MicroGrid Resilience-Oriented Scheduling: A Robust MISOCP Model

This article introduces a Robust Mixed-Integer Second Order Cone Programming (R-MISOCP) model for the resilience-oriented optimal scheduling of microgrids (MGs). This is developed for MGs that are islanded due to a scheduled interruption from the main grid, where minimizing both operational costs and load shedding is critical. The model introduced presents two main benefits. Firstly, an accurate second order cone power flow model (SOC-PF) is used, which ensures global optimality. Through a comparison with a piecewise linear power flow model on a modified IEEE 33 bus network, it is demonstrated that failure to accurately model power flow equations, can result in a significant underestimation of the operational cost of almost 12%. Secondly, uncertainty is modelled using a robust approach which allows trade-offs between the uncertainty that a MG operator is willing to tolerate, and performance. In this article, performance criteria considered are operational cost and load shedding. Market price, demand, renewable generation and islanding duration are considered as uncertain variables. Results show that by controlling the budget of uncertainty, the MG operator can achieve an almost 20% reduction in the operating cost, compared to a fully robust schedule, while achieving 0% probability of shedding more demand than expected.


MISOCP
Mixed-integer second order cone programming.

PLS
Probability of load shedding. T HE REQUIREMENT to meet zero CO 2 emissions in the electricity sector, restructuring of the electricity business, and technological developments in microgeneration, have set a new paradigm of power systems in modern societies [1], [2]. Advances in distributed generation units (microturbines, fuel cells, etc.) and distributed storage devices (energy storage systems (ESS), batteries, etc.) have formed the broader class of distributed energy resources (DERs). The connection of DERs at medium or low voltage distribution networks have increased the flexibility of network stakeholders over the past decades, shaping new concepts for the future smart grid; including the concept of MicroGrids.
MicroGrids (MG) are medium or low voltage distribution networks with distributed generation units, energy storage devices, and flexible loads [3]. A MG can be operated in an autonomous or non-autonomous way, forming two modes of operation: the islanded and the grid-connected mode. In the grid-connected mode, MGs operate connected to the distribution grid or simply main grid. In the islanded mode, supply from the main grid is interrupted, and the MG depends only on the on-site DERs in order to satisfy the required demand; if demand exceeds generation, loads need to be curtailed.
Resilience represents the ability of a power system to supply the demand in the face of an event that has a high impact and a low probability of occurring [4]. This type of events occurs when the main grid is not available, either due to a scheduled interruption, e.g., an upstream maintenance, or a foreseeable natural disaster such as a hurricane [4]- [6]. MGs present a practical solution to enhance power system resilience by decreasing the probability of load shedding [4]. Operating in islanded mode, the MG can supply local demand in case of a high-impact-low-probability event, when supply from the main grid may be interrupted from several minutes to several hours.
MG resilience merits are well-acknowledged by the academic community [7], [8]. In recent years, there is an increasing interest to provide a framework for the mathematical modelling of the resilience-oriented MG optimal scheduling problem. A literature review of relevant studies is the underlying theme of the following subsection.

B. Literature Review
In order for MGs to enhance power system resilience, this calls for methods that can handle disconnection and re-connection of a MG when power from the main grid is interrupted for an extended period of time. An early attempt clearly tailored towards this path, was presented by A. Khodaei in 2014 in [4]. This study proposed an operational framework for a MG which operated in islanded mode for an extended period of time due to interruption of supply from the main grid, and was introduced as the resiliency-oriented MicroGrid optimal scheduling. The proposed operational framework was mathematically formulated as a robust optimization problem. However, power flow equations were not included in the model, and robustness was treated in a conservative fashion; assuming maximum expected demand and minimum expected renewable generation at all times.
The study in [5], proposed a two-stage stochastic linear optimization model. In this model, unintentional islanding, load, pool price, and EV schedule were considered as random variables. Although power flow equations were included in the model, the formulation used did not have an accurate representation of network losses, which can result in a more optimistic outcome regarding the cost of operation. Additionally, the knowledge of the probability distributions of random variables is required in stochastic optimization. The latter can be a great advantage for a problem with well-known historical data. However, this is a limitation for modelling islanding uncertainty due to an interruption of power supply from the main grid, as this belongs to high-impact/lowprobability events (HILP) for which historical data is rare [6], [9].
The study in [6], presented a two-stage robust optimization model. Load, market price, and the scheduled time of the MG islanding event were considered as uncertain parameters in this model. However, power flow equations were approximated.
The studies in [10], [11], presented resilient operation strategies for an AC/DC MG and a multi-energy MG respectively. The proposed models were mathematically formulated as twostage robust optimization problems. In these studies, power flow equations were not considered at all in the network model, which creates the risk of overloading network lines, i.e., violating voltage or line current limits.
Finally, the study in [12], presented a model to enhance resilience using islanded operation of MGs focusing on the event of inadequate on-site generation to feed all MG loads. The model was mathematically formulated as a nonlinear programming problem. However, the proposed model was not formulated as a convex model, which meant that only a local optimal solution was guaranteed.
To conclude the above: the scientific community has proposed models for optimal scheduling of MGs that either lacked an accurate representation of power flow equations, or did not take into consideration power flow equations in their models. Furthermore, previous studies did not incorporate all prevailing uncertainties (such as pool price and islanding event uncertainty) which are necessary to form the resiliency-oriented scheduling for MGs (for further reading see the analysis in [6] and references therein). Finally, proposed models that accurately consider power flow equations, are non-convex, and therefore can only guarantee a local optimal solution.

C. Contribution and Organization of the Article
This research, proposes a Robust Mixed-Integer Second Order Cone Programming (R-MISOCP) model for the resilience-oriented optimal scheduling of MGs. The developed model captures the benefits of both convexity and robustness.
The contributions of this work are summarized as follows.
1) The Second Order Cone Power Flow (SOC-PF) model proposed in [13] is used to model power flow equations. This formulation is an exact convex approximation of the power flow model. To evaluate the performance of the R-MISOCP model, a detailed comparison is provided with a model that uses a piecewise linear power flow model (PWL-PF) which has been used in relevant MG studies. 2) Robust optimization is used to model uncertain data.
Data uncertainty is considered in market price, demand, renewable generation (PV generation) and islanding duration. The robust approach of [14] is employed. This approach allows the MG operator to control the trade-off between tolerance of uncertainty and operating performance, using a parameter called the budget of uncertainty. In this article, performance is assessed based on both cost of operation and load shedding. In terms of the first contribution, computational experiments show that an AC power flow model that fails to accurately account for power flow equations, can result in a significant underestimation of both the operational costs, and the curtailed demand, and consequently lead to significantly different scheduling decisions. Regarding the second contribution, computational experiments show that, for the MG under study, the MG operator can adjust the budgets of uncertainty, and achieve a sizable reduction in the day-ahead operational costs, compared to a fully robust (conservative) approach, while having a 0% probability of shedding additional loads than expected.
The remainder of this article is organized as follows. Section II, presents the employed robust approach. Section III, presents the proposed R-MISOCP model and the underlying deterministic model. Section IV, shows computational experiments and discussion. Section V, presents the conclusions of this research.

II. MODELLING UNDER DATA UNCERTAINTY
Optimization problems can be formed using data that are not known with certainty. Robust optimization (RO) is an approach to optimization under uncertainty, where the solution is immunized against any realization of the uncertain parameters which belong to a deterministic interval [15]. The motivation and goals of this approach are twofold. First, in RO, data uncertainty is not stochastic and distribution information is not needed. Second, the RO formulation of an important class of optimization problems (such as linear and secondorder cone programming problems [15]) is computationally tractable.
In the day-ahead scheduling problem for MGs, being fully robust, i.e., considering the worst possible case of uncertainty, may be too conservative, in the sense that too much of the performance may be sacrificed in order to be able to tolerate any possible perturbation of the uncertain data. For the resilience-oriented MG day-ahead scheduling considered in this article, performance criteria include both operational cost and load shedding levels. The robust approach of [14] is employed, where the trade-off between tolerance of uncertainty and model performance can be controlled with a parameter , called the budget of uncertainty. This parameter allows the MG operator to control the cost of operation and load shedding levels, while being robust against possible data perturbations with a very high probability.
In this article, the MG day-ahead scheduling problem is formulated as an MISOCP optimization problem with uncertain data. The robust approach of [14] is employed to the problem (1a)-(1e), where data uncertainty only exists in the linear constraints (1b). The robust model that is proposed in Section III is formulated accordingly.
Consider the following deterministic MISOCP problem: Assume that the coefficients λ ij are uncertain, and each entry is a bounded and symmetric random variable Employing the robust approach of [14], the robust counterpart of problem (1a)-(1e) becomes as follows.
, and z i , p ij , y j decision variables that result from the robust formulation of [14].

A. Description
This study presents a model for the optimal scheduling problem of MGs that are islanded for an extended and uncertain period of time. The proposed model is mathematically formulated as a robust mixed-integer second-order cone problem. Uncertain parameters are modelled using the robust optimization approach proposed in [14], as presented in the preceding section.
The model has binary and continuous decision variables. Binary decision variables represent: unit commitment decisions, charging/discharging state of the energy storage system (ESS), charging/discharging state of the electric vehicle (EV) parking lot, and the state of connection between the MG and the main grid (grid-connected/islanded). Continuous decision variables represent: scheduling of MG assets (DGs, charged/discharged power of ESS and EV parking lot), imported power from the main grid, bus voltages, power flows, line losses, ESS state-of-charge, loads (fixed and curtailed), and auxiliary decision variables as a result of the robust formulation.
The objective of the R-MISOCP model is to minimize the MG cost of operation. The cost of operation includes: DG generation cost (using DG cost functions), start-up/shutdown costs, the cost to buy power from the main grid, and the cost for load shedding when (during the islanded operation) there is insufficient generation to feed the demand. Constraints represent: the SOC-PF model, branch current limits, unit commitment decisions, ramp-up/down limits, the ESS model, the EV parking lot model, and upper/lower limits of variables (namely of fixed loads, curtailable loads, bus voltage, DGs and grid power). The EV parking lot is modelled as an aggregated EV, using the set of linear constraints (9a) -(9h) [5], [16], where constraints (9a), (9b) represent the arrival and departure SOC respectively. The EVs follow the CC to CV (Constant Current to Constant Voltage) charging method for Li-ion batteries, presented in the Appendix of [16]. Data for the parameters related to the EV parking lot are presented in Table VIII-Appendix A of this manuscript.
The following subsections present the mathematical formulation of the underlying deterministic model, and the proposed robust model, respectively.

B. Deterministic Model (D-MISOCP)
The deterministic model is mathematically formulated as an MISOCP problem and is presented in equations (3) subject to Power flow equations [13] x ESS , y ESS binary EV parking lot model [5], [16] Fixed and curtailed demand limits [13] (14b)

A. Case Study and Modelling Environment
The test network used is the modified IEEE 33 bus radial distribution network following DER positioning of [5], presented in Fig. 1. Network data is extracted from [17]. DGs, ESSs, EV parking lot data are extracted from [5]. This data is given in detail in Appendix A of this article. Data granularity is 10-minute time intervals over a 24-hour scheduling horizon, i.e., 6×24 = 144 periods. The nominal values of the uncertain parameters are presented in Fig. 2; namely market price, renewable generation, demand, and islanding event. Total 24-hour demand is 206 MWh. Market price, renewable generation (PV), and demand uncertainty are set to ±10% [4]. The islanding event takes place at 5pm-8pm with an hour window of uncertainty, i.e., (5pm±30 minutes)-(8pm±30 minutes). GAMS IDE environment and MOSEK solver are used for optimization problems [18]. Figures and secondary codes are produced in MATLAB R2017a and R2018a. Numerical experiments of the R-MISOCP need less than 30 seconds to run, using a desktop with an Intel Core i5-6600 CPU at 3.30 GHz and 32 GB of RAM.
In the following subsections, computational experiments are performed in order to: a) demonstrate the impacts that accuracy in power flow modelling has on scheduling decisions, and b) study the effects of adjusting the budgets of uncertainty to achieve reductions in operational cost while minimizing the probability of load shedding.

B. Power Flow Model: A Comparative Study
In this subsection, the proposed R-MISOCP model, which uses the second order cone power flow formulation of [13], is compared with an optimal MG scheduling model that uses a piecewise linear power flow formulation. The latter will be referred to as the COMP model.
As this subsection focuses on the power flow model formulation, the COMP model is compared to the R-MISOCP model results for all equal to zero. The COMP model is described by the same equations as the R-MISOCP and only differs in the power flow model formulation, i.e., the COMP model does not include equations (4a)-(4h).
To preserve the comparison within day-ahead scheduling for MGs that are islanded for an extended time period, the power flow model formulation of [5] is chosen for the COMP model. The power flow model of [5] is mathematically formulated as a piecewise linear model according to [19], and the linearization follows the algorithm proposed in [20]. The power flow equations for the COMP model are formulated following these studies (i.e., [5], [19], [20]). The assumption of [5], that the voltage angles between adjacent buses (θ i − θ j ) range within ±10 • , is preserved. The power flow formulation used for the COMP model is presented in Appendix B of this article, in equations (24a)-(24g). The remainder of this section, focuses on the impact that the power flow formulation has on the schedules and the operational costs produced by the R-MISOCP and the COMP model.
Simulation results show that the COMP model underestimates the value of network losses, compared to the R-MISOCP model. In particular, for a demand of 206 MWh, the R-MISOCP model calculates that the network losses are equal to 8.1 MWh, and the COMP model calculates that the network losses are equal to 0.0046≈0 MWh. Therefore, the R-MISOCP produces a generation mix that is 8.1 MWh higher than the COMP model (or 3.9% of the total demand). The different  calculation in network losses has also resulted in the two models calculating different schedules for the generation units and therefore different operational costs. More specifically, the R-MISOCP operational cost is £12 925, whereas the COMP operational cost is £11 443; which means, that the R-MISOCP is 11.47% more expensive than the COMP model, for the MG under study. A detailed comparison between the two models in terms of their schedules and operational costs follows below.
In Fig. 3, the main grid schedules for the R-MISOCP and COMP models are shown. During times 00:00am-06:00am, when market price is low, both models draw high levels of power from the main grid. However, a visible difference is observed between the two models (particularly during 00:00am-06:00am), as the R-MISOCP schedules a lower amount of power to be drawn from the main grid than the COMP model.
During the same period, the R-MISOCP model also prioritizes dispatching of the DGs as, although they are more expensive, they are electrically closer to the load at the given time window (Fig. 4). This is a result of the R-MISOCP model having a more accurate calculation of network losses. The demand during 00:00am-06:00am, at buses 8, 13, 16 and 25 is: 8.40 MWh, 2.52 MWh, 2.52 MWh, and 17.63 MWh respectively. Amongst all DGs, the schedule for DG at bus 8 differs significantly between the two models; this is attributed to the fact that bus 8 has a high demand, and it is located before the DGs at buses 13 and 16 (see Fig. 1). During the rest of the day, DG schedules remain relatively similar between the two models, as: the DG cost is lower or very close to the market price cost, DGs have reached their upper limits (Fig. 4 - Table VI), and demand is at its highest levels (Fig. 2). Fig. 5 presents the ESS schedules for the R-MISOCP and COMP model. The ESSs are at buses 19 and 26. Bus 19 and bus 26, are both located close to the main grid bus (Fig. 1); no other generator is located between the ESSs and the main grid. ESS schedules also vary due to the calculated losses especially when discharging. In the COMP model, the dominant factor for dispatching the ESSs is market price. However, in the R-MISOCP model, the ESS schedules are affected by the occurrence of the islanding event as well (especially when discharging). This difference is noticeable between ≈3:00pm and 8:00pm (i.e., before and during the islanding event). Fig. 6 presents the EV parking lot schedule for the R-MISOCP (continuous line) and the COMP (dotted line) model. The EV parking lot is located at bus 25, and EVs arrive at 8:30am and depart at 5:30pm (shown with two thick vertical lines at Fig. 6). Simulation results show that both models schedule the EVs according to the levels of the PV generation at bus 25 (PV generation is shown with a yellow line on the right axis of Fig. 6). More specifically, both models discharge the EVs when the PV generation is very low: which is shown after 3:00pm until 5:30pm. The R-MISOCP model mainly charges the EVs when the PV generation takes place (shown from around 12:00pm until 3:00pm with a blue line in the negative values of Fig. 6). However, the COMP model does not schedule EV charging between 8:30am and 5:30pm.
The operational cost (for both models) is the summation of: the main grid cost, the dispachable generators' cost, the load shedding cost, and the unit commitment cost. This means that the operational cost is affected, and also affects the schedules produced by the generation units. These costs are shown in Figs. 7-9 for both the R-MISOCP and the COMP model. In these figures, the left axis (blue color) presents the cumulative costs, and the right axis (red color) presents the costs per time-step.
More specifically, the fact that the COMP model schedules more power from the main grid (Fig. 3) is reflected on the cumulative main grid cost, shown in Fig. 7. Furthermore,  the R-MISOCP model uses the dispatchable generators more than the COMP model, especially during 00:00am-06:00am (Fig. 4), which is also shown in the cumulative cost of DGs in Fig. 8.
The difference in the calculation of network losses also resulted in a lower level of load shedding by the COMP model, compared to the R-MISOCP model (Fig. 9). In particular, for a demand of 33.89 MWh during the time of the islanding (5:00pm-8:00pm), the R-MISOCP sheds 1.89 MWh and the COMP model sheds 0.2 MWh. During 5:00pm-8:00pm, the R-MISOCP calculates that network losses are equal to 1.55 MWh, whereas the COMP model calculates that losses are equal to 0.002 MWh. For a load shedding cost of c Shed i,t =£600/MWh, the cumulative cost due to load shedding,    for both models, is shown in Fig. 9. Finally, the unit commitment cost is the same for both models, and equal to £280.1. This is expected as, for both cases, DGs start-up at 00:00am and operate continuously throughout the day (Fig. 4).
To conclude, this subsection presents a comparative study between two models that use different power flow formulations; namely, the R-MISOCP and the COMP model. Computational experiments show that the incorporation of network losses in the power flow model can impact both the operational cost and calculation of generation schedules. For the MG under study in particular, this has resulted in the following main differences between the two models. First, in terms of the operational costs, the R-MISOCP is 12% more expensive than the COMP model, since accounting for network losses in the R-MISOCP results in increased generation and increased load shedding. Second, in terms of the day-ahead schedules, the COMP model schedules, are mainly driven by the cost of generation, whereas the R-MISOCP schedule calculations, as well as the cost of generation, are also sensitive to the electrical location of the generation units.

C. Uncertainty Modelling: R-MISOCP Results
The aim of the resilience-oriented R-MISOCP model is to minimize operational costs. Operational costs include DG generation cost, unit commitment costs, main grid import costs, and the cost for load shedding. Load shedding takes place during the islanded operation if there is insufficient on-site generation to supply the demand. The four sets of uncertain data with their respective budgets of uncertainty are the following: market price with M ∈ [0, 144], demand with For any value of the uncertain parameters within the boundaries defined by these budgets of uncertainty, the MG operator is guaranteed that operational costs and load shedding will not exceed the values calculated by the R-MISOCP model. For example, if D t = 0, then no uncertainty in the values of the demand is considered, and there is no guarantee that the operational costs and load shedding will not exceed the values calculated by the R-MISOCP model. If D t = 1, then it is guaranteed that for any value of the demand between D P it and D P it + 10% D P it ∀i ∈ B , t ∈ T , the operational costs and load shedding will not exceed the values calculated by the R-MISOCP model.
The ranges of the budgets of uncertainty for market price, demand, and renewable generation, depend on the number of uncertain parameters per constraint, as shown in [14] or Section II of this article. For example, the budget of uncertainty for market price takes values in D t ∈ [0, 144], since there are up to 144 uncertain parameters of the market price in each constraint (16) (as this study uses 10-minute data, that result in 144 time steps). The range of the budget of uncertainty for the islanding event depends on the uncertainty in the duration of the islanding event, and represents the number of uncertain time periods. For example, if I = 6, the islanding event duration is increased by six 10-minute time periods, i.e., the islanding event occurs between 4:30pm and 8:30pm, instead of 5:00pm-8:00pm.
However, considering the upper limits of the budgets of uncertainty can lead to conservative solutions. Therefore, the simulations below focus on how to adjust the budgets of uncertainty in order to reach a desirable trade-off between: the level of uncertainty considered, and the day-ahead operational cost and load shedding levels.
In this subsection, the appropriate levels of the parameters are evaluated according to the probability of underestimating operational cost (PoU) and the probability of load shedding (PLS). PoU is the probability that the actual cost of operation, when considering data perturbations within the full range of uncertainty, will exceed the day-ahead operational cost calculated by the R-MISOCP model. Respectively, PLS is the probability that the actual load shedding will exceed the load shedding calculated by the R-MISOCP model, again when considering data perturbations in the full range of uncertainty. In this article, loads are shed at a cost during the islanding period. However, the MG operator can choose the budgets of uncertainty appropriately in order to ensure that this will be limited while preserving an overall low operational cost.
First, the sensitivity of the optimal day-ahead operational cost and load shedding against the four uncertain sets of data are tested. Optimal day-ahead operational cost and load shedding range from: £12 925 with 1.89 MWh, which represents <1% of total 24-hour demand, when accounting for no uncertainties (all = 0); up to £16 551 with 5.89 MWh, which represents <3% of total 24-hour demand, when accounting for the full range of uncertainty (fully robust case). Therefore, the MG operator is presented with a significant range of options regarding trade-offs between the tolerance of uncertainty, and the day-ahead operational cost and load shedding levels.
According to Table I, two of the parameters that profoundly impact the day-ahead cost of operation and the load shedding are: the D t for demand uncertainty, and I for islanding duration uncertainty. As M and RG t do not largely affect the performance for this case, it is chosen to tolerate for these the maximum uncertainty, i.e., the maximum budgets of uncertainty are assigned for these parameters.
The levels of PoU and PLS are calculated for 21 combinations of D t − I (Tables II, III). The PoU and PLS are calculated by running 10 000 Monte Carlo simulations for each combination. Each iteration of the Monte Carlo simulations runs a power flow model where the inputs are: the R-MISOCP schedules (for the DGs, ESSs, and EV parking lot), and the random yields of demand spanning within its minimum and maximum limits (i.e., D P it ±10% D P it ∀i ∈ B , t ∈ T ), for I = {0, 3 or 6}. PoU is calculated as the fraction of the  number of times that the actual operational cost (produced by the Monte Carlo iterations) exceeds the R-MISOCP day-ahead operational cost, over the total number of iterations. Similarly, PLS is calculated as the fraction of the number of times that the load shedding (produced by the Monte Carlo iterations) exceeds the R-MISOCP load shedding, over the total number of iterations. Power flow simulations are run using the software package of MATPOWER [22]. For the 21 combinations of D t − I , the R-MISOCP day-ahead operational cost and load shedding values are presented in Tables IV and V  respectively, and the PoU and PLS results in Tables II and III respectively.
In order to account for data perturbations in all the range of uncertainty, the MG operator would operate the MG in a fully robust case for a cost of £16 551 (Table IV), with a PoU = 0% probability of exceeding this cost during the actual operation (Table II), and a PLS = 0% probability of shedding more loads than 5.89 MWh (Tables III and V (Table V) with a probability which is also 0% (Table III), for a significantly lower operational cost of £13 310, with an also 0% probability of exceeding this cost during the actual operation (Table II).

V. CONCLUSION
This article presents a Robust Mixed-Integer Second Order Cone Programming model for the day-ahead scheduling of MicroGrids that are islanded for an extended and uncertain period of time due to a power interruption from the main grid. The R-MISOCP model holds the benefits of both convexity and robustness.
In terms of the power flow formulation used in this work, it is shown that incorporation of a non-accurate AC power flow model (that does not account for network losses), can result in a sizable underestimation of the day-ahead operational cost; as accounting for network losses in the power flow model, results in higher generation and increased load shedding. For the MG under study, comparing the proposed model with a model that uses a piecewise linear power flow formulation, this underestimation is found to be 11.47%. Furthermore, it is shown that when network losses are not taken into account, the scheduling decisions are mainly driven by the cost of generation units; whereas, when accounting for network losses, the scheduling decisions are sensitive to both the cost of generation units and their electrical location within the network. In terms of the uncertainty modelling, it is shown with the R-MISOCP model that the MG operator can achieve a significant trade-off between tolerance of data perturbations of the uncertain data, and performance. In particular, computational experiments show that by adjusting the budgets of uncertainty, the MG operator can achieve a 19.58% reduction in the day-ahead operational cost, compared to a fully robust schedule. Moreover, the R-MISOCP model can guarantee 54.33% lower load shedding compared to a fully robust schedule, while preserving a 0% probability of additional load shedding during the actual operation.  [5] Two general points to discuss are in order. Firstly, the proposed approach is designed for MGs. However, it can also be used for radial distribution networks with distributed energy resources and Active Distribution Networks. Furthermore, the proposed method is applicable to business-as-usual case studies, with no islanding event, by setting I t = 0 ∀ t. Secondly, the proposed R-MISOCP model can be reformulated to a mixedinteger linear programming problem using the -polyhedral approximation of [23] with a very high accuracy. This ability can be found very useful, particularly for studies where, mathematically, a linear model formulation is preferable.

