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Energy losses in a distribution line with distributed generation based on stochastic power flow
Electric Power Systems Research 81 (2011) 1986–1994
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Energy losses in a distribution line with distributed generation based on stochastic power flow Antonios G. Marinopoulos a,b,∗ , Minas C. Alexiadis b , Petros S. Dokopoulos b a b
ABB AB, Corporate Research, SE-72178 Västerås, Sweden Power Systems Laboratory, Department of Electrical and Computer Engineering, Aristotle University of Thessaloniki, 54124 Thessaloniki, Greece
a r t i c l e
i n f o
Article history: Received 26 March 2010 Received in revised form 30 May 2011 Accepted 17 June 2011 Available online 16 July 2011 Keywords: Distributed generation Losses Monte Carlo simulation Photovoltaic power systems Power distribution
a b s t r a c t This paper proposes a methodology for stochastic power flow in a distribution line with dispersed photovoltaic (PV) penetration. Both load and PV generation are stochastic processes. The methodology uses a probabilistic model for load demand based on measured data and an extensive stochastic modeling of PV units’ power production based on historical meteorological data and commercial available PV panels. Annual power flow simulations are performed to evaluate loss reduction resulting for different penetration levels and siting strategies of PV into an urban radial distribution feeder. Finally, a cost index for the losses is defined taking into account System Marginal Price data. Results may be of interest for dimensioning, siting and cost allocation in distribution systems with dispersed generation. © 2011 Elsevier B.V. All rights reserved.
1. Introduction Electric power systems (EPS) have been originally designed based on the unidirectional power flow. Nevertheless, in the last years the concept of distributed generation (DG) has led to new considerations concerning the distribution networks (DN) [1]. The penetration of DG may impact the operation of a DN in both beneficial and detrimental ways. Some of the positive impacts of DG are possibly: voltage support, power loss reduction, support of ancillary services and improved reliability, whereas negative ones regard protection coordination, dynamic stability and islanding. In order to maximize benefits and minimize problems, technical constraints concerning the interconnection of DG units and their penetration levels are being adopted worldwide. Furthermore, the presence of DG in the deregulated market has raised new regulatory issues, concerning financial incentives, cost allocation methods, generation management techniques, etc. [1–13]. Numerous researchers have dealt with the issue of penetration of DG into DNs. A group of articles optimizes sizing and/or siting of DG units in order to succeed maximum benefits, such as maximum loss reduction or reliability, minimum cost, etc. For example in [14], an optimal power flow (OPF) method is used to effectively maximize network capacity for the connection of DG. In [15] the
∗ Corresponding author at: ABB AB, Corporate Research, Västerås 72178, Sweden. Tel.: +46 21324140. E-mail addresses: [email protected], [email protected] (A.G. Marinopoulos). 0378-7796/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.epsr.2011.06.006
authors suggest analytical approaches for optimal siting of DG for both radial and networked systems to obtain minimum losses. In [16] optimal siting of DG under several technical constraints is examined. A multi-objective (MO) optimization procedure using a genetic algorithm (GA) to minimize cost for network upgrading, purchased energy, energy losses and energy not supplied is presented in [17]. Another MO approach is presented in [18], where impact indices are proposed for evaluating power losses, voltage profile, current capacity of lines, and fault levels. In [19,20] analytical approaches for DG penetration and siting are assessed for different DG technologies, whereas in [21] a fuzzy inference system is used to achieve optimal DG placement. The benefits arising from DG penetration are nowadays becoming even more important in the deregulated electricity market, since producers will eventually demand to be paid for their beneficial impact [2,3,9]. In order to treat all customers in a fair manner and be able to compensate or charge them accordingly, the distribution system operator (DSO) will have to calculate their contribution to fixed and running cost more accurately and, most important, over a long period of use, based on power flow calculations. However, the above mentioned papers use power flow analysis either for a certain loading condition or for a few specific scenarios (e.g. seasonal loadings) based on measured data or default test cases [3–5,9,10,14–21]. Other authors propose probabilistic analysis but mainly for reliability studies and/or for the transmission system [22–28]. In general, probabilistic (or stochastic) power flow analysis has been widely used but not combined with time varying loads. In [29] the authors use Monte Carlo simulations for DG in low voltage
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(LV) grids to examine the impact on voltage limits. However, they focus only on three typical seasonal loadings (winter, midseason, summer) and do not investigate losses. Stochastic modeling with Monte Carlo is also used in [30], however it is applied on the yearly discount rate and the fuel prices and the work regards least cost planning. Until now, only a few studies attempt to evaluate DG impact on loss reduction using stochastic loading conditions, but even then they do not take into account long time simulations. In specific, authors in [6,7] deal with voltage rise and power losses reduction from widespread photovoltaics, limited in two typical days for summer and winter; while in [31] hourly power flow simulations are performed for a period of one day. However, cost is not considered. The present work introduces stochastic processes for both series of loads and PV generation, for any period. Also cost is considered based on stochastic System Marginal Prices. The paper is structured as follows. In Section 2 the general problem and the proposed methodology used for the stochastic analysis are presented. Section 3 describes the test case in which the methodology is applied, along with the performed simulations. Finally, the overall results of the simulations with the arising discussion and some general conclusions are presented in Sections 4 and 5, respectively.
2. Description of the problem and the stochastic processes: methodology analysis In this section the proposed stochastic procedure will be presented for the time-varying loads and PV production. Furthermore, using the output of power flow analysis the cost of losses is estimated based on stochastic values of the System Marginal Price (SMP). The analysis concerns a Medium Voltage Distribution Line with nominal voltage U1 , which feeds loads over distribution transformers at low voltage U2 , as shown in Fig. 1. Photovoltaic Generators are connected to low voltage, as well. There are m transformers connected at m corresponding coupling points (common points of coupling) to the line. Each transformer i, for i = 1, . . ., m, serves aggregated loads and photovoltaic generations. The objective is to estimate, for a period of time, the losses and their cost, for stochastic load, PV production and SMP. This is done by executing load flow calculations for each of the below mentioned time intervals. A time period T, e.g. a year, is considered, subdivided in a sequence of equal time intervals tj for j = 1, . . ., n, e.g. 1 h. Time step tj is followed by the next time step tj+1 and so on, while in each step load and photovoltaic generation are considered as state variables. In case of deterministic values usually the term “time series” is used, while for stochastic values the term “stochastic processes” is used. Here, in each time step and for each coupling point the values of load and photovoltaic generation will be formed by adding to a deterministic value a stochastic part that follows different random distributions.
1987
For each coupling point i and time interval tj four variables have to be initially considered: Pi,j = Si,j · cos ϕi,j
load active power : load reactive power :
Qi,j = Si,j · sin ϕi,j
PV active power generation : PV reactive power generation :
PPV,i,j = i · Rj QPV,i,j = PPV,i,j · tan ϕPV
(1) (2) (3) (4)
where i = 1, . . ., m is the coupling point; j = 1, . . ., 17,520 is the 30min time step of a one-year period; ϕi,j is the load angle; Rj is the irradiation at time step tj ; i is a timely constant factor for transforming the irradiation to electrical power; cos ϕPV is the constant power factor of the inverters of the photovoltaic generators. In the test case studied, reactive power of PV generators is assumed zero. The remaining variables, Pi,j , Qi,j and Rj can be considered stochastic variables, meaning they have a deterministic and a stochastic part. Concerning the processes of the stochastic parts, two basic requirements are followed: 1. They are statistically independent. 2. They are Markov chains, so that the state tj+1 does not depend on previous state tj . As long as the stochastic processes meet the above requirements, their values can be calculated using a Monte Carlo method. To proceed with these simulations, we need to further study the time series of load and irradiation. 2.1. Load In the present work, the stochastic active and reactive powers needed for power flow are derived from the stochastic values of apparent power and the power factor of the load. The probabilistic modeling of loads, especially residential ones, is well justified by the fact that electricity demand is largely a stochastic process exhibiting diversity, acknowledged and illustrated in many papers [23–25,28,32,33]. The mean value of the load, different for each time step tj , was determined from historical measurements data of the total current Ij at the feeding end of the line, provided by the Greek Public Power Corporation (PPC). Then, the mean value of the load current Ii,j at any coupling point i is assumed to be proportional to the rated power of transformer i. Thus, for the stochastic process, the mean apparent power at the coupling point i at time tj is defined as: S¯ i,j =
√
3 · U1 · I¯i,j =
√
3 · U1 · Ij ·
S
mN,i
S i=1 N,i
(5)
where SN,i is the rated power of transformer i; I¯i,j is the mean value of the load current Ii,j at the coupling point i; Ij is the total current at the feeding end of the line; U1 is the voltage level. The corresponding deviation for each random distribution is considered to be a function of the mean value: i,j = f (S¯ i,j )
(6)
Thus, in case of normal distribution, the apparent power of load at the coupling point i at time interval tj can be expressed as:
sn Si,j ∼N S¯ i,j , S¯ i,j · 100 Fig. 1. Power flow general configuration for a distribution line during time interval tj . Pi,j , Qi,j are the stochastic loads, PPV,i,j , QPV,i,j the photovoltaic generation, and Zti the impedance of the ith transformer.
2
(7)
where Si,j is the stochastic apparent power of the load; S¯ i,j is the mean apparent power, as calculated from (5); sn is the standard deviation in % of the mean value S¯ i,j .
