# Pp OPF - Pandapower Implementation of Three-phase Optimal Power Flow Model
††thanks: This work has been supported in part by the European Structural and
Investment Funds under KK.01.2.1.02.0042 DINGO (Distribution Grid
Optimization) and in part by the FLEXIGRID project from the European Union’s
Horizon 2020 research and innovation programme under grant agreement No
864579. This paper reflects the FLEXIGRID consortium view and the European
Commission is not responsible for any use that may be made of the information
it contains.

Tomislav Antić  Faculty of Electrical Engineering
and Computing
University of Zagreb
Zagreb, Croatia
<EMAIL_ADDRESS>Andrew Keane  UCD Energy Institute
University College Dublin
Dublin, Ireland
<EMAIL_ADDRESS>Tomislav Capuder  Faculty of Electrical Engineering
and Computing
University of Zagreb
Zagreb, Croatia
tomislav.capuderfer.hr

###### Abstract

Challenges in the planning and operation of distribution networks caused by
the integration of distributed energy resources (DERs) create the need for the
development of tools that can be easily used by system operators, industry,
and the research society but are also easily upgraded with new
functionalities. The full implementation of one such open source tool, named
pp OPF (pandapower Optimal Power Flow), is presented in this paper. Pp OPF is
the tool used for three-phase optimal power flow (OPF) calculations and it is
based on already existing functionalities of pandapower, a Python library for
power system calculations. The developed tool enables the possibility to use
both power-voltage and current-voltage formulation in different OPF problems,
such as determining the photovoltaics (PVs) hosting capacity in three-phase
distribution networks, the problem on which the functionality of the developed
tool is tested. Additionally, the accuracy of pp OPF is verified by comparing
the results with ones obtained by pandapower power flow calculation for the
same set of input values. The open-source implementation allows further
upgrades, the addition of new functionalities, and the creation of new case
studies relevant to the planning and operation of distribution networks.

###### Index Terms:

distribution networks, distributed energy resources, pandapower, three-phase
optimal power flow

## Nomenclature

###  -A Sets and Indices

  * $N$

Set of all nodes

  * $P$

Set of all phases of the node

  * $T$

Set of all time intervals

  * $i,j$

Begin and end node of the element, $i,j\in N$

  * $n$

Observed node, $n\in N$

  * $slack$

Slack node

  * $p$

Phase of the node, $p\in P$

  * $t$

$t\in T$

###  -B Parameters

  * $\begin{bmatrix}Z_{012}\end{bmatrix}$

Sequence impedance matrix

  * $\begin{bmatrix}A\end{bmatrix}$

Transformation matrix

  * $P_{i,p,t\ l}$

Active power of load $l$

  * $Q_{i,p,t\ l}$

Reactive power of load $l$

  * $U_{min/max}$

Minimum and maximum voltage magnitude

  * $VUF_{n,max}$

Maximum value of voltage unbalance factor

  * $S_{max,ij}$

Maximum apparent power of element $ij$

###  -C Variables

  * $U_{n,p,t}^{re/im}$

Real/imaginary part of voltage at node $n$, phase $p$, and time period $t$

  * $U_{n,p,t}^{re/im}$

Real/imaginary part of voltage at node $n$, phase $p$, and time period $t$

  * $P_{ij,p,t}$

Active power between nodes $i$ and $j$, of phase $p$, and at time period $t$

  * $Q_{ij,p,t}$

Reactive power between nodes $i$ and $j$, of phase $p$, and at time period $t$

  * $P_{n,p,t\ l/g}$

Active power of load $l$/generator $g$ connected to the node $n$, phase $p$,
at time period $t$

  * $Q_{n,p,t\ l/g}$

Reactive power of load $l$/generator $g$ connected to the node $n$, phase $p$,
at time period $t$

##  I Introduction

The share of low-carbon (LC) technologies has been continuously increasing.
Despite the positive effect it has on the environment and the end-users’
financial, the integration of LC units is often uncoordinated which can lead
to different technical problems in the operation of distribution networks,
including overvoltage and undervoltage [1], increased voltage unbalance [2],
and overloading of lines and transformers [3].

