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Two stage risk based decision making for operation of smart grid by optimal dynamic multi-microgrid
Electrical Power and Energy Systems 118 (2020) 105791
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Electrical Power and Energy Systems journal homepage: www.elsevier.com/locate/ijepes
Two stage risk based decision making for operation of smart grid by optimal dynamic multi-microgrid☆ M. Zadsara, S.Sina. Sebtahmadia, a b
T
⁎,1
, M. Kazemib, S.M.M. Larimia, M.R. Haghifama
Faculty of Electrical and Computer Engineering, Tarbiat Modares University, Jalal Ale Ahmad, Nasr, Tehran, Iran Faculty of Electrical Engineering, University of Shahreza, Shahreza, Iran
A R T I C LE I N FO
A B S T R A C T
Keywords: Active distribution network Distributed energy resource Emergency operation Energy storage system Multi-microgrid Renewable energy resource
Resect severe economic losses caused by distribution system equipment outage have highlighted the importance of improving the system resiliency and reliability. In active distribution networks (ADNs), the distributed energy resources (DERs) managed by dynamic isolated microgrids in contingency mode provide an alternative approach to enhance the system resiliency and continue supplying critical loads after equipment outage. How to incorporate this ADNs capability into a short-term DERs scheduling is a challenging issue. In response to this challenge, in this paper, a two stage risk based decision making framework for operation of ADNs is proposed to coordinate 24-h DERs’ scheduling and outage management scheme, in a way to be immune against contingency by creating optimal dynamic multi-micrigrid, micro-turbine and energy storage island operating and load shed plan. The first stage is the normal operation condition that operation cost should be minimized considering of the uncertainties of renewable resources, the electricity price and customers loads. The second stage is the operation in contingency condition. In the second stage, the main objective is to keep the shed load at minimum level. Numerical results and sensitivity analysis from modified 33-bus IEEE network are further discussed to demonstrate the efficiency of the solution approach.
1. Introduction Active distribution networks (ADNs) is formed owing to the addition of distributed energy resources (DER) into the traditional electrical distribution systems [1]. Complicated technical constraints and economic decisions related to ADNs, requires a new management system. This management system, which is known as distributed management system (DMS). It is a hardware and software organization that is responsible for making decisions about the optimal operation of the system [2]. Proposing various methods for increasing the applicability of DMS have been widely investigated in [3–9]. An energy management strategy for hybrid energy storage system is proposed based on the Pontryagin’s minimum principle in [3]. A new energy management strategy and a novel probabilistic index for evaluating operation efficiency is proposed in [4] for performing in multi-microgrid system by various DERs. A dual-horizon rolling scheduling model for planning and operation of ADNs is proposed in [5]. In [6,7], the optimal energy
management strategies are proposed to response the renewable resources uncertainties. In energy management strategy proposed by [9], the two optimization levels are proposed, which consist of the microgrid centralized controller and local controllers and the distribution network in order to response load uncertainties. In the referred papers, the proposed methods for DMS are designed for the normal operation. Additionally, DMS should be designed in a way to be able of handling emergency situations by outage management system (OMS) application by providing a robust operation plan. Robust operation refers to the ability of the system to withstand events in an efficient manner, while ensuring the least interruption in the supply of electricity and enabling a quick recovery and restoration to the normal operation state [10]. Considering the importance of this topic, several studies have been conducted in this regard [11–13]. A fully decentralized multi-agent system is presented to handle the distribution system restoration (DSR) problem, considering incorporation of DERs in [11]. An agent-based paradigm for self-healing protection systems is introduced in [12], using the graph theory-based expert system. A new DSR procedure is
☆
Grant numbers: Iran’s National Elites Foundation – Bright Spark. Corresponding author. E-mail addresses: [email protected] (M. Zadsar), [email protected] (S.S. Sebtahmadi), [email protected] (M. Kazemi), [email protected] (S.M.M. Larimi), [email protected] (M.R. Haghifam). 1 The paper’s supervisor. ⁎
https://doi.org/10.1016/j.ijepes.2019.105791 Received 10 August 2019; Received in revised form 11 December 2019; Accepted 17 December 2019 0142-0615/ © 2019 Elsevier Ltd. All rights reserved.
