Load Area model accuracy in distribution systems

Electric Power Systems Research 143 (2017) 321–328

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Electric Power Systems Research journal homepage: www.elsevier.com/locate/epsr

Load Area model accuracy in distribution systems G.M. Casolino ∗ , A. Losi Dipartimento di Ingegneria Elettrica e dell’Informazione “M. Scarano”, Università degli studi di Cassino e del Lazio Meridionale, Via G. Di Biasio 43, 03043 Cassino, FR, Italy

a r t i c l e

i n f o

Article history: Received 16 May 2016 Received in revised form 15 September 2016 Accepted 17 October 2016 Keywords: Load Area Model accuracy Distribution network representation

a b s t r a c t Modern distribution systems evidence an increasing complexity, stemming from the necessity of new and updated monitoring and control functionalities. A heavy information burden is associated with complexity, which requires an appropriate information treatment. In this view, a relevant simplification of information can be achieved by representing distribution networks by sub-networks, called Load Areas (LAs). The LA concept and its possible implementations have been already proposed. The paper proposes a general formulation of LA, with an exhaustive explanation of the approximation introduced by the model; also, the reduction of information burden is evidenced. For a given study network, the results in different configurations highlight the impact of the proposed model on the accuracy of the LA representations, and also the relevant reduction of the amount of information required to describe the network, and a reduction of computational burden as well. © 2016 Elsevier B.V. All rights reserved.

1. Introduction The sustainability, convenience, and security of the electrical industry require flexible, accessible, reliable, and economical power grids [1]. The entire electricity chain is changing to fulfil these requirements. The distribution grids are deeply changing too; their architecture is increasingly flexible so as to give benefits to all their users [2]. Focus is on the role of consumers, the connectivity among areas, the exploitation of network resources, and even the use of the electrical network to sell other types of services. In distribution control centres, a huge amount of information results from the new and/or updated monitoring and control functionalities [3,4]. It should be noted that not all the available data have to be treated in details; a significant reduction of the burden could be obtained by a proper selection of the relevant information [5]. Some recent proposals go in that direction, by introducing an alternative approach which tries to model distribution networks in a way similar to the transmission ones [6]; other proposals investigate the possibility of grouping part of the network for monitoring purposes [7]. In this connection, it is of primary importance to recognize the elements strictly necessary for a correct yet compact representation of the electrical system. To this end, a useful idea, originated from the project ADDRESS [8], is the one of identifying

∗ Corresponding author. E-mail address: [email protected] (G.M. Casolino). http://dx.doi.org/10.1016/j.epsr.2016.10.044 0378-7796/© 2016 Elsevier B.V. All rights reserved.

different sub-networks of distribution system, called Load Areas (LAs), by evaluating the influence of prosumers’ injections on the relevant network constraints [9]. Upon this view, a LA is seen as a set of prosumers whose power injection has a similar impact on the operating conditions of the distribution grid. The simplified network representation deriving from the LA approach can support and foster the adoption of a variety of recently proposed methods concerning load management in smart grids [10], autonomous yet interacting sub-networks [11], implementation of Demand Response programs [11–13]. Different aspects concerning LAs have been already examined: representation in [14], radial equivalent circuit in [15], application in a Medium Voltage Control Center in [16,17], adoption of specialized methods for radial networks to get compact representation in [18]. In this paper a general formulation of LA is proposed, with an exhaustive explanation of the approximation introduced by the model; also, the related reduction of information burden is evidenced (see Section 4). For a given study network, the results in different configurations highlight the impact of the proposed model on the accuracy of the LA representations, and also the relevant reduction of the amount of information required to describe the network, and a reduction of computational burden as well. 2. Load Area – the concept The concept of LA was introduced in the ADDRESS project [8] and further investigated in [14,15,17,18]; it focuses on the participation of small consumers, connected to a low voltage grid, to the power

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G.M. Casolino, A. Losi / Electric Power Systems Research 143 (2017) 321–328

Fig. 1. Example of a distribution system organized in Load Areas [9].

