# Distribution Network Reconfiguration Including OLTC Limits and Variable Shunt Reactors

Aghyles Graine, Jean-Paul Gaubert, Didier Laraillet

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## To cite this version:

Aghyles Graine, Jean-Paul Gaubert, Didier Laraillet. Distribution Network Reconfiguration Including OLTC Limits and Variable Shunt Reactors. The 26th European Conference on Power Electronics and Applications (EPE 2025), Groupement de recherche du CNRS Systèmes d’énergie électrique dans leurs dimensions sociétales (GDR SEEDS); Entreprises pour l’environnement (Association EpE), Mar 2025, Paris, France. 10.34746/epe2025-0317. hal-05075583

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Submitted on 20 May 2025

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Distributed under a Creative Commons Attribution 4.0 International License
# Distribution Network Reconfiguration Including OLTC Limits and Variable Shunt Reactors

Aghyles Graine<fnref n="1" />,² ; Jean-Paul Gaubert<fnref n="1" /> ; Didier Larraillet²

¹Poitiers University ; ²SRD

15 Rue de l'Hôtel Dieu ; 78 Av. Jacques Cœur

Poitiers, France

graine.aghyles@gmail.com

https://www.srd-energies.fr

## ACKNOWLEDGMENT

This work was supported by ANRT, Région Nouvelle-Aquitaine, and SRD as part of the LabCom @LIENOR project, France.

The authors would like to thank Mr. F. Petit from SRD for his valuable comments and suggestions.

*Index Terms*—Smart grids, Power system, Distribution of electrical energy, Distributed generation, TSO-DSO coordination.

*Abstract*—Networks often suffer from overvoltage problems. The On-Load Tap Changers (OLTCs) may not always be able to regulate the voltage. Distribution network reconfiguration and variable shunt reactors can help mitigate these issues. In this paper, OLTCs are modeled and incorporated into the optimization. The results show an improvement in the accuracy.

## I. INTRODUCTION

Modern Distribution Networks (DNs) experience more overvoltage issues, due to, among others, the increased share of undergrounded cables and penetration of Distributed Generation (DGs). In the High Voltage-Medium Voltage (HV-MV) substations, On-Load Tap Changers (OLTCs) are used in the transformers, in order to keep the secondary voltage (MV voltage) within allowed limits. In some cases, the secondary voltage cannot be maintained within the permissible limits. This is known as the saturation of the OLTC, i.e., the tap position is at its limit but is still unable to bring the voltage within the acceptable range. The Distribution Network Reconfiguration (DNR) problem or the Variable Shunt Reactors (VSRs) can be used to mitigate this phenomenon. The DNR is an optimization problem, where the decision variables are the state of

the switches of the network (binary variables)<fnref n="1" />, and various types of objective function can be considered, e.g., loss minimization. It allows us to obtain a new topology that is better with respect to the considered objective function. VSRs are, as the name suggests, shunt reactors in DN, that have a variable reactive power output. Thus, in the case where the OLTCs can be saturated, it is important to model them and to include them in the optimization problem, so that both the DNR and VSRs can be combined in order to effectively regulate the secondary voltage.

Various formulations of the DNR exist in the literature. Reconfiguration methodologies are classified into five groups: classical methods, heuristic methods, metaheuristic methods, hybrid methods, and methods based on machine learning [1]. For classical methods: in [2], a Mixed Integer Quadratic Programming (MIQP) model is used to formulate the dynamic DNR. In [3], a Mixed Integer Linear Programming (MILP) model is proposed, for the simultaneous optimal DNR and optimal placement of DGs. As for meta-heuristics methods: [4] can be cited, where a quantum particle swarm algorithm is used to find the topology that minimizes power losses. In [5], the authors propose a convex model for the optimal service restoration of power distribution networks after a permanent fault, which is a type of DNR, but the OLTC is modeled as an ideal transformer.

In this paper, it is considered that it is not really necessary to know (during the optimization) which tap position is chosen by the OLTC, and that what matters is that the voltage at the secondary side of the HV-MV

<footnote n="1">¹There can be more decision variables to the DNR problem.</footnote>
transformers is kept within the allowed limits. The tap position can be determined once the optimization is finished. This subtle distinction allows us to not have to add the tap positions as part of the decision variables, we simply take the extreme tap positions to write the OLTC constraints. The model of the non-ideal HV-MV transformers (thus, including the OLTCs, the voltage drops across the impedances, as well as the power losses of the transformer), is indirectly considered through a linearization technique, followed by a transformation to increase the accuracy of the model, and combined with the DNR and the VSRs. The voltage variation of the HV buses is also considered, and some information about the Transmission Network (TN) must be known to the Distribution System Operator (DSO), thus calling for a coordination between the Transmission System Operator (TSO) and the DSO (called the TSO-DSO Coordination).

