::: IEEEkeywords D̉istributed Energy Resources, Electric Vehicle Charge Stations, Cross-phase power flow, Power Quality, Voltage Imbalance Factor :::
the loss and the operational cost of distribution networks is a critical goal for distribution system operators (DSOs). Distribution networks, by nature, are unbalanced in terms of voltages and currents because of having untransposed lines [@paranavithana2009global] in distribution feeders and integration of single-phase generators and loads with highly probabilistic behavior. While voltage imbalance mostly impacts the costumers' power quality, the current imbalance is one of the main causes of loss increase in distribution networks. Mitigation of current and voltage imbalance sources is a serious technical requirement and has drawn the attention of researchers for many years. We review the approaches revolving around the voltage and current imbalance compensation in the following.
Based on [@kong2017three] we can decompose the origins of the imbalance into twofold classes: structural and random. The method focusing on the structural aspect includes network reconfiguration [@borozan1997; @morton2000efficient; @dall2014sparsity; @ding2015feeder; @fu2018toward] and phase balancing [@zhu1998phase; @chen1999optimal; @siti2007reconfiguration]. These techniques, however, do not have real-time flexibility and continuous controllability for compensating the random component of imbalance. Static transfer system (STS) based methods are proposed to compensate for this [@ding2018per]. This method may impose switching transients and is less flexible than power electronics devices with continuous controllability. Device-based compensation has been broadly investigated in the literature and proved outstanding capability in mitigating the current imbalance [@valderrama2001reactive; @pires2004unbalanced; @salmeron2004compensation; @xu2007voltage]. These techniques were rarely implemented in the past but their integrations are abruptly on the rise due to the proliferation of Distributed Energy Resources (PVs) and Electrical Vehicle Charge Stations (EVCSs).
Local control strategies for leveraging distributed energy PVs for imbalance compensation have been discussed in [@singh2010grid; @weckx2015load]. These approaches, as expected, are highly effective but they still lack utilization of full capability of flexible power electronics resources due to the control with incomplete local information status. In the wake of cyber-physical distribution networks enabled by the vast deployment of communication systems, the operational strategies running over the network model with complete system-wide information turn into a real practice [@dall2012optimization; @robbins2015optimal; @araujo2013three; @su2014optimal; @nguyen2018exact; @sheshyekani2019participation]. These methods accommodate the controllability of PV resources and seek to minimize the operation cost in the presence of distribution network imbalance. Voltage imbalance is tackled as a subsidiary objective function in the optimization problem.
All of the methods mentioned in last paragraph mainly revolve around three-phase or single-phase PVs but non of them implements the cross-phase capability for the three-phase inverters in their models. In addition, they are either proposing a non-convex model which compromises the computational performance, or if they propose a convex model, it is only valid in radial topology and due to the rectangular model requirement for those simplifications, the highly non-linear non-convex VUF equation is ignored. Therefore, in this paper a novel feasible three-phase optimal power flow that can also include the above motioned overlooked aspects is tailored which includes these main features:
- Introducing a Convex (QP) three-phase optimal power flow that models bus voltages in polar form with magnitudes and angles and uses linearized load flow equations and capacity constraints that minimizes an objective function including total cost and a polynomial data driven approximation of VUF which is computationally fast and scalable because of convexity and can also be implemented in heavily meshed three-phase distribution networks.
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- Participating the ever-growing three- and single-phase EV chargers alongside with three- and single-phase PVs in optimal power flow operation not only to avoid their possible harmful operation but also exploiting the full capacity of installed devices to reduce cost and improve system conditions.
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- Introducing Cross-phase operation in three-phase inverter models that can exploit the unused three-phase inverter capacity in terms of active and reactive power in different operational and non-operational modes in order to improve system conditions and reduce the cost even more.
Various sub-methods and scenarios are created, tested, and compared in IEEE 33 bus radial and 192 bus heavily meshed test systems. The provided results fully compare the scenarios, sub-methods, Inverter devices, and the two topologies in terms of cost, loss and VUF. The results Also create a starting point to address these critical questions for DSOs; 1) How much should an inverter or load owner pay for making system unbalanced. and 2) How much should be paid for an inverter owner for enabling the cross-phase operation.