Electrical Power and Energy Systems 118 (2020) 105791
M. Zadsar, et al.
Nomenclature
β ηiES γs κ i, s , t
Sets
i, j, s, t DsF , T ΩB , ΩS , ΩMT , ΩES ΩMaster , MT
Index of buses, scenarios, time Fault period in scenario s, Horizon time Set of network’s buses, uncertainty scenarios Set of micro-turbine buses, energy storage buses Set of buses connected to master micro-turbine
κiRamp
Risk- aversion coefficient Charging/discharging efficiency of battery storage in bus i Probability of scenario s Customer damage weight factor in scenario s in bus i at time t Ramp rate limit of the micro-turbine located at bus i [kW/ min]
Variables Voltage angle difference between bus i and bus j in scenario s in time t The auxiliary variable used in CVaR calculation η PiD, s,,shed / QiD, s,,shed Active/reactive load curtailment in bus i in scenario s t t at time t [kW/kVar] dch Pich Charge/discharging power of the battery storage located , t / Pi, t at bus i at time t [kW] PiG, t / QiG, t Active/Reactive power generated by the micro-turbine located at bus i at time t [kW] PijP, s, t / QijQ, s, t Active/reactive power of the line connecting bus i to bus j in scenarios at time t [kW/kVar] Psex, t / Qsex, t Active/Reactive power transacted with the upstream network in scenario s at time t [kW/kVar] Ss The auxiliary variable used for CVaR calculation in scenario s SOCi, t State of charge of the battery storage located at bus i at time t Voltage magnitude of bus i in scenario s in time t [kV] Vi, s, t Charging/discharging state of battery storage in bus i in ψich ,t scenario s at time t υij, s, t Binary variable indicating the connection status of the line connecting bus i to bus j in scenario s at time t
δij, s, t
Parameters
aiG , biG , ciG Micro-turbine cost function coefficients Customer damage cost in scenario s at bus i at time t CDFi, s, t [$/kWh] PijP, max / QijQ, max Maximum active/reactive power capacity of the line connecting bus i to bus j [kW/kVar] Pich, max Maximum charge rate of battery storage in bus i [kW] Pidch, max Maximum discharge rate of battery storage in bus i [kW] Predicted active power output of renewable energy rePiR, s, t sources in bus i in scenario s at time i [kW] PiD, s, t / QiD, s, t Predicted active/reactive load demand in bus i in scenario s at time t [kW] PiMT , max / QiMT , max Maximum active/reactive power capacity of the micro-turbine located at bus i [kW/kVar] ri, j / x i, j Line resistance/reactance connecting bus i to j P sub, max / Q sub, max Maximum active/reactive power capacity of substation [kW/kVar] SOCimin Minimum energy limit of battery storage in bus i SOCimax Maximum energy limit of battery storage in bus i Vimin/ Vimax Minimum/Maximum voltage limit in bus i [kV] V Set Voltage magnitude of slack bus i [kV] WiES Capacity of battery storage in bus i [kWh] α Confidence factor of CVaR method
However, in the first category, only the set of switches status are evaluated considering DERs’ generation level as fixed parameters; and, in the second category, the DERs’ optimum generation level is determined assuming a predetermined multi-MG configuration plan. This paper covers the gap of the literature by proposing a model handling both of the decision variable sets consist of DERs scheduling and switches status in contingency mode, simultaneously. This point of view has been also investigated in [24,25]. In [24,25] the real-time operation model is proposed, in which the post-contingency operation strategy (consist of DERs’ scheduling and MG formation) is determined based on post-contingency conditions in order to reduce load shed and improve system’s resiliency. Thus, in [24,25], only an outage scenario is investigated. However, in this paper, the 24-h scheduling plan has been determined in order to reduce system interruption risks due to several outage scenarios. Therefore, the coordinating adaptive dynamic mutiMG and DERs’ islanding operation in the proposed model improves system’s efficiency and reliability. The proposed model compromises between operation cost and operation risk in ADNs. In order to achieve a resilient adaptive distribution network, different items are performed including dynamic Multi-MG configuration, DGs inslanded operation mode, load shedding plan and energy storage systems. One of the advantages of using storage systems is that some of the operation risks are covered in normal and contingency modes. In order to reduce the interruption risks in distribution systems, storage systems should be fully charged to be ready to feed the islanded loads in the contingency condition [25]. However, in the normal operation condition, the storage system is charged and discharged periodically to minimize the operation cost of purchasing energy from the upstream network [3]. These operation strategies are against each other and a balanced strategy is required to cover both of the operational cost and