system markets. A LA is a part of the network made of nodes whose injection has a similar impact on the distribution grid key operating constraints, and the lines connecting them. A whole distribution system can be seen as a composition of LAs (see Fig. 1). The clustering of nodes in LAs is obtained in three steps (the reader is referred to [14,17] for more details).

with ˙ representing the n – vector of the eigenvalues of Y˙ (˙ h is its ˙ a n × n matrix whose columns are the correh–th component),  sponding eigenvectors, and D{xh } represents the diagonal matrix whose elements along the principal diagonal are the components of vector x. If all the eigenvalues are distinct and non-zero, the L2 −norm of the node voltage vector can be expressed as:

2.1. Selection of key constraints Grid key constraints (under/overvoltage, loading) are selected based on historical data collected in the Distribution Management System, possibly in conjunction with load flow analysis.

n  1 ˙T 1 ˙T Sk J, Sh J  

U = Y˙ −1 J  = 

h=1

The impact of each nodal injection on the selected constraints is evaluated. 2.2.1. Loading constraints For each loading constraint, in a radial network the list of all the nodes downstream of the involved component is built with simple graph navigating techniques; the impact of a nodal injection is set to one if it is in the list, to zero otherwise.

∂˙ h ⎢ ∂Y˙ 11 ⎢

⎢. . S˙ h = ⎢ ⎢.

⎢ ˙ ⎣ ∂h ∂Y˙ nb 1

..

.

...

∂˙ h ∂Y˙ 1nb ⎥ ⎥

.. .

∂˙ h ∂Ynb nb

J = Y˙ U ,

2.3. Clustering nodes

˙ ˙ −1 , Y˙ = D{ ˙ h }

(2)

⎥ ⎥, ⎥ ⎥ ⎦

(4)

and k is the index of the element of ˙ with minimum modulus. Each matrix element in (4) represents the sensitivity of the h–th eigenvalue to the corresponding element of the nodal admittance matrix [19]. The classification of the grid nodes can be obtained on the basis of |S˙ k (i, i)| normalized to the maximum: Skr (i) =

where Y˙ is the n × n nodal admittance matrix (n is the number of nodes); it can be expressed as:

(3)

⎤

...

2.2.2. Voltage constraints The impact of each nodal injection on the voltages can be computed in a comprehensive way upon the conditions set by the Inherent Structure Theory of Networks [19,20,9]; the impact of a nodal injection on any nodal voltage is described by its impact on the voltage dominating component. Nodal voltages, U, and currents injected in the nodes, J, are related by (matrix-vector notation applies): (1)

|˙ k |

where S˙ h is the h-th eigenvalue sensitivity matrix,

⎡

2.2. Impact of nodal injections on the grid constraints

˙ h

|S˙ k (i, i)| . maxi |S˙ k (i, i)|

(5)

The value of Skr (i) is a score of the impact of the variation of the load at i–th node on the nodal voltages [9].

The nodes with comparable values of impact factors are grouped for the two different issues of loading and voltage; then, the groupings are combined into LAs.

G.M. Casolino, A. Losi / Electric Power Systems Research 143 (2017) 321–328

323

⎡ ⎤





J de U de Y˙ de,de Y˙ de,in ⎥ = . ⎦+

Je

⎢ J = Y˙ U ⇒ ⎣ 0

Y˙ in,de

J in

0

Y˙ in,in

(7)

U in

From (7), with a Gaussian elimination we can write: −1 J eq = Y˙ eq U eq + Y˙ de,in Y˙ in,in J in − J de ,

where

(8)



Je

J eq =

,

0

U eq = U de (9)

−1 ˙ Y˙ de,de − Y˙ de,in Y˙ in,in Yin,de .

Y˙ eq =

Injected currents can be expressed as:



J=D

Fig. 2. Example of LAs [9].

1) Loading: For each loading constraint, all the nodes with a unitary impact factor for that constraint are grouped in the so-called Overload Load Area (OLA) (see Fig. 2). 2) Voltage: For the voltage issue, clusters are obtained that form the so-called Voltage Load Areas (VLAs) (see Fig. 2). Nodes are ordered based on the value of Skr (i) (5); two consecutive nodes in this ordering, m and n, are considered to belong to different VLAs if |Skr (m) − Skr (n)| > Sl ,

(6)

where Sl is an appropriate threshold [9]. 3) Overall: The OLAs and the VLAs are sets of grid nodes; LAs are obtained by the intersection of these sets so that all the nodes of the distribution system belong to a LA and each node belongs to only one LA [9] (see Fig. 2).

