using plug-in electric vehicle eets to support the ... · eets to support the integration of...

Loukas Theodoulou

Using plug-in electric vehiclefleets to support the integration ofincreasing photovoltaic capacity

Semester Project

Power Systems Laboratory (PSL)Swiss Federal Institute of Technology (ETH) Zurich

Supervision

Marina Gonzalez VayaOlivier Megel

Prof. Dr. Goran Andersson

June 2013

Acknowledgement

For the completion of my thesis I would like to thank the doctoral students of PowerSystem Laboratory (PSL), Marina Gonzalez Vaya and Olivier Megel for their out-standing support and guidance that they provide me during my semester thesis.Without their help, the fulfilment of this work would have been impossible.

Special thanks to Swissgrid for the Swiss transmission grid data and the Institutefor Transport Planning and Systems of the ETH Zurich for the transport simulationdata. Also I would like to thank, MeteoSwiss for the Solar forecast data.

Sincere thanks, to all the professors of the Power System Laboratory, who managedin the most extraordinary way, to impart their knowledge through my semesterproject; Also, through lectures and discussions which have developed my criticalthinking and ability to address scientific problems.

i

Abstract

Nowadays, two recent phenomena pose new challenges in the energy sector. Thefirst one is the Renewable Energy Sources (RES) and the second one is the intro-duction of Plug-in Electric Vehicles (PEVs). The latter, have a dual role in thepower sector. On one hand, they are mobile loads and on the other hand, they playthe role of distributed storage resources. This thesis deals with the interaction ofPEVs and the increasing photovoltaic capacity. Specifically, a study was done onhow does the PEVs charging affect the curtailment of solar power, how are chargingpatterns affected by solar power in-feed and how can PEVs help to compensate solarforecast error.

The methodology followed can be described by three modules: the photovoltaicmodule (PV), the random generation scenarios module and the smart-charging algo-rithm module. The PV module is an empirical model [1], which formulates the solarAC power output of a weather station by the means of empirical equations basedon thermal, electric and optical characteristics of different solar modules. Inputs ofthis model are the solar irradiance (actual and forecast), the ambient temperature,the wind speed, the longitude, latitude and the elevation angle of a solar weatherstation. Output of this model are the actual/forecast AC power, which are inputsto the second module. Forecast generation scenarios module has historical/time-series data of AC power output as input and generates random scenarios for a daywith maximum/minimum solar production, respectively. Moreover, these randomscenarios are the inputs of “smart-charging algorithm”. The goal of this algorithm isthe generation cost-minimization, while satisfying the network constraints and theconstraints imposed by PEVs. This algorithm is used in two case scenarios. Thefirst one has as an objective function the generation cost-minimization and the sec-ond one, except the generation cost-minimization, introduces additional constraintsfor a random forecast error compensation. The data obtained by these two scenar-ios help to answer the three question mentioned in the first paragraph. The PEVsfleets were simulated as virtual batteries, whose characteristics, depend on the typ-ical driving patterns for Switzerland. These driving simulations were obtained froma transport simulation called MATsim [2].

From the simulation of the two smart charging scenarios we found out that PEVsincrease the total demand (load) and they reduce the amount of solar power, whichmust be curtailed. This is the case, because an amount of the excess solar powerproduction is stored in PEVs (load shifting). Furthermore, charging patterns aredepended on the season, if there are RES connected to the grid and also in thetotal RES in-feed power magnitude. During summer (days with maximum solarproduction) and especially when the solar in-feed power is increased, the most ofthe charging occurs during the hours of solar production (valley filling) while charg-ing during night hours, is significantly decreased. On the other hand, during winter(days with minimum solar production) the charging occurs mostly during nighthours because the price of electricity is cheaper due to the lower demand. Fur-

ii

thermore, PEVs help to compensate a solar forecast error. For example, when theforecast solar output is higher than the actual output, PEVs will supply the systemwith the stored energy in their batteries (V2G mode). On the other hand, whenthe forecast solar output is lower than the actual power, the system has an energysurplus. This surplus can be stored in PEVs batteries (load shifting).

iii

Contents

Acknowledgement i

Abstract ii

1 Introduction 1

2 Sandia photovoltaic model 32.1 Inputs-outputs of SANDIA photovoltaic model . . . . . . . . . . . . 32.2 Dependencies of SANDIA PV module’s output. . . . . . . . . . . . . 4

3 Analysis of forecast error 53.1 Comparison of RMSE when forecast solar irradiance data are ob-

tained by two different approaches. . . . . . . . . . . . . . . . . . . . 53.2 Comparison of solar AC power RMSE when night hours are includ-

ed/excluded, respectively. . . . . . . . . . . . . . . . . . . . . . . . . 63.3 Comparison of solar AC power RMSE when forecast horizon change. 7

4 Temporal and geographical structure of PV power production andforecast error modeling 84.1 Temporal and geographical structure of PV power output . . . . . . 84.2 Generating forecast scenarios for a specific day with maximum and

minimum solar production, considering the total power in-feed in thenetwork. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

5 Methodology 135.1 Smart charging scheme . . . . . . . . . . . . . . . . . . . . . . . . . . 13

6 Results 176.1 Charging of PEVs and solar power curtailment. . . . . . . . . . . . . 176.2 Charging patterns and how they affected by solar power in-feed. . . 186.3 PEVs and forecast error compensation. . . . . . . . . . . . . . . . . . 19

7 Sunmary - Conclusion 247.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247.2 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

A SANDIA PV model functions 28

iv

Chapter 1

Introduction

Nowadays, due to climate change and the increase of environmental pollution, sci-entists and society have started to invest and promote alternative ways, both in theenergy, and in the transportation sector such as the replacement of traditional carswith Plug-in Electric Vehicles (PEVs) and introducing Renewable Energy Sources(RES) in the generation mix. Until now, energy production was primarily based onfossil fuels such as coal and oil. However, due to the fact that their energy reservesstart to decline, the electricity energy mix according to some studies will changeduring the next years [3].

According to these studies, the future generation mix will consist more of renewableenergy technologies (RES) like wind and solar power. This helps in CO2 reduction,but causes an increase of non-usable capacity due to the stochastic dependence onweather [3]. The increase of non-usable capacity, introduces concerns in security ofsupply and the increase of RES in the generation mix causes the curtailment of solarpower, when the power generation is more than the demand or network capacityinsufficient. A plausible solution could be the increase in capacity reserves, or theuse of gas power plants which can compensate energy fluctuation quickly, but withhigh marginal cost [3]. This thesis investigates how the interaction of the increas-ing solar capacity with the PEVs affects the curtailment of solar power, how thecharging patterns of PEVs are affected from solar output and how this interactionhelps in solar forecast error compensation.