Despite the high number of commercial software used for distribution networks
analyses, recent digitization and more often exploitation of computing
techniques in power systems planning and operation has led to the development
of open source tools in high-level programming languages, such as MATPOWER [4]
or pandapower [5]. A large shortcoming in using these tools is that they are
often only suitable for analyses of distribution networks in which consumption
and generation patterns are known in advance. As mentioned before, the
uncoordinated integration of LC units creates numerous challenges for system
operators. As a way of battling these problems, the distribution system
operators now more than ever reach for optimization tools as opposed to the
conventional simulation-based approach, making optimal power flow algorithms
essential in everyday operation. Even though OPF models are well-known and
widely used in different power system problems, e.g., unit commitment [6],
their use in distribution and especially low voltage (LV) networks require
modification and transition towards three-phase four-wire models. Three-phase
OPF models have numerous applications, including optimal control of devices
[7], congestion mitigation [8], calculation of operating envelopes [9], etc.
Three-phase optimal power flow problems can be formulated as current-voltage
[10] or power-voltage [11] and can be modeled as exact, non-convex problems
[10] or they can be linearized [9] or relaxed [12].

Recent developments of open source software include the implementation of a
three-phase OPF model as a Julia package [13] and integration three-phase OPF
model with OpenDSS software [14]. Despite the recent developments in
pandapower, including the creation of new elements and introduction of new
functionalities, e.g., three-phase power flow and state estimation [15] or the
availability of harmonic power flow simulations [16], using pandapower Python
library in three-phase OPF problems is still not possible.

To overcome the detected pandapower shortfall, in this paper we propose an
implementation of non-linear, current-voltage, and power-voltage three-phase
OPF formulations using existing pandapower and newly added functionalities
necessary for modeling the wanted problem. The developed tool named pp OPF
(pandapower Optimal Power Flow) is open source and can be used by accessing
[17]. The functionality of the tool is tested on the problem of defining
photovoltaics (PVs) hosting capacity in the real-world LV network. The
accuracy of the tool is verified against the results calculated by pandapower
power flow simulations for the same input data. Additionally, open source
development allows further modification of pp OPF so it can be applied in
other planning and operation problems or extended with some additional
functionalities. It is important to emphasize that the presented mathematical
model and defined case study on which the model was tested are not novel
contributions. However, to the best of the authors’ knowledge, this is the
first time that pandapower was used for the implementation of a three-phase
OPF mathematical model. Additionally, problems that are solved by exact non-
linear models are often non-scalable and require a significant computational
burden in order to be solved. In this paper, an introduction to such problems
and one possible method for overcoming such problems is presented next to the
verification and the analysis of the case study results.

The rest of the paper is organized as follows: Section II presents the
connection to pandapower by using some of the already developed
functionalities. In Section III, the mathematical model of the power-voltage
formulation is given. The results of a defined case study and the comparison
of the results with ones obtained by pandapower are presented in Section IV,
while the conclusions and future work are given in Section V.

##  II Connection to pandapower

Pandapower is developed as an open source tool for power system modeling,
analysis, and optimization with a high degree of automation. Since the new
tool presented in this paper is used for analyses and optimization problems
that have not yet been implemented in pandapower, the only functionality that
is used is the modeling of distribution networks by using pandapower for
defining different elements necessary for creating the representation of the
observed distribution network, while all other necessary functions are built
on top of that. Used elements are an external grid, transformers, lines, and
loads. An external grid is used to define the per-unit voltage at the slack
node. The assumption in the presented three-phase OPF model is that the
voltage of the slack node is symmetrically distributed over phases, as shown
in eq. (1).

$\begin{gathered}U_{slack,a}=U_{slack}\\\
U_{slack,b}=U_{slack}\angle-120^{\circ}\\\ U_{slack,c}=U_{slack}\angle
120^{\circ}\end{gathered}$ (1)

Transformers are defined with parameters needed for the calculation of their
impedance, such as voltage ratio, rated power, and short-circuit voltage.
Lines are created with parameters including but not limited to resistance and
reactance for zero and positive sequence systems, length, and maximum current.
After creating these elements, already existing pandapower functionalities are
used for calculating their impedance and placing them in the impedance matrix
$\begin{bmatrix}Z_{012}\end{bmatrix}$ containing the calculated impedance of
zero, positive, and negative sequence systems. By using transformation matrix
$\begin{bmatrix}A\end{bmatrix}$ and its inverse
$\begin{bmatrix}A\end{bmatrix}^{-1}$, three-phase impedance matrix
$\begin{bmatrix}Z_{abc}\end{bmatrix}$ is calculated.

