Electrical Power and Energy Systems 57 (2014) 141–147

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Electrical Power and Energy Systems

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A methodology to calculate maximum generation capacity in lowvoltage distribution feeders

0142-0615/$ - see front matter � 2013 Elsevier Ltd. All rights reserved.http://dx.doi.org/10.1016/j.ijepes.2013.11.047

⇑ Corresponding author. Tel.: +31 224565171.E-mail address: ioulia.papaioannou@ec.europa.eu (I.T. Papaioannou).

1 The views expressed are purely those of the authors and may not in anycircumstances be regarded as stating an official position of the European Commission.

Ioulia T. Papaioannou ⇑, Arturs PurvinsEuropean Commission, DG JRC, Institute for Energy, Postbus 2, 1755 ZG Petten, The Netherlands1

a r t i c l e i n f o

Article history:Received 19 November 2012Received in revised form 18 October 2013Accepted 25 November 2013

Keywords:Renewable energy sourcesDistributed generationDistribution gridLow voltage feederPhotovoltaicHosting capacity

a b s t r a c t

The aim of this article is to present an easily applicable methodology to calculate maximum distributedgeneration (DG) capacity in radial low-voltage feeders. The methodology indicates the highest capacitythat can be installed at a fixed point in the feeder for which the voltage is kept within the permissiblelimits in critical scenarios, i.e. high generation and low load. The concept is based on the main findingsof previous studies identifying the points of connections where the voltage may be exceeded. The provi-sional voltage profile of a line is strongly related to the topology of DGs along the line. The methodologycan be applied in any radial feeder with or without existing DG installations.

� 2013 Elsevier Ltd. All rights reserved.

1. Introduction surements such as grid reinforcement, reactive power provision,

One of the EU’s biggest challenges towards 2050 is cuttinggreenhouse gas emissions by 80–95% [1]. The electricity generationsector is one of the main contributors to achieve this commitmentby reaching near-complete decarbonisation [2]. In this context,renewable energy sources (RES), mainly wind and solar, will pro-vide about half of the electricity generated in the EU by 2050 [3].Moving towards a carbon-free society requires considerable effortto reach the target whilst maintaining the quality and security ofsupply high. The EU is launching funding mechanisms to supportactivities that push forward key policy priorities towards renew-able energy generation, including the development of methodsand tools for network integration of distributed renewable re-sources [4].

Maximum capacity of distributed generation (DG) sources islimited by line loading and by voltage rise—issues already tackledin literature [5,6]. One of the major challenges at high DG penetra-tion is to secure uninterruptable energy supply. This can be prob-lematic in critical scenarios of low demand and high generation,causing reverse power flows in the low voltage (LV) distributionfeeders. In these scenarios, according to [7,8] the voltage limitcan be exceeded, leading to disconnection of the generators. Thus,one of the key elements under high DG deployment is the increaseof the hosting capacity of the lines. This can be realised by mea-

active power curtailment or limiting the feed in [9–12]. These mea-surements have been proven costly or inconvenient to the endusers [13,14]. Thus the first decision making step should be the cal-culation of the maximum DG capacity [15]. A precise calculation oflocal hosting capacities has been addressed in [16]; however, thiscalculation should be feeder and topology specific [17,18], becausethe maximum capacity of a new DG is strongly depended on thepoint where is going to be connected along the feeder.

Recognising the importance in finding the maximum hostingcapacity of a LV feeder, the present article introduces an easy appli-cable tool for a straightforward estimation with no need of compli-cated calculations, e.g. test and trial or neural networks [19,20].The proposed methodology calculates the maximum DG capacityin a fixed point in radial LV feeders taking into account the proper-ties of the line and the already connected DGs. The methodology isbased on an analytical approach described in [21]. As research isfocusing on the measures to increase the hosting capacity, thepresent article answers a very targeted question which should apriori been considered: What is the maximum capacity of a newDG to be installed considering (i) specific feeder characteristics,(ii) the installation point of the new DG and (iii) already installedDGs, which leads the line to exhaust its capacity and thus afore-mentioned measurements should be taken afterwards?