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In case of uniform distribution the load Si,j will be uniformly distributed between the values:
down Si,j = S¯ i,j · 1 −
su 100
su up Si,j = S¯ i,j · 1 + 100
(8)
(9)
down is the lower limit of the uniform distribution interval; where Si,j up
Si,j is the upper limit of the uniform distribution interval; su is a parameter in % of the mean value S¯ i,j , defining the length of the uniform distribution. In this point it has to be noted that, in the proposed methodology a global constraint is imposed so that the annual energy consumption of all loads remains the same for every Monte Carlo simulation. This is obviously done in order to be able to compare the results for the annual energy losses later, since a change in the load would lead to an intrinsic change of the losses. The constraint can be expressed from the below equation, which states that the total energy consumption is kept constant: Eyear =
m j
Pi,j = ct
(10)
i=1
where Pi,j is the active power demand of node i; Eyear is the total annual energy consumption of the feeder. Similar practices concerning the modeling of electric loads have been also used by other authors [23–25,28,29]. A typical assumption of some authors is that the power factor of the load has a constant value. In the present work, the power factor cos ϕi,j is considered to be a random variable uniformly distributed between an upper and a lower value, set before each simulation. A similar approach was practiced in [6,7]. Furthermore, the proposed algorithm, as explained later, offers also the possibility of keeping the power factor constant at any desirable value. From the above analysis it is clear that, the load in each node is recalculated at every time step. First of all, its mean value changes, since it is determined from the current at the beginning of the line, and following that change the deviation changes, as well. In both cases of normal and uniform distribution the standard deviation is constantly a percentage of the mean value. The above assumption is based on the fact that in times of low load demand the stochastic analysis provides random values with small variation, while in times of high load demand the variation is greater. This has also been described in previous works referring to load modeling [27,33]. 2.2. Photovoltaic generation The photovoltaic power depends on the irradiation Rj (W/m2 ), which is common at each point of coupling, since it is assumed to refer to the same geographical area considering solar radiation, but different for every time interval tj . However, the output in electric power depends also on the PV installation, i.e. panel size, inclination and ambient temperature. Also, the operating point of the inverter plays a role, determining the efficiency of energy transformation. The value of irradiation at time step tj is calculated in a stochastic way based on irradiation measurements over many years. This work has been done using methods described in the Data Base METEONORM© [34]. In specific, hourly values for the air temperature and mean irradiance of global horizontal radiation are calculated, based on historical data from 1961–2005 for temperature and 1981–2000 for global radiation. The method combines extended databases and algorithms and applies interpolation to calculate monthly average data for any desired location in the
world. Starting with monthly global radiation values, first the daily values, then the hourly values are generated stochastically. The stochastic models generate intermediate data having the same statistical properties as the measured ones. Thus, for every hour of the year and for a given locality one can obtain a stochastic value of irradiation. Now, the calculation of the stochastic value of PV power PPV,i,j is possible, depending on the PV installation, the number of panels, their inclination and the ambient temperature. In the present paper the software PVSYST© was used for the calculation of the transformation factor i , which determines the injecting power of PV for a given irradiation, for time step tj and coupling point i. PVSYST© software uses extensive meteorological data and PV systems components databases to calculate the power production of a PV installation [35], taking also into account the operating point of the associated inverter. Since the above results are calculated in a p.u. basis, in this work a 20 kW installed power PV unit was used as a base, and its power production curve was calculated during a period of one year. Interpolation was used for the PV production to obtain also intermediate values for 30-min intervals, as load profiles were in a 30-min basis, as well. Finally, for the reactive power QPV,i,j generated from the PV units, the power factor cos ϕPV was assumed constant and equal to one, i.e. we assumed that PV units do not control reactive power. The approach of modeling a PV unit as a PQ node with unity power factor is quite common in the literature [6,7,10,14,19,20]. It is noted that, PV units are assumed to be connected to the LV side of the Distribution Transformers (DTs) in parallel with the loads (see Fig. 1). This supposes that each PV unit cannot exceed the limit of 100 kW installed capacity, according to the directive of the Greek PPC for the interconnection of DG to the LV distribution grid. However, more than one PV units can be connected simultaneously to the same transformer as long as their total installed capacity does not exceed the transformer rated power. 2.3. System Marginal Price calculation In order to have a cost estimation of the losses, a cost index cj is used as following: cj = Plj · SMPj · tj
(11)
where Plj are the power losses for time interval tj in MW; SMPj is a stochastic value of the System Marginal Price at time interval tj in D /MWh; tj is the time step in hours. SMP for time step tj is considered as stochastic variable, with a random distribution, based on historical data from the market, provided by the TSO. For tj = 0.5 the annual cost index c (in D ) is defined as:
17,520
c=
cj
(12)
j=1
2.4. Flowchart of complete process Fig. 2 shows the flowchart of the proposed methodology. Branch and bus data along with load current profiles from the measurements at the feeding point of the line are read. Other inputs include the distribution to be used for the random load modeling, the installed capacity of PV units (penetration), their distribution across the feeder (siting), and their power production series as calculated earlier from PVSYST© . The different scenarios, which are going to be simulated, concern the power and siting of PV units, and are going to be thoroughly explained in Section 4. Finally, the time period of the simulation and the total number of itera-
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Then, for each node i containing PV units connected, the total load demand at time interval tj is calculated as: Pd,i,j = Pi,j − PPV,i,j
(13)
where Pd,i,j is the total active power demand of node i; Pi,j is the load active power generated randomly for node i; PPV,i,j is the total power production of all PV units connected to node i. The total reactive power demand Qd,i,j for each node i is equal with Qi,j , since it was assumed that PV units do not produce any reactive power. Using the calculated P–Q values for all nodes and the input data for all branches, the power flow problem is solved with a standard Newton–Raphson method. For the power flow, a MATLABTM [36] power system simulation package named MATPOWER was used [37]. The m-files were appropriately modified in order to be included in the proposed method. Power losses for each branch and for the feeder, as well as node voltages are stored after each power flow. At this point, stochastic variable SMPj is read and combined with power losses providing the cost index cj . After the simulation for the whole year is completed, values of energy losses, node voltages, and cost index are again calculated and stored. After that, the whole process is repeated again and again for different sets of random values for load demand and PV production and the overall results are being analyzed using simple statistical methods. 2.5. Output databases The above methodology provides for each time step tj and for each PV scenario results from many M Monte Carlo iterations (e.g. in this work M = 50), which can be used for statistical processing or for further analyses. Variables calculated are: - node voltages - branch losses - cost of branch losses. In the present work, mean annual energy losses and cost of the feeder and of each branch individually are used to evaluate the different penetration and siting scenarios for the PV units. Also voltage profiles for all nodes may be used to impose technical limitations for PV penetration. In this work, since the scenarios did not include very high penetration levels, no voltage violations were observed. This was verified by a function in the algorithm, which was checking the voltage of each node in every time step, in order not to exceed the voltage range allowed in DNs, i.e. ±5% of the nominal voltage. 3. Study case Fig. 2. Flowchart of stochastic power flow method.
tions using random loading for each time (Monte Carlo simulations) are given. After all the input data have been read, the Monte Carlo simulations begin for one scenario. The program runs a series of iterations, each of them containing as many power flows as defined previously. In case of an annual simulation, for each iteration 17,520 sequential power flows are performed, one for each 30-min interval. The process is as follows: for the time step tj of iteration s (s = 1, . . ., M, see Fig. 2) the program generates randomly values for the load demand of each node, following a given distribution pattern. Active and reactive power demands are calculated from (1) and (2), respectively, either for constant or for random power factor.
The analysis was applied on a Medium Voltage (MV) distribution line in Greece. 3.1. Description of the radial feeder The area chosen for simulation lies in the centre of Thessaloniki, a city in northern Greece. This area has a relatively high load density with many low voltage residential and commercial customers. The distribution feeder, Fig. 3, is fed by a 150/20 kV transformer. The 20 kV terminals of that transformer are considered as an infinite bus. Five feeders are connected to the 20 kV side of the transformer, however only one has been taken into consideration for this study. The feeder studied has a total length of 6.23 km and consists of underground cables of various lengths. The type of the cables is 3 × 240 mm2 NAEKBA,
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Fig. 3. General layout of the test feeder.