## II. MATHEMATICAL FORMULATION

### A. OLTC

To facilitate the understanding, the OLTC constraints are first presented without the transformation. Then, the transformation is applied and explained.

As the tap position of the OLTCs is a discrete variable, it is not possible to achieve exact voltage regulation. We thus give an allowed voltage interval that the OLTCs have to meet :

$$U_{mv\_ref} - tol_{oltc} \leq U_{mv} \leq U_{mv\_ref} + tol_{oltc} \quad (1)$$

**Where :** $U_{mv\_ref}$ : voltage reference for the OLTC ; $tol_{oltc}$ : voltage tolerance of the OLTC.

For ideal transformers, the transformer ratio can be defined as follows :

$$a = \frac{N_{hv}}{N_{mv}} = \frac{U_{hv}}{U_{mv}} \quad (2)$$

**Where :** $a$ : transformer ratio ; $N_{hv}/N_{mv}$ : number of turns in the primary/secondary winding ; $U_{hv}/U_{mv}$ : voltage at the primary/secondary winding.

In this case, the OLTC is considered to be located in the primary (HV) side of the transformer. Thus, when the tap position increases, $N_{hv}$ also increases as well, leading to an increase in $a$. And if we suppose that $U_{hv}$ is fixed : it means that $U_{mv}$ decreases. Thus, plotting

$U_{mv}$ as a function of $a$ is a decreasing curve. For non-ideal transformers, $U_{mv}$ as a function of $a$ also decreases.

Thus, in the case of an overvoltage in the secondary side of the transformer, the most favorable case is when the tap position is at its maximum :

$$a = a_{max} \quad (3)$$

**Where :** $a_{max}$ : maximum value of the transformer ratio, obtained when the tap position is at its maximum. The reason behind that is because doing so will put the voltage at the secondary side of the transformer at its lowest level. Then, the voltage computed this way is :

$$U_{mv\_a\_max} = U_{mv}(a = a_{max}) \quad (4)$$

**Where :** $U_{mv\_a\_max}$ : voltage at the secondary side of the transformer, when the transformer ratio is at its maximum.

Since the issue here is with overvoltage, the upper bound of (1) must be respected. To address this issue, the following constraint is added :

$$U_{mv\_a\_max} \leq U_{mv\_ref} + tol_{oltc} \quad (5)$$

In the case of an overvoltage, for the solution to be feasible, at least in terms of the model<fnref n="2" />, the final topology and power dispatch must respect the constraint (5).

The same type of reasoning can be performed in the case of an undervoltage issue : When $a$ decreases, $U_{mv}$ increases.

In the case of an undervoltage in the secondary side of the transformer, the most favorable case is when the tap position is at its minimum. This is because doing so will put the voltage at the secondary side of the transformer at its highest level. Thus, the lower bound of (1) must be respected. In terms of equations, we obtain :

$$a = a_{min} \quad (6)$$

$$U_{mv\_a\_min} = U_{mv}(a = a_{min}) \quad (7)$$

$$U_{mv\_ref} - tol_{oltc} \leq U_{mv\_a\_min} \quad (8)$$

<footnote n="2">²Not necessarily in terms of the Power Flow (PF), but at least in terms of the model, as it makes an error due to the linearization, as it will be explained below.</footnote>
**Where :** $a_{min}$ : minimum value of the transformer ratio, obtained when the tap position is at its minimum
; $U_{mv\_a\_min}$ : voltage at the secondary side of the transformer, when the transformer ratio is at its minimum.

For the rest of the paper, only the overvoltage case will be considered. Thus, we first have to set the tap position of every OLTC at its maximum position.

$$ \forall i \in N_{OLTC} : \\ tap_pos(u) = tap_pos_{max}(u) \quad (9) $$

**Where :** $N_{OLTC}$: set of the HV-MV transformers with OLTC; $tap_pos(u)$: tap position of the HV-MV transformer $u$; $tap_pos_{max}(u)$: maximum tap position of the HV-MV transformer $u$.