developed based on graph models and a knapsack problem formulation using the dispersed generation availability in [13]. The above-mentioned researches do not consider some advantage of ADNs’ options such as scheduling of distributed generation (DG) and energy storage system resources in island operation mode. According to the IEEE standard 1547.4, dividing ADNs into multiMGs and island operating is an effective method to improve system reliability during the contingency condition [14]. The idea of optimal switching with the purpose of MG creation under emergency condition is investigated in [15–23]. Several studies which is conducted in this area can be categorized in two groups. The first group of researches have proposed methods for switching and forming of multi-MGs during contingency condition with different objectives energy not supplied and non-served load [16–21], as well as the number of switching [15]. In this category, scheduling of DERs are not involved in the planning problem and it is assumed that their optimal level of generation are predetermined. Moreover, the only decision variable of this category is the status of switches. The second category [22,23], the multi-MG configuration are predetermined and the planning problem is solved from the perspective of DERs’ optimum generation level. Across the first category, this group concerns about the scheduling of DERs and switches status are not a decision variable here. The simplifying assumptions made in these categories are could affect the optimally of their results. For example, the predetermined configuration assumed in the second category, is not necessarily the optimal one at the time of fault occurrence. The optimal configuration can be changed depending on the fault location, DERs generation level, the renewable resources available power and etc. Furthermore, the optimal value for both of the decision variable sets, i.e., the set of switches status and the set of generation levels of DERs, should be simultaneously evaluated. 2
Electrical Power and Energy Systems 118 (2020) 105791
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2.1. Stage 1: Normal operation scheduling
operational risk. This paper provides an operation model of energy storage systems with the aim of compromising the operational risk of a ADN versus operating costs considering the ADNs’ capabilities including self-healing. Thus, considering the capability of DERs’ islanding operation, especially energy storage system, by multi-MG formation in contingency mode in the evaluating the optimal 24-h operation plan is the most important contribution of this paper. Also, in this paper a new viewpoint of energy storage systems and their applications in smart distribution systems is presented. The energy storage systems are used as reserve resources in contingency mode modeled by the second stage. Considering this point of view, a minimum level of state of charge (SOC) can be evaluated as the required reserve capacity of the distribution systems. This paper proposes a framework, in which the minimum SOC of energy storage systems can be calculated. The main contributions of the paper are highlighted bellow:
In this operation condition, a 24-h scheduling plan is determined. The decision variables at this level are the scheduling of MT generation units, charging/discharging of battery storage systems and the energy level transacted with the bulk power system. The objective of this optimization problem is to minimize the operation cost of the AND in a 24-h period. Various sources of uncertainty can affect the scheduling problem of normal condition, such as, the generation of renewable resources, the electricity prices and the demand level. In this section, the uncertainty of the mentioned resources are modeled through scenario tree method [26]. The conditional value at risk (CVaR) is used to measure and control the risk level of the decision making process. According to the above-mentioned explanation, the optimization problem of the normal condition can be formulated as follow. 2.1.1. Objective function The objective function has been formulated as follow,
1. A two-stage resilient operational framework of ADNs that considers compromising between operation cost and operational risk due to equipment outage and renewable resources uncertainties is proposed. 2. For calculating the operational risk, renewable resources uncertainties and equipment outage has been modeled by scenario tree method and sequential Monte Carlo method, respectively. In the contingency mode the second stage is proposed, where the optimal Multi-MG structure is calculated by new graph-based algorithm in order to reduce load shed cost. 3. Compared with models presented in the literature, the proposed model lead system operator to schedule DERs in 24-h horizon by taking into account ADNs’ capabilities such as adaptive control and protection and dynamic multi-MG operation in contingency mode. 4. In the proposed model, coordinating the islanding operation of DERs, especially the energy storage system, and an adaptive dynamic Multi-MG in contingency mode improves the system’s efficiency and reliability. 5. This paper by proposing a risk-based decision making platform has investigated that how neutral-risk and risk-averse operation strategy could overshadow the scheduling of DERs, especially energy storage systems, under uncertainties of equipment outages, renewable resources, loads, and electricity price. 6. This paper proposed a framework, in which the minimum SOC of energy storage systems can be calculated as the required reserve capacity of the distribution systems in order to achieve an adaptive resilient operation plan.