Table 1 Distribution transformers characteristics. Type (no.) (acc. Fig. 3)
Sr (kVA)
uk (%)
r (%)
I0 (%)
PFe (kW)
A (17) B (1) C (1) D (1)
630 1000 250 500
5.24 6 5 5.2
1.12 1.13 1.11 1.12
2 2 2 2
1.3 1.65 0.65 1.1
with R = 0.15 /km, X = 0.108 /km and maximum thermal current Imax = 310 A. Twenty 20/0.4 kV distribution transformers with a total rated power of 12 MVA are feeding the various customers in LV. The rated values of the four different types of DTs, such as the rated power Sr , the short-circuit voltage uk , the p.u. resistance r, the open-circuit current I0 , and the iron-core losses PFe , are given in Table 1. 3.2. Description of the simulations All the above data concerning the feeder and the distribution transformers were used as inputs into the program. Afterwards, a number of simulations for various scenarios were performed using stochastic data for loads and PV power production. First of all, a yearlong power flow was conducted using the mean measured values for load current, a typical power factor for the load of 0.95 lagging, and no PV units connected. The measurements as obtained by the Greek PPC were processed for one year, 15/11/2006 to 14/11/2007. This deterministic power flow is going to be used as reference for comparison with stochastic power flows, in order to evaluate the impact of stochasticity. Regarding the probabilistic power flows, again without any PV units, Monte Carlo method is used to provide deviations from the deterministic load series following either uniform or Gaussian distribution with various parameters. Afterwards, mean daily profiles of load demand, PV energy production, and SMP are presented. These profiles describe the basic characteristics of the examined case, provide useful information and lead to valuable conclusions. The main set of annual simulations concerns various scenarios for different cases of PV units’ siting and penetration levels. A final set of simulations is also performed to assess the specific power production profile of PV systems. 4. Simulations and results 4.1. Deterministic versus stochastic load power flow Both deterministic and stochastic load power flows are performed in order to estimate yearly energy losses. The deterministic case uses the processed measured load series. On the contrary, for the stochastic power flows fifty different Monte Carlo simulations are performed for each case of different PDF and deviation, producing fifty alternative results for energy losses. The range of the
results and their average value are presented in Tables 2 and 3, for normal and uniform PDF respectively. It has to be noted that, for each Monte Carlo simulation the total load energy consumption remains the same, see Eq. (10), however the variation in load distribution is different. One would expect a larger deviation among the results, especially for as few as fifty simulations. However, energy losses are actually the sum of 17,520 values of power losses, thus for such a large number of uncorrelated random values the result will be very close to the mean value. From Tables 2 and 3 it is clear that, the larger the standard deviation of the distribution is, the higher the energy losses of the feeder will be, which is also proven mathematically is presented in Appendix A. It is reminded that, the standard deviation of normal PDF and the length of uniform PDF are determined from (7)–(9), respectively. Thus, parameter sn coincides with the standard deviation, however su is actually the half of the uniform distribution’s length as a percentage of the mean value. For example, su = 20% means that stochastic variable Si,j is uniformly distributed between 80% and 120% of its mean value. Therefore, an immediate comparison between normal and uniform distributions according to parameters sn and su should be avoided. However, due to the linear relationship between su and the standard deviation of the uniform PDF, a rough comparison can be made for qualitative reasons. Finally, uniform PDF sets specific limits for the stochastic variable, whereas for normal PDF and a large number of samples, extreme values may show up. Thereupon, due to these peak values and based on the analysis performed in Appendix A, higher losses are expected when using normal PDF for the stochastic loads. The above result can be very useful to the DSO when calculating the expected energy losses and the impact of a DG unit interconnection. Using historical data and performing statistical analyses the DSO could estimate the deviation expected for the load demand, thus the increase in expected calculated losses, as well. In other words, the proposed methodology outweighs the results of a respective deterministic modeling, since it takes in mind the stochastic nature of many variables, otherwise considered constant. Therefore, the DSO would be sure to stand in the safe side, regarding the ways to do the cost allocation, however it is not in the scope of this paper to investigate these issues. 4.2. Mean daily profiles Based on measurements, the mean daily profiles of load demand and PV production, as well as SMP for the same time period, as taken
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Table 2 Annual energy losses without PV in MWh – deterministic vs. stochastic power flow with normal PDF. Deterministic power flow
183.475
Standard deviation in % of mean value
Stochastic power flow with normal PDF
10% 20% 30% 40% 50%
Min
Mean
Max
184.29 187.04 191.86 197.99 207.32
184.41 187.37 192.13 199.01 207.74
184.50 187.64 192.53 199.48 208.56
Table 3 Annual energy losses without PV in MWh – deterministic vs. stochastic power flow with uniform PDF. Deterministic power flow
183.475
Half length in % of mean value
10% 20% 30% 40% 50%
from historical data provided by the Hellenic TSO [38], are shown in Figs. 4–6, respectively. The maximum load demand for the examined feeder is presented in winter and it is this season that mostly determines the total load profile, as seen in Fig. 4. While summer presents a typical load profile with two peak periods in the morning and the evening, in winter (in spring and autumn, as well, but less obvious) the peak demand occurs between 11:00 p.m. and 03:00 a.m. This is due to the fact that, in the studied area there is a high penetration of night heat storage, which absorbs high amounts of electric energy during the night. Thus, it contributes in a peak demand during these hours, even higher than the midday peak, as its operation is almost concurrent in all customers. Fig. 5 presents the PV production curve, which typically follows the variations of solar radiation in Greece. Thus, summer production has the highest peak, and the daylight interval is the longest. Finally, Fig. 6 shows the SMP, where winter presents the highest value and spring the lowest. The profile is almost the same for all seasons, except from winter when even in the night hours the price does not fall below 50 D /MWh.
Fig. 4. Mean daily profile of load demand for the examined feeder.
Stochastic power flow with uniform PDF Min
Mean
Max
183.70 184.61 186.13 188.28 191.05
183.79 184.77 186.41 188.64 191.54
183.86 185.01 186.89 189.04 192.08
Fig. 5. Mean daily profile of power production for a 3 MW PV unit.
Fig. 6. Mean daily profile of SMP.
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Fig. 7. Schematic illustration of the four topologies for PV units’ siting.
4.3. PV siting and penetration scenarios The siting of PV units was based on four different topologies. Topology A considers uniformly distributed PV units in all the DTs across the feeder. Topology B assumes three installation points for the PV units in the beginning, the middle and the end of the feeder. In topology C the first installation points were DT nos. 10 and 11 (in the middle of the feeder) and additional capacity of PV units was installed to adjacent DTs symmetrically. Finally, in topology D the first installation point was the DT at the end of the feeder and additional PV units were installed to adjacent DTs towards the beginning of the feeder. The above mentioned topologies of the PV units are shown schematically in Fig. 7 for a random penetration level. As the penetration level increases, the installed capacity of these units will also increase and upon reaching the rated power limit of the transformer more units will be installed in the adjacent nodes. Different penetration levels were used for each of the above topologies of PV units. The penetration level was defined as the ratio of the total installed capacity of PV units to the total loading capacity of the feeder. In total, five different penetration levels were considered, i.e. 10–50% in steps of 10%. The combination of the four different topologies with the five penetration levels provided twenty different scenarios. Even though the above scenarios were simulated for both uniform and normal distribution functions, regarding load demand, in the following results only the latter will be illustrated. This was considered to be more suitable to model the random load demand for each node, since in the examined feeder the loads are already aggregated, thus tend to present a normal rather than uniform distribution. The respective results, in case of uniform distribution, present similar qualitative behavior and differ only in quantitative manner, as expected from the previous analysis (see Tables 2 and 3). In Fig. 8 the annual energy losses of the feeder are illustrated for random load demand for each node with normal PDF and standard deviation 20% of the mean value. The profiles of energy losses curves for a range of standard deviation until the case of 50% of the mean value are exactly the same, with only quantitative differences. It has to be noted that, the losses in Fig. 8 are the mean values resulting from fifty different Monte Carlo simulations for a whole year. The deviation of the results did not exceed ±0.5% for reasons explained earlier in the paper. Fig. 8 shows that the best topology for maximum loss reduction is A, i.e. PV units uniformly distributed across the feeder. However, annual energy losses as a function of PV penetration level present a U-shape trajectory for all topologies. Losses start to decrease sharply for low PV penetration, whereas after a specific point (different for each topology) they start to increase with higher penetration levels. This point seems to be a bit higher than
Fig. 8. Annual energy losses. Mean values from 50 different simulations for random load demand with normal PDF and 20% standard deviation.
50% penetration for topology A, around 50% for topology B, 40% for topology C and 35% for topology D. It is also important to notice that, excluding topology A that is in general the best one, for every penetration level there is a different topology leading to maximum loss reduction. The above results are presented in Table 4, as % loss reduction in respect to the base scenario (without PV). Along with loss reduction lred , reduction cred of annual cost index is also given (mean value). As observed, the tendency in cost reduction is the same with losses, however the percentages are little higher. This is explained by the fact that PV units contribute in loss reduction during daytime, and mostly around noon, when SMP presents its peak value. From the above results some general conclusions can be drawn. First of all, uniform distribution of PV units across the feeder provides the maximum loss reduction. However, the uniform installation of PV units is sort of an ideal scenario and in reality almost impossible to occur. Thus, the best realizable siting of PV units seems to be the installation in three different points symmetrically. However, for some penetration levels other topologies may be even better, i.e. the best siting is penetration dependent. Moreover, for every topology there is a maximum penetration level, above which the energy losses start to increase and for extreme cases would lead to greater values, even higher than the base scenario (feeder without PV). Finally, cost reduction (due to loss reduction) resulting from PV penetration seems to be important, since PV units produce their power in high SMP hours. Finally, a last set of simulations was conducted in order to evaluate the effect of DG production profile. In specific, a virtual type of
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Table 4 Annual loss & cost reduction in % of base scenario. Topology
Penetration level 10%
A B C D
20%
30%
40%
50%
lred
cred
lred
cred
lred
cred
lred
cred
lred
cred
9.21 6.69 5.08 5.70
9.52 6.97 5.35 5.99
16.11 10.11 8.62 9.32
16.70 10.60 9.10 9.83
20.95 10.41 10.78 11.06
21.79 11.07 11.47 11.79
23.65 13.44 13.11 11.33
24.69 14.31 14.03 12.23
24.17 13.57 11.46 10.19
25.40 14.41 12.53 11.26
I0 of the stochastic currents. To prove this, we introduce n quantities d1 , d2 , . . ., dn defined as follows: Ii = I0 + di ,
i = 1, . . . , n
(A1)
Obviously, the mean value of di for i = 1, . . ., n is zero: n
di = 0
(A2)
i=1
Using (A1) and (A2):
n
i=1
Ii2 =
n
2
(I0 + di ) =
i=1
n i=1
(I02 + di2 + 2I0 di ) = n · I02 +
n
di2
(A3)
i=1
Therefore n
Fig. 9. Comparison between PV units and virtual DG units operating in constant power, both producing the same annual energy.
DG unit was used, capable of operating during the whole year with constant power output, producing the same amount of energy as a PV unit. For the comparison of the two cases, the best topology, i.e. topology A, was used and 50 different power flows were simulated for each penetration level with random load profiles with normal PDF and standard deviation 20% of the mean value. The results are illustrated in Fig. 9, from which it is concluded that a DG unit has a better impact on loss reduction when producing constant power. Thus, as far as energy loss reduction is concerned, DG will have the maximum positive impact when, topologically it is uniformly distributed (i.e. uniform siting) and chronically it produces power in a uniform way (i.e. constant power production).
5. Conclusion A methodology for stochastic power flow studies in distribution networks is proposed, showing the impact of random load profiles on energy losses calculation. Using this methodology, various scenarios concerning PV penetration and siting in a radial distribution feeder are simulated. Results have shown that losses calculated with stochastic power flow are higher than the corresponding values of a deterministic power flow. Uniformly distributed dispersed generation may lead to optimal siting concerning losses in a distribution line.