In order to compute the voltage $U_{mv\_a\_max}$, we perform a linearization around the initial OP :

$$ \forall u \in N_{src} : \\ U_{mv\_a\_max}(u) = U_{mv\_a\_max}^{0}(u) + \sum_{\forall i \in N_{src}} \left. \frac{\partial U_{mv\_a\_max}(u)}{\partial P(i)} \right|_{initial OP} (P(i) - P^{0}(i)) + \sum_{\forall i \in N_{src}} \left. \frac{\partial U_{mv\_a\_max}(u)}{\partial Q(i)} \right|_{initial OP} (Q(i) - Q^{0}(i)) \quad (10) $$

**Where :** $N_{src}$: set of the secondary nodes of the HV-MV transformers, considered as MV sources; $U_{mv\_a\_max}^{0}(u)$: initial voltage at the bus $u$ (secondary side of an HV-MV transformer), when the tap position is at its maximum (computed using a PF); $P^{0}(i)/Q^{0}(i)$: initial active/reactive power injected in the node $i$ (that is also the secondary side of an HV-MV transformer)

The partial derivatives of (10) are obtained using the forward finite difference method.

In this case, the computation of the partial derivatives with respect to the active power involves slightly increasing the active power consumption of the load at bus $i \in N_{src}$ by a small amount $\delta P$. After that, a PF is performed to obtain the new values of the($U_{mv\_a\_max}(u), \forall u \in N_{src}$). Then, we compute the partial derivatives for all the other secondary buses of the HV-MV transformers as follows 3:

$\forall u \in N_{src}$:

$$ \frac{\partial U_{mv\_a\_max}(u)}{\partial P(i)} = \frac{U_{mv\_a\_max}(u) - U_{mv\_a\_max}^{0}(u)}{\delta P} \quad (11) $$

Then, the power consumed by the load at bus $i$ is put back to its original value, and the load connected to the next bus $i \in N_{src}$ is slightly increased, then the partial derivatives are computed using (11), and so on, until all the partial derivative values are obtained:

$$ ( \frac{\partial U_{mv\_a\_max}(u)}{\partial P(i)}, \forall u \in N_{src}, \forall i \in N_{src} ). $$

The same type of procedure is performed for the partial derivatives with respect to the reactive power, except that this time, the reactive (instead of the active) power of the loads are slightly increased by an amount $\delta Q$, and the partial derivatives are computed as follows:

$\forall u \in N_{src}$:

$$ \frac{\partial U_{mv\_a\_max}(u)}{\partial Q(i)} = \frac{U_{mv\_a\_max}(u) - U_{mv\_a\_max}^{0}(u)}{\delta Q} \quad (12) $$

As can be seen in (10), the constraint is linear. From this equation, it can also be seen that, for a bus $u$: the decision variable $U_{mv\_a\_max}(u)$ does not only depend on $P(u)$ and $Q(u)$, but also depends on the power of the other buses $i \in N_{src}$ (this is the general case, but it is obviously different if some partial derivatives are null). For instance, in Fig. 1 (a) (where bus 0 represents the HV external grid, bus 1 and 2 represent HV nodes, and bus 3-6 represent MV nodes): the buses of the TN form a triangle (bus 0, bus 1, bus 2), and $U_{mv\_a\_max}(3)$ depends on $P(3)/Q(3)$ as well as $P(5)/Q(5)$. Also, what happens in the DN connected to one transformer will affect the flow of power in the HV lines, thus having an impact on the other transformer.

<figure><img src="image_3.png" /><figcaption>Fig. 1. Test networks.</figcaption></figure>

<footnote n="3">³ Meaning that after increasing one load, the effect on voltage is computed for all other buses in $N_{src}$.</footnote>
**Note**: to be able to compute the partial derivatives, a certain degree of cooperation between the TSO and the DSO is necessary, as the DSO needs specific data about the TN: the voltage value of the HV external grid, TN's topology, and the impedance values of the HV lines.

We will now **introduce the transformation** of the constraints (5) and (10). Since the voltage $U_{mv\_a\_max}(u)$ does not vary linearly with the power $P(i)/Q(i)$, the linearization is obviously going to cause an error in the computation of the voltage, as the solution deviates from the initial OP. As it has been said earlier, what matters in this optimization framework is not to obtain the value of $U_{mv\_a\_max}(u)$, rather, it is to force the final network<fnref n="4" /> to respect the constraint (5). Thus, we define the simple power function:

$$g(x) = x^n \quad (13)$$

**Where**: $g$: power function used for the transformation of the constraints; $n$: real number (parameter that must be defined before the optimization, not a decision variable).