⎛ min (1 − β ) ⎜ ∑ ⎝t∈T +
∑ s ∈ Ωs
ai + bi PiG, t + ci (PiG, t )2
∑ i ∈ ΩMT
⎛ γs ⎜ ∑ ⎝t∈T
CiShed , s, t +
∑ i ∈ ΩB
∑ t∈T
⎛ 1 +β ⎜ 1 − α ⎝
∑ s ∈ Ωs
⎞⎞ Csex, t ⎟ ⎟ ⎠⎠
⎞ γs Ss − η⎟ ⎠
(1)
D,shed CiShed , s, t = κ i,s,t CDFi,s P i,s,t
(2)
Csex, t
(3)
=
πs, t Psex, t
The objective function (1) consists of two parts, the operation cost and therisk index measured by CVaR The operation cost is made up of three terms; (1) The operation cost of MTs, (2) The cost or benefit of transacting power with the upstream network and, (3) The load interruption cost. The interruption cost,that are presented in (2), is evaluated from the second stage optimization, and then it will be sent back to the first stage as an input parameter. The power exchange cost or benefit are presented in (3). 2.1.2. Risk model constraints Considering the probability density function of exchange cost, interruption cost, and α as a parameter indicating the right tail probability of that function, CVaR α is defined as the expected value in the worst 100(1 − α )% of the probability density function. The CVaR term has been considered as a linear formulation by introducing additional variables and constraints [27] are presented in (4) and (5).
The paper is organized into five sections; Section 2introduces the problem formulation; Section 3describes the proposed solution methodology, Section 4presents the numerical results, and Section 5concludes and closes this paper.
⎛ ⎜∑ ⎝t∈T
2. Problem formulation
∑
CiShed , s, t +
i ∈ ΩB
∑ t∈T
⎞ Csex, t ⎟ − η ⩽ Ss ∀ s ∈ ΩS ⎠
Ss ⩾ 0 ∀ s ∈ ΩS
In this paper, a framework for short-term scheduling of smart grids is provided, in a way to be immune against outages of distribution elements. This purpose can be achieved by making decision considering two different conditions. The first one is the normal operation condition in which, the operation cost should be minimized. The second one is the operation during contingency condition. In the second condition, the main objective is to keep the shed load at minimum level. Decisions made in each conditions, i.e., normal and contingency conditions, can affect the other one. Thus, the decision making frameworks for each condition should be designed in a way to reach a compromise. In this section, the decision making formulation for each condition is provided.
(4) (5)
2.1.3. Upstream power exchange constraints The limit of energy transmission from upstream network is presented in (6) and (7).
− P Sub,max ⩽ Psex, t ⩽ P Sub,max − Q Sub,max ⩽ Qsex, t ⩽ Q Sub,max
∀ s ∈ ΩS , t ∈ T ∀ s ∈ ΩS , t ∈ T
(6) (7)
2.1.4. Energy storage constraints The battery storage system operation constraints are provided by (8)–(11). In this paper, the time interval (Δt ) for determining the 24-h scheduling plan is set to one hour. 3
Electrical Power and Energy Systems 118 (2020) 105791
M. Zadsar, et al. ch ch,max 0 ⩽ Pich ∀ i ∈ ΩES , t ∈ T , t ⩽ ψi, t Pi
(8)
ch dch,max 0 ⩽ Pidch ∀ i ∈ ΩES , t ∈ T , t ⩽ (1 − ψi, t ) Pi
(9)
