Probabilistic simulation framework for balanced and unbalanced low voltage networks

Electrical Power and Energy Systems 82 (2016) 439–451

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Electrical Power and Energy Systems journal homepage: www.elsevier.com/locate/ijepes

Probabilistic simulation framework for balanced and unbalanced low voltage networks Vasiliki Klonari a,⇑, Jean-François Toubeau a, Zacharie De Grève a, Olgan Durieux b, Jacques Lobry c, François Vallée a a b c

Electrical Engineering Department, University of Mons, 31 Bd. Dolez, 7000 Mons, Belgium Smart Metering Department of the Belgian DSO ORES, Rue Antoine de Saint-Exupéry, 14, 6041 Gosselies, Belgium General Physics Department, Polytechnic Faculty, University of Mons, 9 Rue de Houdain, 7000 Mons, Belgium

a r t i c l e

i n f o

Article history: Received 19 January 2015 Received in revised form 23 March 2016 Accepted 29 March 2016

Keywords: Distributed power generation Probabilistic simulation Photovoltaic systems Smart meters Voltage unbalance Three-phase power flow

a b s t r a c t In Low Voltage (LV) distribution networks, the high volatility of distributed photovoltaic (PV) generation has a severe impact on the variation of operation indices, in steady state conditions. During periods with high PV injection and low demand, LV feeders are more and more subject to overvoltage events and temporary PV units’ cut-offs. As a result, the delivered power quality is affected and network operational expenses increase. Moreover, the income of the PV owner is decreased due to the loss of generated energy. For efficiently addressing such operational issues, long term observability analytics of the LV network are required. Distribution System Operators (DSOs) currently deploy such studies in a deterministic manner, focusing on ‘‘worst-case” hypothesis, without considering the uncertainty of nodal power injection and consumption. This approach can lead to over restrictive decisions and costly technical solutions. For refining DSO strategies to the variability of network states, probabilistic methods are highly recommended. In this context, this paper presents a Monte-Carlo (MC) framework that simulates the steady operation of the LV network by elaborating user-specific smart metering (SM) measurements. The presented framework integrates a complete three-phase power flow algorithm that can analyse most possible LV network configurations, balanced and unbalanced, considering nodal power injections and consumptions as random variables of each network state. Such unbalanced power flow algorithms had not up to now been linked with probabilistic analysis using network-specific SM readings. For demonstrating the interest of the proposed framework, the latter is used to simulate several configurations in an existing LV feeder with high PV integration and SM deployment. Ó 2016 Elsevier Ltd. All rights reserved.

Introduction In recent years, the ongoing development of distributed photovoltaic (PV) generation in Low Voltage (LV) networks raises a number of issues that need to be investigated before taking decisions on further developments to come. Which strategy should Distribution System Operators (DSOs) adopt for allowing a lean transition from the ‘‘concentrated” to the ‘‘distributed” generation model without compromising the network stability, the quality of distributed power and a maximal performance of PV units? Which will be the cost to pay to compensate for PV related problems? Which technical strategies to deploy for fairly allocating distribution costs to all users? ⇑ Corresponding author. Tel.: +32 485688173. E-mail address: [email protected] (V. Klonari). http://dx.doi.org/10.1016/j.ijepes.2016.03.045 0142-0615/Ó 2016 Elsevier Ltd. All rights reserved.

For answering such questions, long term observability analytics of the LV network are necessary. Given the high volatility of distributed generation and consumption loads, new models need to be developed for addressing the induced uncertainties [1]. The understanding of the actual situation in LV distribution has been lately facilitated by the sparsely deployed SM devices while their large deployment is considered as the next important milestone for power distribution. (Pseudo) measurements deployed at several locations over a long period, with updating latencies of 10–15 min, can be exploited for a reliable and information-rich observability of the network, useful to many different systems across distribution utilities. Compared to real-time telemetered data, such long term measurements are more suitable for technoeconomic analysis and network development studies. Focusing on long term observability of the LV network, the present paper develops a comprehensive framework that elaborates such metering

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Nomenclature MV/LV PV SM TDP Einj,i Econs,i Pinj,i Pcons,i fi m l q E h i Z Y 012 y0z ; y1z ; y2z

Medium Voltage/Low Voltage Photovoltaic Smart Meter Typical Day Profile total quarter-hourly (15-min) injected energy at node i [kW h] total quarter-hourly (15-min) consumed energy at node i [kW h] peak injected power at node i during a 15-min interval [W] peak consumed power at node i during a 15-min interval [W] time repartition factor of Einj,i or Econs,i iteration index of the Monte-Carlo algorithm iteration index of the power flow algorithm index of the simulated 15-min interval of the day (with q = 1:96) convergence error of the power flow algorithm [V] sequence components admittance matrix [S] zero, positive and negative sequence admittance matrix [S]

