Representation of Voltage-Controlled Devices in Reduced Modeling of Distribution Networks

The number of distributed energy resources is continuously increasing, which makes the study of distribution systems increasingly demanding; a reduced yet accurate network modeling could be beneficial in terms of computational effort, and of information treatment burden as well. With a traditional approach, the presence of numerous voltage-controlled devices limits the possibility of obtaining a significantly reduced representation of the network, due to the need to keep the voltage of the nodes to which they are connected. In this paper, we propose two approaches for obtaining and solving the reduced model of a distribution network with numerous devices with voltage-controlled power injection. The application of the two approaches to four test cases highlights the acceptability of the approximations introduced by the reduction of the model and the computational aspects of the solution of the reduced model. The results put into evidence the practicability of the proposed approaches, as well as the associated computational benefits.

INDEX TERMS Distribution network reduced modeling, unbalanced systems, voltage-controlled devices.

NOMENCLATURE
The principal symbols and names used in the paper are: Fixed share of the active and reactive power injection by the k-th prosumer in the h-th category.
Prosumers' injected current at the φ-th phase of the i-th bus. M k , N k Square matrices at the k-th iteration.
Categories of prosumers whose reactive power depends on active power or on voltage, respectively. P h , Q h Active and reactive powers injection of the h-th category. P h,k , Q h,k Active and reactive power injections by the k-th prosumer in the h-th category. P inj i_φ , Q inj i_φ Active and reactive power injections in the φ-th phase of the i-th bus. P L Active power consumption of the pure load prosumer's category. R Suffix: Reduced. U Nodal voltages.

U i_φ
Voltage of the φ-th phase of the i-th bus. U node, hk Voltage amplitude of the node where the k-th prosumer of the h-th category is connected to the grid. Y Admittance matrix.

I. INTRODUCTION
The study of distribution systems is becoming more and more demanding, as models grow in complexity and dimension, mainly due to the increasing number of distributed generators and the exploitation of prosumers' flexibility. In the planning of distribution systems, new approaches are adopted which require a huge amount of operations simulations [1]. Real-time simulations are carried out to study the operational behavior of physical devices without interfering with the actual distribution grid [2]. New monitoring and control functionalities make available a large amount of information, which must be adequately treated [3], [4]. In all cases, benefits can accrue from a reduced (yet accurate) grid modeling, as it allows to reduce the computational time and the burden of information treatment. The approach of Load Area (LA) [5], [6] can be adopted: LAs are subgrids identified as clusters of nodes whose power injection has a similar impact on the network operating conditions; a LA can comprise a whole distribution grid [7]. In each LA the relevant busses can be selected and a reduced network model can be obtained for the LA [8].
Starting from [9], network reduction is regularly revisited; recently, attention is paid to the simplification of the modeling of distribution networks [1], [2], [10]- [14]. The nodes where the reactive (and possibly even the active) injection is voltage-controlled should be retained, for a correct representation of the control laws. This makes it difficult to get a reduced model with good computational speed-ups when the number of such voltage-controlled devices is significant [15]. Recently, some methods have been presented that address this issue; in this paper, new approaches to the problem are proposed.
Compared to [16], in this study no optimization problem is solved at each time step to obtain the aggregate control curve for all voltage-controlled devices represented in an aggregated node, and individual voltage-controlled devices are explicitly represented. In [15] the voltage of the to-be-reduced nodes with voltage-controlled power injection devices are obtained by means of an estimation through sensitivities, while in the present study the voltages are explicitly obtained.
In this paper, two methods to represent the voltagecontrolled power injection in a reduced modeling of distribution systems are proposed, which are useful to take into account any type of voltage-based control law for power injection of devices installed in the to-be-reduced nodes. In one method, the modeling of voltage-controlled devices is the usual one, and computational benefit can derive from the newly proposed algorithm. In the second method, a new modeling of volt-var control laws installed in the to-be-reduced nodes is proposed together with a specific algorithm.
This work contributes to the problem of representing the units with voltage-controlled power injection in network reduction mainly with: • straightforward treatment of voltage-var control laws; • no need of introducing equivalent reactive generation; • little need for matrix inversions; • relevant reduction of computational effort. The details of the methods are presented, and the reduced relationships among relevant voltages/currents are obtained. The methods are applied to four test cases of different sizes, highlighting the computational aspects related to the reduced network modeling and solution.