SOCi, t = SOCi, t + Pich ,t
SOCimin
⩽ SOCi, t ⩽
2.1.9. Load shed constraints Constraints (25) and (26) guarantees that bus voltage magnitude is within predefined range.
0.9 ⩽ Vi, s, t ⩽ 1.1
ηiES Δt
− Pidch ,t
WiES
SOCimax ,t
Δt
∀ i ∈ ΩES , t ∈ T
ηiES WiES
∀ i∈
ΩES ,
t∈T
(10)
Vi, s, t = V
(11)
In the previous subsection, the operation in normal condition is discussed. It is seen that one of the most important objective’s items is the amount of shed load, which can be evaluated from the second stage optimization problem. The input of the second stage is the optimum operation determined by the first stage. Moreover, the second stage optimization problem, uses the output of the first stage optimization problem, i.e., the generation level of MTs, the energy transactions with the upstream network and the battery storage systems schedules, as the base point of its analysis. Thus, switching for creating the optimal adaptive multi-MG configuration is calculated in stage 2 based on operation condition in stage 1. In order to model the outage conditions in this stage, the probabilistic nature of elements failure should be involved in the optimization problem. Another source of uncertainty can be introduced as the failure of elements, including their failure time and their repair duration. In this way, the uncertain parameters of this problem can be categorized into two groups. The first group is the uncertainty associated with the wind and solar generations, energy prices, and customer loads. These sources of uncertainties are modeled in the first level of optimization problem using scenario tree method [24]. The second group is the uncertainty of elements failure, which should be modeled in the second level optimization problem. The nature of this group of uncertainty is different from the uncertainties of the first group, i.e., uncertainties of the first level. The elements failure include uncertainties of the location, occurrence time, and clearance time (repair or replace time) of the failure. Therefore, sequential Monte Carlo methods presented in [29] have been used to model the uncertainty of this group. An illustrative example for the generated scenarios according to (27) and (28) presented in [29] that models failure of n elements is depicted in Fig. 1.
(12)
QijQ, s, t = −αijL (δi, s, t − δj, s, t ) + βijL (Vi, s, t − V j, s, t )
−
PijP,max
⩽
PijP, s, t
⩽
PijP,max
∀ ij ∈ ΩB , s ∈ ΩS , t ∈ T
− QijQ,max ⩽ QijQ, s, t ⩽ QijQ,max ∀ ij ∈ ΩB , s ∈ ΩS , t ∈ T In this model αijL and calculate as αlL = rl/(rl2
βijL
(13) (14) (15)
are distribution line characteristic that are
+ xl2) and βlL = xl /(rl2 + xl2) ,respectively [28].
2.1.6. Power balance in each bus Constraints of the active and reactive power balance in each bus are presented in (16)–(19). dch PiG, t + PiR, s, t + Pich , t − Pi, t −
∑
υij, s, t PijP, s, t
j ∈ ΩB
=PiD, s, t − PiD, s,,shed ∀ ij ∈ ΩB , s ∈ ΩS , t ∈ T t
QiG, t −
∑
(16)
υij, s, t QijQ, s, t = QiD, s, t − QiD, s,,shed t
j ∈ ΩB
∀ ij ∈ ΩB , s ∈ ΩS , t ∈ T Psex, t −
∑
(17)
υ1j, s, t P1Pj, s, t = P1,Ds, t − P1,Ds,,shed t
j ∈ ΩB
∀ j ∈ ΩB , s ∈ ΩS , t ∈ T Qsex, t −
∑
(18)
υ1j, s, t Q1Qj, s, t = Q1,Ds, t − Q1,Ds,,shed t
j ∈ ΩB
∀ j ∈ ΩB , s ∈ ΩS , t ∈ T
(19)
∀ i ∈ ΩMT , t ∈ T
QiMT ,min ⩽ QiG, t ⩽ QiMT ,max
∀ i ∈ ΩMT , t ∈ T
|PiG, t − PiG, t − 1| ⩽ κiRamp PiMT ,max
∀ i ∈ ΩMT , t ∈ T
1 TTF = − ln (U1) λ
(27)
1 TTR = − ln (U2) μ
(28)
At this level of optimization, different types of corrective action can be used to restore the system with the minimum level of load shedding. These actions in a smart distribution system can be listed as: (1) optimal switching for dynamic multi-MG structure creation in accordance with the fault location, (2) using the stored energy in the energy storage
2.1.7. MTs’ constraints The MT generation operation constraints including their maximum active and reactive power capacity and their ramp rate limits are provided by (20)–(22), respectively. In constraint (20), the minimum limit of MTs output (PiMT,min ) is assumed set to be zero.
0 ⩽ PiG, t ⩽ PiMT ,max
(20) (21) (22)
2.1.8. Load shed constraints Load shedding constraint are presented in (23) and (24).
0 ⩽ PiD, s,,shed ⩽ PiD, s, t ∀ i ∈ ΩB , s ∈ ΩS , t ∈ T t
(23)
0 ⩽ QiD, s,,shed ⩽ QiD, s, t ∀ i ∈ ΩB , s ∈ ΩS , t ∈ T t
(24)
(26)
2.2. Stage 2: Operation during contingency conditions
PijP, s, t = βijL (δi, s, t − δj, s, t ) + αijL (Vi, s, t − V j, s, t )
∀ ij ∈ ΩB , s ∈ ΩS , t ∈ T
δi, s, t = 0
∀ i = 1, s ∈ ΩS , t ∈ T
2.1.5. Lines flow constraints Line flow equations of the distribution system are imposed to the problem by (12)–(15).
∀ ij ∈ ΩB , s ∈ ΩS , t ∈ T
set
(25)
Fig. 1. Uncertainty model of equipment failure. 4
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M. Zadsar, et al.
upstream network (30), load flow equations, and operation constraints including the thermal capacity constraints of the network lines (31), the load shedding limits (32), and the network bus voltage limits (34).
systems. Considering these corrective actions, the optimization problem of the second level can be formulated as follows.
min
∑ ∑
CiShed , s, t
t ∈ DsF i ∈ ΩB
(29) 3. Metaheuristic-based solution methodology
s.t.
(6) − (7)
∀ t ∈ DsF
(12) − (17)
∀ ij ∈ ΩB , t ∈ DsF
(31)
(18) − (19)
∀ i = 1, j ∈ ΩB , t ∈ DsF
(32)
(23) − (25)
∀ i ∈ ΩB , t ∈ DsF
(33)
(26)
∀ i = {1, Ωmaster , MT }, t ∈ DsF
The formulation presented in this paper is nonlinear and nonconvex due to the complex nature of the problem. The two-stage structure of the optimization problem and the existence of complex binary constraints associated with radial configuration of the network and multi-MG creation constraints, make it difficult to use mathematical approaches to solve this problem. In addition, at this level of complexity, heuristic methods with their high ability to handle the complicated binary constraints can result in a better solution. Among the heuristic methods, particle swarm optimization (PSO) method is used in this paper, because of simple concept, easy implementation, robustness to control parameters, and computational efficiency when compared with mathematical algorithm and other heuristic optimization techniques [30]. The proposed framework for optimization is presented in Fig. 2. According to the proposed algorithm, variable set in stage 1 in a physical 2-dimensional search space (variable and horizon time) are generated using the PSO method for all particles. For each particle, PSO optimization procedure is performed on normal network structure in case of normal condition. This part of optimization process solves the
(30)
(34)
This objective function (29) is the interruption cost of the customer loads based on the time varying customer damage function during the contingency mode. It should be noted that the optimization problem of the second stage is performed for a specific scenario. In a specific scenario, all the stochastic variables of the problem, including the renewable resources generation, energy prices, customers’ load, outage location, outage time occurrence, and the repair time, are determined. Therefore, the index of the scenario used in the equation of this section is merely for a specific scenario. The constraints in the second-stage optimization problem include the power exchange limit with the
Fig. 2. Optimization algorithm and proposed method framework. 5
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M. Zadsar, et al.
optimization problem are: (1) the operation scheduling of MT and energy storage systems obtained in the first stage, (2) the stochastic data of uncertain parameters includes the renewable resources generation and the faults scenario data, i.e., the fault locations, their occurrence and repair times. The main outputs of the second stage optimization problem are: (1) the optimal switching for muli-MG forming, and (2) load shedding plan. υij, s, t variables in ((16)–(19)indicate the connection status of the line connecting bus i to bus j in scenario window s at time t (connect = 1, disconnect = 0). Thus, at each time of each scenario window where the equipment outage has not occurred, these variables shall be considered based on the basic network’s configuration under normal operation conditions. With the outage occurring in any hour of the scenario window, the these variables are controlled in stage 2 in order to create the optimal multi-MG formation. The second stage optimization problem has a complex structure due to the existence of complicated binary constraints associated with Multi-MG forming through optimal switching actions. In addition, the ac power flow constraints, radial structure constraints, the presence of MT with frequency control capability in each MG, increase the complexity of the second stage optimization problem. Due to this level of complexities, the second stage optimization problem is solved by a full search method. In this paper, the spanning forest search algorithm presented in [24] is used to form MGs through switching actions. This method ensures that the MGs structure in the post-contingency period is radial and there is at least one MT in each MG with capability of voltage and frequency control. In this method, all possible switching states are generated to create feasible multi-MG structures. The operating constraints are checked using ac load flow, and the objective function of the second stage is calculated in each structure. The best structure of multi-MG is chosen as an optimal operation strategy in contingency mode. Certainly, the run time of the second stage will be increased by increasing the number of controllable switches in the network. In the second stage, the method presented in [15] is used to simplify the network graph and limit possible responses in order to improve the algorithm’s efficiency. After assessing all scenarios, the objective function of the first stage can be calculated. Then, all particles’ position and velocity, i.e., the first layer problem variables, are updated in the PSO optimization algorithm [30]. This process continues until the convergence criteria are met.
Table 1 MTs’ data. Bus
15 18 19 25 29
aG
bG
cG
($)
($/kW)
($/kW2)
P MT , max (kW)
P MT , min (kW)
Factor
0.025 0.039 0.023 0.010 0.010
0.071 0.055 0.062 0.075 0.061
0.00005 0.00006 0.00005 0.00006 0.00005
400 375 50 350 350
0 0 0 0 0
0.95 0.95 0.95 0.95 0.95
Power
Table 2 ESs’ data. Bus
5 14 20 33
WES (kWh)
P ch, max (kW)
P dch, max (kW)
Initial
200 150 150 200
100 75 75 100
100 75 75 100
0.95 0.3 0.5 0.1
SOC
Fig. 3. Electricity price.
4. Numerical results The modified IEEE 33-bus network that was presented in [24], is used for the numerical studies. This network includes 33 load points, 37 lines, and 5 lines with normally open switches for maneuvering on the network [31]. The MTs and ESs data are presented in Table 1 and Table 2, respectively. Also, the electricity price is shown in Fig. 3. All test data are available online at https://www.dropbox.com/s/ o08911dagpo5qlq/InputData.xlsx?dl=0. The failure of distribution lines are considered in this paper. As noted before, wind and solar generation, customer load, and electricity price uncertainties are considered by generating scenarios based on the discrete probability distribution function. According to the five states for each of this stochastic variables, a total of 625 uncertain scenarios are generated. Therefore, the number of uncertain scenarios has been reduced to 30 in order to reduce the computational time while maintaining accuracy using the probability distance scenario reduction method [32] in this study. Meanwhile, in addition to the uncertain scenarios, the Monte Carlo simulation scenarios for the modeling of the network equipment failure greatly increase the time of the calculations. In these circumstances, the number of scenarios should be selected to reduce the computational time while maintaining the accuracy of calculations. Taking into account 100,000 scenarios (days) for modeling network failure, the variance of the objective function in a 50times calculation is less than 1, which is reasonable. According to the proposed model, the coefficient β in the objective function models the
Fig. 4. MT sources and expected purchased power from upstream network in risk-neutral scenario. Table 3 values of various parameters of the objective function
MT operation cost Expected exchange cost Expected interruption cost Total cost Risk index
Neutral-risk
Risk-averse
2144.6 3349.70 6.29 5500.6 2340.1
3116.04 2516.83 5.11 5639.2 1839.6
first stage formulated by (1)–(26). In scenarios involving failures, the outage management scheme is implemented in contingency mode. In fact, this part of optimization process solves the second stage optimization problem, in which the shed load is minimized by taking remedial actions explained in Section 2.2. The inputs of the second stage
6
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Fig. 5. ESs SOC and charging/discharging in risk-neutral scenario.
operator risk-taking. Therefore, the operator strategy is investigated by changing this parameter. The program of network operation in two modes of neutral risk (β = 0) and risk aversion (β = 1) is presented and analyzed. The results are then presented for different scenarios. 4.1. Neutral risk operation strategy In this scenario the β parameter is set to zero. The results for MT resources and the expected value of purchasing electricity from the upstream network, optimal operation of the ES systems, and the state of charge for each ES system from this operating scenarios is shown in Figs. 3 and 4, respectively. According to Table 3, the sum of operation costs and risk index in the neutral-risk scenario are 5500.6 $ US and 2340.1 $ US, respectively. Based on Fig. 4, in the case of neutral risk, the amount of power purchased from the upstream network can be
Fig. 6. MT sources and expected purchased power from upstream network in risk-averse scenario.