2. Methodology

The proposed methodology is developed to calculate maximumdistributed generation (DG) capacity in low voltage (LV) feeders in

142 I.T. Papaioannou, A. Purvins / Electrical Power and Energy Systems 57 (2014) 141–147

a specific node. This will determine the maximum capacity of anew DG, which can be installed, keeping power quality of the fee-der in the acceptable range. It is applied with the aid of an analyt-ical approach presented in [21]. This calculation is performed forcritical scenario, when there is high DG generation and low load.In critical scenario under high DG deployment voltage upper limitmay be violated. Thus the proposed methodology ensures that thevoltage is within the limits under these conditions. In any othercondition, the voltage along the line is lower. A single-line LV fee-der, as depicted in Fig. 1, is used to explain the methodology. Thepresence of loads and their values (household consumption) alongthe feeder does not change the procedure. To simplify the method-ology, the maximum capacity of a new DG to be installed is as-sumed to refer to a DG operating in unity power factor. Reactivepower of already connected DGs (if some or all of them operatein power factor other than one) is included in the methodology(see Eq. (1), QDG�i). Lastly, after applying the methodology, the load-ing of the line with the calculated maximum capacity should bechecked. In case the thermal limit of the feeder is exceeded, thecalculated maximum capacity of the new DG should be recalcu-lated as it will be explained later (Section 2.1).

In order to calculate the maximum capacity of the new DG in aspecific node, i.e. j + 1, of a LV feeder with N nodes, the followingequation is obtained from Eq. (11) in [21]:

PDG�max�jþ1¼XN

i¼jþ1ðPLOAD�iÞ�

XN

i¼jþ2ðPDG�iÞþ

XN�1

i¼jþ1ðPLOSS�i;iþ1Þ

�Ujþ1DUj;jþ1�Lj;jþ1X0

PNi¼jþ1ðQ LOAD�iÞþ

PNi¼jþ2ðQ DG�iÞ

� �Lj;jþ1R0

ð1Þ

where node j + 1 is the node in which the maximum capacity of thenew DG is calculated, N is the total number of nodes, PDG�max�j+1 isthe maximum capacity that can be installed in the node j + 1, PLOAD-�i is the active load at node i, PDG�i is the rated active power of the DGsystems already installed along the line, PLOSS�i,i+1 are the losses be-tween nodes i and i + 1, Uj+1 is the voltage of the node j + 1, DUj,j+1 isthe difference of the voltage between the successive nodes j andj + 1, Lj,j+1 is the distance between the successive nodes j and j + 1,X0, R0 are the reactance and resistance of the LV line per meter,QLOAD�i is the reactive load at node i, QDG�i, is the reactive power ofthe DG at node i in case DG operates in a power factor other thanone. For the subject DG to be connected at the node j + 1, the reac-tive power is considered zero.

In order to solve Eq. (1), the following assumptions areintroduced:

� Losses, PLOSS�i,i+1, along the line are relatively small [21] and areignored. The calculated capacity will be smaller but within safelimits.

Fig. 1. Single-line diagram

� Since critical scenario conditions of low demand and high gen-eration are considered, loads can be assumed as a minimumhousehold consumption (e.g. the refrigerator: 0.2 kW/phasewith cosu = 0.7 [21]) and that the DGs operate at their ratedpower, thus PLOAD�i, PDG�i, QLOAD�i and QDG�i are known.