data for evaluating the most important steady operation indices. The latter are outlined in the following paragraphs. Regarding steady state conditions, the main concern in LV networks with distributed PV generation is voltage profile. The volatility of PV generation has a severe impact on the variation of voltage profile. During periods of high PV injection and low demand, reverse power flows towards the head of the feeder become more frequent and lead to voltage rise towards the end of the feeder. If the r.m.s. voltage at a certain PV node exceeds the upper limit suggested in the EN 50160 standard, an overvoltage event takes place and the PV inverter must be temporarily cut off [2,3]. A loss of generated PV power is therefore induced which means a loss of income for the PV owner. Besides, this hard curtailment of PV generation deteriorates the delivered power quality, due to significant voltage and current transients, and accelerates the degradation of inverters [4]. Voltage magnitude variation is not the only concern in LV networks with PV units. In three-phase LV networks, unbalanced single-phase loading and generation lead to unequal voltage magnitudes over the three phases. The increased current in the neutral conductor, due to unbalance, results in neutral-point shifting which is disadvantageous for the voltage profile. Voltage unbalance adversely affects network elements and connected equipment [5,6] while the network induces more losses and heating effects. Although restrictions on unbalance of assets are imposed since the 1950s [7], it has always been a challenging issue which nowadays emerges due to the on-going single-phase connection of distributed generation (DG) sources. In case of urban grids the maximum loading current (and the induced congestion risk) is the main source of power quality issues given that they supply many households in a small area; the transformer capacity and the number of connected users are high and the length of lines is short. At the same time, reverse power flows due to high PV injection can significantly increase line losses and operational expenses (OPEX) that the DSO needs to cover. Finally, other operational indices such as the DG impact on protection failures or network equipment ageing should also be treated with a long term view.

CDF MC DG %VUF Vnom Vinitial,abc

Cumulative Distribution Function Monte-Carlo distributed generation Voltage Unbalance Factor (%) nominal voltage of the MV/LV transformer [V] matrix of initial nodal voltages (phase components) [V]

Vinitial,012 matrix of initial nodal voltages (sequence components) [V] Sload,abc matrix of initial nodal loads (phase components) [VA] Sload,012 Slateral Iload,abc Iload,012 Vabc V012

matrix of initial nodal loads (sequence components) [VA] total transited power by each unbalanced lateral (per phase) [VA] matrix of initial nodal currents (phase components) [A] matrix of initial nodal currents (sequence components) [A] matrix of computed nodal voltages (phase components) [V] matrix of computed nodal voltages (sequence components) [V]

Nowadays, DSOs are called to safeguard a stable and secure power supply in all possible demand conditions while fostering the massive integration of DER generation. As a result, the adoption of a streamlined planning approach for analysing the current energy system becomes urgent and the necessity of leaving behind deterministic worst case design is highlighted. Indeed, it is in the economic interest of the DSO and of the network user that power supply relates to normally expected conditions rather than to extreme cases [3]. A large variety of commercial and noncommercial algorithms are nowadays available for deploying long term analysis of LV networks. The vast majority of them apply a deterministic approach. The software user models the network by deterministically defining the parameters that influence its operation, considering most of the times worst case scenarios. The purpose is to ensure 100% security of the system. For example, the steady state analysis for determining the overvoltage risk in an LV feeder usually considers that each PV unit injects power equal to its rated power and each supply point consumes its lowest expected load. Based on such extremely rare cases, voltage magnitudes near PV nodes result very high compared to the situation that usually occurs. As a result, the DSO determines the maximum acceptable PV power that a feeder can host in a very restrictive manner since this one heavily depends on the feeder’s voltage margin. Naturally, this approach may lead DSOs to high initial investments with low amortization rates as well as to very restrictive decisions in terms of DER hosting capacity. However, if one leaves behind deterministic worst case approach, a new challenge appears. Is there a reliable and accurate way for simulating the most usual operating conditions of LV networks? How can one choose which network states to focus on? The present study investigates whether the above questions regarding LV network modelling could be efficiently answered by applying probabilistic analysis taking advantage of the available SM measurements. For examining this hypothesis, a careful literature review has been initially deployed in Section ‘State of the art’. The main contribution of this paper is presented in Section ‘Approach and contributions of this study’. Practically, a novel probabilistic simulation framework is presented. First of all, unlike

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existing probabilistic methods, the presented framework is conceived for using user-specific 15-min SM measurements, recorded in the studied networks. The proposed methodology was initially constructed upon available SM data (recorded over a period of 3 years in LV feeders in Belgium). Nevertheless, the algorithm’s design makes it easily transposable to any LV feeder with SM measurements of nodal energy flows, with any averaging time step (usually 10 or 15 min in European LV networks). Secondly, the developed framework can analyse most possible configurations of LV networks (balanced/unbalanced, with/without mutual effects between phases) and to address in an overall manner all important steady operation indices. Indeed, existing contributions addressing the LV network usually focus on one network configuration (balanced or unbalanced) and on a specific operation index. Beyond the analysis of the actual situation, the framework can be used for the long term techno-economic evaluation of different mitigation solutions to operational problems [8,9] and can be further extended for incorporating PHEVs charging or demand response consideration and other solutions. The objective is to provide answers for the cost-effective and secure operation of LV networks. Finally, the last contribution of this paper reflects the following argument; although there is an extensive number of unbalanced power flow methodologies already presented in the literature, none of them had been up to now linked to a probabilistic framework using network-specific SM measurements. The paper is organised as follows. Section ‘State of the art’ outlines the actual scientific contributions regarding the probabilistic analysis of LV networks and the consideration of multi-phase configurations. In the same section, previous developments of the authors are presented and the innovative contribution of this paper is highlighted. Section ‘Approach and contributions of this study’ thoroughly explains the overall contributions regarding the previously discussed issues. Section ‘Simulation of an existing low voltage feeder’ presents the implementation of the proposed framework to analyse an existing LV feeder in Belgium. The impact of a reactive power control scheme on voltage variation is, among other things, evaluated. Finally, Section ‘Conclusion’ discusses the conclusions and the future objectives of the authors.