All loads in a distribution network are created using pandapower
functionalities, together with other elements relevant to the representation
of the network’s mathematical mode. Since pp OPF is used in the multi-temporal
optimization of three-phase power systems, loads need to be defined with the
active and reactive power of each phase in each time period. Since the use of
pandapower in multi-temporal simulations of a three-phase network is not
straightforward, a load curve that defines the change of the loads’ value
during the observed period is additional input in the tool. Based on the
objective function and the goal of the optimization problem, additional
elements representing DER such as PVs or electric vehicles (EVs) can be added
to the formulation. These elements can be defined with a fixed value or can be
defined as variables whose power needs to be determined during the solving of
the defined problem.

##  III Three-phase OPF mathematical model

After defining a three-phase distribution network with all of its elements
using pandapower, its parameters are used in the developed OPF formulation.
The pp OPF tool integrates both power-voltage and current-voltage
formulations. OPF models are formulated using Python programming language and
Pyomo optimization framework. However, in this paper, only the mathematical
model of a power-voltage formulation is presented, while both formulations are
tested and verified on case studies presented in Section IV.

Equations used for constraining values of voltage magnitude and voltage
unbalance factor (VUF) are presented with eqs. (2) and (3).

$|U_{min}|^{2}\leq|U_{i,p,t}|^{2}\leq|U_{max}|^{2}$ (2)

$VUF_{n}=\frac{|U_{2,n}|}{|U_{1,n}|}=\frac{|U_{n,a}+a^{2}\cdot U_{b,n}+a\cdot
U_{c,n}|}{|U_{n,a}+a\cdot U_{b,n}+a^{2}\cdot U_{c,n}|}\leq VUF_{n,max}$ (3)

Active and reactive power flow between nodes $i$ and $j$ is described with
eqs. (4) and (5). Additionally, power flow through each element cannot be
larger than the maximum power which is defined as an input parameter of a
transformer or a line in the observed network. This constraint is given with
eq. (6).

$\begin{gathered}P_{ij,p,t}=\sum_{q\in\\{a,b,c\\}}(U_{i,p,t}^{re}\cdot
U_{i,q,t}^{re}+U_{i,p,t}^{im}\cdot U_{i,q,t}^{im})\cdot G_{ij,pq}+\\\
\sum_{q\in\\{a,b,c\\}}(U_{i,p,t}^{im}\cdot U_{i,q,t}^{re}-U_{i,p,t}^{re}\cdot
U_{i,q,t}^{im})\cdot B_{ij,pq}-\\\ \sum_{q\in\\{a,b,c\\}}(U_{i,p,t}^{re}\cdot
U_{j,q,t}^{re}+U_{i,p,t}^{im}\cdot U_{j,q,t}^{im})\cdot G_{ij,pq}-\\\
\sum_{q\in\\{a,b,c\\}}(U_{i,p,t}^{im}\cdot U_{j,q,t}^{re}-U_{i,p,t}^{re}\cdot
U_{j,q,t}^{im})\cdot B_{ij,pq}\end{gathered}$ (4)

$\begin{gathered}Q_{ij,p,t}=-\sum_{q\in\\{a,b,c\\}}(U_{i,p,t}^{re}\cdot
U_{i,q,t}^{re}+U_{i,p,t}^{im}\cdot U_{i,,tq}^{im})\cdot B_{ij,pq}+\\\
\sum_{q\in\\{a,b,c\\}}(U_{i,p,t}^{im}\cdot U_{i,q,t}^{re}-U_{i,p,t}^{re}\cdot
U_{i,q,t}^{im})\cdot G_{ij,pq}+\\\ \sum_{q\in\\{a,b,c\\}}(U_{i,p,t}^{re}\cdot
U_{j,q,t}^{re}+U_{i,p,t}^{im}\cdot U_{j,q,t}^{im})\cdot B_{ij,pq}-\\\
\sum_{q\in\\{a,b,c\\}}(U_{i,p,t}^{im}\cdot U_{j,q,t}^{re}-U_{i,p,t}^{re}\cdot
U_{j,q,t}^{im})\cdot G_{ij,pq}\end{gathered}$ (5)