3. Load Area – the representation Once the nodes of the LA are identified (see Section 2.3), the lines connecting them can be easily recognized; they all form an electrical sub-network. Various modelings do exist for an electrical network; one of them is based on nodal injections and nodal voltages [see (1)]. A compact representation of the LA network would retain the border nodes, and for radial networks also some additional internal nodes to preserve the radiality (the reader is referred to [14,17] for more details). The other nodes of the LA can be removed by a Gaussian elimination, as in the following; in the radial case, specialized methods could be exploited (i.e. by means of forward and backward algorithms) [18]. Here, we consider the total complex power injected into the nodes. Similar prosumers in a given geographical region can be grouped in categories (residential, small commercial, etc. [21]); in Appendix A it is shown how to express the power injected into the nodes as a function of the active powers injected by the prosumer categories. Let J e represent the current injections into the edge nodes, that model the connection of LA with the rest of the distribution grid. With the partition of grid nodes into describing nodes ‘de’ (edge and retained internal), and (other) internal nodes ‘in’, from (1) we can write:

1 ˆ Ui



ˆ C,

(10)

where C˙ is the n-vector of the injected complex powers. In Appendix A it is shown how to express the complex injected powers as functions of the active powers injected by prosumer categories: Cˆ =

nc 

A˙ h (Ph ) Ph ,

(11)

h=1

where nc is the number of prosumer categories, Ph is the overall active power injection by the h-th category, and A˙ h (Ph ) is the hth n-vector of complex functions which relate the conjugate nodal power injections to the overall active power of the h-th category. With (10) and (11), Eq. (8) becomes: {12} J eq

 −1 = Y˙ eq U eq + Y˙ de,in Y˙ in,in D

 −D

1 ˆ Ueq,i

 nc

1

 nc

ˆ in,i U

A˙ h (Ph ) Ph

h=1

 A˙ h (Ph ) Ph

h=1



in

,

(12)

de

where {12} points to the model equation; note that model (12) has no approximation, but it still contains voltages U in , so it is not a truly equivalent model. Let * denote a reference case, and assume that nodal voltages do not change too much with respect to this reference case: ∗

U≈U ,

(13)

when powers do change: Ph = Ph∗ + Ph ,

for anyh = 1, . . ., nc .

(14)

With (13), from (12) we obtain the (linear) equivalent model



{15} J eq

=

−1 Y˙ eq U eq + Y˙ de,in Y˙ in,in D

 −D

1

 n c 

ˆ∗ U in,i

h=1

 n  c  1 . A˙ h (Ph ) Ph ∗

ˆ U eq,i

h=1



A˙ h (Ph ) Ph in

(15)

de

Due to (13), the results obtained with model (15) will contain errors when Ph = / 0, for any h = 1, . . ., nc . We try reducing the errors by considering again the actual voltages, but only of the retained nodes in order to have a truly equivalent model. One such possibility is introducing the actual voltages of the retained nodes

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Fig. 3. Simple network.

in the second and third terms of the rhs of (15), as follows:



{16} J eq

= Y˙ eq U eq + D

 −D

D

= Y˙ eq U eq + D

1 ˆ U eq,i

−1 Y˙ de,in Y˙ in,in D

 

∗ ˆ eq,i U



−D



ˆ eq,i U

ˆ eq,i U



∗

ˆ eq,i U

1

 nc

∗

ˆ in,i U

 n  c  1 ˙ Ah (Ph ) Ph ∗

ˆ eq,i U

∗ ˆ eq,i U





ˆ eq,i U

 nc

h=1

−1 Y˙ de,in Y˙ in,in D

 A˙ h (Ph ) Ph

h=1

1

 nc

∗

ˆ in,i U

.