The remainder of this paper is organized as follows. Section @ formulates the proposed optimization problem. The computation implementation is described in @. Section @ shows the parameter tuning while the results and the corresponding discussion are presented in section @. Finally, conclusive remarks are presented in Section @.
The optimization problem is cast based on an OPF model with a single objective function and multiple physical and operational constraints including the imbalance limits. The convex form of the OPF model is employed here and the new constraints are convexified to be in line with the whole model trait.
Three-phase power balance equations consisting of PV injection, EV interaction, and load consumption are expressed as:
::: center $$\label{Q_ij_phigamma} \begin{split} \forall i\in N,\varphi\in\phi, \hskip 80pt\ Q_{i_{inject}}^{\varphi}+Q_{i_{PV}}^{\varphi}=Q_{i_{ld}}^{\varphi}+Q_{i_{EV}}^{\varphi}+ \displaystyle\sum_{j|(i,j)\in L}\displaystyle\sum_{\gamma|\gamma\in\phi}Q_{ij}^{\phi\gamma} \end{split}$$ :::
It is implicitly assumed in the framework that if a bus does not have a
connected device (MV transformer, PV or EVCS) the corresponding active
and reactive power related to that device is forced to be zero. In
addition, the self and mutual impedance of all phases are also accounted
in the three-phase active and reactive power balance equations. In
([P_ij_phigamma]{reference-type="ref"
reference="P_ij_phigamma"}) and
([Q_ij_phigamma]{reference-type="ref"
reference="Q_ij_phigamma"}) the
In order to have a convex optimization model and tackle the problem via one of the efficient convex solvers, the linearized forms of (1) and (2) for single-phase distribution network are taken from [@yuan2016novel] and extended to three-phase unbalanced network. Equations ([replaced_P_ij_phigamma]{reference-type="ref" reference="replaced_P_ij_phigamma"}) and ([replaced_Q_ij_phigamma]{reference-type="ref" reference="replaced_Q_ij_phigamma"}) denote branch active and reactive power losses and are defined based on the linearized power flow expression captured in ([branch_active_power]{reference-type="ref" reference="branch_active_power"})-([linearized_Qij2]{reference-type="ref" reference="linearized_Qij2"}). As justifiable assumptions in distribution networks, it is assumed that the voltage magnitudes are around 1 p.u. and the angle differences of successive nodes are close to zero.
$$\label{linearized_Pij1} \hat{P {ij-1} ^{\varphi\gamma} }=\frac {r{ij}^{\varphi\gamma}x_{ij}^{\varphi\gamma}} {r_{ij}^{\varphi\gamma2}+ x_{ij}^{\varphi\gamma2}} . \frac {V _{i} ^{\varphi} - V _{j} ^{\varphi}} {x _{ij} ^{\varphi\gamma}} = k _{ij-1} ^{\varphi\gamma} . \frac {V _{i} ^{\varphi} - V _{j} ^{\varphi}} {x _{ij} ^{\varphi\gamma}}$$
$$\label{linearized_Pij2} \hat{P {ij-2} ^{\varphi\gamma}}=\frac {x{ij}^{\varphi\gamma2}} {r_{ij}^{\varphi\gamma2}+ x_{ij}^{\varphi\gamma2}} . \frac {\theta _{i} ^{\varphi} - \theta _{j} ^{\varphi}} {x _{ij} ^{\varphi\gamma}} = k _{ij-2} ^{\varphi\gamma} . \frac {\theta _{i} ^{\varphi} - \theta _{j} ^{\varphi}} {x _{ij} ^{\varphi\gamma}}$$
$$\label{linearized_Qij1} \hat{Q {ij-1} ^{\varphi\gamma}}=\frac {-r{ij}^{\varphi\gamma}x_{ij}^{\varphi\gamma}} {r_{ij}^{\varphi\gamma2}+ x_{ij}^{\varphi\gamma2}} . \frac {\theta _{i} ^{\varphi} - \theta _{j} ^{\varphi}} {x _{ij} ^{\varphi\gamma}} =- k _{ij-1} ^{\varphi\gamma} . \frac {\theta _{i} ^{\varphi} - \theta _{j} ^{\varphi}} {x _{ij} ^{\varphi\gamma}}$$
$$\label{linearized_Qij2} \hat{Q {ij-2} ^{\varphi\gamma}}=\frac {x{ij}^{\varphi\gamma2}} {r_{ij}^{\varphi\gamma2}+ x_{ij}^{\varphi\gamma2}} . \frac {V _{i} ^{\varphi} - V _{j} ^{\varphi}} {x _{ij} ^{\varphi\gamma}} = k _{ij-2} ^{\varphi\gamma} . \frac {V _{i} ^{\varphi} - V _{j} ^{\varphi}} {x _{ij} ^{\varphi\gamma}}$$
As a result, ([P_ij_phigamma]{reference-type="ref" reference="P_ij_phigamma"}) and ([Q_ij_phigamma]{reference-type="ref" reference="Q_ij_phigamma"}) can be replaced with ([replaced_P_ij_phigamma]{reference-type="ref" reference="replaced_P_ij_phigamma"}) and ([replaced_Q_ij_phigamma]{reference-type="ref" reference="replaced_Q_ij_phigamma"}):
::: center $$\label{replaced_P_ij_phigamma} \begin{split} \textit{P}{i{inject}} ^{\varphi} + P {i{PV}} ^{\varphi} = P {i{ld}} ^{\varphi} + P {i{EV}} ^{\varphi} \ +\displaystyle\sum_{\ell|(i,j)\in L}\displaystyle\sum_{\gamma|\gamma\in\phi}\hat{P_{ij}^{\varphi\gamma}} \end{split}$$ :::
$$\label{replaced_Q_ij_phigamma} \begin{split} Q_{i_{inject}} ^{\varphi} + Q {i{PV}} ^{\varphi} = Q {i{ld}} ^{\varphi} + Q {i{EV}} ^{\varphi} \ +\displaystyle\sum_{\ell|(i,j)\in L}\displaystyle\sum_{\gamma|\gamma\in\phi}\hat{Q_{ij}^{\varphi\gamma}} \end{split}$$
Capacity of distribution lines should guarantee the secure operation of the system. So, it can be written as:
In perspective of optimization problem parameters and variables, the equation ([line_capacity_constraint]{reference-type="ref" reference="line_capacity_constraint"}) is nonlinear and nonconvex. Therefore, linear approximation using piecewise linearization method [@akbari2016linear] is applied to maintain the linear and convex form of the problem and can be shown as:
::: center $$\label{linearized_line_capacity_constraint} \begin{split} \forall h, h\in N, h\leq H, \forall(i,j) \in \ell, \forall\varphi,\varphi \in \phi \hskip 20pt \ \left(\sin\left(\frac{360^{\circ}h}{H}\right)-\sin\left(\frac{360^{\circ}}{H}(h-1)\right)\right) P_{ij} ^{\varphi \varphi}-\ \left(\cos\left(\frac{360^{\circ}h}{H}\right)-\cos\left(\frac{360^{\circ}}{H}(h-1)\right)\right) Q_{ij} ^{\varphi \varphi} \ \leq S_{ij}^{max} \times \sin\left(\frac{360^{\circ}}{H}\right) \end{split}$$ :::
The hard margin constraint voltage magnitude constraint is implemented as:
In
([Voltage_magnitude_constraint]{reference-type="ref"
reference="Voltage_magnitude_constraint"}) the
It should be mentioned that the hard margin implementation of highly non-linear VUF constraint is impossible in the convex optimization platform. Therefore the two options could be hard margin linear VUF approximation and soft margin quadratic approximation in a weighted-sum multiple objective optimization model. It is later discussed in 4.1{reference-type="ref" reference="Reg"} that the latter is a more accurate approach, and therefore it is used as a VUF constraint basis in the presented formulation. The quadratic approximation of VUF can be shown as:
Without loss of generality, the minimum operational cost of the system is sought here as ([objective_function]{reference-type="ref" reference="objective_function"}).
The active power injection of PVs and reactive power support of EVCSs are assumed to be costless. The objective functions can be readily tailored for meshed and multiple input distribution networks connected to MV systems with different cost terms. The objective function can also imply either minimization of the cost or maximization of firm benefit in the highly PV penetrated, and reverse power flow enabled systems. It is quite straightforward to replace it with maximum social welfare, maximum firm benefit, etc., or even proceed with multiple objectives, for the purpose of this paper the multi objective objective function is presented as ([objective_function_VUF]{reference-type="ref" reference="objective_function_VUF"}).
The weight coefficient
EVCSs are considered both single- and three-phase in distribution
systems. While they can flexibly provide reactive power based on system
optimization, their consumed active power is assumed to be constant and
adjusted based on the customer charging preference and the battery and
inverter capability in a certain period of time. Nevertheless, they can
provide some flexibility over consumed active power in each time
interval thanks to their schedulability. Because we are targeted to
perform the optimization in the near real time scope to handle random
phenomena, we prefer to ignore this capability at least in the scope of
this study. The formulation of a single-phase EVCS in phase
The variable
$$\label{three_phase_EVCSs_model1} \forall i \in N, \hskip 2pt \displaystyle\sum {\varphi \in \phi} P{i_{EV}}^{\varphi} = P_{i_{EV}}^{sch}$$
Therefore, corresponding
As same as the EVCSs, PVs are installed as three-phase and single-phase
in distribution networks. While the three-phase PVs can contribute to
the balancing and voltage magnitude control, the single-phase
configurations may be harmful to the system power quality when used
uncontrolled. Therefore, PV management not only can help the
optimization problem to have enough flexibility to satisfy its
objective, but also it is necessary for maintaining the power quality
standards. The single-phase PV in phase
With the linear approximation, ([PV_model_3]{reference-type="ref" reference="PV_model_3"}) can be replaced by:
The corresponding
Finally, the
The input Data structure is built and developed based on the MATPOWER structure [@zimmerman2010matpower]. The Matpower test case structure is modified from single line balanced to three phase unbalanced in order to be able to add the three-phase unbalanced test cases to the case library. For the purpose of this paper, two unbalanced standard systems are added to Matpower library; IEEE 33 Bus [@baran1989network] and the 192 LV buses of the IEEE 342-bus test system [@schneider2014ieee]. The two systems are chosen in such away that cover different voltage levels, bus numbers, topology type (meshed or radial) and number of slack buses. Assuming the symmetrical configuration of its lines and cables, the IEEE 33-bus system is modified to be balanced three-phase system and then the loads are forced to be unbalanced by load intensity factor which will be defined later in the paper. However, the 192 LV buses of the IEEE 342-bus test system is inherently unbalanced and heavily meshed and connected to the MV system from different nodes in order to maintain reliability and satisfy the high load density. For the purpose of this research, single-phase and three phase PV and EVCs with different penetration levels are added to the test cases. Fig. 1{reference-type="ref" reference="33_Bus_system"} and [192_Bus_system]{reference-type="ref" reference="192_Bus_system"} demonstrate the PV and EV locations of the mentioned systems respectively. It worth to mention that the location of EVs and PVs are occasionally overlapped to create a more intense test environment for the proposed method. The details of the test system parameters are elaborated in table 1{reference-type="ref" reference="table2"}.
{#33_Bus_system width="9cm"}
::: center ::: {#table2} System 33-Bus 192-Bus
L-n Voltage 12.6kv 120
Min Voltage 0.9 p.u 0.9 p.u
Max Voltage 1.1 p.u 1.1 p.u
Cost Coefficient 3 3
Cost Coefficient
: System parameters :::
:::
The proposed method is clustered into Five sub-methods presented in table 2{reference-type="ref" reference="table1"} to form an ablation study. The Fixed method is regular OPF solver in Matpower extended to three phase systems. Consequently, three-phase EVs and PVs are considered as three single-phase models. The OPF method is the linearized form of the Fixed method solved in CVX. The OPF VUF method is the OPF method that consideres the VUF constraint. Similarly, the Crossphase method is the OPF with enabling the crossphase capability of PVs and EVs. Finally the CrossPhase VUF method integrates all the contributions of this paper in one method. Both Matpower and CVX solvers are run in the the MATLAB 2022a, Intel Core i5 8250U (1.6-1.8 GHz) CPU, and 8 GB DDR3-1333 (667 MHz) memory Environment.
::: center ::: {#table1} Sub-Method Solver Equations
Fixed Matpower (1-4), (13), (15), (19), (21), (23), (27), (29) OPF CVX (5-12), (14-15), (19), (21), (23), (27), (30) OPF VUF CVX (5-12), (14-15), (18), (20-21), (23), (27), (30) Crossphase CVX (5-12), (14-15), (19), (21-22), (23-24), (26-28), (30-31), (33) Crossphase VUF CVX (5-12), (14-15), (18), (20-22), (23-24), (26-28), (30-31), (33)
: Sub-methods used for ablation study ::: :::
In order to test the individual and combined contribution of each inverter type, three scenarios are considered as Base(NC), PV-noon(NC), Ev-night(NC), and PV-EV-Afternoon. In addition, unlike the Fixed, OPF and OPF-VUF methods, Crossphase and Crossphase OPF methods have the ability to exploit the active power capacity of the inverters in their non operation hours. In other words, non operation hours for PVs is night, which is day in case of EVs Accordingly, three more scenarios are added as Base(C), PV-noon(C), Ev-night(C), to elevate the contribution of each inverter type in non operational hours. So, capacity of the inverters can be used in non operational hours by proposed method.
All of the scenarios consists of eight time intervals each having a different load imbalance level as shown in Fig. [2](#Load Imbalance){reference-type="ref" reference="Load Imbalance"}. The load imbalance (Active and Reactive) is getting worse linearly in each interval while maintaining a constant total load consumption in order to obtain standardized load flow results.
::: center {#Load Imbalance width="8cm"} :::
In ([VUF_apr_simplified]{reference-type="ref"
reference="VUF_apr_simplified"}) the
{#VUF_reg width="8cm"}
In order to enabling a quantified trade-off between cost and sum of VUFs
the Pareto method is used. For having a 2D Pareto Front, it is assumed
that all buses have the same weight and
{#prt width="8cm"}
This section compares the effectiveness of the presented sub-methods and scenarios in four aspects; 1) total cost, 2) voltage imbalance, 3) total loss, and 4) line capacity utilization. For each of the mentioned aspects, first, the 2{reference-type="ref" reference="table1"} sub-methods are compared in the most complete scenario, PVEV afternoon. Then the 3.3{reference-type="ref" reference="scenario_section"} scenarios are compared for the state-of-the-art method, CrossPhase VUF.
Total cost of the systems for the sub-methods of Table. 2{reference-type="ref" reference="table1"} is evaluated for the fifth time interval in Table.3{reference-type="ref" reference="Table9"}. Assessing the table proves that using a cross-phase enabled sub-method reduces the total cost of the systems in both test systems. As it is visible, utilizing the other sub-methods is not so profitable compared to the . This cost reduction is achieved by exploiting the capacity of PV and EV inverters during operational and non-operational hours in Crossphase and Crossphase VUF method. Fig. 5{reference-type="ref" reference="Cross"} shows a comparison between the apparent, active, and reactive power reference of the three-phase EV charger installed on bus number 88 of the IEEE 192 bus system for both Crossphase and OPF methods during operational and non-operational hours. As it can be shown on the figure, the inverter capacity utilization in operational mode is 5% more when crossphase is enabled. Moreover, in non-operational hours, crossphase enables the inverter to circulate the active power by consuming power from phase c, and transferring it to phase b to compensate the induced load imbalance shown in figure [2](#Load Imbalance){reference-type="ref" reference="Load Imbalance"}. Also, as shown in Tables. [Table11]{reference-type="ref" reference="Table11"}, the total cost of the Crossphase VUF is lesser in the scenarios that enables the inverters to contribute in non-operational hours than the scenarios that turn the inverters off during non-operation. This resultant crossphase flexibility, leads to a better optimal solution in terms of cost. It should be mentioned that due to the meshed topology of IEEE 192 bus, enabling crossphase is not guaranteed as in the IEEE 33 bus radial system. This behaviour is related to possible current imbalance propagation to several main lines in meshed grids which is further discussed in 5.3{reference-type="ref" reference="secloss"}. Another result that can be obtained from the tables. [Table11]{reference-type="ref" reference="Table11"} is that although both systems have more PVs in terms of capacity and numbers installed, EVs show a better non-operational crossphase contribution for cost reduction.
Sub-methods that contain a VUF constraint, experience a higher total cost because of the additional term in their objective functions. It can be seen that in both systems intensifying the load imbalance will increase system cost in VUF constrained sub-methods. The cost difference between the VUF-constrained and non constrained sub-methods, is the price that should be paid for compensation of the VUF.
::: center ::: {#Table9}
System Fixed OPF OPF-VUF Crossphase Crossphase-VUF
33-Bus 346.54 346.54 351.72 341.29 344.55
192-Bus 127.19 127.19 147.22 127.19 142.07
: Total cost of the IEEE systems during " PV-EV-afternoon" scenario during 5th time interval ::: :::
::: center ::: {#VUFstandar}
Scenarios Base (C) Base (NC) EV-Night(C) EV-Night(NC) PV-Noon(C) PV-Noon(NC)
33-Bus 361.87 367.81 369.94 373.40 337.35 339.27
192-Bus 155.19 155.19 171.19 171.19 126.94 129.92
: Total cost of the IEEE 33- and 192-Bus systems while implementing Crossphase VUF for different scenarios during 5th time interval ::: :::
{#Cross width="8cm"}
Figure 6{reference-type="ref" reference="VUF33m"} and 7{reference-type="ref" reference="VUF192m"} compares the results of the VUF for each bus in all time intervals with the mentioed sub-methods of table. 2{reference-type="ref" reference="table1"}. The results shows the superiority of crossphase VUF in reducing the VUF in comparison with others. Also, the crossphase show reduction in VUF howeve, the effect of VUF constraint is more than crossphase individully. The OPF and Fixed method that are only different in their lineararity of constraints and solver show a very severe VUF especially in intense time intervals.
The results for the Scenarios other than PVEV afternoon are summarized in table. 5{reference-type="ref" reference="VUFstandar"}. In this table only the results for the most extreme condition (time interval 8) are shown. According to the table, however the standard could not be maintained in this worst case scenario, the crossphase contribution of EVs and PVs were significantly effective in reducing the nonstandard conditioned buses. Also results reveal that unlike the cost, in terms of VUF reduction, crossphase contribution of PVs in both Base (C) and EV Night (C) are more effective than crossphase contribution of EVs such that in IEEE 33 bus system they were able to reach a standard solution. The results also reveals that the crossphase capability is more effective in radial 33 bus than heavily meshed 192 bus system.
{#VUF33m width="8cm"}
{#VUF192m
width="8cm"}
::: center ::: {#VUFstandar}
Test system Base (NC) Base (C) PV-Noon (NC) PV-Noon (C) EV-Night (NC) EV-Night (C)
33-Bus 76 0 24 6 69 0
342-Bus 75 43 46 41 71 44
: Percentage of the Buses of IEEE 33 and 192-Bus system violating the
IEEE 1159 standard during different scenarios (
Total power loss of the test systems for different sub-methods are pointed out in Tables. 6{reference-type="ref" reference="Table5"} and 7{reference-type="ref" reference="Table6"}. Analyzing Table.6{reference-type="ref" reference="Table5"} clears that by increasing load imbalance intensity, total power loss of the IEEE 33-Bus system experiences an upward trend. Due to the constant three-phase load power in all intervals, it can be concluded that the the increase in loss is solely a result of increase in the line imbalance. Before discussing the relation between line imbalance and crossphase, it needs to be mentioned that the crossphase opeartion for the cost and VUF minimization can have two forms: 1) allocating power of the inverter in order to meet the unbalanced load demand. 2) transferring flowing power probably generated from a collection of single-phase PVs in one phase to another. The first form will locally compensate the line imbalance of the lower side and can have a balancing effect on the upper main line in radial systems. The latter however will exploit the line capacity to transfer a generated single phase power to the upper side loads or loads in the other phases and therefore my not guarantee a balanced upper main line balance. By taking the above mentioned into consideration, it can be seen that on table. 6{reference-type="ref" reference="Table5"} when the loads are nearly balanced (intervals 1-3) crossphase VUF compromises the the line imbalance and the loss to reach a better VUF and cost, however in the highly unbalanced situations, the crossphase VUF minimizes the VUF and cost without any need to compromise the loss. Therefore in these cases both current and voltage imbalance are minimized alongside with the cost. Analysing the results of table. 7{reference-type="ref" reference="Table6"} with the same manner reveals that in the presented heavily meshed grid, the loss in crossphase VUF is more than crossphase in all time intervals which means that the line imbalance most be compromised to reach a better VUF. This behaviour is like the behaviour of the radial grid between (1-3) intervals. Therefore it can be concluded that the voltage and current imbalance are not always inlined.
The scenario based results are presented in tables 8{reference-type="ref" reference="Table7"} and 9{reference-type="ref" reference="Table8"}. According to 33 Bus system results, non-operational contribution of EVs and PVs both worked in favor of the system loss reduction in all intervals which illustrates the superiority of cross phasing in radial topology in terms of loss. In analysing different conclusions can be made. First, comparing the results of Base (NC) and (C) for interval 1 where there is no single-phase generation reveals that attempt to reduce VUF and cost will propagate line imbalance due to meshed topology. This result shows that the claim made for the first form of cross-phasing is not valid in meshed networks. Second, in comparing results of EV Night (NC) and (C), where the cross phasing is dine by PVs and a lot of single-phase generation exists, loss reduction will not occur before interval 7 which supports the claim made in the previous paragraph about the second form of cross-phasing. On the other hand, PV noon (NC) and (C) shows that only EV cross-phasing in full presence of PVs was able to reduce the loss in all time intervals successfully.
::: center ::: {#Table5}
Hour Fixed OPF OPF-VUF Crossphase Crossphase-VUF
1 4.88 4.87 4.68 4.86 5.01
2 4.98 4.97 4.81 4.92 5.05
3 5.15 5.14 5.13 5.06 5.16
4 5.39 5.37 5.35 5.26 5.32
5 5.70 5.69 5.60 5.54 5.56
6 6.08 6.07 5.91 5.88 5.86
7 6.53 6.52 6.29 6.29 6.22
8 7.06 7.04 6.75 6.78 6.66
: Total power loss in IEEE 33-Bus system with different dispatching methods (kW) ::: :::
::: center ::: {#Table6}
Hour Fixed OPF OPF-VUF Crossphase Crossphase-VUF
1 14.15 14.40 15.51 14.42 14.80
2 14.47 14.72 16.11 14.72 14.82
3 15.00 15.
70D6
25 16.62 15.24 15.14
4 15.74 16.00 17.16 15.95 15.99
5 16.70 16.94 17.85 16.86 17.29
6 17.87 18.07 18.72 17.92 18.53
7 19.25 19.43 19.81 19.29 19.79
8 20.85 21.00 21.09 20.81 20.97
: Total power loss in IEEE 192-Bus system with different dispatching methods (kW) ::: :::
::: center ::: {#Table7} Hour 1 2 3 4 5 6 7 8
Base(C) 5.22 5.20 5.09 5.37 5.69 5.83 5.98 6.11
Base (NC) 5.31 5.41 5.59 5.83 6.15 6.53 6.99 7.51
EV-Night (C) 5.87 5.74 5.66 5.78 6.31 6.66 6.49 6.76
EV-Night (NC) 6.27 6.06 6.03 6.17 6.38 6.66 7.01 7.40
PV-Noon (C) 2.43 2.68 3.12 3.35 3.32 3.07 3.89 4.48
PV-Noon (NC) 2.80 2.79 2.85 2.96 3.23 3.67 4.46 5.06
: Total power loss of the 33-Bus system with crossphase-vuf method during different scenarios (kW) ::: :::
::: center ::: {#Table8} Hour 1 2 3 4 5 6 7 8
Base(C) 16.72 16.85 17.17 17.79 18.53 19.35 20.31 21.38
Base (NC) 16.48 16.79 17.32 18.06 19.00 20.16 21.53 23.11
EV-Night (C) 19.85 20.02 20.38 21.01 21.77 22.64 23.57 24.69
EV-Night (NC) 19.60 19.77 20.17 20.79 21.59 22.58 23.77 25.18
PV-Noon (C) 12.52 12.70 12.98 13.64 14.93 16.12 17.30 18.46
PV-Noon (NC) 12.68 12.87 13.32 14.24 15.68 16.97 18.40 19.77
: Total power loss of the 192-Bus system with crossphase-vuf method in different scenarios (kW) ::: :::
In this paper an exact three-phase optimal power flow that includes both voltage angle and magnitude (AC load load flow) that also works in meshed networks is introduced to be able to optimally manage the per phase active and reactive power references of single and three phase PVs and EVs. The consideration of voltage angle and magnitude enabled the optimal power flow to include the VUF as an operational constraint/objective. In addition, in order to guarantee a good computational performance, all the optimal power flow and constraints are linearized and VUF constrained is approximated with a second degree multi-variable polynomial by regression over an inclusive data set in the operational range of voltage magnitudes and angles. This considerations led to a quadratic formulation of three-phase optimal power flow that can be solved by fast convex solvers. In addition, the cross-phase capability of three-phase EVs and PVs are implemented into formulations to fully exploit the inverter capacity to minimize cost and fix the VUF. This crossphase capability can also be profitable in non-operational modes of PVs and EVs. Later on, in order to better integrate VUF and cost minimization, a Pareto front trade-off has been made. In order to show the the effect of both including VUF constraint and cross-phase capability in operational and non-operational modes, multiple sub-methods and scenario are tested in terms of total cost, total loss, and VUF reduction in IEEE 33 bus radial and IEEE 192 heavily meshed systems.
The method comparison results revealed that using a cross-phase enabled sub-methods reduces the total cost of the systems in both test systems and the Sub-methods. This reduction is the cost that an inverter owner saves for DSO by enabling the cross-phase capability. Also sub-methods containing a VUF constraint, experience a higher total cost because of the additional term in their objective functions while showing a superiority reducing the VUF. Therefore, the cost difference between the VUF-constrained and non constrained sub-methods, is the price that should be paid for compensation of the VUF. In terms of loss, by increasing load imbalance intensity, total power loss of the systems which is closely related to line imbalance due to normalized implementation of load imbalance experienced an upward trend with increase in load imbalance. When the loads were nearly balanced crossphase VUF compromised the the line imbalance to obtain a better VUF and cost, however in the highly unbalanced situations, the crossphase VUF minimized the VUF and cost without any need to compromise the line imbalance. In terms of scenario comparison, results for multiple scenarios showed that enabling non-operational cross-phase leads to a better optimal solution in terms of cost and is significantly effective in reducing the VUF.
Finally, as a device based comparison, EVs show a better non-operational cross-phase contribution for cost reduction. however, in terms of VUF reduction, it is vise versa.