II. MODELING OF PROSUMERS
The modeling of the power injections by prosumers in three-phase unbalanced systems is presented; it is based on the modeling proposed in [13], [14], with the addition of prosumers with voltage-controlled reactive power injection.

Active power
The active power injection by the k-th prosumer in the h-th category can be assumed to be a fixed share of the active power injection of the whole category: The reactive power injection by the same prosumer can be expressed as a function of either its active power injection or of the voltage at its terminals, depending on the characteristics of the prosumer. Reactive power as a function of active power The reactive power injection depending on the active power injection can be put as: the details of such relationships for different prosumers' categories can be found in Appendix A.
For the subsequent development it is useful to put (2) as Reactive power as a function of nodal voltage As the penetration of DG increases, grid codes may require that they assist in voltage regulation. A volt-var control scheme is realized through a fixed control curve programmed in the DG inverter [18], [19], so that: For any given category in the N B set, it can be assumed that all prosumers behave according to the same control law the details of functions g h (U node,hk ) can be found in Appendix B. Moreover, the rated value of reactive power of any prosumer in a given category is a known share of the total rated reactive power of its category: From (4)-(6) it follows that: where Let π i_φ,h,k represent the connection of the prosumers to the grid within the given LA: is connected in the i-th grid bus to the φ-th phase, and to ν h,k phases in total; 0 otherwise.
From (1), (9), the active power injection in the φ-th phase of the i-th bus is: with The expression for the reactive injected power is different based on the prosumers' category.
For category N A , from (3), (9) it is where For category N B , from (7), (9) where From Equations (10), (12) and (14), it is For the subsequent development, the value of each voltage is assumed to remain close to that of a reference case for all nodes:Ū Combining (16) and (17) yields

III. REDUCED MODELING OF NETWORK
The phase voltages at the LA's edge busses and the currents injected there from the outside into the LA are easily recognized as relevant quantities to represent the LA. In addition, the phase voltages of other, specific busses within the LA may be deemed relevant for monitoring and control purposes; for example, the phase voltages of transformers equipped with OLTC or of DG units operating under voltage control. In the following, relevant nodes/busses are called describing. Let us focus on the DG plants operating upon volt-var control; the dependence of their reactive injection on the voltage would require to consider the voltages of the busses they are connected to as relevant quantities in the representation of the LA. The number of such plants connected to distribution systems, in particular PV with inverters, is continuously and significantly increasing. It is straightforward to keep all the busses to which such DG plants are connected as 'describing'; however, the size of the network model is quite large and possibly not significantly reduced compared to the whole model. It is therefore desirable to have a model that represents the aforementioned dependence without maintaining all DG busses as describing.
Nodal injected currents are the sum of two contributions: the prosumers'currents (18) and the currents coming from outside the LA, which account for the connection of the LA to the remainder of the system. Only for e nodes there are external injections, while for all other nodes the external injection is zero; recalling that e nodes are included in d nodes and then in D nodes, it can be written:

1) REMOVAL OF THE in NODES
From (18) and (20), it is: Let (21) be written as With a Gaussian elimination ofŪ in with (22.2), the (22.1) becomes: A completely reduced representation based on d nodes only as in [13], [14] is not possible here, due to the reactive injection at the q nodes depending on the voltage, in a highly non-linear way. Any representation of the network that gives evidence to d nodes has to take into account this issue for q nodes. In the following, we first derive a model encompassing all D nodes as describing; then, we propose a model with only d nodes as describing, which embeds an approximated linear dependence of reactive generation on the voltage of q nodes and is corrected by means of additional fictitious variables to account for the non-linear dependence.