Then, we write:

$$\forall u \in N_{src} :$$

$$\tilde{U}_{mv\_a\_max}(u) = g(U_{mv\_a\_max}(u)) = U_{mv\_a\_max}^{n}(u) \quad (14)$$

The function $g$ can be applied to (5), but for the transformation to be valid, since $U_{mv\_a\_max}(u)$ is positive, for the case $n > 0$, we have<fnref n="5" />:

$$\forall u \in N_{src} :$$

$$\tilde{U}_{mv\_a\_max}(u) \leq (U_{mv\_ref} + tol_{oltc})^n \quad (15)$$

As for the case $n < 0$, we obtain:

$$\forall u \in N_{src} :$$

$$(U_{mv\_ref} + tol_{oltc})^n \leq \tilde{U}_{mv\_a\_max}(u) \quad (16)$$

Since the case with $n = 0$ is meaningless (would obtain $\tilde{U}_{mv\_a\_max} = 1$), it is not considered.

For the transformation of the constraint (10): the value of the new decision variable $\tilde{U}_{mv\_a\_max}(u)$ also changes when the powers $P(i)/Q(i)$ change. Thus, we once again perform a linearization around the initial OP:

$$\forall u \in N_{src} :$$

$$\tilde{U}_{mv\_a\_max}(u) = \tilde{U}_{mv\_a\_max}^{0}(u) + \sum_{\forall i \in N_{src}} \left. \frac{\partial \tilde{U}_{mv\_a\_max}(u)}{\partial P(i)} \right|_{initial OP} (P(i) - P^{0}(i)) + \sum_{\forall i \in N_{src}} \left. \frac{\partial \tilde{U}_{mv\_a\_max}(u)}{\partial Q(i)} \right|_{initial OP} (Q(i) - Q^{0}(i)) \quad (17)$$

**Where**: $\tilde{U}_{mv\_a\_max}^{0}(u)$ : initial value of $\tilde{U}_{mv\_a\_max}(u)$

The computation of the new partial derivatives is done exactly the same way as performed in (11) and (12), except that the function $g$ is applied to the voltages obtained by the PFs. Thus, we obtain:

$$\forall u \in N_{src} :$$

$$\frac{\partial \tilde{U}_{mv\_a\_max}(u)}{\partial P(i)} = \frac{\tilde{U}_{mv\_a\_max}(u) - \tilde{U}_{mv\_a\_max}^{0}(u)}{\delta P} \quad (18)$$

$$\frac{\partial \tilde{U}_{mv\_a\_max}(u)}{\partial Q(i)} = \frac{\tilde{U}_{mv\_a\_max}(u) - \tilde{U}_{mv\_a\_max}^{0}(u)}{\delta Q} \quad (19)$$

**Note**: instead of having the same $n$ for all the OLTCs, it is actually possible to use a specific $n(u)$ for each OLTC $u \in N_{src}$, but this is not done in this paper for the sake of simplification.

The value of $n$ should be chosen so as to minimize the error on the computation of $U_{mv\_a\_max}$. Once the optimizer converges to a solution, the inverse function of $g$ can be used to obtain the value of $U_{mv\_a\_max}$:

$$g_{inv}(x) = x^{1/n} \quad (20)$$

$$U_{mv\_a\_max}(u) = g_{inv}(\tilde{U}_{mv\_a\_max}(u)) \quad (21)$$

**Where**: $g_{inv}$ : inverse function of $g$.

Then, a PF is performed on the final network, and the error can be computed by comparing $U_{mv\_a\_max}$

<footnote n="4">⁴Defined by both the optimized topology and the power dispatch of the VSR.</footnote>

<footnote n="5">⁵This can easily be understood by plotting $\tilde{U}_{mv\_a\_max}$ as a function of $U_{mv\_a\_max}$ for different signs of $n$.</footnote>
<figure><img src="image_4.png" /><figcaption>Fig. 2. Absolute value of the linear correlation between V3 and Q3 for the network of Fig. 1 (a).</figcaption></figure>

obtained by the model to its PF version (considered as the ground truth).

Even if the initial OP is known before the optimization, determining the right value of *n* is not trivial, as the computed voltage depends on many variables, and the power deviations ($P(i) - P^0(i)$) and ($Q(i) - Q^0(i)$) are obviously not known in advance.

For the rest of this paper, we make the hypothesis that, to minimize the error when computing $U_{mv\_a\_max}(u)$, special attention must be given to the reactive power flowing through it $Q(u)$. That being said, the method can be presented as follows: Starting from the initial OP, we make a certain vector (called $Q_{vector}$) including both negative and positive values, describing how much we want to deviate in terms of reactive power, with respect to the initial OP$^6$. All the tap positions are put at their maximum.

For a $u \in N_{src}$: we sequentially add each value of $Q_{vector}$ to modify the reactive power consumed/generated$^7$ at the bus $u$, perform a PF, and add the obtained voltage at bus $u$ to a vector (named $V_{vector}(u)$). Once the iteration over all the elements of $Q_{vector}$ is finished, the reactive power at bus $u$ is put back to its original value. This procedure is performed for all the OLTCs ($\forall u \in N_{src}$).

Thus, at the end of this procedure: we obtain one vector containing the reactive power variation, and one voltage vector for each OLTC. Taking the example

of Fig. 1 (a): **we would obtain two voltage vectors**: $V_{vector}(3)$ representing the voltage at bus 3 when $Q(3)$ is modified, and $V_{vector}(5)$ representing the voltage at bus 5 when $Q(5)$ is modified.

The above procedure was done without involving any *n*. The next step is to now create a vector containing the values of *n* that we want to experiment (named $n_{vector}$)$^8$. Then, for each *n* element of that vector, and for each OLTC ($\forall u \in N_{src}$), the function *g* is applied to $V_{vector}(u)$, and the linear relationship between that transformed vector $g(V_{vector}(u))$ and $Q_{vector}$, is measured using the Pearson correlation coefficient, calculated through the *pearsonr* function from the *stats* module in the SciPy library [6].

The results of the linear correlation between the voltage vector and reactive power vector, for each *n* of $n_{vector}$, at the bus 3$^9$, can be seen in Fig. 2. The curve for bus 5 also looks very similar. For the two buses, the curve is not at its maximum when *n* = 1 (no transformation). Rather, the maximum is obtained for *n* = 2.9798 (for both buses). Thus, we call this value $n_{opt}$: $n_{opt} = 2.9798$, which we consider now to be the optimal value of *n* for the network of Fig. 1 (a), with respect to minimizing the error during the computation of $U_{mv\_a\_max}$.

## B. DNR

The DNR formulation presented in this section is closely aligned to the one used in [7] (using the loss-less power balance model). In this paper, the objective is to minimize the total losses. Certain constraints have to be respected$^{10}$: (i) the final topology must be radial (no loop), and each non-source node must be connected to exactly one source; (ii) the MV voltage at every bus must stay within predefined limits; (iii) the thermal limits of the lines and transformers must be respected; (iv) the power limits of the VSR must be respected; (v) the OLTC must be able to regulate the voltage of the secondary bus of the transformers.

Let us consider an oriented graph $G = (N, A)$ representing the DN **where**: $N$ : set of Medium Voltage (MV) nodes, and $A$ : set of MV arcs ; $N_{non\_src} \subset N$ : set of non-source MV nodes (defined as $N_{non\_src} = N - N_{src}$) ; $N_{DG} \subset N$ :

<footnote>$^8$In this example, *n* varies from -5 to 5, with a total of a 100 evenly spaced values.</footnote>

<footnote>$^9$After applying an absolute value function to the result, since what matters here is the magnitude of the correlation, not its sign.</footnote>

<footnote>$^{10}$Other type of constraints can be added, depending on the network.</footnote>

<footnote>$^6$In this example, the power varies from -7 to 7 (MVAR), with a total of a 100 evenly spaced values.</footnote>

<footnote>$^7$Depending on the sign of the element of $Q_{vector}$.</footnote>
set of MV DG nodes; $N_{zi} \subset N$: set of MV zero injection nodes; $N_{VSR} \subset N$: set of MV nodes with VSRs; For an MV node $u$, $N^{in}(u) \subset N$: set of (MV) direct predecessors of $u$, and $N^{out}(u) \subset N$: set of (MV) direct successors of $u$.

In this paper, the VSRs are modeled as continuous variables (which is an approximation). The optimization problem is formulated as follows:

1) The objective function:

$$ \min Z = \frac{1}{U_{mv\_ref}^2} \cdot \sum_{(u,v) \in A} \left( R(u, v) \left[ y p(u, v)^2 + y q(u, v)^2 \right] \right) \quad (22) $$

2) Radiality constraints:

Taken from [8] and slightly modified:

$$ \sum_{(u,v) \in A} d(u,v) = n_{bus} - n_{src} \quad (23) $$

3) Voltage constraints:

$$ yfs(u) + \sum_{v \in N^{in}(u)} f(v, u) - \sum_{v \in N^{out}(u)} f(u, v) = feq(u) \quad (24) $$

4) Branches' power limit:

$$ f_{min} d(u,v) \le f(u,v) \le f_{max} d(u,v) \quad (25) $$

5) Sources' power limit:

6) $\forall (u, v) \in A$:

$$ Us_{min} d(u, v) \le yUs(u, v, u) \le Us_{max} d(u, v) \quad (26) $$

$$ Us_{min} d(u, v) \le yUs(u, v, v) \le Us_{max} d(u, v) \quad (27) $$

$$ Us_{min} (1 - d(u, v)) \le Us(u) - yUs(u, v, u) \le Us_{max} (1 - d(u, v)) \quad (28) $$

$$ Us_{min} (1 - d(u, v)) \le Us(v) - yUs(u, v, v) \le Us_{max} (1 - d(u, v)) \quad (29) $$

7) $\forall u \in N_{src}$:

$$ Us(u) = U_{mv\_ref}^2 \quad (30) $$

8) $\forall u \in N_{non-src}$:

$$ \sum_{v \in N^{out}(u)} (yUs(u, v, u) - yUs(u, v, v)) + \sum_{v \in N^{in}(u)} (yUs(v, u, u) - yUs(v, u, v)) = 2 \left[ \sum_{v \in N^{out}(u)} R(u, v) yp(u, v) + X(u, v) yq(u, v) - \sum_{v \in N^{in}(u)} R(v, u) yp(v, u) + X(v, u) yq(v, u) \right] \quad (31) $$

9) $\forall (u, v) \in A$:

$$ -S_{max}(u, v) d(u, v) \le yp(u, v) \le d(u, v) S_{max}(u, v) \quad (32) $$

$$ -S_{max}(u, v) d(u, v) \le yq(u, v) \le d(u, v) S_{max}(u, v) \quad (33) $$

10) $\forall u \in N_{src}$:

11) $-S_{max-source}(u) \le yps(u) \le S_{max-source}(u)$ (34)
6) VSR constraints :

$\forall u \in N_{VSR}$:

$$0 \le Q_{VSR}(u) \le Q_{max-VSR}(u) \quad (36)$$

7) Power balance constraints :

$\forall u \in N$:

$$yps(u) + \sum_{v \in N^{in}(u)} yp(v, u) - \sum_{v \in N^{out}(u)} yp(u, v) = \quad (37)$$

$$P_l(u) - P_{dg}(u)$$

$$yqs(u) + \sum_{v \in N^{in}(u)} yq(v, u) - \sum_{v \in N^{out}(u)} yq(u, v) = \quad (38)$$

$$Q_l(u) - Q_{dg}(u) + Q_{VSR}(u)$$

8) OLTCs' constraint :

$\forall u \in N_{src}$:

$$\begin{aligned} \tilde{U}_{mv\_a\_max}(u) = \tilde{U}_{mv\_a\_max}^0(u) &+ \sum_{\forall i \in N_{src}} \left. \frac{\partial \tilde{U}_{mv\_a\_max}(u)}{\partial P(i)} \right|_{initial OP} (yps(i) - yps^0(i)) \\ &+ \sum_{\forall i \in N_{src}} \left. \frac{\partial \tilde{U}_{mv\_a\_max}(u)}{\partial Q(i)} \right|_{initial OP} (yqs(i) - yqs^0(i)) \end{aligned} \quad (39)$$

$\forall u \in N_{src}$:

$$\tilde{U}_{mv\_a\_max}(u) \le (U_{mv\_ref} + tol_{oltc})^n$$

With:

$$yps(u) = yqs(u) = 0, \forall u \in N_{non-src}.$$

$\text{freq}(u) = 1, \forall u \in N_{non-src}$, if $\text{Peq}(u) < 0 \cup \forall u \in N_{zi}$; $\text{freq}(u) = 0$, for all other buses; $\text{yfs}(u) = 0, \forall u \in N_{non-src}$.

Peq(u)/Qeq(u) : net consumed active/reactive load at bus u which is computed as (power of Load(u) - power of DG(u)) for both active

and reactive power, and equals zero for the source buses.

Equation (15) is re-written in this subsection. It is chosen because $n > 0$.

Where : Z : objective function ; d(u,v) : binary variable that represents the state of the switch in the arc (u,v) : its value is 1 for a close switch, and 0 for an opened switch ; nbus : number of elements in the set N ; nsrc : number of elements in the set Nsrc ; yfs(u) : fictitious power injected at source u (continuous variable) ; f(u,v) : fictitious power flowing through the arc (u,v) (continuous variable) ; freq(u) : fictitious power consumed at node u ; fmin/fmax : minimum / maximum fictitious flow ; Us(u) : voltage squared at bus u ; yUs(u,v,u) : if the arc (u,v) is ON, this variable represents the voltage squared at bus u, otherwise, it is null ; Usmin/Usmax : minimum/maximum MV voltage squared ; R(u,v)/X(u,v) : resistance/reactance of the arc (u,v) ; yp(u,v)/yq(u,v) : active/reactive power flowing through the arc (u,v) ; yps(u)/yqs(u) : active/reactive power that is injected at source u (with the loss-less DNR model) ; yps0(u)/yqs0(u) : initial active/reactive power that is injected at source u (with the loss-less DNR model) ; Smax(u,v)/Ssource-max(u) : apparent power limit of the arc (u,v)/of the source u ; QVSR(u) : reactive power consumed by the VSR at bus u ; Qmax-VSR(u) : maximum reactive power that can be consumed by the VSR at bus u.