Appendix A. We consider a period consisting of a sequence of n successive equal intervals of time. The aggregated Joule losses for a stochastic current flow I1 , I2 , I3 , . . ., In in a conductor are always greater than the losses caused by currents I1 , I2 , I3 , . . ., In equal to the mean value
Ii2 ≥ n · I02
(A4)
i=1
Assuming conductor resistance equal to 1 and interval of time 1 s the above inequality constitutes the desired proof. Furthermore, the difference between losses is increasing with the variance, i.e. n d2 /n. i=1 i References [1] T. Ackermann, G. Andersson, L. Soder, Distributed generation: a definition, Electr. Pow. Syst. Res. 57 (3) (2001) 195–204. [2] J.A. Pecas Lopes, N. Hatziargyriou, J. Mutale, P. Djapic, N. Jenkins, Integrating distributed generation into electric power systems: a review of drivers, challenges and opportunities, Electr. Pow. Syst. Res. 77 (9) (2007) 1189–1203. [3] J. Mutale, G. Strbac, S. Curcic, N. Jenkins, Allocation of losses in distribution systems with embedded generation, IEE Proc.-Gener. Transm. Distrib. 147 (1) (2000) 7–14. [4] P. Chiradeja, R. Ramakumar, An approach to quantify the technical benefits of distributed generation, IEEE T. Energy Conver. 19 (4) (2004) 764–773. [5] P. Chiradeja, Benefit of distributed generation: a line loss reduction analysis, Presented at Transmission and Distribution Conference and Exhibition: Asia and Pacific, IEEE/PES, 2005, pp. 1–5. [6] M. Thomson, D.G. Infield, Impact of widespread photovoltaics generation on distribution systems, IET Renew. Power Gen. 1 (1) (2007) 33–40. [7] M. Thomson, D.G. Infield, Network power-flow analysis for a high penetration of distributed generation, IEEE T. Power Syst. 22 (3) (2007) 1157–1162. [8] R.A. Walling, R. Saint, R.C. Dugan, J. Burke, L.A. Kojovic, Summary of distributed resources impact on power delivery systems, IEEE T. Power Deliver. 23 (3) (2008) 1636–1644. [9] J. Deuse, S. Grenard, D. Benintendi, P.J. Agrell, P. Bogetoft, Use of system charges methodology and norm models for distribution system including DER, in: Proc. Int. Conf. on Electricity Distribution, CIRED 2007, Paper No. 76, Vienna, May 21–24, 2007. [10] S. Kotamarty, S. Khushalani, N. Schulz, Impact of distributed generation on distribution contingency analysis, Electr. Pow. Syst. Res. 78 (9) (2008) 1537–1545. [11] S.A. Papathanassiou, A technical evaluation framework for the connection of DG to the distribution network, Electr. Pow. Syst. Res. 77 (1) (2007) 24–34. [12] G. Pepermans, J. Driesen, D. Haeseldonckx, R. Belmans, W. D’haeseleer, Distributed generation: definition, benefits and issues, Energy Policy 33 (6) (2005) 787–798. [13] P.P. Barker, R.W. De Mello, Determining the impact of distributed generation on power systems. I. Radial distribution systems, in: Proc. IEEE Power Engineering Society Summer Meeting, vol. 3, 2000, pp. 1645–1656. [14] G.P. Harrison, A.R. Wallace, Optimal power flow evaluation of distribution network capacity for the connection of distributed generation, IEE Proc.-Gener. Transm. Distrib. 152 (1) (2005) 115–122.
1994
A.G. Marinopoulos et al. / Electric Power Systems Research 81 (2011) 1986–1994
[15] C. Wang, M.H. Nehrir, Analytical approaches for optimal placement of distributed generation sources in power systems, IEEE T. Power Syst. 19 (4) (2004) 2068–2076. [16] A. Keane, M. O’Malley, Optimal allocation of embedded generation on distribution networks, IEEE T. Power Syst. 20 (3) (2005) 1640–1646. [17] G. Celli, E. Ghiani, S. Mocci, F. Pilo, A multiobjective evolutionary algorithm for the sizing and siting of distributed generation, IEEE T. Power Syst. 20 (2) (2005) 750–757. [18] L.F. Ochoa, A. Padilha-Feltrin, G.P. Harrison, Evaluating distributed generation impacts with a multiobjective index, IEEE T. Power Deliver. 21 (3) (2006) 1452–1458. [19] V.H.M. Quezada, J.R. Abbad, T.G.S. Roman, Assessment of energy distribution losses for increasing penetration of distributed generation, IEEE T. Power Syst. 21 (2) (2006) 533–540. [20] N. Acharya, P. Mahat, N. Mithulananthan, An analytical approach for DG allocation in primary distribution network, Int. J. Elec. Power 28 (10) (2006) 669–678. [21] A. Lakshmi Devi, B. Subramanyam, Optimal dg unit placement for loss reduction in radial distribution system—a case study, ARPN J. Eng. Appl. Sci. 2 (6) (2007) 57–61. [22] M.V.F. Pereira, M.E.P. Maceira, G.C. Oliveira, L.M.V.G. Pinto, Combining analytical models and Monte-Carlo techniques in probabilistic power system analysis, IEEE T. Power Syst. 7 (2) (1992) 265–272. [23] X. Long, W. Yan, Z. Lu, Monte Carlo method for line losses evaluation in transmission network, in: Proc. Int. Conf. on Power System Technology, October 22–26, 2006, pp. 1–5. [24] G.K. Stefopoulos, A.P. Meliopoulos, G.J. Cokkinides, Probabilistic power flow with non-conforming electric loads, Int. J. Elec. Power 27 (9–10) (2005) 627–634. [25] A.P.S. Meliopoulos, G.J. Cokkinides, X.Y. Chao, A new probabilistic power flow analysis method, IEEE T. Power Syst. 5 (1) (1990) 182–190. [26] A. Sankarakrishnan, R. Billinton, Sequential Monte Carlo simulation for composite power system reliability analysis with time varying loads, IEEE T. Power Syst. 10 (3) (1995) 1540–1545. [27] X. Li, Y. Li, S. Zhang, Analysis of probabilistic optimal power flow taking account of the variation of load power, IEEE T. Power Syst. 23 (3) (2008) 992–999. [28] A.P.S. Meliopoulos, X.Y. Chao, G. Cokkinides, R. Monsalvatge, Transmission loss evaluation based on probabilistic power flow, IEEE T. Power Syst. 6 (1) (1991) 364–371. [29] F. Demailly, O. Ninet, A. Even, Numerical tools and models for Monte Carlo studies of the influence on embedded generation on voltage limits in LV grids, IEEE T. Power Deliver. 20 (3) (2005) 2343–2350. [30] N. Vaitheeswaran, R. Balasubramanian, Stochastic model for optimal selection of DDG by Monte Carlo simulation, in: Proc. Int. Conf. on Power Elec. Drives and Energy Systems, 2006, pp. 1–5.
[31] W. El-Khattam, Y.G. Hegazy, M.M.A. Salama, Investigating distributed generation systems performance using Monte Carlo simulation, IEEE T. Power Syst. 21 (2) (2006) 524–532. [32] D.H.O. McQueen, P.R. Hyland, S.J. Watson, Monte Carlo simulation of residential electricity demand for forecasting maximum demand on distribution networks, IEEE T. Power Syst. 19 (3) (2004) 1685–1689. [33] A. Capasso, W. Grattieri, R. Lamedica, A. Prudenzi, A bottom-up approach to residential load modeling, IEEE T. Power Syst. 9 (2) (1994) 957–964. [34] METEONORM© Version 6.0, October 2007, http://www.meteonorm.com. [35] PVSYST©Version 4.33, October 2008, http://www.pvsyst.com. [36] MATLAB® Version 7.0, May 2004, http://www.mathworks.com. [37] R.D. Zimmerman, C.E. Murillo-Sanchez, D. Gan, “MATPOWER: A MATLABTM Power System Simulation Package,” ©1997–2007 Power Systems Engineering Research Center (PSERC), School of Electrical Engineering, Cornell University, Ithaca, NY 14853, Version 3.2, September 2007, http://www.pserc.cornell.edu/matpower. [38] http://www.desmie.gr. Antonios G. Marinopoulos was born in Thessaloniki, Greece, in July 1980. He received his Dipl.-Eng. degree in 2003 and the Ph.D. degree in 2009, both from the Department of Electrical and Computer Engineering at the Aristotle University of Thessaloniki, Thessaloniki, Greece. He is currently a Research and Development Engineer in ABB AB, Corporate Research, Västerås, Sweden. His research interests are in the field of power system analysis and simulation, power system integration of renewable energy sources, and distributed generation. Minas C. Alexiadis was born in Thessaloniki, Greece, in July 1969. He received his Dipl.-Eng. degree in 1994 and the Ph.D. degree in 2003, both from the Department of Electrical and Computer Engineering at the Aristotle University of Thessaloniki, Thessaloniki, Greece. He is currently a Lecturer with the Department of Electrical and Computer Engineering at the Aristotle University of Thessaloniki, Thessaloniki, Greece. His research interests include distributed generation, renewable energy sources and artificial intelligence applications in power systems. Petros S. Dokopoulos was born in Athens, Greece, in September 1939. He received his Dipl. Eng. degree from the Technical University of Athens, Athens, Greece, in 1962 and the Ph.D. degree from the University of Brunswick, Brunswick, Germany, in 1967. From 1962 to 1967, he was with the Laboratory for High Voltage and Transmission, University of Brunswick. From 1967 to 1974, he was with the Nuclear Research Center, Julich, Germany, and from 1974 to 1978, he was attached to the German team of the Joint European Torus in Oxfordshire, UK. From 1978 to 2006 he was Full Professor with the Department of Electrical Engineering, Aristotle University of Thessaloniki, Thessaloniki, Greece. He is currently Professor Emeritus with the same department. His scientific fields of interest are dielectrics, power switches, power generation (conventional and fusion), transmission, and distribution and control in power systems.