State of the art Probabilistic analysis of low voltage networks In LV feeders, the magnitude of the supply voltage for an individual user at any instant is determined both by the individual user’s demand and by the simultaneous demands of other network users. Since the demands of every network user and the degree of coincidence between them constantly varies, so does the supply voltage at each user’s coupling point with the network [3]. Based on this consideration, EN 50160 deals with the voltage characteristics in statistical or probabilistic terms. It gives recommendations that, for a percentage of measurements (e.g. 95%) over a given time, the value must be within the specified limits. Under normal operating conditions, 95% of the 10 min mean r.m.s. values of the supply voltage over weekly periods shall be within the range of Vnom ± 10%; similarly 100% of the 10-min mean r.m.s. values of the supply voltage shall be within the range of Vnom ± 15%. Also, under normal operating conditions, 95% of the 10-min mean r.m. s. values of the negative phase sequence component of the supply voltage over weekly periods shall be within the range of 0–2% of the positive phase sequence component. The notion of probabilistic analysis of the power system can be traced back to the 1960s. Some thirty years later, probabilistic models specifically tailored to the distribution network were

441

introduced [10–14]. Since then, many research institutes worldwide have presented related contributions and few of the existing commercial tools have been adapted to integrate probabilistic analysis of the distribution network. A big share of these models deploy stochastic scenario analysis regarding the number, the size and the position of loads or DG units while each network state is still analysed by deterministically defining power injection and consumption values. Such contributions are out of the scope of the present study. The following literature review addresses probabilistic methodologies that simulate the uncertainty of network state due to volatility and uncontrollability of nodal power injections and consumptions. Several contributions deploying long term analysis and evaluation of the distribution network use probabilistic methods, based on analytical or numerical approach. In this context, paper [15] highlights the increased amount of information that can be obtained by applying (analytical) probabilistic load flow analysis in a distribution feeder with wind turbines versus a deterministic type of analysis. In a similar vein, a planning tool developed in a quasi-sequential Monte Carlo (MC) environment (numerical method) was proposed in contribution [16]. The conclusion was that the consideration of hourly probabilistic behaviour of PV generation led to a computed hosting capacity much lower than the one computed with a deterministic load flow. However, this paper only considered entire correlation scenarios between load and PV generation. Papers [17–22] evaluate different scenarios of spatial correlation for PV generation and load. Paper [23] applies a quasi-sequential MC algorithm to perform a reliability analysis of smart distribution networks. The repeated iterations involved in MC methods usually cause long execution times. Despite this fact, they are often preferred because of their implementation simplicity and their flexibility regarding the number of random variables [24]. All previously mentioned methodologies aim to analyse the impact of time varying loading parameters mainly focusing on voltage and current magnitudes. Indeed, a large number of probabilistic models for LV networks can be found in literature however only few of them present three-phase power flow (PF) analysis and voltage unbalance consideration. Paper [25] presents a MC algorithm which assesses voltage unbalance at the secondary output of an LV transformer using correlated Gaussian random variables to represent active and reactive power. Three-phase power data recorded at the MV/LV transformer are used but the random loading parameters at every node of the feeder are not specifically accounted for. Moreover, distributed PV units are not considered in that model. Paper [26] links a probabilistic analysis method to a three-phase power flow which simulates PV injection based on meteorological data of solar irradiation. Similarly, [27] presents a probabilistic methodology for estimating voltage unbalance in MV distribution networks when monitoring data are not available and [28] presents a methodology using both real and pseudo measurements to statistically estimate voltage unbalance in distribution networks. The previously mentioned contributions study the LV network with a long term statistical approach. In real-time state estimation, several studies propose solid probabilistic models for efficiently incorporating uncertainty by elaborating real time measurements [29–32]. As pointed out in [33], the long term observability approach opposed to the purely passive one (consisting of managing problems in the operational phase or in short-term time frames) would allow for interaction between the distribution network’s different timeframes. Indeed, real-time analysis is insufficient for the long term observability of the LV network and for the evaluation of certain technical decisions and solutions with a long term strategic view. For this reason, the present study focuses on long term analytics of the LV network deployed with the use of

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SM measurements. The objective is to develop tools that can directly and specifically address the planning and network development DSO timeframe. For deploying long term LV network analytics, several contributions propose sound probabilistic models based on solar irradiation data for PV injection and meteorological data for consumption loads. Such data are rarely directly accessible to the DSO and most of them do not consider the efficiency of the PV cells. Among the previously mentioned studies, few cases exploit sparse measurements for developing representative models for some of the systems’ variables. Despite this lack of data, most of these contributions demonstrated that the probabilistic approach can lead to an increased amount of information and to much less conservative decisions. However, none of these contributions examined the hypothesis of applying long term probabilistic estimation of LV feeders that addresses, in an overall manner, all operation indices (nodal voltage magnitudes and unbalance, current values, captured renewable energy, line losses, etc.) by taking advantage of network- and user-specific long term SM measurements. So far, the hypothesis that distribution utilities have an interest to leave behind deterministic approach has not investigated the great opportunities that will appear with the large deployment of nodal SM measurements. Prompted by these considerations, the authors of this study initially developed in [34] a MC simulation tool that uses 15-min SM data for computing the voltage profile along a radial LV feeder with PV units. In that algorithm, no consideration was taken for the fluctuations of the voltage at the MV/LV transformer. In [35], the initial probabilistic tool was complemented for considering another random variable in each network state that is the voltage at the output of the MV/LV transformer. Moreover, in [35], individual userspecific statistical distributions of power injection and consumption were assigned to each node. Practically, two random variables were assigned to each node, each one of them statistically represented by the respective user-specific SM dataset. Thus, the algorithm can create statistical profiles (Cumulative Distribution Functions (CDFs) of probability) to represent the uncertainty of voltage at the head of the feeder (based also on 15-min SM readings) and the variability of nodal energy flows, between different time steps and different users. Considering phase unbalance in power flow calculations Regarding power flow computations, the MC algorithms of [34,35] could only model the network as a perfectly balanced system. In case of multi-phase LV configurations, this consideration hardly reflects the actual situation since the vast majority of residential PV units are connected in single-phase mode, distributed in a heavily unbalanced manner over phases. As a result, an important load unbalance is present in most three-phase LV networks. To address this argument, the present paper puts together all previous developments with a complete generalised three-phase power flow algorithm which, depending on the analysis objectives, can consider or not load unbalance as well as mutual effects between phases. Thanks to this new development, the proposed probabilistic framework can evaluate, in a long term observability context, the most important indices that affect steady operation of LV networks (voltage indices, current indices, voltage unbalance) and the induced network costs (due to line losses, hard curtailment of renewable energy and other). For considering the existing unbalance in power flow computations, various studies have been conducted [36–42]. In several cases, the LV network consists of a main three-phase line with different two-phase or single-phase loads and also unbalanced laterals per phase. Papers [36,37] present a power flow method that allows solving the main three-phase network based on the

decoupled positive, negative, and zero sequence networks. The unbalanced laterals are solved using the forward/backward method in phase components. The advantage of this power flow algorithm is that it can be easily adapted to cover multiple configurations of LV networks, with multiple three-phase, two-phase and/or single-phase loads and laterals, without requiring long elapsed time. Moreover, depending on the studied case, it can be particularized to independent single-phase lines or to balanced three-phase networks. Finally, the algorithm is structured in a way that allows modelling voltage control schemes, that are time varying and locally implemented at various nodes. This is a great advantage for performing long term observability studies of the LV network. Indeed, such studies should also address the impact of local network state aware technical strategies. The considered power flow algorithm has been more than once presented in the related literature [36,37,43], with minor adaptations, but none of these studies links it with probabilistic analysis considering nodal power flows and voltage at the MV/LV transformer as random variables. The present paper adapts the discussed power flow algorithm to a probabilistic framework for evaluating several operation indices with a long term view. Approach and contributions of this study The developed probabilistic framework for the long term observability of LV networks is graphically outlined with the flowchart of Fig. 1. The 5 principal blocks of the simulation are explained in the following paragraphs. Feeder model and load/PV profiles The 1st block determines the topology of the studied network, its technical and operational characteristics as well as the loads and PV connections configuration. Practically, the position and the type of each node (simple user, PV user, intermediate nodes, lateral root nodes, single-phase or three phase connection, etc.) as well as the network components and lines (physical and electrical parameters) are defined. User nodes are assigned to their respective load/generation statistical profiles and operational constraints with regard to voltage magnitude, unbalance and other operational indices are set. Moreover, in this 1st block, the network balanced or unbalanced configuration is selected among the three principal options that are thoroughly described in Section ‘Power flow analysis of each network state’. The simulation can consider either a balanced three-phase configuration (case (i)), an unbalanced single-phase configuration which does not consider mutual coupling effects between phases (case (ii)), or an unbalanced threephase configuration (case (iii)). This user decision depends on the simulation objectives and on the availability of information regarding the phase connections. The availability of this option differentiates the proposed algorithm from the ones presented in bibliography since any radial LV network can be simulated by a single overall tool. Construction of the ‘‘Typical Day” statistical profiles The 2nd block creates the statistical distributions of timevarying parameters (nodal PV injections/loads and voltage at the MV/LV transformer) in an LV feeder. The latter are considered as the random variables that parameterize each network state. The load/ PV profiles of network users are created by using their respective SM recorded datasets as analytically presented in [44]. The two 15-min resolution datasets of Einj;pv and Eload (variables defined in [44]) are used to build two Typical Day Profiles (TDPs) for each user. The ‘‘typical day” represents and characterises a

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443

Fig. 1. Structure of the probabilistic framework.

selected period (which can be a month, a season, a year or so on). Each TDP reflects the variation that the respective parameter can have at every individual quarter of an hour of a ‘‘typical day”. Random sampling of the different network states The created statistical distributions (CDFs) of the time varying parameters (PV injection, net energy consumption and voltage at the output of the MV/LV transformer) are used for assembling multiple possible network states corresponding to each 15-min time step. Practically, a MC algorithm is applied for randomly sampling the values of the variable parameters at each node of the studied LV feeder as explained in [34]. The assembling of the nodal sampled values defines each network state that will be afterwards analysed by the power flow algorithm. Subsequently, for each state, a feeder with N nodes is characterised by the following variables:

fEloadi ; Einj;pv i ; ti g for nodes i ¼ 2 : N fV MV=LV g for node i ¼ 1

ð1Þ

where Eloadi is the 15-min net energy consumption at node i, Einj;pv ;i is the 15-min PV energy generation at node i, ti is the time repartition factor of the consumed or generated energy at node i and VMV/LV is the voltage at the MV/LV transformer node. In case there are no sufficient data for intra 15-min intermittencies of energy flow, it can be assumed that the power flow is stable during the 15 min of each time step. This means that the time repartition factor ti corresponds to 15 min and is thus equal to 0.25. The power flow analysis of the feeder requires considering each system state as instantaneous and therefore the sampled energy values have to be transformed into instantaneous power values (Einj,pv ? Pinj,pv and Eload ? Pload). Thus, for each node we can consider either power injection or power consumption since both of them cannot be applied simultaneously at an instant. In such a way, the instantaneous power value that represents the power flow at the point of common coupling (PCC) of each user i with the feeder is determined as follows:

Pi ¼

Eloadi  Einj;pv i 0:25

ð2Þ

If Pi is positive the respective user i is instantaneously consuming power from the grid whereas if Pi is negative, the user is instantaneously injecting power into the grid. The probabilistic deployment of this simulation tool relies on the principle that load/PV generation profiles of users are highly time-varying. The generation of the system states is therefore based on a very large number of random combinations of users’ energy flow values. This time-variability induces another variability that concerns the time coincidence of the load profiles of various users. Both arguments are very important when assessing the impact of PV generation on a LV network. Indeed, the consideration of this variability, both in the time axis and regarding users coincidence, makes more realistic the simulation of the network operation. As far as correlation of users is concerned, three options are possible. The 1st one is to sample network states considering all random variables as mutually independent. The 2nd option is to consider users as totally correlated. In this case, one common sampling is done to define the values of their variable parameters. For example, the 15-min injected energy ðEinj;pv i Þ of each user is defined thanks to the sampling of a single common random number (between 0 and 1) that is afterwards applied to the TDP associated to each user. The 3rd option is more complicated and requires more computational time. It practically groups feeder users in clusters (varying between time steps) in function of their respective 15-min energy flow datasets, by applying the clustering methodology that is explained in [45]. Power flow analysis of each network state This paragraph describes the restructure of the probabilistic framework for integrating a complete three-phase power flow algorithm that can address most possible configurations of LV networks. The network configuration option in the 1st block defines the algorithm that will be used for the power flow analysis of all

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network states, in function of the selected configuration (i), (ii) or (iii). If the consideration of a perfectly balanced system is selected, then the load flow analysis is performed according to case (i). In case of three-phase or two-phase system with unbalanced loads/ PV units and with a configuration that allows neglecting mutual coupling effects between phases, the algorithm of case (ii) can be selected. The last and most generic case (iii) covers all the other possible configurations of LV radial feeders, including the ones treated with the simplified and faster cases (i) and (ii). The general network configuration that can be treated by case (iii) is shown in Fig. 2. This last option should be selected in case of an unbalanced three-phase or two-phase main configuration with unbalanced three-phase, two-phase and/or single-phase load/PV connections and unbalanced laterals, where the mutual coupling effect between phases should not be neglected. The three configurations are explained in the next subsections. i. Three-phase main line with balanced three-phase loads/PV connections In this case, the system is analysed as a three-phase balanced radial grid, considering three-phase loads/PV connections as well as three-phase laterals at certain nodes. The applied power flow algorithm is the one already used in the initial version of the probabilistic framework [34]. This algorithm allows the computation of nodal voltage magnitudes, line currents and losses for every system state. As mentioned in the following Section ‘Simulation of an existing low voltage feeder’, it demonstrates a good computational time, converging after a small number of iterations. ii. Single-phase main lines with unbalanced single-phase loads/PV connections and unbalanced single-phase laterals In this case, the distance between phases allows neglecting their mutual coupling effects. Each phase is thus considered independently as a single-phase line with its respective single-phase loads/PV connections and single-phase laterals. This configuration takes into account the loading unbalance between phases but only an independent power flow analysis of each single-phase line is deployed, with the algorithm used in [34]. Of course, the cases that are treated by this option (and by option (i)) could also be analysed with the complete three-phase algorithm of case (iii). However, in case of an LV feeder with many nodes in which mutual coupling effects between phases can be neglected, the single-phase algorithm of [34] might be faster. iii. Three-phase main line with unbalanced single, two- or three-phase loads/PV connections, and unbalanced single, two- or three-phase laterals This last generic case considers all existing unbalances due to loads/PV connections and laterals, as well as mutual coupling

effects between phases. Both previous cases (i) and (ii) can also be treated by this one by adequately factorizing the sequence components impedance matrix. The methodology can also be adapted to radial networks with single-phase or two-phase main line, or with multiple single-phase, two-phase and/or three-phase loads and laterals (Fig. 2). This can be done by adapting the phase and sequence components impedance matrix as well as the loads/PV connections matrix. In case three-phase laterals also exist, they are modelled as part of the main three-phase line. In the present paper, the system is analysed as a three-phase main line with unbalanced single-phase loads/PV connections and unbalanced single-phase laterals, therefore the case of unbalanced two-phase laterals is not explained in detail although it can also be treated by the proposed algorithm. The process starts by constructing the sequence admittance matrix for the main line (flowchart in Fig. 3). In case of a transposed main line, this matrix is in phase variables full and symmetrical. Therefore, the line can be modelled by three uncoupled sequence circuits as presented by the following relation presented in [37]:

h

i

2

y0z

0

6 Z Y 012 ¼ 40 0

y1z 0

0

3

7 0 5; z y2

ð3Þ

where 0, 1, 2 represent respectively the zero, positive and negative sequences, whereas the superscript Z stands for series admittance. h i The shunt admittance matrix Y S012 is not taken into account in this study. At the beginning the probabilistic algorithm samples the voltage value at the slack node (secondary output of the transformer), which is one of the random variables in each system state (sampled from a 15-min dataset recorded by the DSO at the MV/LV transformer). Then the forward/backward load flow process performs the first forward step considering this slack node voltage fixed for the three phases: node V slack ¼ V nom \0 a node V slack ¼ V nom \120 b node V slack c

¼ V nom \240

ð4Þ



where Vnom is the nominal voltage of the transformer and the bold type stands for complex numbers. Fixing this value at the slack node is necessary for the convergence of the forward/backward method. During this first forward step, the nodal phase voltages Vinitial,abc are initialized to the slack node phase voltages. In sequence components, this reads [37]:

V initial;0 ¼ 0\0 V initial;1 ¼ V nom \0

ð5Þ



V initial;2 ¼ 0\0

where the bold type stands for complex numbers and the under bar stands for vectors. However, the analysis of the main line requires the computation of the existing unbalanced (in this case singlephase) laterals. Given that they usually represent small and simple radial networks, a simplified forward–backward method in phase components can be efficiently applied to solve them. This method requires that the voltage at the root node of each lateral has to be fixed, so that the forward–backward procedure converges. However, this value cannot be fixed a priori because it depends on the solution of the main line. For this reason, the main line and all the unbalanced laterals take part in an iterative forward–backward method, named the hybrid power flow method in [36]. The unbalanced laterals are solved with a forward–backward procedure in phase components. The first forward–backward step Fig. 2. General network configuration that can be treated in case (iii).

of the hybrid method gives the power injections Slateral per phase

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445

Fig. 3. Power flow algorithm applied in case (iii).

at each lateral’s root node, which is the total transited power by the respective lateral. In the described case the unbalanced laterals are single-phase lines therefore Slateral is computed per phase by a single-phase algorithm. The computed Slateral replaces each unbalanced lateral at its root node in the main line. During the backward step of the hybrid method, the phase currents due to the nodal loads are computed for each node i of the main line as in [37]:



I load;x;i ¼

S load;x;i V initial;x;i



ðx ¼ a; b or c phase and i ¼ 1 : NÞ

ð6Þ

where the superscript ⁄ stands for the complex conjugate and S load;x;i is calculated for simple nodes with (7) and for lateral root nodes with (8):

S load;x;i ¼ ðPx;i Þ þ jðQ x;i Þ

ð7Þ

S load;x;i ¼ ðPx;i Þ þ jðQ x;i Þ þ S lateral;i

ð8Þ

Active power values Px;i are defined per node per phase with (2) whereas reactive values Q x;i are defined considering constant values for cosucons,x,i = 0.86 and cosuinj,x,i = 1. Once the Iload,abc (phase components) matrix is constructed, it is transformed by means of the Fortescue transformation into the respective Iload,012 (sequence components) matrix. The specified nodal loads for the positive sequence Sload,1 are computed with (9):

S load;1 ¼ V initial;1  ðI load;1 Þ

ð9Þ

where the superscript ⁄ stands for the complex conjugate. At this point, the positive sequence nodal voltages V1 are computed by applying the single-phase load flow algorithm of [34], by considering Sload,1 at all unbalanced (main three-phase line) nodes. The negative and zero sequence voltages are computed by solving the linear systems:

 

 y0Z V 0 ¼ I load;0  y2Z V 2 ¼ I load;2

ð10Þ

Once the V012 (sequence components) matrix is constructed, it is transformed into the respective Vabc (phase components) matrix for the main line. The values of Vi (i = a, b or c) at the root nodes of the laterals are fixed and the single-phase algorithm is once again applied to compute the final nodal voltages of the laterals’ nodes. At this point, the first forward/backward step of the hybrid power flow method is completed and the nodal voltages of the whole network (main line and laterals) are updated with the new values:

V initial;x ¼ V x

ðx ¼ a; b or c phaseÞ

ð11Þ

After each iteration (l) of the power flow algorithm, the convergence error E is calculated for phase x at node i as follows:

   ðlÞ ðl1Þ  Ei;n ¼ V x;i  V x;i ;

x ¼ a; b; c and i ¼ 1 : N

ð12Þ

where N is the number of nodes of the whole feeder. As soon as the error Ei becomes smaller than a given tolerance for each phase at

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each node, the algorithm stops and the last calculated Vabc is compared to the operational limits. The overall structure of the hybrid load flow algorithm adapted for this study is presented in Fig. 3. Indexes of network operation In this last block, indices of network operation are computed. This computation uses the power flow results obtained in the previous block. The choice of the long term observability indices can be easily adapted to the simulation objectives. They are computed separately for each network state while the output datasets are used to construct distributions that represent the long term statistical behaviour of the feeder. The most important indices are listed below: i. Voltage profile (magnitude and unbalance) The phase voltage magnitudes are computed at each node for all the simulated network states. An extensive number of possible loading conditions are therefore examined such that the variation amplitude of the computed magnitudes is realistically estimated. For the total amount of simulated states, operation compliance with the probabilistic criteria of the EN 50160 standard (mathematically expressed by relations (13)) is verified for each node i and phase x:

P ov erv oltage ðV i;x > 1:10  V nom Þ < 0:05 P underv oltage ðV i;x < 0:90  V nom Þ < 0:05 P unbalance ðVUF i > 2%Þ < 0:05

ð13Þ

for i ¼ 1 : N and x ¼ a; b or c phase where V i;x is the voltage magnitude of phase x at node i, VUF i is the Voltage Unbalance Factor at node i and N is the number of nodes in the feeder. If for instance the simulation runs for M iterations, the probability of exceeding the 95-percentile limit of EN50160 for voltage magnitude (%OV Risk) and unbalance (%VU Risk) can be computed for node i as follows:

%OV Riski ¼ 100

! M 96 X 1 X  Network States with V i;x > 1:10Vn 96  M m¼1 i¼1 ð14Þ

%VU Riski ¼ 100 

M 96 X 1 X Network States with %VUF i > 2% 96  M m¼1 i¼1

!