The synergies between PEV and RES have been the subject of some studies[6], [7]and [8]. The outcome of [7] and [8] was that the integration of RES in the powernetwork is possible by the use of PEVs. Most of the studies have as an objectivefunction to minimize the generation cost and/or the Green House Gas emissions(GHG) [9], or to compensate the intermittent RES. Furthermore, most of them donot take into consideration the network constraints, with an exception [10], whichshows that charging management helps reduce RES curtailment. This thesis alsotakes into consideration the network and PEVs end-use constraints.

The problem was formulated as an OPF. Two different cases were studied. In thefirst case the objective was to minimize the generation cost. In the second casethe same objective was pursued, but with additional constraints, to compensate arandom solar forecast error. For the later case, the random scenarios were gener-ated based on a probabilistic model [11], which has as inputs time-series (historical)data of normalized AC solar power output both for actual and forecast data. Thesedata, are generated from weather data by the use of a photovoltaic model [1]. Fur-thermore, the PEVs were formulated as virtual storage, aggregated to the different

1

2

nodes, while the necessary energy and power constraints of virtual storage weredetermined by the use of individual driving patterns. These patterns were obtainedfrom a transport simulation called MATsim [2].

From the simulations of the above two cases we found out that PEVs reduce theamount of solar power which must be curtailed. This is the case, because part ofthe excess solar power production is stored in PEVs (load shifting). Also, anotherfinding is that charging patterns are depended on the season of the year. Forexample, during a day with high solar production, the charging of electric vehiclesoccurs mostly during the hours of solar production, instead of during the nighthours, as is typically expected. On the other hand, during a day with low solarproduction, the charging occurs mostly during night hours. This is because duringthese hours the electricity is cheap due to smaller demand. Furthermore, PEVs helpto compensate a solar forecast error. For example, when the forecast solar output ishigher than the actual output, PEVs will supply the system with the stored energyin their batteries (V2G mode). On the other hand, when the forecast solar outputis lower than the actual power, the system has an energy surplus. This surplus canbe stored in PEVs batteries (load shifting).

Chapter 2

Sandia photovoltaic model

This chapter describes the SANDIA photovoltaic model, which is based on [1] andformulates the AC power output of a weather station after the inverter. It is anempirical model, described by a set of equations based on characteristics of solarmodules such as, electrical, thermal and optical characteristics. Furthermore, thischapter introduces the inputs and output of this model and a more detail descriptionof its functions is given in Appendix A. The function were implemented by PV LIBToolbox [4].

2.1 Inputs-outputs of SANDIA photovoltaic model

Inputs to this model are the hourly Global Horizontal Irradiation (GHI) (w/m2),the hourly ambient temperature of the module (C), the hourly wind speed (m/s2),the longitude and lattidute of the weather stations, as well as the elevation angle ofthe weather station. The output of this model is the AC power output of a specificweather station, after the inverter. A schematic shown the inputs-output of thismodel is given in Figure 2.1.

Figure 2.1: Inputs and output of SANDIA photovoltaic model.

3

4 2.2. Dependencies of SANDIA PV module’s output.

2.2 Dependencies of SANDIA PV module’s out-put.

The AC power output of this model is depended on:

1. The type of PV module and inverter.

2. The number of PV modules connected in series and in parallel.

3. The day of the year, the site location and sun position through the day.

4. The incident irradiation to the plane of array (POA).

5. The cell and module temperature.

A more detail description about these dependencies and how the output changesaccording to these quantities is given in the Appendix A.

Chapter 3

Analysis of forecast error

This chapter includes the analysis of forecast error by the use of Root Mean SquareError (RMSE) indicator. The aim was to investigate the magnitude of the forecasterror when the forecast solar irradiance is obtained by a persistent model (real dataof date d-1 are used as forecast data of day d) and by one day ahead forecast data,obtained from MeteoSwiss. Furthermore, how does the magnitude of RMSE changeswhen night hours are included/excluded and when the time horizon is increased.

3.1 Comparison of RMSE when forecast solar irra-diance data are obtained by two different ap-proaches.

The forecast solar irradiance data were obtained with the two approaches given be-low. The available data are for one year starting from 01/08/2011 until 31/07/2012.

1. One day ahead forecast starting at 15:00 o’clock of each day. These data wereobtained from MeteoSwiss.

2. Persistent model in which the forecast data of a day d is the real data of thethe day d− 1.

The RMSE of forecast solar irradiance of the persistent and of the one day aheadmodel were compared. The RMSE is given by the formula below [5] and is computedfor each weather station in Switzerland.

RMSEirradiance =1√N

√√√√ N∑i=1

(Real (i)− Predicted (i)

Pnorm

)2

(3.1)

Where Pnorm is the maximum irradiance for each weather station. The outcome isshown on Table 3.1.

Comparing the RMSE for each weather station for the two different cases (one dayahead and persistent solar irradiance forecast), it is obvious that the forecast dataof one day ahead case are more precise, comparable with the persistent model, evenif the error is high. For that reason, these data are used in the foregone analysis.

5

63.2. Comparison of solar AC power RMSE when night hours are

included/excluded, respectively.

Table 3.1: RMSE for each station, depending on the two approaches of solar dataforecast derivation.

Weather RMSE(%) RMSE (%)Station persistent model one day ahead model

Tanikon 13.67 12.69Basel 12.90 9.68

La Fretaz 13.32 11.90Davos 13.10 11.62

Interlaken 14.05 9.44Pully 14.35 10.35Sion 11.62 8.94Stabio 13.66 14.35Zurich 14.17 9.72

3.2 Comparison of solar AC power RMSE whennight hours are included/excluded, respectively.

In this section the normalized RMSE error for solar AC power output is computedand compared when we take into consideration the night hours and when not. Theformula for the computation of the RMSE is given below[5] and the correspondingresults are given on Table 3.2.

RMSEACpower =1√N

√√√√ N∑i=1

(PACpredicted (i)− PACreal (i)

Pnorm

)2

(3.2)

Where Pnorm is the maximum real AC power output and is equal to 166.67 W. TheRMSE of AC power for each station respectively, as well as, for aggregate AC poweroutput is shown Table 3.2 .

Table 3.2: RMSE (%) for the AC solar power output when night hours are includ-ed/excluded, respectively.

Weather RMSE(%) RMSE (%)Station night hours included night hours excluded

Tanikon 19.74 33.98Basel 14.54 24.51

La Fretaz 17.97 30.57Davos 18.14 30.12

Interlaken 14.26 23.27Pully 14.85 25.40Sion 14.12 23.22Stabio 20.54 34.20Zurich 14.60 24.11

Aggregated AC power output 16.70 28.04

From the above table is obvious that the RMSE is lower when the night hours areincluded because during night hours, both real and predicted solar output are equalto zero, hence, the error between forecast and real AC power output is equal to zeroas well.