A. MODEL A-ALL THE D NODES AS DESCRIBING NODES
The terms under summation in (23) introduce errors due to (17); the error can be reduced by considering the actual voltage of the D nodes instead of the reference one [7]: equation (25) is the usual load flow equation for the network reduced to the D nodes, as shown in [13]. Approximations still remain due to (17) for the eliminated in nodes, but the VOLUME 9, 2021 related errors are very low and acceptable from a practical point of view [7]. For the subsequent development, it is useful writing (25) as:J

B. MODEL B-ONLY THE d NODES AS DESCRIBING NODES
Let (23) be written as:

1) LINEARIZATION FOR q NODES
For the aims of network reduction and solution, it is useful to express G h (·) in (27) in a different way for the q nodes. Let new variables,δ q,h h ∈ N B , be introduced (see Appendix B for details): which allows obtaining an explicit expression ofŪ q : By substituting (30) in (27.1), we get: Equation (31) is the representation of the whole network with evidence to the d nodes; it is not a completely reduced representation, as the q nodes are still present with their voltages, which must respect (28), (30). The presence of voltage-dependent reactive generation in the q nodes is linearly embedded in the reduced representation (31), and the non-linearities of the volt-var control laws are taken into account throughδ q .
The terms under summation in (31) introduce errors due to (17); the error can be reduced by considering the actual voltage of the d nodes instead of the reference one [7]: equation (33) is a load flow equation for the network reduced to the d nodes [13]. As before, approximations still remain due to (17) for the eliminated nodes, but the related errors are very low and acceptable from a practical point of view.

IV. SOLUTION ALGORITHMS
Various algorithms can be used to solve the models presented in the previous Sect. III, either general purpose or specialized. In all cases, due to the high non-linearity of the volt-var control laws, which have discontinuous derivatives (see Appendix B), the performance of such algorithms can be assessed only with extensive numerical applications.

A. ALGORITHMS FOR MODEL A
For a compact formulation of the algorithms, let functions χ, ψ: formally represent (26.1) and (26.2), respectively. The non-linear system (34) can be solved as a whole with many existing general purpose methods.
A different, iterative approach can be envisaged; at the kth iteration, firstJ d ,Ū d are updated by solving (34.1) with U q unchanged, thenŪ q is updated with a scheme based on (34.2): The expected computational savings derive from the very small dimension of the non-linear problem encompassed in (35.1), while calculations in (35.2) are straightforward (provided that matrix M k does not require any inversion); this is counteracted by the iterative nature of the algorithm.

B. ALGORITHM FOR MODEL B
The dimension of the whole representation (28), (30), (33) is bigger than the dimension of model (26); we can expect that solving it as a whole would require even more computational effort, and then we consider an iterative algorithm only.
For a compact formulation of the algorithm, let functions γ , ζ, η: formally represent (33), (30), and (28), respectively. At the k-th iteration of the iterative algorithm, firstJ d ,Ū d are updated by solving (36.1) withδ q unchanged, thenŪ q is updated through (36.2) still withδ q unchanged, and finallyδ q is updated with a scheme based on (36.3): Also for this algorithm, the expected computational efficiency is due to the very small dimension of the non-linear problem encompassed in (37.1), while calculations in (37.2) and (37.3) are straightforward (provided that matrix N k does not require any inversion); once again, this is counteracted by the iterative nature of the algorithm.