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Electric Power Systems Research journal homepage: www.elsevier.com/locate/epsr
Energy losses in a distribution line with distributed generation based on stochastic power flow Antonios G. Marinopoulos a,b,∗ , Minas C. Alexiadis b , Petros S. Dokopoulos b a b
ABB AB, Corporate Research, SE-72178 Västerås, Sweden Power Systems Laboratory, Department of Electrical and Computer Engineering, Aristotle University of Thessaloniki, 54124 Thessaloniki, Greece
a r t i c l e
i n f o
Article history: Received 26 March 2010 Received in revised form 30 May 2011 Accepted 17 June 2011 Available online 16 July 2011 Keywords: Distributed generation Losses Monte Carlo simulation Photovoltaic power systems Power distribution
a b s t r a c t This paper proposes a methodology for stochastic power flow in a distribution line with dispersed photovoltaic (PV) penetration. Both load and PV generation are stochastic processes. The methodology uses a probabilistic model for load demand based on measured data and an extensive stochastic modeling of PV units’ power production based on historical meteorological data and commercial available PV panels. Annual power flow simulations are performed to evaluate loss reduction resulting for different penetration levels and siting strategies of PV into an urban radial distribution feeder. Finally, a cost index for the losses is defined taking into account System Marginal Price data. Results may be of interest for dimensioning, siting and cost allocation in distribution systems with dispersed generation. © 2011 Elsevier B.V. All rights reserved.
1. Introduction Electric power systems (EPS) have been originally designed based on the unidirectional power flow. Nevertheless, in the last years the concept of distributed generation (DG) has led to new considerations concerning the distribution networks (DN) [1]. The penetration of DG may impact the operation of a DN in both beneficial and detrimental ways. Some of the positive impacts of DG are possibly: voltage support, power loss reduction, support of ancillary services and improved reliability, whereas negative ones regard protection coordination, dynamic stability and islanding. In order to maximize benefits and minimize problems, technical constraints concerning the interconnection of DG units and their penetration levels are being adopted worldwide. Furthermore, the presence of DG in the deregulated market has raised new regulatory issues, concerning financial incentives, cost allocation methods, generation management techniques, etc. [1–13]. Numerous researchers have dealt with the issue of penetration of DG into DNs. A group of articles optimizes sizing and/or siting of DG units in order to succeed maximum benefits, such as maximum loss reduction or reliability, minimum cost, etc. For example in [14], an optimal power flow (OPF) method is used to effectively maximize network capacity for the connection of DG. In [15] the
∗ Corresponding author at: ABB AB, Corporate Research, Västerås 72178, Sweden. Tel.: +46 21324140. E-mail addresses: [email protected], [email protected] (A.G. Marinopoulos). 0378-7796/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.epsr.2011.06.006
authors suggest analytical approaches for optimal siting of DG for both radial and networked systems to obtain minimum losses. In [16] optimal siting of DG under several technical constraints is examined. A multi-objective (MO) optimization procedure using a genetic algorithm (GA) to minimize cost for network upgrading, purchased energy, energy losses and energy not supplied is presented in [17]. Another MO approach is presented in [18], where impact indices are proposed for evaluating power losses, voltage profile, current capacity of lines, and fault levels. In [19,20] analytical approaches for DG penetration and siting are assessed for different DG technologies, whereas in [21] a fuzzy inference system is used to achieve optimal DG placement. The benefits arising from DG penetration are nowadays becoming even more important in the deregulated electricity market, since producers will eventually demand to be paid for their beneficial impact [2,3,9]. In order to treat all customers in a fair manner and be able to compensate or charge them accordingly, the distribution system operator (DSO) will have to calculate their contribution to fixed and running cost more accurately and, most important, over a long period of use, based on power flow calculations. However, the above mentioned papers use power flow analysis either for a certain loading condition or for a few specific scenarios (e.g. seasonal loadings) based on measured data or default test cases [3–5,9,10,14–21]. Other authors propose probabilistic analysis but mainly for reliability studies and/or for the transmission system [22–28]. In general, probabilistic (or stochastic) power flow analysis has been widely used but not combined with time varying loads. In [29] the authors use Monte Carlo simulations for DG in low voltage
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(LV) grids to examine the impact on voltage limits. However, they focus only on three typical seasonal loadings (winter, midseason, summer) and do not investigate losses. Stochastic modeling with Monte Carlo is also used in [30], however it is applied on the yearly discount rate and the fuel prices and the work regards least cost planning. Until now, only a few studies attempt to evaluate DG impact on loss reduction using stochastic loading conditions, but even then they do not take into account long time simulations. In specific, authors in [6,7] deal with voltage rise and power losses reduction from widespread photovoltaics, limited in two typical days for summer and winter; while in [31] hourly power flow simulations are performed for a period of one day. However, cost is not considered. The present work introduces stochastic processes for both series of loads and PV generation, for any period. Also cost is considered based on stochastic System Marginal Prices. The paper is structured as follows. In Section 2 the general problem and the proposed methodology used for the stochastic analysis are presented. Section 3 describes the test case in which the methodology is applied, along with the performed simulations. Finally, the overall results of the simulations with the arising discussion and some general conclusions are presented in Sections 4 and 5, respectively.
2. Description of the problem and the stochastic processes: methodology analysis In this section the proposed stochastic procedure will be presented for the time-varying loads and PV production. Furthermore, using the output of power flow analysis the cost of losses is estimated based on stochastic values of the System Marginal Price (SMP). The analysis concerns a Medium Voltage Distribution Line with nominal voltage U1 , which feeds loads over distribution transformers at low voltage U2 , as shown in Fig. 1. Photovoltaic Generators are connected to low voltage, as well. There are m transformers connected at m corresponding coupling points (common points of coupling) to the line. Each transformer i, for i = 1, . . ., m, serves aggregated loads and photovoltaic generations. The objective is to estimate, for a period of time, the losses and their cost, for stochastic load, PV production and SMP. This is done by executing load flow calculations for each of the below mentioned time intervals. A time period T, e.g. a year, is considered, subdivided in a sequence of equal time intervals tj for j = 1, . . ., n, e.g. 1 h. Time step tj is followed by the next time step tj+1 and so on, while in each step load and photovoltaic generation are considered as state variables. In case of deterministic values usually the term “time series” is used, while for stochastic values the term “stochastic processes” is used. Here, in each time step and for each coupling point the values of load and photovoltaic generation will be formed by adding to a deterministic value a stochastic part that follows different random distributions.
1987
For each coupling point i and time interval tj four variables have to be initially considered: Pi,j = Si,j · cos ϕi,j
load active power : load reactive power :
Qi,j = Si,j · sin ϕi,j
PV active power generation : PV reactive power generation :
PPV,i,j = i · Rj QPV,i,j = PPV,i,j · tan ϕPV
(1) (2) (3) (4)
where i = 1, . . ., m is the coupling point; j = 1, . . ., 17,520 is the 30min time step of a one-year period; ϕi,j is the load angle; Rj is the irradiation at time step tj ; i is a timely constant factor for transforming the irradiation to electrical power; cos ϕPV is the constant power factor of the inverters of the photovoltaic generators. In the test case studied, reactive power of PV generators is assumed zero. The remaining variables, Pi,j , Qi,j and Rj can be considered stochastic variables, meaning they have a deterministic and a stochastic part. Concerning the processes of the stochastic parts, two basic requirements are followed: 1. They are statistically independent. 2. They are Markov chains, so that the state tj+1 does not depend on previous state tj . As long as the stochastic processes meet the above requirements, their values can be calculated using a Monte Carlo method. To proceed with these simulations, we need to further study the time series of load and irradiation. 2.1. Load In the present work, the stochastic active and reactive powers needed for power flow are derived from the stochastic values of apparent power and the power factor of the load. The probabilistic modeling of loads, especially residential ones, is well justified by the fact that electricity demand is largely a stochastic process exhibiting diversity, acknowledged and illustrated in many papers [23–25,28,32,33]. The mean value of the load, different for each time step tj , was determined from historical measurements data of the total current Ij at the feeding end of the line, provided by the Greek Public Power Corporation (PPC). Then, the mean value of the load current Ii,j at any coupling point i is assumed to be proportional to the rated power of transformer i. Thus, for the stochastic process, the mean apparent power at the coupling point i at time tj is defined as: S¯ i,j =
√
3 · U1 · I¯i,j =
√
3 · U1 · Ij ·
S
mN,i
S i=1 N,i
(5)
where SN,i is the rated power of transformer i; I¯i,j is the mean value of the load current Ii,j at the coupling point i; Ij is the total current at the feeding end of the line; U1 is the voltage level. The corresponding deviation for each random distribution is considered to be a function of the mean value: i,j = f (S¯ i,j )
(6)
Thus, in case of normal distribution, the apparent power of load at the coupling point i at time interval tj can be expressed as:
sn Si,j ∼N S¯ i,j , S¯ i,j · 100 Fig. 1. Power flow general configuration for a distribution line during time interval tj . Pi,j , Qi,j are the stochastic loads, PPV,i,j , QPV,i,j the photovoltaic generation, and Zti the impedance of the ith transformer.
2
(7)
where Si,j is the stochastic apparent power of the load; S¯ i,j is the mean apparent power, as calculated from (5); sn is the standard deviation in % of the mean value S¯ i,j .