Chapter 3. Analysis of forecast error 7

3.3 Comparison of solar AC power RMSE whenforecast horizon change.

In this section the forecast solar irradiance data were obtained from MeteoSwiss fordifferent forecast horizons. Specifically, all the forecast were done at 15:00 of eachday and the forecast horizon is changed from 4-6 hours ahead, to 7-9 hours aheadand to one day ahead. The RMSE [5] was calculated for the aggregated AC poweroutput, including the night hours, and the results are shown on Table 3.3.

Table 3.3: Comparison of solar AC power RMSE when forecast horizon change.

Time horizon RMSE(%) (including night hours)

4-6 hours ahead 16.587-9 hours ahead 16.60

The results from the above table shows that the RMSE does not change significantlywhen the forecast horizon is increased. For example, when the time horizon is 4-6hours ahead the RMSE is equal to 16.58% and when the forecast horizon is one dayahead the RMSE is equal to 16.70%. Hence, there is no big change.

Chapter 4

Temporal and geographicalstructure of PV powerproduction and forecast errormodeling

This chapter is divided into two main sections. The first one deals with the temporaland geographical analysis of PV Power production, and the second one deals withthe extension of an existing probabilistic wind forecast error tool with the main aimto generate samples of forecast errors.

4.1 Temporal and geographical structure of PVpower output

The main aim of this section, is the assignment of a weather profile to each districtand each district to the closest node of the network. Based on this, we computethe total solar power in-feed in the network and the percentage of the total in-feedpower which correspond to each network node.

The above goal was achieved, by following the steps below.

1. Definition of the power profile for each weather station. This is possible bythe use of SANDIA PV model.

2. Assignment of each district to the closest weather station, in order to definethe power profile of each district.

3. Weight the contribution of each district to the total power in-feed, accordingto residential & industrial area of each district. The reason that we choseto weight the contribution of each district with respect to its area is becausewe assumed that the installed capacity is correlated with roof top area, andthereby with industrial and residential area.

weight factor =Districtsurface[ha]

Totalsurface[ha]where, (4.1)

8

Chapter 4. Temporal and geographical structure of PV power production andforecast error modeling 9

Districtsurface is the surface of each district (residential & industrial) andTotalsurface is the total surface of all districts together (residential & indus-trial).

4. Assignment of each district to the closest node of the network and computationof the total in-feed power at each bus. This can be done by the use of theformula below.

PowerBUSi =Y

AC nominal

n∑j=0

(weight factor)j ·(district profile)j (4.2)

Where Y is an assumed total Swiss installed capacity, AC nominal is thenominal power used to calculate the power profile of each weather station, iis an indicator of a specific bus, j is an indicator of a specific district, n is thetotal number of districts connected to the bus i, districtprofile is the powerprofile of district j and weight factor is the scaling factor of district j. Thenominal power of each PV plant depends on the type and the number of PVmodules which were used. For all the stations, the type and the number ofPV modules were the same. Specifically, 2 rows of 9 modules in series wereused. According to the previous statement the nominal power of the referenceprofile given by the following formula.

AC nominal = 2× 9× 166.6667W = 3000W (4.3)

5. Computation of the aggregated power in the network as the sum of the totalin-feed power of each bus of the network, as described by the formula below.

Totalpower =n∑

j=0

PowerBUSj (4.4)

Where j is an indicator of a specific bus, n is the total number of buses in thenetwork and PowerBUSj is the total in-feed power at bus j.

The above description is depicted also in Figure 4.1.

4.2 Generating forecast scenarios for a specific daywith maximum and minimum solar produc-tion, considering the total power in-feed in thenetwork.

In this Section a method to generate probabilistic forecast scenarios will be describedand is based on [11]. By the use of this method random forecast scenarios for extremedays of the year could be generated. Important fact is that we analyse only the errordue to the weather forecast. However, in practice the function which describes theformulation of the solar power output from weather data is also subject to errors.The inputs and outputs of this method are depicted in Figure 4.2.As it shown from the above figure inputs to this model are time series of predictedand real AC power output, both normalised by the maximum AC power output.The normalization is done by the following formulas.

NormrealAC =Real AC power output

Maximum AC power output(4.5)

104.2. Generating forecast scenarios for a specific day with maximum and minimum

solar production, considering the total power in-feed in the network.

Figure 4.1: Assignment of weather profile to each district and each district to theclosest node of the network.

Figure 4.2: Model with inputs/outputs for the forecast scenarios generation.

NormpredictedAC =Predicted AC power output

Maximum AC power output(4.6)

Furthermore, the night hours (00:00-06:00) and the hours for which the solar powerproduction is almost equal to zero (15:00-23:00), were extracted from the input dataset.

The method adapted from [11] can be described by the following steps.

1. Computation of quantile regression coefficients β for each hour between thereal and forecast data so that the following probability to satisfied.

Pr(Observeddata < β × forecastdata) = quantile tau (4.7)

Where quantile tau are given quantiles.

2. The coefficients β are multiplied with the forecast data of each hour, of eachday, in order to estimate the distribution of the power output. These coeffi-cients β are depicted in Figure 4.3.

Chapter 4. Temporal and geographical structure of PV power production andforecast error modeling 11

Figure 4.3: β coefficients from the quantile regression.

3. Using the distribution obtained in step 2, the samples are transformed intoa multivariate Gaussian random variable, for which, the covariance matrix iscalculated.

4. The final step is the extraction of N samples of the multivariate Gaussian ran-dom variable and their conversion back to the solar power output distribution,from which the forecast error samples can be computed.

The above description is demonstrated in Figure 4.4.

Figure 4.4: Random forecast scenarios generation.

The figure demonstrates the four previously mentioned steps. First, the quantileregression coefficients β for each hour between the real and forecast data are com-puted and then they are multiplied by the forecast data of each hour of each day.This result is given by the blue dots in the first graph for a given hour. Each blue dotrepresents the result of β × forecast. To obtain the estimated cumulative distribu-tion function we interpolate in order to construct the red line. The extrapolations,were computed by the use of exponential tails.

With the inverse of the distribution function, we transform the realisation distribu-tion samples into a variable that is uniformly distributed. By having a sample of the

124.2. Generating forecast scenarios for a specific day with maximum and minimum

solar production, considering the total power in-feed in the network.

uniform distribution we can find the Gaussian distribution value corresponding tothe uniform distribution. This can be done with the aid of the second graph wherethe cumulative distribution function (cdf) of Gaussian variable is depicted. Thenby having the Gaussian variables the covariance matrix can be computed. Then Nsamples of the Gaussian random variable can be obtained. Finally, by following theinverse procedure as before, we could compute the forecast power distribution forwhich the forecast errors can be computed.

Chapter 5

Methodology

5.1 Smart charging scheme

This Section explains the basic idea of the smart charging algorithm in order toachieve a cost effective charging. Also, all the necessary constraints, which arebased on [12], are given.