ð15Þ where M is the number of simulated system states for each 15min time step (M = number of MC iterations) and 96 is the number of 15-min time steps in a day. Similarly, the probability of having instantaneous cut-off of the PV unit (as soon as the node voltage exceeds 1.15Vnom [2,3]) can be determined. The percentage Voltage Unbalance Factor (%VUF) is computed in this paper by applying the IEEE definition that is also used in the EN 50160 standard:

%VUF ¼

Vn  100 Vp

V ab þ a  V bc þ a2  V ca 3 V ab þ a2  V bc þ a  V ca ; Vn ¼ 3  2 a ¼ 1\120 and a ¼ 1\240

subject to : V p ¼

ð16Þ ð17Þ

ð18Þ

ii. Current magnitudes, line losses and other indices The maximum current capacity of the lines is determined considering values suggested by the DSO or standard tables like the one in [46]. Compliance with upper limits is verified in all simulated states. The line segments with the highest current magnitudes are identified thanks to the probabilistic simulation. The statistical distribution of total energy losses in the feeder is constructed characterising monthly periods. The cost of line losses can be monetized as energy consumption that is not directly included in the network use tariff. In most European countries, the DSO has a clear economic interest in reducing line losses, especially in countries that have adopted a ‘‘Revenue cap” methodology framework for the use of the distribution network. In a similar way, statistical distributions for captured or curtailed renewable energy can be constructed, per node or for the entire feeder. Moreover, the amount of reverse flows can be determined in a statistical manner. Simulation of an existing low voltage feeder The simulated LV feeder This section presents the application of the probabilistic framework for analysing a real LV feeder, with distributed PV generation and long term SM measurements. The topology and technical parameters of the feeder are presented in Appendix A (Fig. A1). All three system configurations (i), (ii), (iii) (Section ‘Power Flow analysis of each network state’) are computed. In configuration (i), loads and PV units are considered to be connected in balanced three-phase mode although in reality they are connected in unbalanced single-phase mode (as in Fig. A1). In configurations (ii) and (iii), the real single-phase connections are considered. The load and the PV unit of each node are connected to the same phase. The convergence of the computed indices has been achieved with M = 1000 iterations, namely with the simulation of 1000 ‘‘typical days” for each month. For the simulation of each month, the respective SM input is used for sampling the 15-min network states (SM measurements recorded during the respective month of years 2013 and 2014). Lastly, a reactive power control scheme is modelled for configuration (iii), for demonstrating how such a mitigation solution could be technically evaluated in the proposed tool. The computed operation indices for the studied network are presented in the following paragraphs. The total elapsed time (per simulated month) is in the range of 450 s for configuration (i), 950 s for configuration (ii) and 1800 s for configuration (iii). For most of the network states, the power flow algorithm converged after 2–3 iterations in configuration (i), 3–4 iterations in configuration (ii) and 7–8 iterations in configuration (iii), with a tolerance equal to 106 V in all cases. Results and discussion i. Voltage magnitude The simulated voltage magnitude variation at node 14, for two consecutive April days, is shown in Fig. 4 for configurations (i), (ii) and (iii). The range of voltage variation is wider in configuration (i) because in the other two configurations the same nodal injections and consumptions are distributed, in a balanced manner, over the three phases. Comparing configurations (ii) and (iii), one can easily observe that since mutual coupling effects are considered in configuration (iii), the voltage profile of each phase is affected by the loading parameters of the other 2 phases, which is not the case in configuration (ii). Another interesting remark concerns the

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Fig. 4. Voltage magnitude variation at node 14 for configurations (i), (ii) and (iii).

accuracy of configuration (iii); in case configuration (ii) is applied to solve a three-phase LV feeder in which phase coupling effects are not negligible, the obtained outputs concerning voltage margin (Fig. 4(ii) and (iii)) might be conservative or misleading, depending on the repartition of generation and loads over the three phases. In this specific case, the repartition of loads and generation among phases is quite balanced and as a result the coupling between phases reduces the range of voltage variation.

Regarding overvoltage, Table 1 presents the overvoltage risk at all PV nodes computed with relation (14) for the month of April. According to the outputs of configuration (i), the 1st condition of relation (13) (EN 50160 standard) is violated at most PV nodes (>5%). This is not the case in the unbalanced configuration. Given that generation and loads are distributed over the three phases in a balanced manner, the over voltage risk results much lower. Configuration (iii) computes that there is no overvoltage risk in

Table 1 Overvoltage risk at all PV nodes for configurations (i), (ii) and (iii). PV node

4

5

6

8

10

11

12

13

14

(i) (ii) phase A (ii) phase B (ii) phase C (iii) phase A (iii) phase B (iii) phase C

3.712% 0% 0% 0% 0% 0% 0%

6.073% 0% 0% 0% 0% 0% 0%

6.422% 0% 0% 0% 0% 0% 0%

9.26% 0% 0.031% 0% 0% 0% 0%

9.573% 0% 0.067% 0% 0% 0% 0%

9.701% 0% 0.085% 0% 0% 0% 0%

9.877% 0% 0.1188% 0% 0% 0% 0%

9.995% 0% 0.1271% 0% 0% 0% 0%

9.999% 0.0125% 0.1292% 0% 0% 0% 0%

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Fig. 5. (a) CDFs of probability of phase voltage magnitudes at node 14 for the month of April, in configuration (iii), (b) CDFs of probability of phase voltage magnitudes in the entire feeder for the month of April in configuration (iii) (c) CDFs of probability of %VUF for configurations (ii) and (iii), for the month of April.

the feeder during the month of April, based on the probabilistic analysis of SM input recorded in 2013 and 2014. Configuration (ii) computes a minor overvoltage risk at phase B, independently of the phase connection of the PV unit. This can be explained by the fact that the biggest PV unit (10 kV A at node 5) is connected to phase B. Since there is no coupling between phases, the impact of this big generator is not distributed over the three phases as in configuration (iii). With a long term view, Fig. 5(a) and (b) presents statistical distributions for the behaviour of the feeder in the month of April, for configuration (iii). According to the outputs, the voltage magnitude at node 14 will most probably be within the range of 212–243 V in the vast majority of system states (based on the probabilistic analysis of the available SM input) (Fig. 5(a)). Regarding the entire feeder, voltage magnitudes result to be in the range of 212–245.5 V for the vast majority of possible network states in April (Fig. 5 (b)). Comparing these statistical indices to the voltage ‘‘snapshots” (Fig. 4), one can clearly identify the wider range of voltage magnitude variation in configuration (ii). Indeed, the voltage snapshots of configuration (ii) show that voltage often takes the value of 250 V at node 14, which is never the case in configuration (iii), according to the long term statistical analysis. Voltage unbalance. Regarding %VUF, Fig. 5(c) demonstrates that in April the value of %VUF in the entire feeder does not exceed the

Fig. 6. %VUF variation at nodes 5 (PV 10 kV A) and 14 (PV 5 kV A) for configuration (iii).

EN 50160 standard limit in any network state in configuration (iii). In configuration (ii), the limit is exceeded in around 2% of the simulated network states which means that relation (13) is not violated (<5%). Taking a more rigorous view on %VUF, Fig. 6 shows the %VUF variation at nodes 5 (PV 10 kV A) and 14 (PV 5 kV A) in configuration (iii), for five consecutive days in April. At node 5, %VUF

V. Klonari et al. / Electrical Power and Energy Systems 82 (2016) 439–451

shows an important rise during PV injection hours whereas at node 14 voltage unbalance is mostly reduced during the same hours. At node 14, the highest %VUF values appear during peak consumption hours and this rise is much higher than the one induced by the PV injection at node 5. Thus, although singlephase PV units might often be a source of voltage unbalance in three-phase LV feeders, single-phase loads might affect unbalance in a much more severe manner. In such cases, the integration of single-phase PV units can also contribute to unbalance reduction during PV injection hours, as in the case of node 14 in the studied feeder. Consequently, the integration of single-phase PV units with a balanced repartition over the three phases might be useful in reducing phase unbalance. However given that unbalance is time varying due to time varying loads/generation, network state aware control schemes might be much more effective in many cases [9]. The fact that voltage unbalance is mostly affected in the studied feeder by single-phase loads rather than single-phase PV units, is also highlighted in Fig. 7. This figure shows that higher %VUF values are slightly most possible to happen in January compared to August or to April when PV injection is much higher. The range of %VUF values in April is wider than in January however the probability to get higher values is extremely low (<0.5%) (see Fig. 7). Line losses. For demonstrating the cost-effectiveness of deploying long-term measurements in the LV network and analysing it with a probabilistic approach, a statistical evaluation of line losses is performed for configuration (iii). This evaluation focuses on total energy losses along the lines of the feeder during high PV injection hours. For this evaluation, a worst case deterministic approach would only consider one system state (the worst), which is more likely to occur in the period between 12:00 AM and 18:30 PM on a typical July day (based on the available historic data for the feeder). Applying the deterministic approach, total energy losses during PV injection hours (in one day) would result equal to 5.8 kW h. The proba-

Fig. 7. CDFs of probability of %VUF in the entire feeder for January, April and August.

449

bilistic approach and the consideration of SM measurements demonstrated that total energy losses in the feeder vary significantly, depending on the system state. Notably, in 80% of the simulated days, the respective total energy losses during high PV injection hours (12:00 AM to 18:30 PM) are lower than 5.5 kW h. Accounting for this low probability of worst case scenarios taking place simultaneously for all feeder users, the DSO could manage a less conservative and more cost-effective network analysis and development. Reactive power control for voltage variation mitigation Voltage variations could be compensated with a control scheme allowing the optimal adjustment of the PV inverter reactive power as the one recommended in the Italian norm [47]. For demonstrating how such a control scheme can be evaluated by the proposed framework in the studied feeder, the reactive power generated by PV inverters at each node has been set according to the Q/V function of [47], in each network state. Configuration (iii) was again analysed and a reduction of voltage magnitudes of phase B is achieved, thanks to the action of the control. The voltage profile of the other two phases is hardly affected. As Fig. 8 shows, the range of voltage magnitudes in the entire feeder is between 224 V and 243 V if reactive power control is applied while the respective one for the base scenario (no reactive power control) is between 226 V and 246 V. Given that the application of the control induces much more line losses, during PV injection hours (Fig. 8), a trade-off between voltage profile improvement and maximum acceptable losses should be found.

Conclusion Passing from a deterministic to a probabilistic approach for deploying long term observability analytics of LV networks is a very promising strategy. This paper investigates the hypothesis of analysing LV feeders by deploying MC simulation that uses as input feeder- and user-specific SM energy flow measurements. In this purpose, a comprehensive probabilistic framework that can analyse a wide range of different network configurations (balanced and unbalanced) is presented. The proposed framework practically performs offline state estimation of multiple network states considering nodal power flows and voltage at the MV/LV transformer as the set of random variables that define each network state. The values of these variables are sampled from SM datasets recorded in the studied networks. The simulation of a real LV feeder demonstrated that the use of the proposed framework allows very information-rich analytics of the network which can lead DSOs to cost-effective and less conservative strategic decisions, compared to the commonly applied deterministic approach.

Fig. 8. CDFs of phase voltage B and of total energy losses during PV injection hours for the base scenario (no reactive power control) and for the reactive power control scenario.

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Fig. A1. The simulated LV feeder (conductors colour code as in IEC 60446 standard).

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