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In case of uniform distribution the load Si,j will be uniformly distributed between the values:
down Si,j = S¯ i,j · 1 −
su 100
su up Si,j = S¯ i,j · 1 + 100
(8)
(9)
down is the lower limit of the uniform distribution interval; where Si,j up
Si,j is the upper limit of the uniform distribution interval; su is a parameter in % of the mean value S¯ i,j , defining the length of the uniform distribution. In this point it has to be noted that, in the proposed methodology a global constraint is imposed so that the annual energy consumption of all loads remains the same for every Monte Carlo simulation. This is obviously done in order to be able to compare the results for the annual energy losses later, since a change in the load would lead to an intrinsic change of the losses. The constraint can be expressed from the below equation, which states that the total energy consumption is kept constant: Eyear =
m j
Pi,j = ct
(10)
i=1
where Pi,j is the active power demand of node i; Eyear is the total annual energy consumption of the feeder. Similar practices concerning the modeling of electric loads have been also used by other authors [23–25,28,29]. A typical assumption of some authors is that the power factor of the load has a constant value. In the present work, the power factor cos ϕi,j is considered to be a random variable uniformly distributed between an upper and a lower value, set before each simulation. A similar approach was practiced in [6,7]. Furthermore, the proposed algorithm, as explained later, offers also the possibility of keeping the power factor constant at any desirable value. From the above analysis it is clear that, the load in each node is recalculated at every time step. First of all, its mean value changes, since it is determined from the current at the beginning of the line, and following that change the deviation changes, as well. In both cases of normal and uniform distribution the standard deviation is constantly a percentage of the mean value. The above assumption is based on the fact that in times of low load demand the stochastic analysis provides random values with small variation, while in times of high load demand the variation is greater. This has also been described in previous works referring to load modeling [27,33]. 2.2. Photovoltaic generation The photovoltaic power depends on the irradiation Rj (W/m2 ), which is common at each point of coupling, since it is assumed to refer to the same geographical area considering solar radiation, but different for every time interval tj . However, the output in electric power depends also on the PV installation, i.e. panel size, inclination and ambient temperature. Also, the operating point of the inverter plays a role, determining the efficiency of energy transformation. The value of irradiation at time step tj is calculated in a stochastic way based on irradiation measurements over many years. This work has been done using methods described in the Data Base METEONORM© [34]. In specific, hourly values for the air temperature and mean irradiance of global horizontal radiation are calculated, based on historical data from 1961–2005 for temperature and 1981–2000 for global radiation. The method combines extended databases and algorithms and applies interpolation to calculate monthly average data for any desired location in the
world. Starting with monthly global radiation values, first the daily values, then the hourly values are generated stochastically. The stochastic models generate intermediate data having the same statistical properties as the measured ones. Thus, for every hour of the year and for a given locality one can obtain a stochastic value of irradiation. Now, the calculation of the stochastic value of PV power PPV,i,j is possible, depending on the PV installation, the number of panels, their inclination and the ambient temperature. In the present paper the software PVSYST© was used for the calculation of the transformation factor i , which determines the injecting power of PV for a given irradiation, for time step tj and coupling point i. PVSYST© software uses extensive meteorological data and PV systems components databases to calculate the power production of a PV installation [35], taking also into account the operating point of the associated inverter. Since the above results are calculated in a p.u. basis, in this work a 20 kW installed power PV unit was used as a base, and its power production curve was calculated during a period of one year. Interpolation was used for the PV production to obtain also intermediate values for 30-min intervals, as load profiles were in a 30-min basis, as well. Finally, for the reactive power QPV,i,j generated from the PV units, the power factor cos ϕPV was assumed constant and equal to one, i.e. we assumed that PV units do not control reactive power. The approach of modeling a PV unit as a PQ node with unity power factor is quite common in the literature [6,7,10,14,19,20]. It is noted that, PV units are assumed to be connected to the LV side of the Distribution Transformers (DTs) in parallel with the loads (see Fig. 1). This supposes that each PV unit cannot exceed the limit of 100 kW installed capacity, according to the directive of the Greek PPC for the interconnection of DG to the LV distribution grid. However, more than one PV units can be connected simultaneously to the same transformer as long as their total installed capacity does not exceed the transformer rated power. 2.3. System Marginal Price calculation In order to have a cost estimation of the losses, a cost index cj is used as following: cj = Plj · SMPj · tj
(11)
where Plj are the power losses for time interval tj in MW; SMPj is a stochastic value of the System Marginal Price at time interval tj in D /MWh; tj is the time step in hours. SMP for time step tj is considered as stochastic variable, with a random distribution, based on historical data from the market, provided by the TSO. For tj = 0.5 the annual cost index c (in D ) is defined as:
17,520
c=
cj
(12)
j=1
2.4. Flowchart of complete process Fig. 2 shows the flowchart of the proposed methodology. Branch and bus data along with load current profiles from the measurements at the feeding point of the line are read. Other inputs include the distribution to be used for the random load modeling, the installed capacity of PV units (penetration), their distribution across the feeder (siting), and their power production series as calculated earlier from PVSYST© . The different scenarios, which are going to be simulated, concern the power and siting of PV units, and are going to be thoroughly explained in Section 4. Finally, the time period of the simulation and the total number of itera-
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1989
Then, for each node i containing PV units connected, the total load demand at time interval tj is calculated as: Pd,i,j = Pi,j − PPV,i,j
(13)
where Pd,i,j is the total active power demand of node i; Pi,j is the load active power generated randomly for node i; PPV,i,j is the total power production of all PV units connected to node i. The total reactive power demand Qd,i,j for each node i is equal with Qi,j , since it was assumed that PV units do not produce any reactive power. Using the calculated P–Q values for all nodes and the input data for all branches, the power flow problem is solved with a standard Newton–Raphson method. For the power flow, a MATLABTM [36] power system simulation package named MATPOWER was used [37]. The m-files were appropriately modified in order to be included in the proposed method. Power losses for each branch and for the feeder, as well as node voltages are stored after each power flow. At this point, stochastic variable SMPj is read and combined with power losses providing the cost index cj . After the simulation for the whole year is completed, values of energy losses, node voltages, and cost index are again calculated and stored. After that, the whole process is repeated again and again for different sets of random values for load demand and PV production and the overall results are being analyzed using simple statistical methods. 2.5. Output databases The above methodology provides for each time step tj and for each PV scenario results from many M Monte Carlo iterations (e.g. in this work M = 50), which can be used for statistical processing or for further analyses. Variables calculated are: - node voltages - branch losses - cost of branch losses. In the present work, mean annual energy losses and cost of the feeder and of each branch individually are used to evaluate the different penetration and siting scenarios for the PV units. Also voltage profiles for all nodes may be used to impose technical limitations for PV penetration. In this work, since the scenarios did not include very high penetration levels, no voltage violations were observed. This was verified by a function in the algorithm, which was checking the voltage of each node in every time step, in order not to exceed the voltage range allowed in DNs, i.e. ±5% of the nominal voltage. 3. Study case Fig. 2. Flowchart of stochastic power flow method.
tions using random loading for each time (Monte Carlo simulations) are given. After all the input data have been read, the Monte Carlo simulations begin for one scenario. The program runs a series of iterations, each of them containing as many power flows as defined previously. In case of an annual simulation, for each iteration 17,520 sequential power flows are performed, one for each 30-min interval. The process is as follows: for the time step tj of iteration s (s = 1, . . ., M, see Fig. 2) the program generates randomly values for the load demand of each node, following a given distribution pattern. Active and reactive power demands are calculated from (1) and (2), respectively, either for constant or for random power factor.
The analysis was applied on a Medium Voltage (MV) distribution line in Greece. 3.1. Description of the radial feeder The area chosen for simulation lies in the centre of Thessaloniki, a city in northern Greece. This area has a relatively high load density with many low voltage residential and commercial customers. The distribution feeder, Fig. 3, is fed by a 150/20 kV transformer. The 20 kV terminals of that transformer are considered as an infinite bus. Five feeders are connected to the 20 kV side of the transformer, however only one has been taken into consideration for this study. The feeder studied has a total length of 6.23 km and consists of underground cables of various lengths. The type of the cables is 3 × 240 mm2 NAEKBA,
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Fig. 3. General layout of the test feeder.