A˙ h (Ph ) Ph

h=1

de



 in

 A˙ h (Ph ) Ph

h=1

in

(16)

de

3.1. Improving model accuracy To evidence the advantage of adopting model (16) instead of model (15), let us consider the network of Fig. 3; the part encircled with the dotted line will be represented as a LA, taking node 1 as the describing node while node 2 is internal. Let us assume that there is only one prosumer category (nc = 1), which injects active power only, P; phasors are all aligned with each other, and we can write: J1 =

1∗ U1

J2 =

P,

2∗ U2

Fig. 4. Errors in simple network.

P, 1∗ + 2∗ = 1,

1∗ , 2∗ > 0.

(17)

For the case under exam, it can be easily seen that in (8) and (9) it is: −1 Y˙ de,in Y˙ in,in = −1,

Y˙ eq = 0

(18)

and models (12) and (15) become, respectively, {19}

Jeq

{20}

Jeq

=− =−

2∗ U2

P−

1∗

P,

(19)

∗ P.

(20)

U1

2∗

1∗

U2

U1

∗P −

The voltage of the describing node, Ueq = U1 (see Fig. 3), can be expressed as: U1 = E0 − R0 Jeq ,

(21) {19}

for any model [notice that (21) is not a direct expression for U1 due to the (19) for {20}

the one of U1

19} Jeq ].

,

is the correct one, while

is approximated; the error

 {20}  {19}  ε{20} = U −U 1

The value of

{19} U1

1

(22)

is reported (dashed line) in Fig. 4 in arbitrary units versus the injected power (from maximum load to maximum generation), for three different choices of the reference condition. The error is

obviously zero at the reference operating condition and for zero injection. It is easy to see that the error ε{20} is due to the independence of injected currents from voltage in the model (20). To reduce the error while preserving the nature of the model of being an equivalent model, we should reintroduce the dependence of the current on the voltage (besides the one on the power), but relying on the voltage of the describing node only, U1 . This can be accomplished with the model {23} Jeq

=

U1∗



−

U1

2∗

P U2∗

−

1∗

P U1∗



=−

U1∗ 2∗

P U1 U2∗

−

1∗ U1

P.

(23)

With (23), the dependence of the current on voltage is correctly restored at node 1 (second term in the rhs), while at node 2 it is restored in an approximated but satisfactorily way (first term in the rhs): voltages U1 and U2 differ by no more than 20% in any normal operating condition (including the reference one), and vary with the same sign for a change of injected power. The expression (23) is the same as (16) written for the simple network. In Fig. 4 the error in voltage U1 with model (23)



ε{23} = U1

{23}

{19}

− U1

 

(24)

is shown (continuous line). As expected, it is: ε{23} < ε{20} ,

for anyP = / 0, P = / P∗.

(25)

G.M. Casolino, A. Losi / Electric Power Systems Research 143 (2017) 321–328

32 33

29 30 250

31

28 25 48 47

27

26 25r 23 24

44

49

50

151 300

51

Table 1 Network switches. 111

110 112 113 114

SW6

46

108

45

64

107 109

104 103 450 42 63 41 66 101 102 100 21 40 71 62 197 99 22 39 70 98 38 135 35 69 18 36 SW4 97 19 68 37 SW2 75 20 160160r 74 60 67 73 57 9r SW3 58 85 59 11 14 61 79 72 10 9 610 78 56 55 54 77 2 sourcebus 53 LA3 152 52 76 SW5 8 13 80 84 150 149 1 7 SW1 96 94 34 90 88 92 LA1 LA2 17 12 86 81 87 89 15 91 95 93 3 5 6 82 83 16 4 43

65

105

325

106

Fig. 5. LAs for case SW111100 – default case.

The above analysis cannot be easily generalized; nevertheless, it suggests to adopt model (16) to reduce the errors on voltages. In fact, numerical experiments on different networks show that model (16) actually behaves much better than model (15) (see Section 4 and [14], [17]).