V. CASE STUDIES
The case studies analyze the solution of the models A (34) and B (36), for networks of different sizes; the three algorithms discussed in Sect. IV are investigated.
Standard, available algorithms are adopted to find the solution to non-linear equalities (34), or (35.1), or (37.1). For all of them, function fsolve available in the Matlab suite [20] has been adopted. For (34), also an implementation of the quasi-Newton broyden algorithm [21] in the Matlab environment has been tested; no testing with it for (35.1), or (37.1) has been carried out as it would require a specific implementation, which is outside the scope of this work.
The errors introduced by the reduction of the network and the computational effort required to get the solution are illustrated.
The errors in d node voltages among the investigated methods and the original solution are presented for different values of loads and type and location of DG; the errors are considered admissible if they are smaller than the error due to the smallest tap changer step of OLTCs in the grid [11], which corresponds to 0.006250 p.u. for a 32-step voltage regulator with a range of a ±10%. The computational effort is presented both in terms of computational time and number of iterations.
The IEEE 123-bus network is considered as the base grid for the analysis; it includes both pure loads and DG. Larger networks are obtained by replicating the base grid (from the same origin) to increase the total number of buses while preserving the structure of the original network. In all cases, network data refer to unbalanced three-phase systems and can be obtained from the IEEE Test feeder [27] and the OpenDSS Simulation Tool [22].
The DG is included in two forms; the first one, DG1, models the reactive power as a function of the active one (see Appendix A); the second one, DG2, contributes to voltage regulation with volt-var control law of category B (see Appendix B). If considered, DG1 covers about 15 % of the total nominal load, with a number of DG generators of about 15 % of the load busses located in the 15 % most loaded three-phase busses. DG2 is considered for two cases: case_1, for a total reactive power equal to 10% of the nominal load reactive power, randomly distributed in 10% of the three-phase busses of the network; case_2, where both the above percentages are 20%.

A. BASE GRID
The network scheme of the base grid 123-bus IEEE test grid is depicted in Fig. 1a. The busses 9r, 25r, and 160 model the tap-changer regulators along the lines [22]; all three-phase transformers are assumed wye-wye, with neutrals grounded. The LAs (Fig. 1b) are identified as in [13], to which the reader VOLUME 9, 2021  is referred for more details. The reduced representations of each ot the three resulting LAs is obtained as in Sect. III; the composition of the three reduced LA models leads to a reduced representation of the entire grid made of three busses (150, 149, 80 -red circles in Fig. 1b).  Figures 2a-2b report in log scale the maximum voltage errors of the reduced models versus the total load, with and without DG1, respectively for case_1 and case_2; in all cases the errors result far below the admissibility threshold (red dot-line). The high penetration of DG2 of case_2 does not significantly affect the voltage error; consider that in case_2 there is a high number of q nodes, whose voltage is explicitly computed. It can be noticed that the errors with model A and model B are different depending on the passive load and the penetration of DG; in all cases, the errors stay well below the admissibility threshold. Table 1 reports computational time and number of iterations required to solve the different models with the different algorithms.
For model A, the problem is solved faster by handling the d-q problem in its entirety (34), and broyden performs best. For model B, the computational time is slightly higher than broyden in case_1, while it is about half in case_2. The computational time per iteration in the iterative approaches (35) and (37) is much lower than that of the single iteration required for solving the problem as a whole (34), while the number of iterations is higher.
The comparison of the number of iterations between (35) and (37) is always in favor of the latter.
Model B (36) with algorithm (37) behaves very well; it is apparent that in the base grid the solution time is close to the best for model A (34) for case_1, and it is better when the complexity of the problem grows as in case_2, with an increased number of volt-var control busses.

B. LARGE-SIZE NETWORKS
Numerical tests for large networks are carried out by replicating the base grid 3, 5, and 10 times, from the same origin. The identification and reduced representation of LAs is carried out as for the base grid.
For matrices M k and N k the same values as the base grid are used. No significant difference in the voltage errors emerged among the medium and large grids (see Figs. 2c-2d).
In contrast, the time required to get the solution is strongly affected by the network size, especially for model A solved as a whole. For model A the iterative algorithm (35) performs better than solving the problem as a whole (34), starting from size x3, for fsolve, and from size x5, for broyden; this is due to VOLUME 9, 2021 the smaller computational time per iteration, which becomes relevant for these network sizes.
Concerning model B, it is always the best in terms of computational times, showing growing advantages as the size and the complexity of the network grow.