Applying these assumptions Eq. (1) becomes three unknownparameters function:

PDG�max�jþ1 þ1

Lj;jþ1R0

� �Ujþ1DUj;jþ1

¼XN

i¼jþ1ðPLOAD�iÞ �

XN

i¼jþ2ðPDG�iÞ

þX0PN

i¼jþ1ðQ LOAD�iÞ þPN

i¼jþ2ðQ DG�iÞ� �

R0ð2Þ

Eq. (2) has the following form: PDG�max�j+1 + a Uj+1 DUj,j+1 = b,wherePDG�max�j+1, is the first unknown – the maximum capacity that canbe connected at the node j + 1, Uj+1 is the second unknown – thevoltage at the node j + 1 after the new installation of PDG�max�j+1,DUj,j+1 is the third unknown – the voltage difference between thenodes j,j + 1 after the new installation of PDG�max�j+1.

Thus the voltage at the subject node Uj+1 and the voltage differ-ence of the successive nodes DUj,j+1 need to be calculated. The chal-lenge is to find a way to predict the voltage rise caused by the newDG. The assumption in all cases is that the maximum capacity to beinstalled is limited by the maximum voltage that will eventuallyappear in the line. This voltage defines the maximum DG. The nodewhere the maximum voltage will appear depends on the topologyof the line. Thus in a feeder where DGs are already connected themaximum voltage will appear in the last distant node with a signif-icant DG capacity installed [5]. Significant DG is considered to be acapacity high enough to cause reverse energy flow i.e. from con-sumer to the step-down transformer, in critical conditions (highgeneration and low load). Thus the maximum voltage in the linecan be found at the node where this DG is connected, beyond thej + 1 node.

In order to solve Eq. (2), a distinction should be made betweendifferent feeder topologies, since already connected DG systemsplay a decisive role. Three scenarios are therefore identified as pre-sented in Table 1. The methodology is developed accordingly.

� Scenario A: The new DG is the first DG to be connected alongthe line at a fixed node.� Scenario B: The new DG system is to be connected at a fixed

node. The line has no other DGs connected before the newDG, but at least one DG sited after.� Scenario C: The new DG system is to be connected at a fixed

node. The line has at least one DG connected before the new DG.

of a radial LV feeder.

Table 1Feeder topologies for Scenarios A, B and C.

Fig. 2. Flowchart to calculate maximum capacity of a new DG, Scenario A, B, C, part1 of 2.

I.T. Papaioannou, A. Purvins / Electrical Power and Energy Systems 57 (2014) 141–147 143

After calculating the maximum capacity of the new DG, PDG�max-�j+1, the thermal limit of the feeder should be checked. Under crit-ical conditions (maximum generation and minimum load), wherereverse flow exists, the maximum current appears at the branchx � 1, x, i.e. between nodes x � 1, x. The x is the node where the firstsignificant DG (the closest DG to the transformer) is connectedalong the feeder. This can be approximated with the followingEq. (3):

Ix�1;x ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiPNi¼xðPDG�i�PLOAD�i Þ

� �2þ

PNi¼xðQ DG�i þ Q LOAD�iÞ

� �2

Ux

vuutð3Þ

where Ix�1,x is the current at the branch x � 1, x. Ux is the voltagewhich appears at x after the installation of the PDG�max�j+1 (this valuecan be obtained with Eq. (11) [21]).

In case that Ix�1,x exceeds the thermal limit, a new PDG�max�j+1

should be recalculated, as it will be presented in the methodology(Scenario A).

All the scenarios are explained applying a feeder with 10 house-holds (Fig. 1, N = 10). The three-phase transformer is assumed to bea 250 kV A rated power with secondary windings voltage of 400 V.The single-phase line connected to the one of the three secondarytransformer windings is assumed to be an overhead 35 mm2 line(steel reinforced aluminium conductor, ACSR, R0 = 0.576 X/km,X0 = 0.397 X/km). Such a line is characterized with 224 A rated cur-rent and consequently with a 51.52 kW/phase capacity. The mini-mum household load is assumed 0.2 kW/phase with cosu = 0.7 (i.e.PLOAD�i and QLOAD�i). The upper voltage limit is 10% (Umax = 253 V,phase voltage). Here the maximum capacity of the new DG,PDG�max�j+1, is calculated for the 7th node (j + 1 = 7). The distanceLj,j+1 is considered to be 60 m and the distance L0,j+1 420 m. For eachcalculation, the thermal limit of the line, Imax, should be checked toensure it is within the limits (less than 224 A). As depicted in flow-chart in Fig. 2, these values as well as capacities and operationcharacteristics of existing DGs (i.e. PDG�i and QDG�i) are input data.