Switch

From bus

To bus

1 2 3 4 5 6

13 18 60 97 54 151

152 135 160 197 94 300

32 33

31

29 30 250 28 25 48 47

27

26 25r 23 24

44

49

151 300

51

50

111

110 112 113 114

SW6

46

107

108

45

109

64

104 103 450 41 66 101 102 100 21 40 71 62 197 99 22 39 70 98 38 135 35 69 LA4 18 36 SW4 97 19 68 37 SW2 75 20 160160r 74 60 67 73 57 9r SW3 58 14 85 59 11 61 79 72 10 9 610 78 56 54 55 77 2 sourcebus 53 LA3 152 52 76 SW5 8 13 80 84 150 149 1 7 SW1 96 94 34 90 88 92 LA1 LA2 17 12 86 81 87 89 15 91 95 93 3 5 6 82 83 16 4 43

65

105

42

106

63

4. Study case Fig. 6. LAs for case SW110011 – configuration 5.

The network of Fig. 5 is taken as the study case; network data have been obtained from the IEEE 123-node test feeder [22] and the OpenDSS Simulation Tool [23]. Results for a smaller (IEEE 13-node test feeder) and for a larger network (IEEE 8500-node test feeder) can be found in [14,15,17]. The most relevant aspects of the LA concept and application previously described are tested; attention is paid to the errors introduced by the compact representations. For representing the network, physical bus locations are taken as nodes. Two prosumer categories are considered: pure loads, denoted by suffix ‘L’, and medium/large DG plants with variable power factor [14]. Different configurations of the network are analyzed, considering all loads connected and imposing a radial operation of the grid, so as to observe how the change of topology impacts on the LAs. In each case the LAs are identified according to the procedure recalled in Section 2. The errors on the voltage of the edge nodes of the compact model with either (15) or (16) are evaluated by varying pure loads, with and without DG (randomly distributed in the grid). The error on the voltage is assumed acceptable if it is less than the one due to the smallest tap changer step of OLTCs in the grid, according to [5]; for a 32-step voltage regulator with a range of ±10 %, a one-step tap change corresponds to 0.006250 p.u.

32 33

29 30 250 28 25 48 47

27

26 25r 23 24

44

49

151 300

51

50

111

110 112 113 114

SW6

46

107

108

45

109

64

104 103 450 41 66 101 102 100 21 40 71 62 197 99 22 39 70 98 38 135 35 69 18 36 SW4 97 19 68 37 SW2 75 20 160160r 74 60 67 73 57 9r SW3 58 85 59 11 14 61 79 72 10 9 610 78 56 54 55 77 2 sourcebus 53 LA3 152 52 76 SW5 8 13 80 84 150 149 1 7 SW1 96 94 34 90 88 92 LA1 LA2 17 12 86 81 87 89 15 91 95 93 3 5 6 82 83 16 4 43

65

105

42

106

63

Fig. 7. LAs for case SW111001 – configuration 2.

Table 2 Switch configurations.

4.1. Test grid In the network under exam there are six switches useful for reconfiguration [24] (Table 1); their positions are indicated by a ‘x’ in Figs. 5–8. Assuming a radial operation and the supply to all busses, only 9 configurations are possible out of the possible 64 (26 ). For these 9 configurations, the status of the switches is reported in Table 2, where ‘1’ and ‘0’ indicate the closed and open status, respectively. In the same Table, the default status of the switches is referred to as Configuration 1 (SW111100).

31

→ →

→

→

Configuration

SW1

SW2

SW3

SW4

SW5

SW6

1 (Default) 2 3 4 5 6 7 8 9

1 1 1 1 1 1 1 0 0

1 1 1 1 1 0 0 1 1

1 1 0 0 0 1 0 1 0

1 0 1 1 0 1 1 1 1

0 0 1 0 1 0 1 0 1

0 1 0 1 1 1 1 1 1

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G.M. Casolino, A. Losi / Electric Power Systems Research 143 (2017) 321–328