The fundamental goal of the proposed smart charging is to reduce the cost of charg-ing, while all the network constraints are satisfied. The above approach was im-plemented as an optimal power flow problem (OPF) considering two schemes listedbelow.

1. Scheme 1: only the minimization of generation cost is considered.

2. Scheme 2: apart from the generation cost-minimization, additional constraintsare introduced. These constraints ensure that the virtual battery has thenecessary storage margin to compensate a random solar forecast error.

Both schemes have the same objective function, which is the dispatch of load andgenerators in order to minimize the generation cost. In both cases PEVs were con-sidered as virtual batteries, aggregated to a given network node. The characteristicsof the virtual batteries depend on the driving patterns in Switzerland which wereobtained by a transport simulation called MATSim [2].

The smart-charging algorithm is formulated as follows.

min

P(t)Gi

, P(t)Lj

, E(0)V Bj

,∆ET∑

t=1

∑i

CostGi × P(t)Gi

(5.1)

subject to:

P(t)Lj

= P(t)Lj,r

+ P(t)Lj,c

∀t (5.2)

Where P(t)Gi

is the power produced by generator Gi, P(t)Lj

is the power consumed by

load Lj , P(t)Lj,r

is the reference load without PEVs, P(t)Lj,c

is the charging load, andCostGi is the marginal cost of generator Gi.

The equation 5.1 describes the objective function. According to the objective func-tion generators and loads are dispatched in order to minimize the cost of generation.The objective function is subject to the several constraints. The first one indicates

13

14 5.1. Smart charging scheme

the total load consumption is equal to the sum of the charging and reference load,for each time step.

The next constraint deals with the power balance, where the total produced powermust be equal to the total consumed power. This constraint is given below.

∑i

P(t)Gi

=∑j

P(t)Lj

(5.3)

The limits on generation output, load, line and transformer loading imposes otherconstraints in the OPF. These constraints are given below. The upper and lowerlines depict the upper and lower bounds of variables, respectively.

PGi≤ P

(t)Gi

≤ PGi ∀t, i (5.4)

P(t)Lj,c

≤ P(t)Lj,c

≤ P(t)

Lj,c∀t, j (5.5)

∣∣∣∣∣∑n

Dn,lm ·(P

(t)Gi∈Ωn

− P(t)Lj∈Ωn

)∣∣∣∣∣ ≤ P lm ∀t,m (5.6)

Where Dn,lm is the power transfer distribution factor associated with line or trans-former lm and node n, Ωn is the set of Gi and Lj associated with node n.

Equation 5.6 shows that the loading of a line, or transformer, is calculated withpower transfer distribution factors (PTDF).

The next constraint considers the energy content of the virtual battery, at each timestep t.

E(t)V Bj

= E(t−1)V Bj

+ P(t)Lj,c

·∆t · ηV Bj − E(t)V Bj,d

+ E(t)V Bj,a

∀t, j (5.7)

For each time step t equation 5.7 determines the energy content of the virtual bat-

tery. The energy content (E(t)V Bj

) at time step t is equal to the energy content of the

virtual battery at time step t− 1 (E(t−1)V Bj

), plus the energy contribution of vehicles

arriving at the node where the virtual battery is located (E(t)V Bj,a

), plus the charging

of virtual energy (P(t)Lj,c

·∆t ·ηV Bj ) and minus the energy drop from vehicles depart-

ing from the node where the virtual battery is located (E(t)V Bj,d

). The charging ofthe virtual battery depends on the charging time ∆t, the charging efficiency ηV Bj

and the charging power P(t)Lj,c

.

It is assumed that the energy content of the virtual battery should be exactly thesame after a period of time T and it should be limited by an upper and lower limit.These constraints are given below.

E(0)V Bj

= E(T )V Bj

∀j (5.8)

Chapter 5. Methodology 15

E(t)V Bj

≤ E(t)V Bj

≤ E(t)

V Bj∀j, t (5.9)

Figure 5.1 demonstrates the constraints imposed by 5.9. These constraints showthat the energy content of virtual battery can reach an upper and a lower limit. Thetwo bold blue lines indicate the upper and the lower limit that the virtual batterycould reach. The upper limit indicates that the PEV vehicles are fully charged(100% SOC) and that they cannot store extra energy. On the other hand, the lowerlimit indicates the lower limit of energy content without considering the vehicle togrid mode (V2G). This limit accounts for the discharging due to PEV trips assumingthat at one time during the day the battery will be fully charged. Furthermore,in the figure there is another lower limit which takes into consideration the V2Gmode. This limit considers the additional discharging which will occur in the casethat the forecasted solar power production is higher than the actual production.This indicates that the system has a deficit in active power and this power shouldbe supplied by PEVs by discharging their batteries. This limit is chosen to be equalto 20% of SOC plus the energy necessary before the vehicle departure. The redline indicates the profile of energy content through a day. Its obvious that for somehours of the day the virtual battery is fully charged a fact that does not allow thecompensation of a negative forecast error. As a result, additional constraints wereintroduced. These constraints ensure additional storage margin by introducing ashift ∆E, on the energy content of virtual battery. By doing this we ensure thatbatteries will not be scheduled to reach a full SOC.

Figure 5.1: Constraints show the maximum and minimum energy content of avirtual battery

By the aid of the additional constraints which are given below, a random solarforecast error δ(t) ∈ ∆(t) can be compensated both in terms of available power andenergy capacity. The first constraint deals with the available power at each timestep t in order to compensate the forecast error. The maximum charging power isdetermined by the available charging power, but also by the energy content of thevirtual energy. The minimum power will be zero in the case without V2G modeand minus the available power with V2G mode.

∑j

P(t)Lj,c,υ2g

≤∑j

P(t)Lj,c

+ δ(t) ≤∑j

P(t)

Lj,c,υ2g∀t (5.10)

16 5.1. Smart charging scheme

The other two constraints consider the energy capacity of the virtual battery.

∑j

E(t)V Bj,υ2g

≤∑j

E(t)V Bj

−∆E +

t∑τ=1

δ(τ) ·∆t

η∀t (5.11)

∑j

E(t)V Bj

−∆E +t∑

τ=1

δ(τ) ·∆t · η ≤∑j

E(t)

V Bj∀t (5.12)

The first one ensures that the energy content minus the energy margin ∆E plusthe discharged energy, is higher, or equal to the lower limit of the energy contentof the virtual battery. The other one ensures that the energy content of the virtualbattery minus the energy margin ∆E at each time step, plus the additional powerfrom charging should be less or equal to the maximum energy content.

Furthermore, for both scenarios we assume a PEV penetration of 25%, which corre-sponds to 1 million PEVs. These were assumed to be able to charge and dischargewith a power of up to 11 kW with an efficiency of 90%. Also, the battery capacitiesof 16kWh and 24kWh were used, each assigned to one half of the fleet. Anotherassumption affecting PEV charging patterns is that the solar power has a marginalgeneration cost equal to 0 e /MWh.

Chapter 6

Results

Simulations were performed with a model of the Swiss transmission grid for twoextreme days with minimum and maximum solar production.

The outcome of these simulations helps investigate how PEV charging affects thecurtailment of solar power, how charging patterns are affected by solar power in-feedand finally how PEVs can be used to compensate solar forecast errors. The resultsare demonstrated in the following subsections.

6.1 Charging of PEVs and solar power curtailment.

The solar curtailment occurs when the power generation is more than the demandor the network capacity is insufficient (congestions).Figure 6.1 shows the impact on the curtailment of solar power when the charging

0 5 10 15 20 250

1000

2000

3000

4000

5000

6000

7000

8000

9000

10000

Hours (h)

Pow

er (

MW

)

reference load + charge loadreference loadforecast solar powernoncurtailed power with PEVs fleetnoncurtailed power with no PEVs fleet

Figure 6.1: Curtailment of solar power output: with PEVs and without PEVs.

load of PEVs is taken into consideration and when it is not. The results refer toone day with maximum solar power in-feed during May. The reference load, whichis the load without PEVs and the total load which includes the reference load plus

17

18 6.2. Charging patterns and how they affected by solar power in-feed.

the charging load, are depicted in the figure. A remarkable observation, is that thecurtailment of solar output is reduced when the charging load is considered. Thishappens because a part of the solar output is stored in the PEVs battery, sinceat hours of excess solar power, energy is inexpensive. In contrast, when only thereference load is considered, the curtailment is higher because the energy in-feed bythe solar panels violates network constraints, which also happens when the PEVsare considered, but in a smaller extent.

6.2 Charging patterns and how they affected bysolar power in-feed.

The next issue under consideration is the timing of battery charging. Since thisis affected by the solar power in-feed, a comparison between a day with the max-imum and minimum solar in-feed power is done. The day with maximum solarin-feed power occurs in May and the day with the minimum solar in-feed occurs inDecember. The results are demonstrated on Figure 6.2.

0 5 10 15 20 255000

6000

7000

8000

9000

10000

Hours (h)

Pow

er (

MW

)

Electric vehicle fleet 25% on May

reference loadtotal load with solar in−feed 0GWtotal load with solar in−feed 3GWtotal load with solar in−feed 10GW

0 5 10 15 20 254000

6000

8000

10000

12000

14000

Hours (h)

Pow

er (

MW

)

Electric vehicle fleet 25% on December

reference loadtotal load with solar in−feed 0GWtotal load with solar in−feed 3GWtotal load with solar in−feed 10GW

Figure 6.2: Charging profiles for a 25% PEV penetration, with and without solarin-feed in the network. A day with maximum and minimum solar in-feed power inMay and December, respectively are shown.

The first plot depicts how the total load (reference load + charge load for 25% PEVpenetration) changes for different penetration of solar in-feed for the day with max-imum solar power production in May. In May, the reference load is lower than inDecember because of the warmer weather and therefore less need for heating. Whenthe solar in-feed power is equal to 0GW the charging of PEV occurs during nighthours, because during these hours the price of electricity is lower due to the lowerdemand. When the penetration of solar power increases the charging during nighthours decreases and the charging during the hours of solar production increases.For example, compare the red line where the solar in-feed is 10GW, with the purpleline where the solar in-feed is zero.

Chapter 6. Results 19

The second graph depicts the same concept as above, but for the day with minimumsolar power production in winter. The outcome, is not really clear because as it isshown the charging does not seem to depend on solar power in-feed, and mostlyoccurs during the night hours. This is because the solar production is very low,hence there is no significant charging during the interval of solar production.

The above conclusions are depicted more clearly in Figure 6.3., where the netdemand is shown. The net demand is given by the following formula.

NetDemand = Demand−Non CurtailedPower (6.1)

Then, if the charging load is added to the net demand it becomes obvious that thecharging has a valley filling structure respect to net demand. As the two upperpictures show, during the hours that the solar power in-feed is reaching its peakthe net demand has a valley. During these valley hours, the charging of PEVsoccurs. This is only the case for days with high solar production. In contrast,during winter the results will be different, because the solar production is low. Asa result, the charging occurs during night hours since the demand valley occurs atnight. Independent of solar production.

0 5 10 15 20 255000

6000

7000

8000

Hours (h)

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er (

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)

Infeed solar power 3GW (month MAY)

Net demandNet demand+charge load

0 5 10 15 20 250

5000

10000

Hours (h)

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er (

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)

Infeed solar power 10GW (month MAY)


0 5 10 15 20 250.5

1

1.5x 10

4

Hours (h)

Pow

er (

MW

)

Infeed solar power 3GW (month DECEMBER)


0 5 10 15 20 250.5

1

1.5x 10

4

Hours (h)

Pow

er (

MW

)

Infeed solar power 10GW (month DECEMBER)


Figure 6.3: Charging valley, while solar in-feed power is increasing.

6.3 PEVs and forecast error compensation.

The upcoming figures depict the different constraints and bounds of the virtual bat-tery and how these affect the fleets capability to compensate a solar power forecasterror.

Before analysing each individual graph, a general description will be given in orderto understand what the different bounds represent.

The two blue lines in each plot depict the maximum and minimum energy contentof the virtual battery, respectively. When the energy content of the battery reaches

20 6.3. PEVs and forecast error compensation.

the upper bound this means that the battery is fully charged and no additionalcharging is allowed. Furthermore, the other blue line indicates the minimum energycontent of the virtual battery without considering the vehicle to grid mode (V2G).Basically, this limit is based on the lowest energy content that the battery will reach,if only the discharging related to driving is considered and assuming that batterieswill be fully charged at some point.

The cyan line indicates the minimum energy content of the virtual energy consid-ering the V2G mode. This mode refers to the case when the forecast exceeds theactual power, and vehicles discharge their batteries in order to compensate the fore-cast error. Thus, additional discharging might take place apart from driving-relateddischarging when the PEV is doing a trip. For that reason the lower bound withV2G is based on a minimum SOC of 20% plus the energy needed before departure.

The green line shows the energy content of the virtual battery for each hour andrepresents the case that smart charging does not take into consideration forecasterror compensation (constraints 5.10-5.12 are not included). In contrast the redline shows again the scheduled energy content of the virtual battery but for the casethat a forecast error compensation is taken into consideration.

Moreover, the dotted red lines depict the maximum charging and discharging ofthe virtual battery, obtained from all the possible compensation scenarios. Themaximum charging and discharging were computed as follows:

MaxBound(t) = max(t∑

τ=1

δ(τ)×∆t× η) + Energybaseline(t) (6.2)

MinBound(t) = min(

t∑τ=1

δ(τ)×∆t

η) + Energybaseline(t) (6.3)

The above two equation show the compensation of the forecast error.

The higher dotted red line indicates the maximum level of energy content that canbe achieved from compensating any of the generated forecast error scenarios. Thislevel should not be higher than the maximum energy content of the virtual battery(upper blue line). In contrast, the lower red dotted line indicates the minimumenergy content of the virtual battery that can be achieved from compensating anyof the generated forecast error scenarios. This level should not be lower than theminimum energy content of the virtual battery including the V2G mode (cyan line).

The Figure 6.4 demonstrates the compensation of the forecast of 1 GW solar ca-pacity for the day of maximum solar output and total solar capacity of 3 GW.From the diagram its obvious that during the night hours until early in the morn-ing 00:00-06:00 the forecast error is zero. Hence, there are no deviation from thecompensation baseline. During summer, a forecast error with positive magnitude islarger than the forecast error with negative magnitude. In that case, PEVs wouldsupply the grid by the use of their batteries. This effect is demonstrated with thelower dotted red line where the energy content of battery reaches the lowest value.

Following, Figure 6.5 demonstrates the case of 1GW compensation of the forecasterror in May while the in-feed solar power is equal to 10GW.


0 5 10 15 20 250.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2x 10

4 1GW compensation of the forecast error in MAY and with infeed solar power 3GW

Hours (h)

Pow

er (

MW

)

upper energy limitlower energy limit with V2Glower energy limit no V2Gvirtual energy baselinevirtual energy baseline (compensation)+max chargevirtual energy baseline (compensation)+min chargevirtual energy baseline (compensation)

Figure 6.4: compensation of the forecast error of 1GW PV capacity in the day ofmaximum PV production for a PV capacity of 3GW.

0 5 10 15 20 250.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2x 10

4 1GW compensation of the forecast error on MAY and with infeed solar power 10GW

Hours (h)

Pow

er (

MW

)


Figure 6.5: compensation of the forecast error of 1GW PV capacity in the day ofmaximum PV production for a PV capacity of 10GW.

Referring to the diagram it is obvious again that the magnitude of the forecast er-ror is lying more on positive values, which means the the forecasted solar output ishigher than the actual production. The same observations can be drawn as with theprevious figure. In this case the compensation is still 1 GW but is more attractiveto charge during the day because the electricity is cheaper.

The next two Figures refer to the day with minimum PV power production. Thisday is during December. Figure 6.6 demonstrates the case of 3GW compensationof the forecast error during the day with minimum PV power production, while thein-feed solar power is equal to 3GW.

22 6.3. PEVs and forecast error compensation.

0 5 10 15 20 250.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2x 10

4 3GW compensation of the forecast error on DECEMBER and with infeed solar power 3GW

Hours (h)

Pow

er (

MW

)


Figure 6.6: compensation of the forecast error of 3GW PV capacity in the day ofminimum PV production for a PV capacity of 3GW.

The diagram depicts exactly the inverse case with the day of maximum PV powerproduction. The energy needed to compensate a negative error (forecast output islower than the actual one) is larger than the one needed to compensate a positiveerror. For that reason the virtual energy of the compensation scenario is lower thanthe reference one, in order to have the necessary storage margin to compensate theexcess power.

Furthermore, Figure 6.7 demonstrates the case of 10GW compensation of theforecast error during the day with minimum PV power production, while the in-feed solar power is equal to 10GW.

0 5 10 15 20 250.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2x 10

4 10GW compensation of the forecast error on DECEMBER and with infeed solar power 10GW

Hours (h)

Pow

er (

MW

)


Figure 6.7: compensation of the forecast error on 10GW PV capacity in the dayof minimum PV production for a PV capacity of 10GW.

The same observations as before can be done. The virtual energy of the compensa-


tion scenario is much lower than before, and this is because the storage margin nowmust be higher due to higher solar in-feed. The different thing here is that con-straint 5.11, considering the upper bound of the energy content with compensation,is binding.

Chapter 7

Sunmary - Conclusion

7.1 Summary

Plug-in electric vehicles and the continuously increasing penetration of solar powerin the generation-mix pose new challenges in the energy sector and for that reasontheir interaction is studied. Specifically, this semester thesis investigates how PEVcharging affects the curtailment of solar power, how charging patterns are affected bysolar power in-feed and how PEVs can be used to compensate solar forecast errors.For that reason, an empirical PV model was selected in order to formulate the actualand forecasted solar AC power output according to weather data of different weatherstations in Switzerland and then estimating the solar power production at differentnetwork nodes. By the implementation of a smart-charging algorithm importantconclusion regarding the curtailment of solar AC power output, charging patternsof virtual energy were obtained and a probabilistic solar power forecast model wasused to model the forecast errors to be compensated. The study shows that PEVscould be used as distributed storage devices, which could store the excess energyand reduce the curtailment of solar power in-feed. Moreover, the charging patternof PEVs depends on the season of the year and on the total solar power in-feed inthe network. For example, during days with high solar power production chargingoccurs during the hours of high solar power production, whereas during days withlow solar production charging occurs mostly at night.

7.2 Conclusion

In this section the most remarkable conclusion from the previous simulations willbe mentioned.

1. When PEVs are connected to the grid the total load is increased but thisincrease in the load helps to reduce the curtailment of RES. This is the casebecause a portion of this energy is stored in the batteries of PEV (load shift-ing).

2. The charging pattern is different when there are RES connected to the gridand when there are not. When there are no RES in the grid the chargingoccurs during night because the price of electricity is cheaper due to lowerdemand. In contrast, when RES are connected to the grid, the chargingoccurs both during night but also during the time interval where the solarproduction occurs, since the solar power in-feed reduces the cost of electricity.The charging pattern can be interpreted as a valley filling of the net load.

24

Chapter 7. Sunmary - Conclusion 25

3. The charging patterns therefore depend on solar irradiation. During wintersolar irradiation is typically low, hence the solar power production will below as well. As a result, the charging of PEV will mainly occur during thenight hours. In contrast, during summer where solar irradiation is typicallyhigh, the charging will mainly occurs during hours of solar power productionbecause the power is inexpensive.

4. Also, PEVs could help in compensating forecast errors. When the forecastedsolar production is higher than the actual production then PEVs should com-pensate a positive forecast error. This means additional power is needed,which can be supplied by PEVs. In this case PEVs should be in the Vehicleto Grid mode (V2G), where they deplete their batteries and supply the nec-essary power to the network. On the other hand, when the forecast is lowerthan the actual production we have a negative forecast error, which meansthat there is an excess in energy production. This energy can be stored byPEVs and could be used in a later period of time.

Bibliography

[1] D.L. King, W.E. Boyson, J.A. Kratochvill, “Photovoltaic Array PerformanceModel”, SANDIA Report 3535, Sandia National Laboratories, Albuquerque,2004.

[2] M. Balmer, K. Axhausen, and K. Nagel, “Agent-based demand-modeling frame-work for large-scale microsimulations”, Transportation Research Record: Jour-nal of the Transportation Research Board”, vol. 1985, no. 2006, pp. 125-134,2006.

[3] European Network of Transmission System Operators for Electricity, “SystemAdequacy Forecast 2010 2025 Report”.

[4] PV Performance modeling collaborative, “PV LIBToolbox ”, Web. 10 March2013.

[5] Elike Lorenz, Thomas Scheidsteger, Johannes Hurka, Detlev Heinemann, andChristian Kurz, “Regional PV power prediction for improved grid integration,”25th EU PVSEC WPEC-5, Valencia, Spain, 2010.

[6] P. H. Andersen, J. A. Mathews, and M. Rask, “Integrating private transportinto renewable energy policy: The strategy of creating intelligent recharginggrids for electric vehicles, Energy Policy, vol. 37, no. 7, pp. 24812486, 2009.

[7] H. Lund and W. Kempton, “Integration of renewable energy into the transportand electricity sectors through V2G, Energy Policy, vol. 36, no. 9, pp. 35783587,2008.

[8] W. Short and P. Denholm, “A preliminary assessment of plug-in hybrid electricvehicles on wind energy markets. National Renewable Energy Laboratory, Tech.Rep., 2006.

[9] A. Y. Saber and G. Venayagamoorthy, “Intelligent unit commitment withvehicle-to-grid A cost-emission optimization, Journal of Power Sources, vol.195, no. 3, pp. 898911, 2010.

[10] J. A. Pecas Lopes, F. J. Soares, P. M. R. Almeida, and M. Moreira da Silva,“Smart charging strategies for electric vehicles: enhancing grid performanceand maximizing the use of variable renewable energy resources. in InternationalBattery, Hybrid and Fuel Cell Electric Vehicle Symposium EVS24

[11] P.Pinson, H. Madsen, H. A. Nielsen, G. Papaefthymiou, and Klockl, “FromProbabilistic forecasts to statistical scenarios of short-term wind power produc-tion, Wind Energy,” vol. 12, pp. 51-62, 2009.

[12] Marina Gonzalez Vaya and Goran Andersson, 2013,“Integrating Renewable En-ergy Forecast Uncertainty in Smart-Charging Approaches for Plug-in ElectricVehicles”, Power System Laboratory, ETH Zurich.

26

Bibliography 27

[13] Christiana Honsberg and Stuart Bowden Instructions, “PVCDROM”, Web. 10March 2013.

Appendix A

SANDIA PV modelfunctions

In this section the functions of SANDIA photovoltaic model will be explained. TheSANDIA model is an empirically based model [1] and its versatility and accuracycomes from the fact that all the fundamental equations used within this modelare based on characteristics of different solar cells. Hence, necessary inputs to thismodel are the coefficients of each solar cell.The basic steps of the model are listed and described below.

1. Definition of the PV moduleThe first step is to select and determine the PV module from the data spread-sheet offered by Sandia. The spreadsheet contains an amount of different PVmodules whose specific characteristics are given. For example, some of thegiven coefficients are listed below. For more information about the necessarycoefficients see also [1].

(a) Short circuit current at reference condition in units of amperes.

(b) Maximum power voltage at reference condition in units of volts.

(c) Empirically determined diode factor (dimensionless).

(d) Number of cells in series in a module’s cell string(s).

(e) Number of parallel cell strings in the module.

2. Inverter selectionAfter selecting a PV module, the inverter must be specified. The inverter isalso selected from a list of different inverters. The inverter is necessary forthe inversion of DC output of the photovoltaic module into AC. The invertersize must be specified according the nominal output of the PV power plantthat is being simulated. This is mandatory because if the nominal power ofan inverter is significantly higher than the nominal output of PV module thenthe losses during the inversion will be extremely high. On the other hand,if the nominal power of an inverter is significantly lower than the PV outputthen the inverter will clip out the output of the PV module.

3. Inputs: Irradiance and weather conditionAs mentioned in the beginning, the three most important inputs to the SAN-DIA model are the Global Horizontal Irradiation (GHI) (w/m2), the ambienttemperature of the module (C) and the wind speed (m/s2). These inputsare obtained from MeteoSwiss. They can be either forecast or realization data.

28

Appendix A. SANDIA PV model functions 29

4. Definition of the day of the year, the site location and the sun po-sitionFor the day of the year the function pvl date2doy is defined. This functiontakes as inputs the year, the month and the day and produces a struct of theyear, month and day in the Gregorian calendar.

Furthermore, the site location must be specified. This can be done by the useof pvl makelocationstruct function. This function takes as inputs the latitudeand longitude coordinates in decimal degrees. Latitude convention is positivenorth of the equator and longitude convention is positive east of the primemeridian. Also, an optional input of this function is the altitude which shouldbe given in meters above sea level. By specifying all the inputs this functionconstructs a struct containing the location of the PV station.

Once the location is defined, the sun position should be defined. This is plau-sible by the means of pvl ephemeris function. Inputs of that function arethe site location, the time of the day and the ambient temperature in Celsius.The sun position depends also on the pressure but in this model we assumethat the pressure remains constant and equal to 101325 Pa. Once the inputsare settled, this function uses some equations given below in order to computethe sun position. Before demonstrating the equations, some important terms,which are going to be used are explained.

Local solar Time (LST) and Local Time (LT):Local Solar Time (LST) [13] is defined as when the sun is highest in the skyand the Local Time (LT) is the time of the area under consideration, whichusually varies because of the eccentricity of the earth’s orbit, the time zonesand daylight saving.

Local Standard Time Meridian (LSTM):The Local Standard Time Meridian (LSTM) [13] is a reference meridian usedfor a particular time zone and it is similar to the Prime Meridian (PM),which is used for Greenwich Mean Time (GMT). The LSTM is illustrated inFigure A.1 below.

Figure A.1: LSTM computation

The LST can be computed by the use of the following formula.

LSTM = 150 ×∆TGMT (A.1)

30

where ∆TGMT is the difference of the Local Time (LT) from Greenwich MeanTime (GMT) in hours. Moving on, the Equation of Time (EOT) must bedefined. This equation is given below.

EOT = 9.87 sin(2B)− 7.53 cos(B)− 1.5 sin(B), where (A.2)

B =360

365(d− 81) in degrees and d is the day since the start of the year. For

an example 6th of December means that d would be equal to 340. A graphwhich illustrates the EOT is depicted in Figure A.2

Figure A.2: Illustration of EOT. [13]

The next step is the calculation of the Time Correction factor (TC), whichis given in minutes. The TC is used to correct the LST due to the longitudevariations within different time zones. This factor is computed by the use ofthe following formula.