2.1. Scenario A

In Scenario A, the line has no other DG (Table 2), so the highestvoltage will appear at the subject node, j + 1. The maximum capac-ity that can be installed is thus defined by the voltage limit whichwill appear at the subject node, i.e. Uj+1 = Umax = 253 V. Thus Eq. (2)becomes a two-unknown function. A flowchart showing the calcu-lation steps in Scenario A starts in Fig. 2 and continues in Fig. 3.

In order to calculate DUj,j+1 an approach can be made. The newvoltage profile after installing the new DG can be assumed to belinearly shifted up from U0 at node 0 (secondary windings of trans-former) to 253 V (maximum voltage) at the subject node j + 1.

For this increase, the following equation can be used:

DUj;jþ1 ¼ð253� U0ÞðLj;jþ1Þ

ðL0;jþ1Þð4Þ

where L0,j+1 is the distance between node 0 and node j + 1 (Fig. 1).Using the above and assuming Uj+1 = 253 V, Eq. (2) can be

solved, giving the maximum capacity of the DG system that can

be installed, in this example PDG�max�j+1 = PDG�max�7 = 25.95 kW/phase.

For the specific maximum capacity of the new DG, PDG�max�j+1,the current is also checked so that it does not exceed the thermallimit of the feeder (Imax = 224 A). If the line loading is within thelimit then the maximum capacity of the new DG is the one esti-mated above. If the line loading limit is violated then the maxi-mum capacity should be recalculated. This procedure is shown inFig. 3 (Submodule 1) and it is realised with the aid of Eq. (3) andthe assumption that the current Ix�1,x should be kept equal to orbelow the thermal limit (Imax). Initially Ux is obtained from Eq.(11) [21] assuming the new DG connected. Later Ux is correctedby iterations. In order to calculate the new PDG�max�j+1 from Eq. (3)a two degree function should be solved. Then the obtained valueof PDG�max�j+1 is used in Eq. (11) [21] to calculate the new Ux andan iterative procedure initiates between Eqs. (3) and (11) [21] untilthe difference of the successive obtained values of PDG�max�j+1 is sat-isfactorily small (indicated by a tolerance e).

Finally, in order to validate the methodology the voltage profileof the line with the new DG is calculated with the aid of Eq. (11)[21] and presented in Fig. 4. Indeed, the maximum voltage appearsin the 7th node.

2.2. Scenario B

In Scenario B, a DG system is assumed to be already installed ata node after the 7th, e.g. at the 9th there is significant DG capacity,

Table 2DG capacities per node in Scenarios A, B, C1 and C2.

Scenarios DG capacity per node, kW

PDG�1 PDG�2 PDG�3 PDG�4 PDG�5 PDG�6 PDG�max�7 PDG�8 PDG�9 PDG�10

A 0 0 0 0 0 0 New DG 0 0 0B 0 0 0 0 0 0 New DG 0 14 0C1 0 22 0 0 0 0 New DG 0 0 0C2 0 8 0 0 0 0 New DG 0 14 0

Fig. 3. Flowchart to calculate maximum capacity of a new DG, Scenario A, part 2 of2.

Fig. 4. Line voltage with the maximum capacity of the new DG connected at the 7thnode, Scenario A, B, C1 and C2.