$P_{ij,p}^{re\ 2}+Q_{ij,p}^{im\ 2}\leq S_{max,ij}$ (6)

Besides power flow, the only other constraint needed in the power-voltage
formulation is Kirchoff’s Current Law (KCL) presented with active and reactive
power instead of current. For securing KCL, the following eqs. (7) and (8) are
defined.

$P_{i,p,t\ g}+P_{h\rightarrow i,p,t}=P_{i,p,t\ l}+P_{i\rightarrow j,p,t}$ (7)

$Q_{i,p,t\ g}+Q_{h\rightarrow i,p,t}=Q_{i,p,t\ l}+Q_{i\rightarrow j,p,t}$ (8)

##  IV PVs hosting capacity

PVs or other DERs in general can be single-phase or three-phase connected to
the network. Single-phase connection is easier and cheaper at the LV level but
is limited with relatively low export or import power, e.g., export power in
Croatian LV networks may not be higher than 3.68 kW [18], and can lead to the
increase in voltage unbalance. On the contrary, a three-phase connection
allows the installation of PVs with higher production power and solves the
unbalance problem but the connection is more expensive and complex.

In this paper, two case studies are defined for calculating the hosting
capacity of PVs (generators $g$) in a real-world LV distribution network. In
the first case study, PVs can be only single-phase connected to one phase at a
time, and in the second, the power of PVs is larger than 3.68 kW, and
therefore they are three-phase connected to the network. To obtain an exact,
optimal solution, the presented formulation needs to be extended with binary
variables that will define the exact connection phase in the case of a single-
phase connection or if PVs are single-phase or three-phase connected at a
certain node. Such an approach would uplift the formulation from the non-
linear programming (NLP) problem to the mixed-integer (MI)NLP problem, which
would lead to problems related to the scalability and the computational time
needed for finding an optimal solution. Therefore, both case studies are
defined as NLP problems and solved using the pyomo optimization framework and
the ipopt solver. In the case of a single-phase connection of PVs, their
connection phase is randomly pre-defined in each node. When a three-phase
connection is observed, the power of PVs is distributed over the phases and
PVs can not be single-phase connected. In cases of well-built and resilient
networks, determining the hosting capacity will mostly lead to the connection
of PVs with the larger connection power, which makes the assumption that PVs
are always three-phase connected valid in solving the presented problem. In
both observed case studies, the production power of PVs is constrained with
the curve containing per-unit values scaled according to the largest measured
value of production in the observed network’s location, as shown in eq. (9).

$P_{n,p,t\ PV}\leq P_{PV,max}\cdot P_{PV,t}$ (9)

where $P_{n,p,t\ PV}$ is the variable representing the calculated power of PV
at the certain node $n$, phase $p$, and time period $t$, $P_{PV,max}$ is the
theoretical maximum value of PVs production which is equal to $3.68kW$ in case
of a single-phase and $\frac{500}{3}kW$ in case of a three-phase connection,
and $P_{PV,t}$ is the per-unit value that ensures the value of production that
does not exceed the limit defined by the irradiation and other physical
quantities.

For the presented case study, a real-world Croatian LV network shown in Fig. 1
was created using pandapower. Additionally, measurements collected from the
smart meters were used to define the values of the loads’ active and reactive
power.

Figure 1: A residential LV network

As mentioned before, in the first case, all PVs are connected single phase,
where the connection phase is randomly chosen for each node. Fig. 2 shows the
summarized PV production curve for each phase over time. The curve has the
shape characteristic for the production of PVs, with the characteristic
distortions caused by sudden weather changes that affect the production power.
Together with the shape, values in the PV production curve are similar every
day, with a maximum power of almost $120kW$.