Table 1 Distribution transformers characteristics. Type (no.) (acc. Fig. 3)
Sr (kVA)
uk (%)
r (%)
I0 (%)
PFe (kW)
A (17) B (1) C (1) D (1)
630 1000 250 500
5.24 6 5 5.2
1.12 1.13 1.11 1.12
2 2 2 2
1.3 1.65 0.65 1.1
with R = 0.15 /km, X = 0.108 /km and maximum thermal current Imax = 310 A. Twenty 20/0.4 kV distribution transformers with a total rated power of 12 MVA are feeding the various customers in LV. The rated values of the four different types of DTs, such as the rated power Sr , the short-circuit voltage uk , the p.u. resistance r, the open-circuit current I0 , and the iron-core losses PFe , are given in Table 1. 3.2. Description of the simulations All the above data concerning the feeder and the distribution transformers were used as inputs into the program. Afterwards, a number of simulations for various scenarios were performed using stochastic data for loads and PV power production. First of all, a yearlong power flow was conducted using the mean measured values for load current, a typical power factor for the load of 0.95 lagging, and no PV units connected. The measurements as obtained by the Greek PPC were processed for one year, 15/11/2006 to 14/11/2007. This deterministic power flow is going to be used as reference for comparison with stochastic power flows, in order to evaluate the impact of stochasticity. Regarding the probabilistic power flows, again without any PV units, Monte Carlo method is used to provide deviations from the deterministic load series following either uniform or Gaussian distribution with various parameters. Afterwards, mean daily profiles of load demand, PV energy production, and SMP are presented. These profiles describe the basic characteristics of the examined case, provide useful information and lead to valuable conclusions. The main set of annual simulations concerns various scenarios for different cases of PV units’ siting and penetration levels. A final set of simulations is also performed to assess the specific power production profile of PV systems. 4. Simulations and results 4.1. Deterministic versus stochastic load power flow Both deterministic and stochastic load power flows are performed in order to estimate yearly energy losses. The deterministic case uses the processed measured load series. On the contrary, for the stochastic power flows fifty different Monte Carlo simulations are performed for each case of different PDF and deviation, producing fifty alternative results for energy losses. The range of the
results and their average value are presented in Tables 2 and 3, for normal and uniform PDF respectively. It has to be noted that, for each Monte Carlo simulation the total load energy consumption remains the same, see Eq. (10), however the variation in load distribution is different. One would expect a larger deviation among the results, especially for as few as fifty simulations. However, energy losses are actually the sum of 17,520 values of power losses, thus for such a large number of uncorrelated random values the result will be very close to the mean value. From Tables 2 and 3 it is clear that, the larger the standard deviation of the distribution is, the higher the energy losses of the feeder will be, which is also proven mathematically is presented in Appendix A. It is reminded that, the standard deviation of normal PDF and the length of uniform PDF are determined from (7)–(9), respectively. Thus, parameter sn coincides with the standard deviation, however su is actually the half of the uniform distribution’s length as a percentage of the mean value. For example, su = 20% means that stochastic variable Si,j is uniformly distributed between 80% and 120% of its mean value. Therefore, an immediate comparison between normal and uniform distributions according to parameters sn and su should be avoided. However, due to the linear relationship between su and the standard deviation of the uniform PDF, a rough comparison can be made for qualitative reasons. Finally, uniform PDF sets specific limits for the stochastic variable, whereas for normal PDF and a large number of samples, extreme values may show up. Thereupon, due to these peak values and based on the analysis performed in Appendix A, higher losses are expected when using normal PDF for the stochastic loads. The above result can be very useful to the DSO when calculating the expected energy losses and the impact of a DG unit interconnection. Using historical data and performing statistical analyses the DSO could estimate the deviation expected for the load demand, thus the increase in expected calculated losses, as well. In other words, the proposed methodology outweighs the results of a respective deterministic modeling, since it takes in mind the stochastic nature of many variables, otherwise considered constant. Therefore, the DSO would be sure to stand in the safe side, regarding the ways to do the cost allocation, however it is not in the scope of this paper to investigate these issues. 4.2. Mean daily profiles Based on measurements, the mean daily profiles of load demand and PV production, as well as SMP for the same time period, as taken
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Table 2 Annual energy losses without PV in MWh – deterministic vs. stochastic power flow with normal PDF. Deterministic power flow
183.475
Standard deviation in % of mean value
Stochastic power flow with normal PDF
10% 20% 30% 40% 50%
Min
Mean
Max
184.29 187.04 191.86 197.99 207.32
184.41 187.37 192.13 199.01 207.74
184.50 187.64 192.53 199.48 208.56
Table 3 Annual energy losses without PV in MWh – deterministic vs. stochastic power flow with uniform PDF. Deterministic power flow
183.475
Half length in % of mean value
10% 20% 30% 40% 50%
from historical data provided by the Hellenic TSO [38], are shown in Figs. 4–6, respectively. The maximum load demand for the examined feeder is presented in winter and it is this season that mostly determines the total load profile, as seen in Fig. 4. While summer presents a typical load profile with two peak periods in the morning and the evening, in winter (in spring and autumn, as well, but less obvious) the peak demand occurs between 11:00 p.m. and 03:00 a.m. This is due to the fact that, in the studied area there is a high penetration of night heat storage, which absorbs high amounts of electric energy during the night. Thus, it contributes in a peak demand during these hours, even higher than the midday peak, as its operation is almost concurrent in all customers. Fig. 5 presents the PV production curve, which typically follows the variations of solar radiation in Greece. Thus, summer production has the highest peak, and the daylight interval is the longest. Finally, Fig. 6 shows the SMP, where winter presents the highest value and spring the lowest. The profile is almost the same for all seasons, except from winter when even in the night hours the price does not fall below 50 D /MWh.
Fig. 4. Mean daily profile of load demand for the examined feeder.
Stochastic power flow with uniform PDF Min
Mean
Max
183.70 184.61 186.13 188.28 191.05
183.79 184.77 186.41 188.64 191.54
183.86 185.01 186.89 189.04 192.08
Fig. 5. Mean daily profile of power production for a 3 MW PV unit.
Fig. 6. Mean daily profile of SMP.
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Fig. 7. Schematic illustration of the four topologies for PV units’ siting.
4.3. PV siting and penetration scenarios The siting of PV units was based on four different topologies. Topology A considers uniformly distributed PV units in all the DTs across the feeder. Topology B assumes three installation points for the PV units in the beginning, the middle and the end of the feeder. In topology C the first installation points were DT nos. 10 and 11 (in the middle of the feeder) and additional capacity of PV units was installed to adjacent DTs symmetrically. Finally, in topology D the first installation point was the DT at the end of the feeder and additional PV units were installed to adjacent DTs towards the beginning of the feeder. The above mentioned topologies of the PV units are shown schematically in Fig. 7 for a random penetration level. As the penetration level increases, the installed capacity of these units will also increase and upon reaching the rated power limit of the transformer more units will be installed in the adjacent nodes. Different penetration levels were used for each of the above topologies of PV units. The penetration level was defined as the ratio of the total installed capacity of PV units to the total loading capacity of the feeder. In total, five different penetration levels were considered, i.e. 10–50% in steps of 10%. The combination of the four different topologies with the five penetration levels provided twenty different scenarios. Even though the above scenarios were simulated for both uniform and normal distribution functions, regarding load demand, in the following results only the latter will be illustrated. This was considered to be more suitable to model the random load demand for each node, since in the examined feeder the loads are already aggregated, thus tend to present a normal rather than uniform distribution. The respective results, in case of uniform distribution, present similar qualitative behavior and differ only in quantitative manner, as expected from the previous analysis (see Tables 2 and 3). In Fig. 8 the annual energy losses of the feeder are illustrated for random load demand for each node with normal PDF and standard deviation 20% of the mean value. The profiles of energy losses curves for a range of standard deviation until the case of 50% of the mean value are exactly the same, with only quantitative differences. It has to be noted that, the losses in Fig. 8 are the mean values resulting from fifty different Monte Carlo simulations for a whole year. The deviation of the results did not exceed ±0.5% for reasons explained earlier in the paper. Fig. 8 shows that the best topology for maximum loss reduction is A, i.e. PV units uniformly distributed across the feeder. However, annual energy losses as a function of PV penetration level present a U-shape trajectory for all topologies. Losses start to decrease sharply for low PV penetration, whereas after a specific point (different for each topology) they start to increase with higher penetration levels. This point seems to be a bit higher than
Fig. 8. Annual energy losses. Mean values from 50 different simulations for random load demand with normal PDF and 20% standard deviation.
50% penetration for topology A, around 50% for topology B, 40% for topology C and 35% for topology D. It is also important to notice that, excluding topology A that is in general the best one, for every penetration level there is a different topology leading to maximum loss reduction. The above results are presented in Table 4, as % loss reduction in respect to the base scenario (without PV). Along with loss reduction lred , reduction cred of annual cost index is also given (mean value). As observed, the tendency in cost reduction is the same with losses, however the percentages are little higher. This is explained by the fact that PV units contribute in loss reduction during daytime, and mostly around noon, when SMP presents its peak value. From the above results some general conclusions can be drawn. First of all, uniform distribution of PV units across the feeder provides the maximum loss reduction. However, the uniform installation of PV units is sort of an ideal scenario and in reality almost impossible to occur. Thus, the best realizable siting of PV units seems to be the installation in three different points symmetrically. However, for some penetration levels other topologies may be even better, i.e. the best siting is penetration dependent. Moreover, for every topology there is a maximum penetration level, above which the energy losses start to increase and for extreme cases would lead to greater values, even higher than the base scenario (feeder without PV). Finally, cost reduction (due to loss reduction) resulting from PV penetration seems to be important, since PV units produce their power in high SMP hours. Finally, a last set of simulations was conducted in order to evaluate the effect of DG production profile. In specific, a virtual type of
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Table 4 Annual loss & cost reduction in % of base scenario. Topology
Penetration level 10%
A B C D
20%
30%
40%
50%
lred
cred
lred
cred
lred
cred
lred
cred
lred
cred
9.21 6.69 5.08 5.70
9.52 6.97 5.35 5.99
16.11 10.11 8.62 9.32
16.70 10.60 9.10 9.83
20.95 10.41 10.78 11.06
21.79 11.07 11.47 11.79
23.65 13.44 13.11 11.33
24.69 14.31 14.03 12.23
24.17 13.57 11.46 10.19
25.40 14.41 12.53 11.26
I0 of the stochastic currents. To prove this, we introduce n quantities d1 , d2 , . . ., dn defined as follows: Ii = I0 + di ,
i = 1, . . . , n
(A1)
Obviously, the mean value of di for i = 1, . . ., n is zero: n
di = 0
(A2)
i=1
Using (A1) and (A2):
n
i=1
Ii2 =
n
2
(I0 + di ) =
i=1
n i=1
(I02 + di2 + 2I0 di ) = n · I02 +
n
di2
(A3)
i=1
Therefore n
Fig. 9. Comparison between PV units and virtual DG units operating in constant power, both producing the same annual energy.
DG unit was used, capable of operating during the whole year with constant power output, producing the same amount of energy as a PV unit. For the comparison of the two cases, the best topology, i.e. topology A, was used and 50 different power flows were simulated for each penetration level with random load profiles with normal PDF and standard deviation 20% of the mean value. The results are illustrated in Fig. 9, from which it is concluded that a DG unit has a better impact on loss reduction when producing constant power. Thus, as far as energy loss reduction is concerned, DG will have the maximum positive impact when, topologically it is uniformly distributed (i.e. uniform siting) and chronically it produces power in a uniform way (i.e. constant power production).
5. Conclusion A methodology for stochastic power flow studies in distribution networks is proposed, showing the impact of random load profiles on energy losses calculation. Using this methodology, various scenarios concerning PV penetration and siting in a radial distribution feeder are simulated. Results have shown that losses calculated with stochastic power flow are higher than the corresponding values of a deterministic power flow. Uniformly distributed dispersed generation may lead to optimal siting concerning losses in a distribution line.