As mentioned above, the tap position of the OLTC is not known during the optimization. Thus, we assume during the optimization that the voltage at the MV sources is set to ($U_{mv\_ref}$) (30).

Equation (17) is adjusted to obtain (39) : the power variables are different. This modification is performed to adjust the OLTC formulation with the current DNR formulation, where the power flowing through the transformers $yps(u)/yqs(u)$ do not take into consideration the active and reactive losses of the branches of the DN.

### III. SIMULATIONS AND RESULTS

For the simulations : Gurobi Optimizer is used to solve the optimization problem. The computer has the following characteristics : 11th Gen Intel(R) Core(TM) i5-1135G7 @ 2.40GHz, 2.42 GHz 8.00 GB of RAM. PFs are performed using the Python library called Pandapower [9]. The HV/MV nominal voltage
are respectively set to 90kV/20kV ; the MV voltage limits are ±5% around the nominal value ; the voltage reference for the OLTC is 1.03 p.u. ; the voltage tolerance for the OLTC is 0.01 p.u..

The simulations consist of two parts : a first part where the precision of the proposed method is tested on the network of Fig. 1 (a) **without** DNR, and a second part where the complete formulation (thus **with** DNR) is tested on Fig. 1 (b). In each part, the following two cases are compared :

* case_g : case where the transformation of the constraints is performed

* case_no_g : case where the transformation of the constraints is **not** performed

### A. part 1 : OLTC only

For this part, the data of the network can be found in [10] (test_network_part1).

As said earlier, for this network, we use : $n_{opt} = 2.9798$.

To ease the implementation, we consider that the network has two storage units: one at bus 4 and the other at bus 6. We suppose that the final network deviates from the initial network regarding the power dispatch, as follows:

* consumption at bus 4 : +5 MW / -4 MVAR

* consumption at bus 6 : -5 MW / +9 MVAR

These values are chosen so as to represent an active power consumption exchange between the two substations (total active power for the initial and final network remain the same), and a total increase of the reactive power consumption by +5 MVAR (representing the activation of a VSR). This power exchange is implemented by adding simple constraints to impose those new consumption.

Assigning the tap position at its maximum to evaluate the accuracy of the proposed model : as it can be seen from Tab. I, where the voltage of the final network computed by a PF is compared with the voltage obtained by the models case_g and case_no_g: for bus 3, the error is slightly smaller when using the transformation (0.36946% for case_g vs. 0.39607% for case_no_g), while it is much smaller in the case of bus

<table><caption>TABLE I
RESULTS FOR THE SIMULATION WITH OLTC ONLY (PART 1)
(TAP POSITIONS ARE SET AT THEIR MAXIMUM) (VOLTAGE IN P.U., ERROR IN ABSOLUTE VALUE)</caption><thead><tr><th>Bus</th><th>Final Voltage PF</th><th>Voltage model case_g</th><th>Voltage model case_no_g</th><th>Error case_g (%)</th><th>Error case_no_g (%)</th></tr></thead><tbody><tr><td>3</td><td>0.97710</td><td>0.98071</td><td>0.98097</td><td>0.36946</td><td>0.39607</td></tr><tr><td>5</td><td>0.91267</td><td>0.91662</td><td>0.91958</td><td>0.43280</td><td>0.75712</td></tr></tbody></table>

5 (0.43280% for case_g vs. 0.75712% for case_no_g).

These results demonstrate that transforming the constraints enhances the precision of the model.

### B. part 2 : including the DNR

For this part, the network data can be found in [10] (test_network_part2).

Each MV branch has a switch. A VSR is considered to be placed at bus 7 (implemented by a storage unit that only consume reactive power in the previously mentioned dataset), with a maximum power of $Q_{max-VSR}(7) = 5$ MVAR. Initially, the switch between buses 7 and 8 is open, and all the other switches are closed.

The initial voltage values when the tap positions are at their maximum are 1.0419 p.u. for bus 3, and 1.0321 p.u. for bus 5, meaning that the OLTC at bus 3 is unable to regulate the voltage within the allowed limits, thus requiring the use of DNR/VSR to fix this problem.