TC = 4 (longtitude− LSTM) + EOT (A.3)

The factor 4 in front of parenthesis is because earth is rotating one degreeevery 4 minutes.

By having the TC factor and the LT, it is plausible to compute the LST whichis going to be used in the computation of Hour Angle (HRA). The HRA isthe angular displacement of the sun east or west of the local meridian. Thisis based on the fact that the sun needs 24 hours to travel 360 degrees or 15degrees per minute. The HRA has some predefined values which are givenbelow.

(a) HRA is 0 at solar noon.

(b) HRA is negative in the morning.

(c) HRA is positive in the afternoon.

The regions of the earth are divided into certain time zones, in which noondoes not necessarily correspond to the time when the sun is highest in the sky.For that reason conversions are necessary, otherwise a house one block awayfrom another would actually be different in time by several seconds. Solartime, on the other hand is unique to each particular longitude. In order tocalculate the sun’s position, the local solar time must be found and then theelevation and azimuth angles are calculated. The formulas for computing theLT and HRA are given below.

LST = LT +TC

60(A.4)

HRA = 15 (LST − 12) (A.5)


The final term that must be determined in order to compute the elevation andazimuth angles, which are the cornerstone in solar position calculation, is thedeclination angle. The declination angle is given by the following formula.

δ = 23.45 sin

[360

365(d− 81)

](A.6)

Using of all the above calculations the elevation and azimuth angles can becomputed by the means of the following equations.

Elevation = sin−1 [sin(δ) cos(ϕ) + cos(δ)cos(ϕ)cos(HRA)] (A.7)

Normally the azimuth angle is computed by the following formula, but forsimplicity the azimuth angle was fixed to 180 (facing south).

Azimuth = cos−1

[sin(δ) cos(ϕ)− cos(δ) sin(ϕ) cos(HRA)

cos(α)

](A.8)

Where δ is the declination angle, ϕ is the latitude, HRA is the hour angle andα is the solar altitude angle.

5. Computation of the incidence irradiation to the plane of the array.An assumption is that the panels under investigation will be fixed on the roofof buildings, so no tracking system will be installed. For that reason, eachphotovoltaic panel can be described by a tilt and an azimuth angle. The setvalues for these angles are 45 and 180 respectively. After setting these an-gles the irradiance incident to the plane of the array (POA) can be computed.

In order to compute the POA, the sun position, the array orientation, theirradiance components (direct and diffuse irradiance),the ground surface re-flectivity (known as the Albedo irradiance) and, finally, the irradiance reflectedthrough the sky, must be known. The above can be illustrated by the formulagiven below.

EPOA = ESky Diffuse + EGround Reflect + Eb (A.9)

The first term on the above equation is the ESkyDiffuse which can be com-puted by the use of pvl kingdiffuse function. This function takes as inputsthe tilt angle, the zenith angle, the Global horizontal Irradiance (GHI) andthe Direct Horizontal Irradiance (DHI). By the use of the following formulathe diffuse sky irradiance can be computed.

ESky Diffuse = X1 +X2 where, (A.10)

X1 = DHI × 1 + cos (tiltangle)

2(A.11)

X2 = GHI× (0.012× (Zenithangle)− 0.04)× (1− cos (tiltangle))

2(A.12)

The second term, in the incident irradiance is the diffuse irradiance due to theground, which is known as Albedo irradiance. Especially. this irradiance is aportion of the irradiance that reaches a tilted surface due to ground reflections.It can be computed by the use of pvl grounddiffuse function which has as

32

inputs the tilt angle, the GHI and the Albedo factor. The output of thisfunction is the portion of the GHI which is reflected on the ground and thenit hits the surface of a PV module. This output, can be computed by thefollowing formula.

EGround Reflect =GHI ×Albedofactor × (1− cos (tiltangle))

2(A.13)

Where the Albedofactor assumed to be equal to 0.2.

The last term, in the incident irradiance, is the beam radiation componenton plane array Eb. In order to compute the Eb the Angle of Incidence (AOI)must be defined and this can be done by the use of pvl getaoi function. Eb isgiven by the following formula.

Eb = DNI × cos (AOI) (A.14)

Where DNI is the Direct Normal Irradiance.

6. Module and Cell TemperatureThe temperature plays an important role in the short circuit current, in theopen circuit voltage but also in the maximum voltage and current of photo-voltaic module, respectively. For that reason, this model takes into consider-ation the temperature of the cell and module and precisely models the effectof temperature. By the use of an empirical relationship [1], which is givenbelow, the temperature of a cell and a module can be computed.

Tmodule = E × exp(α+ β × windspeed) + Tα (A.15)

Where:

Tmodule= Back-surface module temperature (C).Tα=Ambient temperature (C).

E=Solar irradiance incident on module surface (W

m2).

Windspeed= Wind speed measured at 10 meter height (m

s).

α=Empirically determined coefficient establishing the upper limit for moduletemperature at low wind speeds and high solar irradiance.β=Empirically determined coefficient establishing the rate at which moduletemperature drops as wind speed increases.

Moreover, the relationship for the cell temperature is given below.

Tc = Tmodule +E

E0×∆T (A.16)

Where:

Tc=Cell temperature inside module (C).Tmodule=Measured back-surface module temperature (C).

E=Measured solar irradiance on module (W

m2).

E0=Reference solar irradiance on module 1000 (W

m2).

∆T= Temperature difference between the cell and the module back surface

at an irradiance level 1000 (W

m2).


All the above data were used as inputs to the pvl sapm function which determines5 points on a PV module’s I-V curve (Voc, Isc, Ix, Ixx, Vmp and Imp). This modelassumes a reference cell temperature of 25C. The resulting points are depicted inFigure A.3 below and these points were used for the computation of DC outputof PV module.

Figure A.3: I-V curve including the computed points. [13]

The DC output of PV module can be computed by the use of the maximum currentand voltage data. By definition the maximum power output can be computed asbelow.

Pmp = Vmp × Imp (A.17)

Where:

Pmp= Maximum output DC power of PV plant.Vmp= Maximum voltage of PV plant.Imp= Maximum current of PV plant.

The Vmp and Imp depend on the number of PV modules that are connected in seriesand in parallel. In order to compute the total voltage and current maximum thefollowing relationships were used.

VmpTOTAL= Array.Ms× Vmp (A.18)

ImpTOTAL= Array.Mp× Imp (A.19)

Where:

Array.Ms= is the number of modules connected in series.Array.Mp= is the number of modules connected in parallel.

After computing the DC output of the PV power plant the pvl snlinverter functionis used. This function determines the AC power output of an inverter given bythe DC voltage, DC power, and appropriate SANDIA grid-connected photovoltaicinverter model parameters. The AC output power, is clipped at the maximumpower output, depending on the nominal power of the inverter.

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