C. SOME CONSIDERATIONS
Case studies evidence that model A (34) and model B (36) show similar accuracy results, but their solution requires different computational efforts. The solution of model A treated as a whole is preferable for small to moderate size grids. In contrast, the iterative algorithms show better performance for big grids, where the low computational time per iteration becomes the dominant factor.
It is also apparent that model B (iteratively solved) has to be preferred, as it generally ensures the best results in terms of computational time, putting together the low number of non-linear equalities to be solved with a low number of iterations. The latter can be explained with the (approximated) linear embedding of voltage-dependent reactive generation for the q nodes in the representation of the network reduced to d nodes.

VI. CONCLUSION
The paper proposes two methods to obtain and solve a reduced model of a distribution system in the presence of numerous devices with voltage-controlled power injection, which limits the degree of network reduction possible with traditional methods.
Focusing on volt-var control, the first method is based on the usual modeling of the control laws and on solving the reduced network model with a specialized iterative algorithm; in the second method, a new representation of control laws is proposed together with a specialized iterative algorithm.
Numerical results are presented for four unbalanced test cases of different sizes, highlighting the acceptability of the approximations introduced by the proposed reduced models and the computational aspects associated with the solution of the reduced network model. The results, presented both with graphs and tables, give clear evidence of the beneficial characteristics of the second method, which is the preferred choice.

APPENDIX A Q-P RELATIONSHIPS
We report here the Q − P relationships already presented in [8], [13], to which the reader is referred for more details.
The active power injection by the k-th prosumer in the h-th category is given by (1), reported here for an easy reference: The general expression for the reactive power injection is (2), also reported here for an easy reference:

A. PURE LOADS
A constant power factor (pf) can be assumed, which allows to write: with apparent meaning of tan ϕ * h,k . From Equations (38) and (40) it is:  [23]. The reactive power injection by the k-th DG plant in the smallsize-DG category is obtained simply as follows: which can be written as (39) with Medium-to-large-size DG plants are connected to MV or HV distribution systems. It is often required that their reactive power injection is made dependent on the active one, so that the plants participate in keeping an acceptable voltage profile. For example, DG plants equipped with static converters should have the pf depicted in Fig. 3 [18].
the pf-to-P dependency is the same for all the plants in the same medium-to-large DG category: From (38) and (46), the reactive power injection by the k-th plant in the medium-to-large size DG category can be expressed as follows:

APPENDIX B VOLT-VAR CONTROL LAWS
Distributed generators may be required to participate in the voltage regulation. This is accomplished by DG inverters which autonomously modify their reactive power based on the local voltage. Volt-var control laws are various, based on either the ability/necessity by the DG to participate into the voltage regulation [19] or the rated power of DG [18]. Figure 4 shows examples of the control laws proposed in [19] for the two DG categories of minimum performance (Category A) and extended requirements (Category B); they have been drawn so as to give evidence to function g h (·) (see (4), (5) 5 shows examples of the control laws proposed in [18] for the rated power of the power park module up to 6 MW or FIGURE 5. Volt-var control proposed in [18]. more than 6 MW . Also in Fig. 5 evidence is given to function g h (·) in (49).

A. LINEARIZATION
Control laws g h (·) are apparently non-linear; they can be expressed as the sum of a linear component and a nonlinear one by introducing some fictious voltages.
New variables δ h ∀h can be defined such that from Figs. 4 and 5 it is apparent that in normal operating condition, 0.9 ≤ U ≤ 1.1, all control laws g h (·) have the same mean slope, s g , and s g = −10 pu pu . In the development of the reduced model in Section III-B1, it is useful to express the voltage-controlled reactive generation (for q nodes) versus voltage phasor instead of voltage amplitude. It can be easily accomplished with: which is (28) with: