# Centralized optimal management of a smart distribution system considering the importance of load reduction based on prioritizing smart home appliances

, Volume 16 (19) – Oct 1, 2022
20 pages

/lp/wiley/centralized-optimal-management-of-a-smart-distribution-system-jQZPnvlmK2
Publisher
Wiley
© 2022 The Institution of Engineering and Technology.
eISSN
1751-8695
DOI
10.1049/gtd2.12570
Publisher site
See Article on Publisher Site

### Abstract

INTRODUCTIONDG resourcesOne of the prominent advantages of smart grids is the optimal management of resources during normal distribution system operation. Smart grids are developed using innovative technologies such as the internet of things (IoT), monitoring, automated control, two‐way communication, and data management. The smart grid's control and communication equipment are designed to significantly impact the distribution system's economic operation at various hours. With these plans, the distribution system can function as an island at various times, sell electricity to the upstream network, or buy electricity from the upstream network with 24‐h planning. This type of operation necessitates adequate distribution system resources. As a result, dispatchable DG resources such as microturbines (MTs) and diesel generators are required. The escalation of climate change‐related problems, an increase in greenhouse gas emissions, and lower emissions from gas‐fired power plants compared to other fuels have contributed to an increase in natural gas‐fired power plants in place of various fossil fuels. Additionally, a smart distribution generation (DG) uses renewable energy sources such as wind and photovoltaic (PV) to mitigate air pollution and greenhouse gas emissions. Non‐dispatchable renewable resources, on the other hand, are used to reduce the impact of non‐dispatchable resources.The essential objectives of optimal distribution system management include minimizing the system's operating costs during normal operation by supplying the maximum load possible to customers and minimizing load reduction. Natural gas power plants emit fewer greenhouse gases than other fossil fuel power plants. Natural gas produces approximately twice the amount of heat as coal [1]. To this end, between 1990 and 2010, the penetration rate of natural gas power plants in Europe increased from 8% to 23% [2]. In recent years, many studies have been conducted on the use of MTs in the distribution system. In [3–8, 18], MTs are used to operate power systems. None of these studies provide a complete model for the unit commitment problem that considers all the limitations of MTs. In [7], a two‐layer algorithm for optimal distribution network operation is presented considering microgrid (MG) and direct load control.In [19–25], an optimal load management strategy using resources such as MTs, storage, and renewable resources is presented.In [17], the DR scheme is employed for optimal load management. While in the DR scheme, only the effects of shiftable loads are examined. In [19, 22, 24–26], intelligent algorithms are used to solve the problem. Due to the lack of guaranteed optimal global solutions, these algorithms are not appropriate. In [20, 21], a linear model is used to solve the presented model, ensuring an optimal solution due to the proposed model's use of the CPLEX solver. However, because the proposed problem lacks load flow constraints, this model cannot be used in a distribution system. In [23], the unit commitment problem is solved using commercial solvers and a non‐linear model without considering the load flow equations. Due to the absence of load flow equations in this model, the degree of non‐linearity is low. As a result, it can resolve the issue quickly. Due to the uncertainty and the addition of load flow equations in this paper, the execution time significantly increases, rendering it unsuitable for use in the distribution system.Demand responseOne of the benefits of the smart grid is utilizing the DR scheme in an emergency or during periods of peak load. The DR scheme is used for economically optimal energy management. In [9–11], lowering the total cost, reducing peak load, and increasing the daily profits of electricity companies are performed by the DR. In [12], unit commitment and economic dispatch problems are solved during the first step to determine local marginal prices. In the second step, the DR program is implemented to minimize the total cost of providing energy to the customers. The total cost includes the cost of electricity purchased from the market and DG units, the cost of incentive payments to customers, and compensation for power outages. In [13], the DR real‐time price‐based management program connected to smart meters is shown. This program runs automatically to determine the equipment's optimal operation. Based on DR priorities, home appliances are classified into interruptible/non‐interruptible and shiftable/non‐shiftable. In [14], smart residential loads are primarily classified into different categories according to DR programs. Additionally, a comprehensive scheme is modelled using the distribution of residential loads and DG resources. The proposed model reduced the cost of electricity consumption, peak load, and peak to valley difference in residential load specifications without causing inconvenience to users. In [15], a distributed algorithm is presented for collecting data from a large number of houses with a mixture of discrete and continuous energy levels. One of the method's distinguishing characteristics is that it decomposes DR non‐convergence and thus enables connection between energy storage (ES) devices, appliances, and distributed energy resources. In none of the studies mentioned above are residential instruments adequately modelled. In [14, 16], smart home appliances are divided into interruptible, shiftable, and adjustable appliances. However, because transmission line losses and load flow constraints are not considered, the proposed model in these papers is inadequate. Load reduction is optimal when it allows the buses’ voltages to return to their allowable range. Nonetheless, without considering these constraints, the load reduction is suboptimal.Main contributionThis study proposes a centralized optimal distribution system management scheme that considers the critical nature of load reduction and allows the system operator to access the loads inside smart homes. In the proposed approach, under normal operating conditions, the network operator is able to reduce loads based on a predetermined prioritization (so that both operating costs and the load reduction in the system are reduced). This method has a significant impact on the distribution system's economic operation. Furthermore, intelligent homes’ controllable loads are classified into three categories based on their importance: shiftable loads with priority 1, interruptible loads with priority 2, and adjustable loads with priority 3.Adjustable loads, such as light bulbs, can reduce consumption by dimming their light in an emergency. If a shiftable load, such as a washing machine, is not operating, its consumption can be transferred to another clock. When these appliances are in operation, they must continue to operate until the work cycle is complete. Additionally, interruptible loads can be reduced at any time. The research employs a variety of dispatchable and non‐dispatchable DG resources.Furthermore, battery storage and load reduction techniques are used to mitigate the effects of renewable energy sources. Increases or decreases in the power of MTs have a maximum limit, and these resources cannot rapidly reach their maximum capacity. Moreover, these resources cannot be activated immediately after being turned off or vice versa.As a result, this article also considers these constraints when discussing unit commitment. The proposed model's objectives fully express the performance of MTs. Furthermore, the objective function considers the planning for the purchase or sale of electricity to the upstream network and the cost of loading for various smart devices inside residential homes. The reason for including pollution costs as one of the objectives of the proposed model is that the presence of MTs in urban areas can result in increased pollution in cities.The limits of active and reactive load flow, the apparent power limit of MTs, and the terms in the objective function relating to the generation and pollution costs of MTs are non‐linear terms in the proposed model. The non‐linear model is converted to a linear model to compare and select the optimal model. These two models are compared, and the model with the best cost, execution time, and load reduction rate is selected. The proposed model can operate in island mode for the majority of the hours due to the DG and storage resources available in the network. Additionally, the system may occasionally sell electricity to the upstream grid. Furthermore, the proposed model purchases electricity from the upstream grid during peak hours. In all of these cases, the system may choose to reduce the load on the smart home based on appliance priority, as load reduction may be more cost‐effective than selling electricity to customers.One of the most significant innovations of this study is the simultaneous review of all the following items, which constitute the article's main characteristics.üDetermination of the optimal unit commitment of DG and storage resources in a smart distribution systemüConsideration of the limitations of MTs’ up/down ramp rates and the MTs’ minimum up/downtime limitsüConsideration of the generation, pollution costs, start‐up, and shutdown costs of MTs in both linear and non‐linear modelsüClassification of the types of loads in smart homes according to their priorityüPresentation of two linear and non‐linear models for the modelling of the unit commitment problemüSensitivity analysis to validate the proposed methodüThe ability of the system to work in the island‐operating and connected grid modesThe second section of this study includes mathematical modelling. The third section contains the numerical results. Finally, the article's conclusions and recommendations for future research are presented.PROBLEM MODELLINGA smart distribution system consists of various DG resources, storage devices, controllable, and uncontrollable loads. Load types in this smart residential community include the following:Adjustable loads such as lightings, refrigerators, and freezers.Interruptible loads such as hairdryers, water coolers, and vacuum cleaners.Shiftable loads such as washing machines, washing machines, and microwaves.Uncontrollable loads such as water heaters, monitors, and laptops.Shiftable, interruptible, and adjustable loads are classified as load reduction with priority 1 (LSP 1), load reduction with priority 2 (LSP 2), and load reduction with priority 3 (LSP 3), respectively. For the system operator, load reduction with a high priority incurs a higher cost than load reduction with a low priority. Additionally, because uncontrollable loads are not eligible for the DR scheme, the network operator must compensate customers more for load curtailment of uncontrollable loads than for smart controllable loads. The load aggregator collects and prioritizes the energy generated by all smart homes. The sum of all adjustable, interruptible, shiftable, and uncontrollable loads is calculated in real time. The load aggregator collects the loads and transmits them to the system controller in real time. The controller receives all system data in real time, including distribution station data, distributable and non‐distributable DG resources, and storage devices.The controller processes the data by utilizing this information and executing the proposed method. The demand‐side controller's output includes the number of homes with load reductions based on the priorities of each bus. Thus, after deciding on the number of load reduction homes, the IoT and smart meters apply these decisions to the types of loads in residential homes. Moreover, the controller controls the amount of charge and discharge of batteries, the output power of MTs, the amount of energy sold to and purchased from the upstream network. This model satisfies constraints on unit commitment and system operation. As a result, the proposed centralized controller manages the demand side. The proposed model's structure is depicted in Figure 1.1FIGUREStructure of smart distribution systemObjective functionAssumptions for optimizing the problem are as follows:The system under study is considered a balanced system.The penetration coefficient of MTs is considered to be one. In other words, MTs cannot operate in overload conditions.Table 1 in Appendix 1 summarizes and generalizes the objective functions of previous studies in the literature reviewed in this subject. The discrepancy between different studies on the objective functions of this issue is depicted in Table 1. According to Table 1, there is no comprehensive objective function that can be used to resolve this problem.1TABLESummary of previous studies on the objective function of the unit commitment problemPrevious studiesProfit from selling electricity to customersProfit from the selling electricity to the upstream networkCosts of power lossesMaintenance costsOperation costs of batteries depreciationCost of power purchased from the upstream networkLoad reduction / load curtailment costsCosts of start‐up and shut down MTsCost of pollutionCost of production[3]––––√√–√–√[4]–––√√√––––[5, 6]––√–––√––√[7]√√–––√√––√[8]√√–––√√–√√[17]√√–√√√√–√√[18]–––––––––√[20]–√––√√√√–√[21]–√–––√––√√[22]–––––√–––√[23]––––√–√––√[24]–––√√√–√√√Proposed method√√–––√√√√√The cost of pollution associated with MTs should be factored into the objective function. Because MTs operate on the consumption side (within the city), one of the world's major problems is urban pollution, which has received scant attention in most articles on the subject. Thus, the cost of load reduction, the cost of load interruption, the cost of purchasing electricity from the upstream network, the profit from selling electricity to customers, and the upstream network, start‐up, and shutdown costs of MTs, as well as the production and pollution costs of MTs, are all considered in the objective function of this paper.Additionally, the cost of power losses in the proposed objective function is low due to the proximity of resources to loads and the island state's low power losses. Due to the low maintenance cost of uncontrollable DG resources, this cost is not factored into the paper's proposed objective function.The unit commitment problem is addressed in this article using two linear and non‐linear models. The linear and non‐linear models are shown in objective functions (1) and (2), respectively. The objective function (1) contains the following terms: the first term represents the production cost of MTs [7], the second term represents the pollution cost of MTs [7], the third term represents the profit from the sale of electricity to customers [7], and the fourth and fifth terms represent the start‐up and shutdown costs of MTs [3]. The costs of LSP 1, LSP 2, LSP 3, and load curtailment are shown in terms (6) to (9), respectively. Furthermore, the 10th term denotes the cost of purchasing power from the network, and the final term denotes the profit earned on power sold to the upstream network [7]. Due to the presence of the first term (MTs production cost [fuel cost]), the objective function (1) becomes a non‐linear model. In some studies, the cost of manufacturing MTs is modelled as a quadratic function. In [8], the computational burden is reduced through a linear model of the production cost function. As a result, a linear term is used in the objective function (2) [8]. Thus, in Equation (2), the objective function (1) is transformed into a linear model. Additionally, some papers consider the pollution cost of MT using a quadratic function, complicating the solution. Therefore, a linear term is used in this paper to account for the pollution cost of MTs in the objective function [8].1min∑t(∑iaiG+biGpi,tG+ciG(pi,tG)2+∑icemiσipi,tG−∑icDpi,tD+∑iSTCi,t+∑iSDCi,t+∑icLS1PLS1i,t+∑icLS2PLS2i,t+∑icLS3PLS3i,t+∑icLS4PLS4i,t)+Ctbλt−Ctsμt$$\def\eqcellsep{&}\begin{array}{l} \min \displaystyle\mathop \sum \limits_t (\mathop \sum \limits_i \left( {{a_i}^G + {b_i}^Gp_{i,t}^G + {c_i}^G{{(p_{i,t}^G)}^2}} \right) + \mathop \sum \limits_i {c^{emi}}{\sigma _i}p_{i,t}^G - \mathop \sum \limits_i {c^D}p_{i,t}^D\;\\[9pt] + \displaystyle\mathop \sum \limits_i STC\left( {i,t} \right) + \mathop \sum \limits_i SDC\left( {i,t} \right) + \mathop \sum \limits_i {c^{LS1}}PLS1\left( {i,t} \right) + \mathop \sum \limits_i {c^{LS2}}PLS2\left( {i,t} \right)\\[9pt] + \displaystyle\mathop \sum \limits_i {c^{LS3}}PLS3\left( {i,t} \right) + \mathop \sum \limits_i {c^{LS4}}PLS4\left( {i,t} \right)) + C_t^b{\lambda _t} - C_t^s{\mu _t} \\[-24pt] \end{array}$$2min∑t(∑icGpi,tG+∑icemiσipi,tG−∑icDpi,tD+∑iSTCi,t+∑iSDCi,t+∑icLS1PLS1i,t+∑icLS2PLS2i,t+∑icLS3PLS3i,t+∑icLS4PLS4i,t)+Ctbλt−Ctsμt$$\def\eqcellsep{&}\begin{array}{l} \min \displaystyle\mathop \sum \limits_t (\mathop \sum \limits_i {c^G}p_{i,t}^G + \mathop \sum \limits_i {c^{emi}}{\sigma _i}p_{i,t}^G - \mathop \sum \limits_i {c^D}p_{i,t}^D\; + \mathop \sum \limits_i STC\left( {i,t} \right) \\[9pt] + \displaystyle\mathop \sum \limits_i SDC\left( {i,t} \right) + \mathop \sum \limits_i {c^{LS1}}PLS1\left( {i,t} \right) \\[9pt] +\,\displaystyle\mathop \sum \limits_i {c^{LS2}}PLS2\left( {i,t} \right) + \mathop \sum \limits_i {c^{LS3}}PLS3\left( {i,t} \right) \\[9pt] + \displaystyle\mathop \sum \limits_i {c^{LS4}}PLS4\left( {i,t} \right)) + C_t^b{\lambda _t} - C_t^s{\mu _t}\\[-24pt] \end{array}$$Constraints of the proposed modellingThe following considers the voltage violation constraint in the distribution system. ∆V denotes the allowable voltage deviation from the nominal value, of which 5% is considered in this paper [7].31−ΔV≤Vi,t≤1+ΔV$$1 - \Delta V \le {V_{{\rm{i}},{\rm{t}}}} \le 1 + \Delta V{\rm{\;}}$$Equations (4) and (5) express the active and reactive AC load flow equations, respectively. These two relationships are highly non‐linear, resulting in the solver reaching an infeasible solution. As a result, these two relationships will become linear in the continuation of the formulation [7].4Pi,t=Vi,t∑jVj,tGi,jcosθij,t+Bi,jsinθij,t$${P_{i,t}} = {V_{i,t}}\;\mathop \sum \limits_j {V_{j,t}}\;\left( {{G_{i,j}}cos{\theta _{ij,t}} + {B_{i,j}}sin{\theta _{ij,t}}} \right)$$5Qi,t=Vi,t∑jVj,tGi,jsinθij,t−Bi,jcosθij,t$${Q_{i,t}} = {V_{i,t}}\;\mathop \sum \limits_j {V_{j,t}}\;\left( {{G_{i,j}}sin{\theta _{ij,t}} - {B_{i,j}}cos{\theta _{ij,t}}} \right)$$where Pi,t${P_{i,t}}$ and Qi,t${Q_{i,t}}$ denote the active and reactive load flow at load point i, respectively. Equations (4) and (5) convert the model into a non‐linear program. Linearization of load flow equations is performed based on Taylor's series and the following assumptions:The voltage values for each bus are always close to one.The angle difference along a line is minimal and sinθk=θk$sin\;{\theta _k} = {\theta _k}{\rm{\;}}$and cosθk=1$cos\;{\theta _k} = {\rm{\;}}1{\rm{\;}}$can be considered.Based on the second assumption, the size of the bus voltage can be considered as follows:6Vi,t=1+ΔVi,t$${V_{i,t}} = {\rm{\;}}1 + \Delta {V_{i,t}}$$where ΔVmin≤ΔVi≤ΔVmax$\Delta {V^{min}} \le \Delta {V_i} \le \Delta {V^{max}}{\rm{\;}}$is an extremely low value, according to the above assumptions, we have7Pi,t=∑j1+ΔVi,t+ΔVj,tGi,j+Bi,j×θij,t$${P_{i,t}} = \mathop \sum \limits_j \left( {1 + \Delta {V_{i,t}} + \Delta {V_{j,t}}} \right)\;\left( {{G_{i,j}} + {B_{i,j}}\; \times \;{\theta _{ij,t}}} \right)\;$$8Qi,t=∑j1+ΔVi,t+ΔVj,tGi,jθij,t−Bi,j$${Q_{i,t}} = \mathop \sum \limits_j \left( {1 + \Delta {V_{i,t}} + \Delta {V_{j,t}}} \right)\;\left( {{G_{i,j}}{\theta _{ij,t}} - {B_{i,j}}} \right)\;\;$$As a result, ΔVi,t$\Delta {V_{i,t}}$ , ΔVj,t$\Delta {V_{j,t}}$, and θij,t${\theta _{ij,t}}{\rm{\;}}$are minimal. The product ΔVi,t.θij,t$\Delta {V_{i,t}}.{\theta _{ij,t}}{\rm{\;}}$can be considered a second‐order expression. Thus, these minor expressions could be omitted. The following equations represent the final linearized load flow equations:9Pi,t=∑j1+ΔVi,t+ΔVj,tGi,j+Bi,j.θij,t$${P_{i,t}} = \mathop \sum \limits_j \;\left( {\left( {1 + \Delta {V_{i,t}} + \Delta {V_{j,t}}} \right)\;{G_{i,j}} + {B_{i,j}}\;.\;{\theta _{ij,t}}} \right)\;$$10Qi,t=∑jGi,jθij,t−1+ΔVi,t+ΔVj,tBi,j$${Q_{i,t}} = \mathop \sum \limits_j \;\left( {{G_{i,j}}{\theta _{ij,t}} - \left( {1 + \Delta {V_{i,t}} + \Delta {V_{j,t}}} \right)\;{B_{i,j}}} \right)\;$$Equations (11) and (12) show the balance of active and reactive power in each bus, respectively.11Pi,t=pi,tG+pi,tE−pi,tD+pi,tWT+pi,tPV$${P_{i,t}} = p_{i,t}^G\; + p_{i,t}^E - p_{i,t}^D + p_{i,t}^{WT} + p_{i,t}^{PV}$$12Qi,t=qi,tG−qi,tD+qi,tWT+qi,tPV$${Q_{i,t}} = q_{i,t}^G\; - q_{i,t}^D + q_{i,t}^{WT} + q_{i,t}^{PV}$$The apparent power of each production source should not exceed the allowable value according to Equation (13) [7]13pi,tG2+qi,tG2≤Simax2$${\left( {p_{i,t}^G} \right)^2} + {\left( {q_{i,t}^G} \right)^2} \le S_{i}^{{max}^2}\;$$where Simax$S_i^{max}$ represents the maximum capacity of each production source. Because Equation (13) is a non‐linear equation, it significantly increases the execution time of the problem. Equation (14) is linearized using a polygon‐based linearization method. The radius of the circle is chosen from the polygons as [29]14Si=Simax2π/n/sin2πn$${S_i} = \;S_i^{max}\sqrt {\left( {2\pi /n} \right)/\sin \left( {\frac{{\;2\pi }}{n}} \right)}$$where n and Si${S_i}$ denote the number of polygon edges for linearization and apparent power of each MT, respectively. Therefore, the linearized Equations (15) to (17) replace the non‐linear Equation (13) [29].15−3pi,tG+Si≤qi,tG≤−3pi,tG−Si$$- \sqrt 3 \left( {p_{i,t}^G + {S_i}} \right) \le q_{i,t}^G \le - \sqrt 3 \left( {p_{i,t}^G - {S_i}} \right)$$16−32×Si≤qi,tG≤32×Si$$- \frac{{\sqrt 3 }}{2} \times \;{S_i} \le q_{i,t}^G \le \frac{{\sqrt 3 }}{2} \times \;{S_i}$$173pi,tG−Si≤qi,tG≤3pi,tG+Si$$\sqrt 3 \left( {p_{i,t}^G - {S_i}} \right) \le q_{i,t}^G \le \sqrt 3 \left( {p_{i,t}^G + {S_i}} \right)\;$$Battery storage contributes to load reduction on the distribution system by storing energy during low load times and discharging during high load times. The following relation expresses the charge and discharge limits of a battery [7].18−Pich,maxλi,t≤pi,tE≤Pidch,maxϕi,t$$- P_i^{ch,max}{\lambda _{i,t}} \le p_{i,t}^E \le P_i^{dch,max}{\phi _{i,t}}$$where pi,tE$p_{i,t}^E$denotes the active power of each ES. The battery can be charged or discharged only in the immediate moment, a limitation stated in Equation (19) [7].19λi,t+ϕi,t≤1$${\lambda _{i,t}} + {\phi _{i,t}} \le 1\;$$The following equation expresses the battery's state of charge (SOC) [7].20SOCi,t=SOCi,t−1−TECiϕi,tpi,tEπd−1+λi,tpi,tEπc$$SO{C_{i,t}} = SO{C_{i,t - 1}}\; - \frac{T}{{E{C_i}}}\left( {{\phi _{i,t}}p_{i,t}^E\pi _d^{ - 1} + {\lambda _{i,t}}p_{i,t}^E{\pi _c}} \right)$$The battery's minimum and maximum SOC are expressed as follows [7]:210≤SOCi,t≤SOCimax$$0 \le SO{C_{i,t}} \le SOC_i^{max}$$The power output of the MTs at time t must be within the operating range given in (22). pg¯(i,t)${\overline {pg} _{( {i,t} )}}$, and pg̲(i,t)${\underline {pg} _{( {i,t} )}}{\rm{\;}}$represent the minimum and maximum time‐dependent operating ranges, respectively. These two variables are not necessarily equal to pgi,tmin,$pg_{i,t}^{min},$ and pgi,tmax$pg_{i,t}^{max}$. The upper operating limit pg¯(i,t)${\overline {pg} _{( {i,t} )}}$ is described in (23) and (24). The upper/lower‐level constraints of MT ramps are modelled as follows [30]:22pg̲i,t≤pgi,t≤pg¯i,t$${\underline {pg} _{\left( {i,t} \right)}} \le p{g_{\left( {i,t} \right)}} \le {\overline {pg} _{\left( {i,t} \right)}}\;$$23pg¯i,t≤pgi,tmaxui,t−zi,t+1+SDizi,t+1$${\overline {pg} _{\left( {i,t} \right)}} \le pg_{i,t}^{max}\left[ {{u_{i,t}} - {z_{i,t + 1}}} \right] + S{D_i}{z_{i,t + 1}}$$24pg¯i,t≤pgi,t−1+RUiui,t−1+SUiyi,t$${\overline {pg} _{\left( {i,t} \right)}} \le p{g_{\left( {i,t - 1} \right)}} + R{U_i}{u_{i,t - 1}} + S{U_i}{y_{i,t}}\;$$25pg̲i,t≥pgi,tminui,t$${\underline {pg} _{\left( {i,t} \right)}} \ge pg_{i,t}^{min}{u_{i,t\;}}$$26pg̲i,t≤pgi,t−1+RDiui,t+SUizi,t$${\underline {pg} _{\left( {i,t} \right)}} \le p{g_{\left( {i,t - 1} \right)}} + R{D_i}{u_{i,t}} + S{U_i}{z_{i,t}}$$where RUi$R{U_i}$ and RDi$R{D_i}$ denote ramp‐up and ramp‐down of MTs, respectively. In Equations (27) and (28), the start‐up and shutdown costs of MTs are represented. These costs are used in the objective function and planning of MTs in (1) and (2) [30].27STCi,t=SUiyi,t$$STC\;\left( {i,t} \right) = S{U_i}\;\;{y_{i,t}}$$28SDCi,t=SDizi,t$$SDC\;\left( {i,t} \right) = S{D_i}\;\;{z_{i,t}}$$where SUi$S{U_i}$ and SDi$S{D_i}$ denote ramp limit for start‐up and shutdown of MTs, respectively. The up and down status of unit i at time t is represented by ui,t${u_{i,t}}$. The start‐up and shutdown of MTs are also determined by yi,t${y_{i,t}}$ and zi,t${z_{i,t}}$, respectively. The minimum up/downtime of MTs are described in (29) and (30), respectively. However, these two equation models result in a non‐linear model. The linearized Equations (31) to (33) are used to linearize the minimum downtime constraint (29). The start‐up and shutdown constraint of MTs is limited by the up and down in the final time, according to Equation (31). Equation (32) shows that the MT turns on or off at time t, and neither state can co‐occur. Equation (33) also indicates that ui,t${u_{i,t}}$, yi,t${y_{i,t}}$, and zi,t${z_{i,t}}$ are three binary variables [30].29yi,t−1−UTi×ui,t−1−ui,t≥0$$\left( {{y_{i,t - 1}} - U{T_i}} \right) \times \left( {{u_{i,t - 1}} - {u_{i,t}}} \right) \ge 0$$30yi,t−1−DTi×(ui,t−ui,t−1)≤0$$\left( {{y_{i,t - 1}} - D{T_i}} \right) \times ({u_{i,t}} - {u_{i,t - 1}}) \le 0$$31yi,t−zi,t=ui,t−ui,t−1$${y_{i,t}} - \;{z_{i,t}} = {u_{i,t}}\; - {u_{i,t - 1}}$$32yi,t+zi,t≤1$${y_{i,t}} + {z_{i,t}} \le 1$$33yi,t,zi,t,ui,t∈0,1$${y_{i,t}},{z_{i,t}},\;{u_{i,t}} \in \left\{ {0,1} \right\}$$where UTi$U{T_i}$ and DTi$D{T_i}$ denote minimum uptime and downtime of MTs, respectively. Equations (34) to (38) represent the minimum uptime constraint of each MT [30].34∑t=1ξi(1−ui,t)=0$$\mathop \sum \limits_{t\; = \;1}^{{\xi _i}} (1 - {u_{i,t}})\; = \;0$$35∑t=kk+UTi−1ui,t≥UTiyi,k,∀k=ξi+1…T−UTi+1$$\mathop \sum \limits_{t\; = \;k}^{k + U{T_i} - 1} {u_{i,t}} \ge U{T_i}{y_{i,k}},\;\forall \;k\; = {\xi _i}\; + 1 \ldots T - U{T_i} + 1$$36∑t=kTui,t−yi,t≥0,∀k=T−UTi+2…T$$\mathop \sum \limits_{t\; = \;k}^T {u_{i,t}} - {y_{i,t}} \ge 0,\;\forall \;k\; = \;T - U{T_i} + 2 \ldots T$$37ξi=minT,UTi−Ui0ui,t$${\xi _i} = \;{\rm{min}}\left\{ {T,\left( {\;U{T_i} - U_i^0} \right){u_{i,t}}} \right\}$$38∑t=1ξiui,t=0$$\mathop \sum \limits_{t\; = \;1}^{{\xi _i}} {u_{i,t}} = \;0\;$$Equations (39) to (41) also indicate the minimum downtime of each MT [30].39∑t=kk+DTi−1(1−ui,t)≥DTizi,k,∀k=ξi+1…T−DTi+1$$\mathop \sum \limits_{t\; = \;k}^{k + D{T_i} - 1} (1 - {u_{i,t}}) \ge D{T_i}\;{z_{i,k}},\;\forall \;k\; = {\xi _i}\; + 1 \ldots T - D{T_i} + 1\;$$40∑t=kT1−ui,t−zi,t≥0,∀k=T−DTi+2…T$$\mathop \sum \limits_{t\; = \;k}^T 1 - {u_{i,t}} - {z_{i,t}} \ge 0,\forall \;k\; = \;T - D{T_i} + 2 \ldots T$$41ξi=minT,DTi−Si01−ui,t=0$${\xi _i} = \min \left\{ {T,\left( {D{T_i} - S_i^0} \right)\;\left[ {1 - {u_{i,t = 0}}} \right]} \right\}\;$$As in (42), the load power in each bus (pi,tD$p_{{\rm{i}},{\rm{t}}}^{\rm{D}}$) is composed of the total power of residential appliances prioritized differently. The load aggregator's task is to pool the power of devices with varying priority levels on different buses. Additionally, the total power of the distribution system is transmitted to the DSM controller in each priority. According to Equation (43), the total load shed by priority 1 in each bus is less than the combined power of all the priority 1 equipment in each bus. Equations (43)–(45) also demonstrate the constraints on LSP 1, LSP 2, and LSP 3. Equation (46) denotes the respective load curtailment limits for uncontrolled loads.42pi,tD=Pload1i,tD+Pload2i,tD+Pload3i,tD+Pload4i,tD$$p_{i,t}^D = Pload1_{i,t}^D\; + Pload2_{i,t}^D + Pload3_{i,t}^D + Pload4_{i,t}^D$$430≤PLS1i,t≤Pload1i,tD$$0 \le PLS1\left( {i,t} \right) \le Pload1_{i,t}^D$$440≤PLS2i,t≤Pload2i,tD$$0 \le PLS2\left( {i,t} \right) \le Pload2_{i,t}^D{\rm{\;}}$$450≤PLS3i,t≤Pload3i,tD$$0 \le PLS3\left( {i,t} \right) \le Pload3_{i,t}^D{\rm{\;}}$$460≤PLS4i,t≤Pload4i,tD$$0 \le PLS4\left( {i,t} \right) \le Pload4_{i,t}^D$$where PLS$PLS$ represents load reduction of each load point. In this paper, two linear and nonlinear models are considered. The non‐linear model includes Equations (2)–(5), (11)–(13), (18)–(30), and (42)–(46). While in the linear model, Equations (1), (3), (9)–(12), (15)–(28), and (31)–(46) are used. The linear model presented in this paper is solved using the powerful CPLEX solver in GAMS software, while the nonlinear model is solved using the CONOPT solver.NUMERICAL RESULTSThe proposed method is validated using a simulation on the IEEE 33‐bus system depicted in Figure 2 [7]. Tables 2 and 3 consist of information on the DG renewable and ES resources used in the studied system, respectively [7]. As outlined in Table 4, electrical appliances in smart homes are classified as controllable loads or uncontrollable loads. Controllable loads are also classified according to their importance: shiftable loads with priority 1, interruptible loads with priority 2, and adjustable loads with priority 3. Because each smart home has a 10‐kW installed capacity, the installed capacity for interruptible loads is 3.1 kW, shiftable loads are 3.5 kW, adjustable loads are 2.6 kW, and uncontrollable loads are 0.8 kW [28].2FIGUREStudied distribution system [7]2TABLENon‐dispatchable units’ information [7]TypeBusSize (kW)WT4, 16, 22, 24, 26, 3030, 85, 60, 50, 50, 70PV3, 9, 17, 2860, 60, 40, 503TABLEES data [7]BusSize (kWh)Minimum SOCInitial SOC51500.20.1141500.20.9202000.20.8332000.20.54TABLEClassification of loads installed in a smart home [28]Non‐controllable loadsControllable loadsPriority 1Priority 2Priority 3TypeSize (W)TypeSize (W)TypeSize (W)TypeSize (W)Water heater500Dishwasher1100Hairdryer1500Freezer50Lap top50Washing machine1000Vacuum cleaner1000Refrigerator50Computer case200Electric mixer200Sewing machine100Lighting500Monitor50Microwave1200Water cooler500Air conditioner2000Sum800350031002600In the case study, the total number of homes shed in each priority can be calculated by dividing the total prioritized capacity in each house by the power shed in each priority. Each smart home in the proposed model will have an installed capacity of 10 kW. Additionally, the devices’ power is prioritized the same way. For instance, the following equation yields the number of homes with LSP 1.47NL1i,tD=pls1i,t3.5$$NL1_{i,t}^D = \frac{{pls1\left( {i,t} \right)}}{{3.5}}\;$$In (47), NL1i,tD${\rm{NL}}1_{{\rm{i}},{\rm{t}}}^{\rm{D}}$ represents the total number of houses shed with LSP 1 (shiftable loads), and pls1(i,t)$pls1( {i,t} )$ denotes the amount of LSP 1. The total power shed in LSP 1 per smart home is 3.5 kW. Tables 5 and 6 provide information on MTs and the costs associated with the objective function's parameters. As illustrated in Figures 3 and 4, the wind and PV generation curves are assumed to be coefficients of their capacity. Additionally, Figures 5 and 6 depict the load multiplier and energy prices, respectively.5TABLEInformation of MTs [3, 4, 7]BusaiG${a_i}^G$, $biG${b_i}^G$(kW)ciG${c_i}^G$(kW2)Start‐up/ shutdown cost ($)Minimum uptime/ downtime (h)Ramp‐up/ down (kW)Initial conditionSize (kW)150.0250.0710.0000832250Off800180.0390.0550.0000432250Off650190.0230.0620.0000332100Off100250.010.0750.0000632250Off750290.010.0610.0000332250Off7506TABLEParameters costs [8, 27, 28]ParameterscLS1${c^{LS1}}$ ($/kWh)cLS2${c^{LS2}}$($/kWh)cLS3${c^{LS3}}$ ($/kWh)cLS4${c^{LS4}}$($/kWh)cG${c^G}$ ($/kWh)cD${c^D}$($/kWh)cemi${c^{emi}}$ ($/Kg)σi${{{\sigma}}_i}$(kg/ kWh)3.532.250.10.30.020.0033FIGUREWind power coefficients [7]4FIGURESolar power coefficients [7]5FIGURELoad coefficients [7]6FIGUREEnergy price in 24 h [7]Case studyThree scenarios are presented in the case study as follows:Scenario 1: Problem‐solving using a linear model without considering the ramp‐up/down and minimum up/downtimes of MTs.Scenario 2: Problem‐solving with a linear model and considering the limitations of ramp‐up/down and minimum up/downtimes of MTs.Scenario 3: Problem‐solving with nonlinear model and considering the limitations of ramp‐up/down and minimum up/downtimes of MTs.Scenario 1The ramp‐up/down and minimum up/downtime constraints of MTs are ignored in this scenario. MTs operate at all capacities throughout the study period. The number of houses with the LSP 3 (adjustable appliances) is depicted in Figure 7. The various colours in this figure indicate the number of buses with reduced loads. According to this figure, the system does not shed any load until 19:00. For instance, at 20:00, bus 16 sheds 11 houses equipped with LSP 3 (adjustable appliances).7FIGURENumber of homes with LSP 3 in different buses in Scenario 1. LSP, load reduction with priority.Additionally, the system lacks LSP 1, LSP 2, and load interruption for uncontrollable loads. Therefore, in this scenario, prioritizing the load yields load reduction in loads with priority 3, while loads with the priority 1 and 2 are not shed.The amount of electricity sold to the upstream network at various hours is depicted in Figure 8. Between the hours of 19:00 and 24:00, the distribution system does not sell any power to the upstream network due to the distribution system's high load. Moreover, the highest amount of electricity sold to the upstream network occurs between 4:00 and 9:00 due to the low load. Considering the MTs and other system resources, no power is purchased from the upstream network in this scenario.8FIGUREThe amount of electricity sold to the upstream network at different hours in Scenario 1The batteries’ charge, discharge, and SOC at various hours are shown in Table 7 and Figure 9 for the four batteries installed in buses 5, 14, 20, and 33, respectively. According to Table 7, batteries are charged during low load hours between 6 and 8:00. However, the battery installed in bus 5 at 1:00 is charged to a minimum charge of 15,789 kW. The batteries are discharged between 22:00 and 24:00 (during the high load hours) and 1:00 to 2:00, as specified in Table 7. Figure 10 depicts the voltage profile at 23:00 to validate Equation (3). The voltage is within the 5% voltage drop range at all load points.7TABLECharging and discharging power of batteries in Scenario 1Charging power of ESs (kW)Discharging power of ESs (kW)TimeES5ES14ES20ES33ES5ES14ES20ES331−15.7––––53.261.6542–––––54.746.3–6––−8.42−8.42––––7−57.3−57.3−80−80––––8−69−69−80−80––––22––––6969808023––––393964–24–––––––649FIGURESOC of batteries in Scenario 1. SOC, state of charge.10FIGUREVoltage profile in Scenario 1Figure 11 shows the load reduction of the entire system in this scenario. As per this figure, no load is shed from 1:00 to 19:00. Nonetheless, some homes shed third priority loads between 20:00 and 24:00 (peak hours of network consumption). The proposed model would be capable of meeting these demands during peak hours. However, because load reduction is more cost‐effective than power generation, priority 3 loads are shed in some houses. The maximum load reduction also occurs at 23:00.11FIGUREThe load reduction of the whole system in Scenario 1Scenario 2In this scenario, modelling is extended to include ramp‐up/down constraints and the minimum up/downtime of MTs. The power outputs of the MTs installed in buses 15, 18, 19, 25, and 29 are depicted in Figure 12. The initial state of all MTs is off at 1:00. Because the minimum downtime required of MTs is 2 h, all MTs are turned off from 1:00 to 2:00. After switching on the MTs based on ramp‐up (100 kW for MT 19 and 250 kW for MTs 15, 18, 25, and 29), the power of the MTs gradually increases on an hourly basis. MT 19 reaches maximum capacity at 3:00, MTs 18, 25, and 29 at 5:00, and MT 15 at 6:00, and then continues to operate at full power until the study period ends. The amount of electricity sold to the upstream grid in Scenario 2 is depicted in Figure 13. Electricity is not sold to the upstream network between 1:00 and 2:00 and between 19:00 and 24:00. Additionally, when comparing Figures 8 and 14, the amount of electricity sold to the upstream network between 6:00 and 18:00 is identical in Scenario 1 and Scenario 2.12FIGUREThe amount of production MTs in Scenario 2. MT, microturbines.13FIGUREThe amount of electricity sold to the upstream network in Scenario 214FIGUREThe amount of electricity purchased from the upstream network in Scenario 2According to Figure 14, electricity is purchased from the upstream network between 1:00 and 2:00 to supply a part of the network's load, as the MTs are not available during these hours. Figures 15 and 16 show the number of houses with LSP 2 and LSP 3, respectively. As shown in Figure 15, LSP 2 only occurred at 1:00. The maximum load reduction occurred on buses 24 and 25 for 35 houses. However, LSP 3 occurred at all hours. Figure 16 illustrates the number of houses assigned a load reduction priority of 3.15FIGURENumber of houses with LSP 2 at 1 o'clock in Scenario 216FIGURENumber of homes with LSP 3 in Scenario 2Therefore, in Scenario 2, prioritizing the load yields load reduction in loads with priorities 2 and 3, while loads with the priority 1 are not shed.Figure 17 depicts the load reduction of the entire system due to LSP 2 and LSP 3. The load is shed at 20:00 to 24:00 and 1:00 to 3:00. As illustrated in Figures 11 and 17, the load reduction of the entire system in Scenario 2 is more significant than Scenario 1. The total load reduction in Scenario 1 is 3050.944 kW less than in Scenario 2, as the limitations of the MTs are considered in Scenario 1.17FIGUREThe amount of load reduction on the entire system in Scenario 2The battery charge is similar to the previous scenario. However, the discharge rate of the batteries is different from Scenario 1. The charge and discharge rates of batteries are shown in Table 8. In contrast to Scenario 1, batteries are discharged between 20:00 and 21:00. Additionally, Figure 18 illustrates the SOC of the batteries. Figure 19 depicts the voltage profile at 23:00.8TABLECharging and discharging power of batteries in Scenario 2Charging power of ESs (kW)Discharging power of ESs (kW)TimeES5ES14ES20ES33ES5ES14ES20ES331−15.7––––53.261.6542–––––54.746.3–6––−8.42−8.42––––7−57.3−57.3−80−80––––8−69−69−80−80––––20–––––6980–21–––––39––22––––6969808023––––393964–24–––––––6418FIGURESOC of batteries in Scenario 219FIGUREVoltage profile in Scenario 2Scenario 3In Scenario 3, a nonlinear model is presented to solve the problem by considering the limitations of MTs. Figure 20 outlines the production capacity of MTs at various hours. All MTs except MT 19 perform differently in this scenario compared to Scenario 2.20FIGUREThe amount of production capacity of MTs in Scenario 2The amount of power purchased from the upstream network is depicted in Figure 21. Given that the MTs are turned off between 1:00 and 2:00, the model is expected to purchase additional power from the upstream network. Nonetheless, the amount of power purchased from the upstream network at various hours is negligible.21FIGUREThe amount of electricity purchased from the upstream network in Scenario 3LSP 1, LSP 2, LSP 3, and load curtailment for uncontrollable loads are depicted in Figures 22 to 25, respectively. In contrast to Scenario 2, the load is shed in priority 1. Moreover, the uncontrolled load is reduced due to insufficient electricity purchased from the upstream network.22FIGURENumber of homes with LSP 1 in Scenario 323FIGURENumber of homes with LSP 2 in Scenario 324FIGURENumber of homes with LSP 3 in Scenario 325FIGURENumber of homes with load curtailment in Scenario 3In this regard, as there is no optimal solution in Scenario 3, all load priorities will be shed. However, in this scenario, lower‐priority loads are shed first, but due to the imbalance in power, loads with higher priorities will be shed as well. (Figure 23 )Figure 26 shows the load reduction of the whole system, which results from the total load reduction and uncontrollable load curtailment. Thus in the non‐linear model, a large load is shed. (Figure 24)26FIGUREThe amount of load reduction on the entire system in Scenario 3In this scenario, battery 5 is charged only at 1:00. Furthermore, none of the remaining batteries are charged. Figures 27 and 28 illustrate the amount of power discharged and the SOC of the batteries, respectively. Battery 5 does not experience a power discharge because battery 5 is at a low SOC and therefore cannot be discharged. According to Figure 29, battery 14 reaches a minimum SOC at 2:00, battery 20 reaches a minimum SOC at 23:00, and battery 33 reaches a minimum SOC at 24:00. Figure 29 illustrates the voltage profile for Scenario 3, where all load points are within the rated voltage range.27FIGUREDischarging power of batteries in Scenario 328FIGURESOC of batteries in Scenario 329FIGUREVoltage profile in Scenario 3Results comparingThe results of the scenarios are compared in Table 9. Because MT constraints are not considered in Scenario 1, the results are superior to Scenarios 2 and 3. However, Scenario 1 is not feasible. Execution time, operation cost, and load reduction are significantly superior in Scenario 2 than Scenario 3. In addition, for a fair comparison between the linear model in Scenario 2 and the non‐linear model in Scenario 3, the linear model distribution strategy in Scenario 2 was applied to the non‐linear model in Scenario 3. The results are shown as Scenario 4 in Table 9. Based on the results, the operational cost for the nonlinear model in Scenario 4 is equal to 3845,115 dollars, which is significantly close to the operational cost of the linear model in Scenario 2 (3602,6721 dollars). However, the computations time in Scenario 4 increases compared to Scenario 2, which is natural. Nevertheless, in relation to the non‐linear model in Scenario 3, the computations time in Scenario 4 is significantly less.9TABLEComparison of scenariosNumber of scenariosExecution time (s)The total cost of operation in 24 h ($)Total load reduction in 24 h (kW)10.06−12098.10651271.41920.063602.67214322.3633407.065105117.275442199.20345.8563845.1154322.363Sensitivity analysisGiven that the proposed linear model in Scenario 2 produces excellent results; thus, sensitivity analysis is performed on the two variables of constraints ramp‐up/down and the initial status of MTs to validate the proposed linear model.Sensitivity analysis on the ramp‐up/down of MTsIn Figures 30 and 31, the ramp‐up/down scales of constraints are decreased and increased, respectively. The effect of varying the scale of this MT limitation on the results is then demonstrated. According to Figure 30, the objective function's cost is constant up to a scale of 0.75. Nonetheless, the system's operating and load reduction costs increase from 0.75 to 0.15. The cost of operation and load reduction remain constant as the scale is decreased from 0.15 to zero. As a result, when ramp‐down is near 250 kW, the overall system's operation and load reduction costs decrease.30FIGURESensitivity analysis on decreasing the scale of the ramp‐down of MTs31FIGURESensitivity analysis on increasing the scale of the ramp‐up of MTsThe range of the increasing rate is linearly increased up to two times in Figure 31. By increasing this constraint, the overall system's operating costs are significantly reduced. However, load reduction begins at Scale 1.2. The operation cost and total load reduction remain constant once Scale 1.3 is reached. Thus, scaling up to 1.3 times the specified range does not affect operating costs and load reduction. Assuming a ramp‐up of 325 kW (1.3 times the defined rate), the total load reduction and operating costs are optimal.Sensitivity analysis on the initial status of MTsSensitivity analysis is performed on the initial status of MTs, as shown in Table 10. In Scenario 2, all MTs are assumed to be off between 1:00 and 2:00. In Table 6, 20 different modes are considered for the initial condition of MTs to investigate the effect of MTs’ presence or absence on the overall system's cost of operation and load reduction. Compared to the other modes, modes 15 and 20 produce the most economical results. Because mode 15 has four MTs and mode 20 has five MTs operating in the early hours, mode 15 is more suitable.10TABLESensitivity analysis on the initial status of MTsMode numberInitial status of MTsOperation cost ($)Load reduction (kW)MT1MT2MT3MT4MT 51000003602.6724322.363210000−2385.8844322.363301000−1141.174322.363400100−1967.7584322.363500010−1967.7584322.363600001−1141.174322.363711000−6953.2573403.875810100−2959.5844322.363910010−7732.3493204.481010001−6953.2573403.8751111100−7526.6513303.8751211010−11102.7451720.851311001−10684.7081918.2631410011−11102.7451720.851511011−12088.2671271.4191610111−11318.9331620.851711101−11103.0961720.4871811110−11318.9331620.851901111−10893.6691818.9892011111−12098.1061271.419CONCLUSIONGiven the importance of loads in the distribution system, this paper presents a scheme known as centralized optimal management of the distribution system. The proposed method allows the system to operate normally to provide the maximum load possible to customers while minimizing load reduction and operation costs. This paper makes use of a variety of dispatchable and non‐dispatchable DG resources, as well as battery storage. In the proposed scheme, the system operator has access to the loads located within the smart home. It is capable of shedding loads according to a predetermined priority list. The proposed model can operate in both a grid‐connected and an island mode. During certain hours, it may even sell electricity to the upstream grid.As a result, the proposed approach significantly impacts the distribution system's economic operation. The unit commitment problem is solved using two linear and non‐linear models. The proposed model encompasses all MTs and their associated operation constraints. The objective function presented here includes costs associated with the generation, pollution, load reduction, electricity purchased from the upstream network, and profit from electricity sales. The linear model outperforms the non‐linear model by a significant margin. The proposed method, which uses smart load handling tools and proper planning for MTs, performs exceptionally well in terms of system economics. Sensitivity analysis is used to validate the proposed model on system variables, which demonstrates the proposed model's proper efficiency in problem‐solving. Auto DR is implemented in the proposed model. In auto DR, the operator reduces power without customer interference. Moreover, homeowners cannot alter the load reduction priorities. In the following works, homeowners can modify their load reduction priorities at any time.FUNDING INFORMATIONNone.CONFLICT OF INTERESTThe consent of all the authors of this paper has been obtained for submitting the paper to the journal ‘IET Generation, Transmission and Distribution’. The authors of this research have completely avoided publishing ethics, plagiarism, and data forgery. There is also no commercial interest in this paper and the authors have not received any payment for their work.DATA AVAILABILITY STATEMENTThe data that support the findings of this study are openly available in:[IET] at [https://doi.org/10.1049/iet‐gtd.2016.0656], reference number [3].[ELSEVIER] at [https://doi.org/10.1016/j.est.2019.101054], reference number [4].[IET] at [https://doi.org/10.1049/iet‐gtd.2016.1783], reference number [7].[IEEE] at [https://doi.org/10.1109/TPWRS.2015.2389753], reference number [8].[IEEE] at https://doi.org/10.1109/TSG.2015.2454436], reference number [27].[ELSEVIER] at [https://doi.org/10.1016/j.enconman.2015.02.042], reference number [28].NOMENCLATUREyi,t/zi,t${\rm{\;}}{y_{i,t}}/{\rm{\;}}{z_{i,t}}$binary variable for start‐up/shutdown of MTspg¯(i,t)/pg̲(i,t)${\overline {pg} _{( {i,t} )}}/{\underline {pg} _{( {i,t} )}}$maximum/minimum output power of MTsΔVi,t/ΔVj,t$\Delta {V_{i,t}}/\Delta {V_{j,t}}$amount of voltage changesCDprofit from the sale of electricityCemicost of pollutionCGcost of productionCLS2cost of load reduction with second priorityCLS3cost of load reduction with third priorityCLS4cost of load curtailmentCsi, tbinary variable determining omitted busesCtb$C_{\rm{t}}^{\rm{b}}$price of purchasing from the upstream networkCts$C_{\rm{t}}^{\rm{s}}$price of selling electricity to the upstream networkBi,j${B_{i,j}}$susceptance of linesECi$E{C_i}$energy storage capacityGi,j${G_{i,j}}$conductance of linesPi,t/Qi,t${P_{i,t}}/{Q_{i,t}}$active/reactive load flowPi,tD/Qi,tD$P_{i,t}^D/Q_{i,t}^D$active/reactive power of demandPi,tPV/qi,tPV$P_{i,t}^{PV}/q_{i,t}^{PV}$active/reactive power of photovoltaicPi,tWT/qi,tWT$P_{i,t}^{WT}/q_{i,t}^{WT}$active/reactive power of wind turbinePich,max/Pidch,max$P_i^{ch,max}/P_i^{dch,max}$maximum energy storage charge/discharge powerPload1i,tD$Pload1_{i,t}^D$total active power for first priority of loadsPload2i,tD$Pload2_{i,t}^D$total active power for second priority of loadsPload3i,tD$Pload3_{i,t}^D$total active power for third priority of loadsPload4i,tD$Pload4_{i,t}^D$total active power of non‐controllable loadsRUi/RDi$R{U_i}/R{D_i}$ramp‐up/down of MTsSOCi,t$SO{C_{i,t}}$state of charge of batteriesSOCimax$SOC_i^{max}$maximum SOC of batteriesSUi/SDi$S{U_i}/S{D_i}$ramp limit for start‐up/ shutdown of MTsSi${S_i}$apparent powerSimax$S_i^{max}$maximum apparent power of MTsUTi/DTi$U{T_i}/D{T_i}$minimum uptime/downtime of MTsUi0/Si0$U_i^0/S_i^0$duration that unit i has been on/off at the beginning of the studied period (end of t = 0)Vi,t${V_{i,t}}$bus voltageaiG,biG,ciG$a_i^G,{\rm{\;}}b_i^G,{\rm{\;}}c_i^G$coefficients of the fuel cost functionpi,tE$p_{i,t}^E$amount of power for charging/discharging batteriespi,tG/qi,tG$p_{i,t}^G/q_{i,t}^G$total amount of active/reactive power generationui,t${u_{i,t}}$binary variable indicating up or down status of MTsθij,t${\theta _{ij,t}}$phase difference between buses voltageλi,t/ϕi,t${\lambda _{i,t}}/{\phi _{i,t}}$binary variable for battery charge/dischargeλt${\lambda _t}$selling power to the upstream networkμt${\mu _t}$purchasing power from the upstream networkπc${\pi _c}$battery charge efficiencyπd${\pi _d}$battery discharge efficiencyσi${\sigma _i}$emission coefficient of an MTΔv$\Delta v$allowable voltage changes in the distribution systemAbbreviationDGdistributed generationDRdemand responseDSMdemand‐side managementESenergy storageIoTinternet of thingsLSPload reduction with priorityMGmicrogridMTmicroturbineParametersPVphotovoltaicSetsSOCstate of chargeVariablesWTwind turbinePLS1$PLS1$first priority of load reductionPLS2$PLS2$second priority of load reductionPLS3$PLS3$third priority of load reductionPLS4$PLS4$load curtailmentSTC/SDC$STC/SDC$start‐up/shutdown cost of MTsTnumber of planning period hoursi,j$i,{\rm{\;}}j\$set of nodeskset of planned period hourstset of timesREFERENCESQiu, J., et al.: Multi‐stage flexible expansion co‐planning under uncertainties in a combined electricity and gas market. 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