APPENDIX A DATA
This Appendix presents the data introduced in Section IV-A. The DG, ESS and EV parking lot technical constraints are shown in Tables VI, VII, VIII respectively. Branch data are shown in Table IX.

APPENDIX B PIECEWISE LINEAR POWER FLOW MODEL (PWL-PF)
This section presents the PWL-PF model formulation of the COMP model, used in Section IV-B. For completeness, the non-convex formulation of the piecewise linear power flow model is first presented, and subsequently the piecewise linear formulation (PWL-PF).
The non-convex formulation of the power flow model formulation used in [5] is shown in equations (23a)-(23b) below.
The term cos (θ it − θ jt ) of equations (23a)-(23b) is approximated by the linear segments shown in (24c), using the algorithm proposed in [20] for the calculation of the terms h n and d n .
where n is the number of linear inequalities used to approximate the cos (θ it − θ jt ). The piecewise linearization of cos (θ it − θ jt ) for |θ it − θ jt | ≤ 10 • (following the study in [5]), is shown in Fig. 10 for seven linear segments. Along with equations (24a)-(24c), the COMP model power flow formulation is also comprised of the following equations: where, (24d) and (24e) are the real and reactive power flow balance respectively, and (24f)-(24g) represent the feeder loss, which is comprised of the summation of the active power injections [5].
Ilias Sarantakos received the Dipl.Ing. degree from