For this network, we obtain : $n_{opt} = 3.0808$.

**Note** : Setting the OLTCs' tap position at their maximum after convergence is performed in this paper only to evaluate the precision of the model. But to compute, e.g., the losses of the solution, then a controller must be included to the network to adjust the tap position of the OLTCs, because the PF calculation only gives the state of the network and cannot select the tap position. This is done in part 2.

We apply the following procedure : when the solver converges, the solution is applied to the network. After that, a discrete controller of the tap position of the OLTC is implemented, and a PF is launched. We obtain :
<table><caption>TABLE II
RESULTS FOR THE SIMULATION INCLUDING THE DNR (PART 2)
(TAP POSITIONS ARE DETERMINED BY THE CONTROLLER)
(VOLTAGE IN P.U., ERROR IN ABSOLUTE VALUE)</caption><thead><tr><th>Bus</th><th>Voltage case_g</th><th>PF</th><th>Voltage case_no_g</th><th>PF</th></tr></thead><tbody><tr><td>3</td><td>1.037</td><td></td><td>1.036</td><td></td></tr><tr><td>5</td><td>1.042</td><td></td><td>1.042</td><td></td></tr></tbody></table>

* the result is the same in terms of the topology for both cases, i.e., the **opening of the switch between bus 5 and bus 6** and closing every other switch

* for both cases, the OLTC connected to the bus 3 is not at its maximum (the tap position is 13 instead of 14), meaning that there is a margin for this transformer, while the tap position of the OLTC that is connected to the bus 5 is at its maximum

* For **case_no_g**: the VSR consumes 4.077 MVAR; losses are reduced by: -53.33 %

* For **case_g**: the VSR consumes 3.951 MVAR; reduction in losses is slightly better for this case, as they are reduced by -54.12 %

We also obtain Tab. II using the same procedure, where we notice that the voltages are very close for the two cases. But the OLTC constraint at bus 5 is violated (the voltage exceeds 1.04 p.u. for both cases). Since the transformer connected to bus 5 is isolated from the rest of the DN, in practice, it would make more sense to disconnect it from the HV network to avoid wasting energy through the no-load losses of the transformer.

Now, assigning the tap positions at their maximum to evaluate the accuracy of the proposed model: as can be seen in Tab. III, for both cases, the PF OLTC constraints are not respected for bus 5 (voltage exceeding 1.04 p.u.), while they appear to be respected in both optimization models. This difference is due to the linearization error<fnref n="11" />. The OLTC problem was transferred from bus 3 to bus 5. For bus 3, the **case_g** is more precise than the **case_no_g** (0.159% vs. 0.209%), while it is slightly worse for bus 5 (0.204% vs. 0.198%). This may be due to the fact that the value of *n* was chosen without giving any consideration to the active power, while the transformer at bus 5 experienced a significant active power deviation of -11 MW.

<table><caption>TABLE III
RESULTS FOR THE SIMULATION INCLUDING THE DNR (PART 2)
(TAP POSITIONS ARE SET AT THEIR MAXIMUM) (VOLTAGE IN P.U., ERROR IN ABSOLUTE VALUE)</caption><thead><tr><th>Bus</th><th>Voltage PF case_g</th><th>Voltage PF case_no_g</th><th>Voltage model case_g</th><th>Voltage model case_no_g</th><th>Error case_g (%)</th><th>Error case_no_g (%)</th></tr></thead><tbody><tr><td>3</td><td>1.020</td><td>1.019</td><td>1.021</td><td>1.021</td><td>0.159</td><td>0.209</td></tr><tr><td>5</td><td>1.042</td><td>1.042</td><td>1.040</td><td>1.040</td><td>0.204</td><td>0.198</td></tr></tbody></table>

**Note:** Generally speaking, once the optimal topology is found, another optimization problem can be formulated (without the presence of topological variables) using a more precise PF model, with the aim of finding the optimal reactive power dispatch of the VSRs.

## IV. CONCLUSION

The objective of this research was to formulate an optimization problem including the DNR, the VSRs, and a non-ideal model of OLTCs, with the aim of eliminating the violation of the voltage constraints at the OLTC level, thereby enhancing the voltage profile. The OLTC constraints were linearized then transformed. The results demonstrate an overall improvement in the precision of the model. Overall, the results demonstrate an improvement in the model's precision, although not all the time.

Future research could include alternative constraint transformation methods, different DNR formulations, various ways to select the *n* parameter used to transform the constraints, as well as simulations on larger-scale TNs and DNs.

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<footnote n="11">¹¹More iterations would likely improve the precision.</footnote>
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