32 33

31

29 30 250 28

49

50

51

151 300

LA3 111 110 112 113 114

SW6

107 108 109 64 25r 44 26 104 23 43 65 105 106 103 24 450 42 63 41 66 101 102 100 21 40 71 62 197 99 22 39 70 98 38 135 35 69 18 36 SW4 97 19 68 37 SW2 75 20 160160r 74 60 67 73 57 9r SW3 58 85 59 11 14 61 79 72 10 9 610 78 56 55 54 77 2 sourcebus 53 LA4 152 52 76 SW5 8 13 80 84 150 149 1 7 SW1 96 94 34 90 88 92 LA1 LA2 17 12 86 81 87 89 15 91 95 93 3 5 6 82 83 16 4 25 48 47

27

46

45

Table 3 Number of physical locations in full and compact representations. Configuration

Full network

With LAs

Reduction

Default 2 5 8

123 123 123 123

3 3 5 4

97.6% 97.6% 95.9% 96.7%

Fig. 8. LAs for case SW011101 – configuration 8.

Out of the 9 possible configurations, some are preferable with respect to some optimality criterion (i.e. maximum reliability, minimum power loss, maximum reliability & minimum power loss). A further reduction follows of the number of configurations to be considered [24]: they are the ones evidenced by a → in Table 2. The identification of the LAs is obtained for all the considered configurations with the procedure of Section 2. With reference to the default configuration (see Fig. 5), the sensitivity analysis (4)–(6) identifies three LA, obtained with a value of 0.003 for Sl (see Fig. 9). Proceeding from the source node, the first nodes encountered are respectively the node 150 (LA1), the node 149 (LA2) and the node 80 (LA3), as it is apparent from Fig. 5. The lines 150-149 and 78–80, which connect respectively LA1 with LA2 and LA2 with LA3, are considered belonging to the corresponding upstream LA. The LA1, LA2 and LA3 count one, two, and one nodes, respectively. The overall grid model based on LA equivalents counts only three nodes in comparison with the 123 of the original grid (see Table 3). The LAs for the other configurations are identified by the same procedure; the results are shown in Figs. 5–8, and summarized in Table 3. The graph of the maximum errors in voltage (modulus) for different values of the (pure) load, with and without DG, are depicted in Figs. 10–13. The maximum values of the errors are also reported in Table 4; they are all of the same order of magnitude, with the exception of

Fig. 10. Max error in voltages – default case [17].

Fig. 11. Max error in voltages – configuration 5.

Fig. 9. Sensitivity analysis – default case [17].

Fig. 12. Max error in voltages – configuration 2.

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327

Future work will deal with the effective treatment of the different LAs deriving from the actual switching in radial networks, and the identification of the physical locations in the case of unbalanced systems. Appendix A. Categories of prosumers Here we recall the modeling of the power injected into the network nodes by means of the active/reactive powers due to prosumer categories proposed in [14,17]. A.1. Prosumers Similar prosumers in a given geographical region can be grouped in categories: residential, small commercial, etc. [21]. The active power injection by the k-th prosumer of the h-th category, Ph,k , is a fixed part, p∗h,k , of the overall active power injection of its category, Ph : Fig. 13. Max error in voltages - configuration 8.

Ph,k = p∗h,k Ph .

The similarity of the behaviour can be extended to include the reactive power injection as follows:

Table 4 Maximum voltage errors [pu]. Model

No DG DG

(15) (16) (15) (16)

(A.1)

Configuration Default

2

5

8

0.0051 0.0006 0.0027 0.0004

0.0044 0.0004 0.0022 0.0004

0.0041 0.0001 0.0023 0.0002

0.0262 NO 0.0012 0.0143 NO 0.0008

∗ Qh,k = fh,k (Ph )p∗h,k Ph .

(A.2)

From (A.1) and (A.2) different categories of prosumers can be modeled: - Pure Load: the injection by the k-th prosumer in the h-th pure-load category is modeled by a constant power factor (pf): ∗ Qh,k = tan ϕh,k Ph,k