144 I.T. Papaioannou, A. Purvins / Electrical Power and Energy Systems 57 (2014) 141–147

PDG�9 = 14 kW/phase (Table 2). This scenario differentiates from theprevious Scenario A as now the maximum voltage after installingthe new DG will be found at the 9th node and not at the 7th. Sothe assumption that Uj+1 = Umax = 253 V is no longer valid. The Umax

value should be reached at the last node (y) with significant DGconnected, i.e. the 9th (y = 9). So, in this scenario, Eq. (2) remainsa three-unknown function.

The key element in this scenario is based on an observationmade by the authors while developing their approach in [21]:The DU for the successive nodes between j + 1 and N remains the sameas before the installation of the new DG at node j + 1. This is due tothe fact that the current in the branch between nodes j + 1 and Nremains constant regardless of the DG capacities in the branch be-tween nodes 0 and j + 1 under conditions of reverse power flow.

The above statement is illustrated graphically in Fig. 5, wherethe voltage profile is presented for three randomly selected DGcapacities at the 7th node (0, 4 and 8 kW/phase), considering14 kW/phase DG at the 9th node. For all these capacities, the DU(inclination of the line) is identical between 7th and 9th node.Installing the new DG does not affect the way the voltage changesfrom the node of the new DG system and beyond.

This property is used to calculate the voltage in the subject nodeafter the installation of the new DG. Thus for calculating Uj+1, the

Pfunction takes into account the nodes beyond the node j + 1 up tothe last significant DG connected:

Ujþ1 ¼ 253�Xy

i¼jþ2ðDUi�1;iÞ ð5Þ

where y is the last node with significant DG installed, y = 9, DUi�1,i isthe voltage difference between nodes i � 1 and i. This is calculated

Fig. 5. Line voltage for various DG capacities connected at the 7th node.

Fig. 6. Flowchart to calculate maximum capacity of a new DG, Scenario B, part 2 of2.

I.T. Papaioannou, A. Purvins / Electrical Power and Energy Systems 57 (2014) 141–147 145

from the initial topology (without the new DG) since it will remainthe same after the installing of the new DG.

Knowing the voltage at the node where the new DG system isplanned to be installed, the DUj,j+1 remains to be calculated. Theapproach here is similar to Scenario A, with two main differences:

Uj+1 as calculated in Eq. (5), is usedThe initial voltage in node 0 is not 230 V but due to the presence

of DGs in the line has already risen. Its value can be obtained by theinitial voltage profile using Eq. (11) [21].

Eq. (4) becomes:

DUj;jþ1 ¼ðUjþ1 � U0ÞðLj;jþ1Þ

L0;jþ1ð6Þ

In the specific example, i.e. 14 kW/phase DG at the 9th node, thecalculated maximum capacity that can be connected at the 7thnode is calculated to be 7.5 kW/phase (Eq. (2)). As regards the lineloading with the new DG, it is within the acceptable limit (lowerthan Imax). As in Scenario A, the flowchart of the Scenario B startsin Fig. 2 and continues in Fig. 6.

Fig. 4 shows the voltage profile along the line with the new DGinstalled, the voltage at the 9th node almost reaches the limit.

2.3. Scenario C

Scenario C can be distinguished in two sub-scenarios C1 and C2:

� Scenario C1: other DG units are connected before the new DGbut no other unit is connected after� Scenario C2: DG units are connected before and after the new

installation.

The calculation steps in Scenario C are described in the follow-ing sub-sections and are depicted in Figs. 2 and 7.

2.3.1. Scenario C1In Scenario C1, the voltage at the subject node j + 1 is assumed

to reach the maximum limit of 253 V (Umax) since there the last DGunit along the line would be connected. Thus Eq. (2) becomes atwo-unknown: PDG�max�j+1 being the first unknown, and DUj,j+1 thesecond. The solution in the Eq. (2) would be the specific combina-tion of PDG�max�j+1 and DUj,j+1 for which, by connecting the specificcapacity of PDG�max�j+1 at node j + 1, gives a voltage of 253 V in the

node j + 1. As above, if the line loading is exceeded, the maximumcapacity of PDG�max,j+1 is determined by the thermal current of theline.

Assuming that a new DG unit is to be installed at the 7th nodeand there is already existing DG of 22 kW/phase at the 2nd node(Table 2), Eq. (2) becomes PDG�max�j+1 – 7320.60 DUj,j+1 = 1362.53.

At the beginning of the calculation the voltage difference DUj,j+1

is set to 0 (see Fig. 7). During each calculation cycle DUj,j+1 is in-creased by Step (e.g. 0.1). Every time DUj,j+1 is increased, PDG�max�j+1

is recalculated (Eq. (2)) and used in Eq. (11) [21] to obtain the newUj+1. This cycle repeats until Uj+1 exceeds 253 V. When this happensthen the maximum capacity to be installed in the specific node,j + 1 is the value of the previous cycle. For the respective examplethis capacity is calculated 18.9 kW/phase. Connecting this DGwould not overload the line giving a voltage profile along the lineas presented in Fig. 4.

2.3.2. Scenario C2In Scenario C2, DG systems are located not only before but also

after the node at which the new DG is planned to be installed.Assuming the 14 kW/phase DG system is connected at the 9thnode, and another one connected at the 2nd node with 8 kW/phasecapacity (Table 2). In this scenario the characteristic of the voltagechange along the line are as in Scenario B. Thus, assuming that thevoltage at the last node with DG, i.e. the 9th, will reach the maxi-mum limit (Uy = Umax) after installing the new DG, Uj+1 can be cal-culated with Eq. (5) (Fig. 7). Then Eq. (2) becomes a two-unknown:PDG�max�j+1 being the first unknown, and DUj,j+1 the second. Now aniterative procedure can be applied as in Scenario C1. The maximumcapacity in the 7th node is calculated 5.4 kW/phase and the new

Fig. 7. Flowchart to calculate maximum capacity of a new DG, Scenario C1 and C2,part 2 of 2.

146 I.T. Papaioannou, A. Purvins / Electrical Power and Energy Systems 57 (2014) 141–147

voltage profile is shown in Fig. 4. There is no line overload hereeither.

2.4. Large scale applications

The methodology is presented for calculating the maximumhosting capacity in a fixed point along a radial LV feeder. In large

scale scenarios, there should be some assumptions so that themethodology can be adapted to the certain conditions. Thus, in amulti feeder network as presented in [17], where several feedersare connected to the same transformer, the voltage of each feedersums up at the transformer secondary windings. This means thatfor estimating the hosting capacity of the one feeder the voltageat the secondary windings should be calculated taking into accountthe influence of the other coupled feeders. Especially, when alongthis multi feeder network DGs are connected, then the voltage atthe transformer in critical scenarios could be already raised.

Moreover, in the case that the maximum hosting capacity needsto be calculated but without having a fixed point of connectionthen the methodology can be utilised with the assumption thatthe maximum DG capacity is hypothetically planned to be con-nected at the last node along the feeder. This is the worst scenarioof DG topology for the voltage distortion [5]. The estimated capac-ity can be placed then in any other node along the feeder assuringthat the voltage will be kept within the limits. For example in a fee-der of N nodes, the maximum DG capacity can be calculated sup-posing that it will be connected in the last node N. The PDG�max�Nthat is obtained can be then distributed in other nodes under thecondition that the algebraic sum of the new DGs connected inthe feeder will not exceed the value of PDG�max�N.

3. Conclusions

The present article proposes an easily applicable methodologyto calculate maximum DG capacity that can be installed in a radialLV feeder at a fixed point. The maximum capacity is limited by theallowable voltage rise along the feeder. Once calculated, the newline loading should be checked again to ensure that the line isnot overloaded. The methodology is based on the analytical ap-proach developed in a previous study of the authors [21]. Thetopology of the line plays a decisive role in estimating the maxi-mum capacity of a new DG. The methodology can be applied forany radial feeder with or without existing installations of DG.

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