Appendix A. We consider a period consisting of a sequence of n successive equal intervals of time. The aggregated Joule losses for a stochastic current flow I1 , I2 , I3 , . . ., In in a conductor are always greater than the losses caused by currents I1 , I2 , I3 , . . ., In equal to the mean value
Ii2 ≥ n · I02
(A4)
i=1
Assuming conductor resistance equal to 1 and interval of time 1 s the above inequality constitutes the desired proof. Furthermore, the difference between losses is increasing with the variance, i.e. n d2 /n. i=1 i References [1] T. Ackermann, G. Andersson, L. Soder, Distributed generation: a definition, Electr. Pow. Syst. Res. 57 (3) (2001) 195–204. [2] J.A. Pecas Lopes, N. Hatziargyriou, J. Mutale, P. Djapic, N. Jenkins, Integrating distributed generation into electric power systems: a review of drivers, challenges and opportunities, Electr. Pow. Syst. Res. 77 (9) (2007) 1189–1203. [3] J. Mutale, G. Strbac, S. Curcic, N. Jenkins, Allocation of losses in distribution systems with embedded generation, IEE Proc.-Gener. Transm. Distrib. 147 (1) (2000) 7–14. [4] P. Chiradeja, R. Ramakumar, An approach to quantify the technical benefits of distributed generation, IEEE T. Energy Conver. 19 (4) (2004) 764–773. [5] P. Chiradeja, Benefit of distributed generation: a line loss reduction analysis, Presented at Transmission and Distribution Conference and Exhibition: Asia and Pacific, IEEE/PES, 2005, pp. 1–5. [6] M. Thomson, D.G. Infield, Impact of widespread photovoltaics generation on distribution systems, IET Renew. Power Gen. 1 (1) (2007) 33–40. [7] M. Thomson, D.G. Infield, Network power-flow analysis for a high penetration of distributed generation, IEEE T. Power Syst. 22 (3) (2007) 1157–1162. [8] R.A. Walling, R. Saint, R.C. Dugan, J. Burke, L.A. Kojovic, Summary of distributed resources impact on power delivery systems, IEEE T. Power Deliver. 23 (3) (2008) 1636–1644. [9] J. Deuse, S. Grenard, D. Benintendi, P.J. Agrell, P. Bogetoft, Use of system charges methodology and norm models for distribution system including DER, in: Proc. Int. Conf. on Electricity Distribution, CIRED 2007, Paper No. 76, Vienna, May 21–24, 2007. [10] S. Kotamarty, S. Khushalani, N. Schulz, Impact of distributed generation on distribution contingency analysis, Electr. Pow. Syst. Res. 78 (9) (2008) 1537–1545. [11] S.A. Papathanassiou, A technical evaluation framework for the connection of DG to the distribution network, Electr. Pow. Syst. Res. 77 (1) (2007) 24–34. [12] G. Pepermans, J. Driesen, D. Haeseldonckx, R. Belmans, W. D’haeseleer, Distributed generation: definition, benefits and issues, Energy Policy 33 (6) (2005) 787–798. [13] P.P. Barker, R.W. De Mello, Determining the impact of distributed generation on power systems. I. Radial distribution systems, in: Proc. IEEE Power Engineering Society Summer Meeting, vol. 3, 2000, pp. 1645–1656. [14] G.P. Harrison, A.R. Wallace, Optimal power flow evaluation of distribution network capacity for the connection of distributed generation, IEE Proc.-Gener. Transm. Distrib. 152 (1) (2005) 115–122.
1994
A.G. Marinopoulos et al. / Electric Power Systems Research 81 (2011) 1986–1994
[15] C. Wang, M.H. Nehrir, Analytical approaches for optimal placement of distributed generation sources in power systems, IEEE T. Power Syst. 19 (4) (2004) 2068–2076. [16] A. Keane, M. O’Malley, Optimal allocation of embedded generation on distribution networks, IEEE T. Power Syst. 20 (3) (2005) 1640–1646. [17] G. Celli, E. Ghiani, S. Mocci, F. Pilo, A multiobjective evolutionary algorithm for the sizing and siting of distributed generation, IEEE T. Power Syst. 20 (2) (2005) 750–757. [18] L.F. Ochoa, A. Padilha-Feltrin, G.P. Harrison, Evaluating distributed generation impacts with a multiobjective index, IEEE T. Power Deliver. 21 (3) (2006) 1452–1458. [19] V.H.M. Quezada, J.R. Abbad, T.G.S. Roman, Assessment of energy distribution losses for increasing penetration of distributed generation, IEEE T. Power Syst. 21 (2) (2006) 533–540. [20] N. Acharya, P. Mahat, N. Mithulananthan, An analytical approach for DG allocation in primary distribution network, Int. J. Elec. Power 28 (10) (2006) 669–678. [21] A. Lakshmi Devi, B. Subramanyam, Optimal dg unit placement for loss reduction in radial distribution system—a case study, ARPN J. Eng. Appl. Sci. 2 (6) (2007) 57–61. [22] M.V.F. Pereira, M.E.P. Maceira, G.C. Oliveira, L.M.V.G. Pinto, Combining analytical models and Monte-Carlo techniques in probabilistic power system analysis, IEEE T. Power Syst. 7 (2) (1992) 265–272. [23] X. Long, W. Yan, Z. Lu, Monte Carlo method for line losses evaluation in transmission network, in: Proc. Int. Conf. on Power System Technology, October 22–26, 2006, pp. 1–5. [24] G.K. Stefopoulos, A.P. Meliopoulos, G.J. Cokkinides, Probabilistic power flow with non-conforming electric loads, Int. J. Elec. Power 27 (9–10) (2005) 627–634. [25] A.P.S. Meliopoulos, G.J. Cokkinides, X.Y. Chao, A new probabilistic power flow analysis method, IEEE T. Power Syst. 5 (1) (1990) 182–190. [26] A. Sankarakrishnan, R. Billinton, Sequential Monte Carlo simulation for composite power system reliability analysis with time varying loads, IEEE T. Power Syst. 10 (3) (1995) 1540–1545. [27] X. Li, Y. Li, S. Zhang, Analysis of probabilistic optimal power flow taking account of the variation of load power, IEEE T. Power Syst. 23 (3) (2008) 992–999. [28] A.P.S. Meliopoulos, X.Y. Chao, G. Cokkinides, R. Monsalvatge, Transmission loss evaluation based on probabilistic power flow, IEEE T. Power Syst. 6 (1) (1991) 364–371. [29] F. Demailly, O. Ninet, A. Even, Numerical tools and models for Monte Carlo studies of the influence on embedded generation on voltage limits in LV grids, IEEE T. Power Deliver. 20 (3) (2005) 2343–2350. [30] N. Vaitheeswaran, R. Balasubramanian, Stochastic model for optimal selection of DDG by Monte Carlo simulation, in: Proc. Int. Conf. on Power Elec. Drives and Energy Systems, 2006, pp. 1–5.
[31] W. El-Khattam, Y.G. Hegazy, M.M.A. Salama, Investigating distributed generation systems performance using Monte Carlo simulation, IEEE T. Power Syst. 21 (2) (2006) 524–532. [32] D.H.O. McQueen, P.R. Hyland, S.J. Watson, Monte Carlo simulation of residential electricity demand for forecasting maximum demand on distribution networks, IEEE T. Power Syst. 19 (3) (2004) 1685–1689. [33] A. Capasso, W. Grattieri, R. Lamedica, A. Prudenzi, A bottom-up approach to residential load modeling, IEEE T. Power Syst. 9 (2) (1994) 957–964. [34] METEONORM© Version 6.0, October 2007, http://www.meteonorm.com. [35] PVSYST©Version 4.33, October 2008, http://www.pvsyst.com. [36] MATLAB® Version 7.0, May 2004, http://www.mathworks.com. [37] R.D. Zimmerman, C.E. Murillo-Sanchez, D. Gan, “MATPOWER: A MATLABTM Power System Simulation Package,” ©1997–2007 Power Systems Engineering Research Center (PSERC), School of Electrical Engineering, Cornell University, Ithaca, NY 14853, Version 3.2, September 2007, http://www.pserc.cornell.edu/matpower. [38] http://www.desmie.gr. Antonios G. Marinopoulos was born in Thessaloniki, Greece, in July 1980. He received his Dipl.-Eng. degree in 2003 and the Ph.D. degree in 2009, both from the Department of Electrical and Computer Engineering at the Aristotle University of Thessaloniki, Thessaloniki, Greece. He is currently a Research and Development Engineer in ABB AB, Corporate Research, Västerås, Sweden. His research interests are in the field of power system analysis and simulation, power system integration of renewable energy sources, and distributed generation. Minas C. Alexiadis was born in Thessaloniki, Greece, in July 1969. He received his Dipl.-Eng. degree in 1994 and the Ph.D. degree in 2003, both from the Department of Electrical and Computer Engineering at the Aristotle University of Thessaloniki, Thessaloniki, Greece. He is currently a Lecturer with the Department of Electrical and Computer Engineering at the Aristotle University of Thessaloniki, Thessaloniki, Greece. His research interests include distributed generation, renewable energy sources and artificial intelligence applications in power systems. Petros S. Dokopoulos was born in Athens, Greece, in September 1939. He received his Dipl. Eng. degree from the Technical University of Athens, Athens, Greece, in 1962 and the Ph.D. degree from the University of Brunswick, Brunswick, Germany, in 1967. From 1962 to 1967, he was with the Laboratory for High Voltage and Transmission, University of Brunswick. From 1967 to 1974, he was with the Nuclear Research Center, Julich, Germany, and from 1974 to 1978, he was attached to the German team of the Joint European Torus in Oxfordshire, UK. From 1978 to 2006 he was Full Professor with the Department of Electrical Engineering, Aristotle University of Thessaloniki, Thessaloniki, Greece. He is currently Professor Emeritus with the same department. His scientific fields of interest are dielectrics, power switches, power generation (conventional and fusion), transmission, and distribution and control in power systems.