Figure 2: Production of single-phase PVs

In the second case, PVs were three-phase connected to the network. Fig. 3 does
not show the power of each phase but only cumulative power at each node, since
the PVs are symmetrically connected, i.e., power at every phase is the same.
The results show that the three-phase connection allows the significantly
higher capacity of installed PVs in the same network, with the production
power at several time periods of almost $3500kW$. The results also confirm and
make valid the assumption that none of the PVs are single-phase connected,
which is relevant in avoiding MINLP formulations of a given problem.

Figure 3: Production of three-phase PVs

The results clearly indicate that in order to maximize the share of LC units,
a three-phase connection should be encouraged but a detailed study on finding
the optimal location and size of PVs in the network is outside the scope of
the paper. Moreover, it is already well-investigated in the existing
literature. The main goal of this paper was to test the functionality of the
implemented tool on one possible example. Additionally, the presented case
study is used as an introduction to the potential methods that can be used in
the improvement of the scalability of the problems solved using three-phase
OPF formulations.

Since both current-voltage and power-voltage formulations are implemented from
scratch, it is necessary to verify the results of OPF calculations
Verification is made by comparing the results obtained by the pp OPF with the
results obtained by pandapower PF calculation for the same network and set of
values. Table I shows the comparison of voltage magnitude calculated with pp
OPF and pandapower.

TABLE I: Comparison of results: pandapower and pp OPF

|  | Maximum
---
error (p.u.)
| Minimum
---
error (p.u.)
| Average
---
error (p.u.)
| Median
---
error (p.u.)
| RMSE
---
(p.u.)
| Current-voltage
---
formulation
0.0624 | 0.0000 | 0.0024 | 0.0000 | 0.0086
| Power-voltage
---
formulation
0.0472 | 0.0000 | 0.0074 | 0.0055 | 0.0097

As it can be seen from the comparison, neither of the calculated errors is of
a significant value, with the maximum error lower than 0.07 p.u., which can be
characterized as an outlier value and with even lower values of other
calculated errors. Calculated errors clearly indicate that the presented tool
can be used in different problems in three-phase distribution networks.

##  V Conclusions and future work

Recent global challenges emphasize the need for green energy transition and a
continuous increase in the share of installed LC units. The role of end-users
in the energy transition is becoming more significant and even though their
investment in PVs, EVs, and other LC units is widely encouraged, their
integration within the LV network is often uncoordinated, creating technical
problems for system operators. To successfully mitigate these challenges,
system operators need to rely on new tools and methods which are able to help
in making the decisions that will maximize the share of installed LC units and
simultaneously avoid major technical problems. One such tool is pp OPF, a
three-phase OPF-based tool presented in this paper. pp OPF is developed as an
extension of pandapower, an already developed open source tool for power
system simulations. Both power-voltage and current-voltage OPF formulations
were added on top of existing pandapower functionalities.

Following the development and implementation, pp OPF has been verified and
tested on the real-world case study. In the defined case study, a real-world
three-phase LV network was used in order to calculate its hosting capacity,
both in cases of single-phase and three-phase connection of PVs. The results
of the simulation were expected and emphasize the need for the encouragement
of a three-phase connection. A more detailed analysis of the results is
outside the scope of this paper since the main objective was to present one of
the potential functionalities of the developed tool. Additionally, a specific
case study was used in order to warn the community about the potential
problems in using MINLP formulation and to suggest a method that reformulates
the problem into an NLP problem.

The output of the calculation (production power of PVs) and input data
including network characteristics and active and reactive power of the passive
loads were used for the comparison of voltage magnitudes calculated by using
pp OPF and pandapower. The results show that there are no significant
differences between voltages calculated by the two different tools, with the
maximum error detected as an outlier value and together with the values of
other calculated errors, it clearly shows that there are no obstacles in using
the pp OPF tool in other optimization problems in three-phase distribution
networks.

Further research includes the investigation of other potential applications of
the presented tool, more detailed research on the methods that can be used in
the improvement of the computational time needed for finding the optimal
solution, methods for removing binary variables from MINLP problems,
comparison of solutions obtained by different formulations (e.g, exact and
linearized), and extension of optimal power flow formulation with power
quality constraints, e.g., harmonic constraints.

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