part by China State Key Lab.of Power System (SKLD19KM09),and in part by State Grid Corporation of China (1400-202024222A-0-0-00).

Declaration of Competing Interest

We declare that we have no conflict of interest.