Table 5 Power flow computation times. Full network

With LAs

0.062 s

0.003 s

with apparent meaning of (A.1), we obtain ∗ Qh,k = tan ϕh,k p∗h,k Ph ,

the Configuration 8, in which a higher error is detected (please note the different scale). The model (15) provides errors quite higher compared with the model (16). The errors are in general admissible for both cases, apart from Configuration 8, in which the model (15) presents errors higher than the acceptability threshold (marked with NO in Table 4); for this case the error with model (16) remains admissible. As regards computational times, the power-flow on the original representation and the one based on LAs were compared. On a desktop computer, equipped with an Intel© CoreTM i7 CPU Q 720 with 8 GB of RAM run under Windows 7 64-bit operating system, the computational times needed to solve the networks are shown in Table 5; a relevant reduction is observed. The results from the test cases confirm the viability of the proposed approach; the errors on voltages are low, and always acceptable. Also, a relevant reduction in the number of element to be considered was observed, as well as of the computational times. 5. Conclusion This paper proposes a general formulation of LA; an exhaustive explanation of the approximation due to the reduced modelling of LA is presented, together with the modification introduced to reduce the errors of the representation. The results in different configurations of a study network highlight the impact of the proposed model on the accuracy of the LA representations, a relevant reduction of the amount of information required to describe the network and also a reduction of computational burden.

(A.3) ∗ . tan ϕh,k

Putting together (A.3) and

(A.4)

which expresses the proportional relation between the single reactive power injection (in the h–th category) and the active power injection of the whole category. - Distributed generation: Also the active power injection by the k–th DG plant in the h–th DG category can be put in the form (A.1); it indicates that in a given region the DG plants of a given category (for example, PV or wind) generate active power in a similar manner. A distinction can be made for small and large plants. Small plants – Small-size DG plants (in Italy, up to 100 kW and connected to the LV grid) are normally characterized by a unitary pf [25]. In this case (A.2) is simply obtained as: Qh,k = 0.

(A.5)

Medium and large plants – Medium- to large-size DG plants are connected to MV or HV distribution systems; for them, it is often required that the reactive power injection is dependant on the injected active power, so as to allow their contribution to the voltage regulation. The pf-to-P dependency of any DG plant is expressed versus the per-unit active power injection by the plant (see [26]); we have: p∗h,k Ph Ph,k P = ∗ r = hr r Ph Ph,k ph,k Ph

(A.6)

since the rated value of the active power injection by the h–th whole category, Phr , is the sum of the corresponding values of the r . From (A.6) we derive single plants, Ph,k pfh,k (Ph,k ) = pfh (Ph ),

(A.7)

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G.M. Casolino, A. Losi / Electric Power Systems Research 143 (2017) 321–328

that expresses a single, common pf-to-P relationship for the whole h-th DG category. Based on (A.7), it can be easily shown that (A.2) can be put as: Qh,k = tan(arccos(pfh (Ph ))p∗h,k Ph = fh∗ (Ph )p∗h,k Ph .

(A.8)

A.2. Nodal injections Let i,h,k represent the connection of the prosumers to the grid: i,h,k =

⎧ ⎨ 1 if the k − th prosumer of the h − th category ⎩

is connected to the i − th grid node, 0

(A.9)

otherwise.

For the i–th grid node, from (A.1), (A.2), (A.9) it is: Pi =

nc nh  

i,h,k Ph,k =

h=1 k=1

Qi =

nc nh  

nh nc  

Ph

h=1

i,h,k Qh,k =

=

k=1

nh nc  

h=1 k=1 nc 

i,h,k p∗h,k =

Ph

h=1

nc 

∗ i,h Ph ,

(A.10)

h=1

∗ i,h,k fh,k (Ph )p∗h,k

k=1

∗ i,h (Ph )Ph ,

h=1

where Pi and Qi are the active and reactive power injections, respectively, nc is the number of categories, nh is the number of prosumers ∗ and ∗ (P ) are in the h–th category, while the expressions for i,h i,h h apparent. Finally, the conjugate complex power injected into the i-th node of the grid is: Cˆ i = Pi − j Qi =

nc 

∗ ∗ (i,h − j i,h (Ph ))Ph =

h=1

nc 

A˙ h (Ph )Ph ,

(A.11)

h=1

where A˙ h represents the h-th n-vector of complex functions whose i-th component, A˙ i,h (Ph ), is: ∗ ∗ − j i,h (Ph ). A˙ i,h (Ph ) = i,h

(A.12)

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