Fig. 7. ESs SOC and charging/discharging in risk-averse mode. 7
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resources and the expected value of purchasing electricity from the upstream network, optimal operation of the ES systems, and the state of charge for each ES system from this operating scenarios is shown in Fig. 6 and Fig. 7, respectively. According to Table 3, the total operation costs and risk index in risk-averse case, are 5639.2 $ US and 1839.6 $ US, which shows a 2.5% increase and 21.39% decrease. In contrast the risk-natural scenario, a risk-averse operator prefers to provide security benefits and reduce network risks. Due to the high interruption costs of customers, the operation strategy has led to a reduced network interruption. The MT resources are operated at their maximum power as are shown in Fig. 6 due to the probability of failure occurrence at any time of day as well as taking into account the ramp rate of MTs, in most hours of the day. Furthermore, reducing the power purchased from the upstream network and increasing the participation of DERs in power generation presents the operator with a lower electricity price risk.
Fig. 8. Network storage capacity index.
Table 4 the charging/discharging of ES systems to electricity price correlation coefficient ES Unit
Neutral-risk
Risk-averse
Comparison (%)
ES5 ES14 ES20 ES33
0.305 0.3218 0.396 0.2304
0.1476 0.1383 0.207 0.2012
−51.6 −57 −47.7 −12.7
4.3. Sensitivity analysis and comparison Exploring the operation strategy of battery storage systems provides an impressive result. The optimal operation strategy is expected to reduce the network interruption risk, keeping the SOC at a minimum level for storage systems. Hence, the operation scheduling is performed in such a way that the storage systems SOC does not get lower than the minimum SOC. However, according to Fig. 7, the optimal strategy for reducing interruption risk is not to maintain the minimum SOC for storage units. In contrast, the operator strategy to reduce interruption risk is to maintain the stored energy for the network based on the results of the proposed model. The battery charging/discharging scheduling is carried out in such a way as to preserve the amount of stored energy in the network while increasing the level of storage systems generation and share of DERs so that the operator is less affected by energy price risk. To illustrate this, the index is calculated as the Network Storage Capacity (NSC) which, shown as follow,
Table 5 the NSC index and the expected value of purchased power to electricity price correlation coefficient
NSC Index Power Exchange
Neutral-risk
Risk-averse
Comparison (%)
−0.4237 −0.438
−0.29832 0.5985
−29.5 -
NSCt =
Table 6 Risk index and correlation of purchased power from the upstream network and NSC index to electricity prices
β
Total Cost ($)
Risk Index
Power Exchange and Electricity Price Correlation
0 0.25 0.50 0.75 1
5500.6 5603.43 5619.23 5636.16 5639.2
2340.1 1856.9 1844.3 1839.62 1839.6
−0.438 0.5732 0.5894 0.5954 0.5958
NSC Index and Electricity Price Correlation −0.4237 −0.3004 −0.2992 −0.2986 −0.2983
∑ES ∈ ΩES SOCES, t WES ∑ES ∈ ΩES WES
(35)
This index represents the amount of available storage capacity in each hour relative to the total storage capacity of the system. The index value is shown for both risk-neutral and risk-averse operators in Fig. 8. As can be seen in the case of neutral risk, the value of the index is a function of energy price, therefore, during high energy price hours, the network storage decreases, and it increases during low energy price hours. In contrast, the index is at a high level most of the time in risk-averse mode. It is noteworthy that, the index decreases in the last hours of operation in Fig. 8. As it is expected, failure probability decreases during the last hours of operation; moreover, increases in energy prices during the last hours of operation caused the operator to use the full capacity of the network during the final hours of the operating period. To numerically evaluate the subjects discussed in this section, the correlation coefficient method is used. Table 4 shows a comparison of the correlation coefficient of the battery charging/discharging scheduling relative to the electricity price in both risk-neutral and riskaverse modes. As can be seen, the correlation index is positive in both cases. This suggests an increase in the level of resource generation with an increase in energy prices in both cases. Further, the correlation index in risk-averse mode for ES5, ES14, ES20, and ES33 sources are 51.5%, 57.7%, 47.7%, and 12.7% smaller, respectively, compared to riskneutral mode. This is more evident in Table 5, where the correlation coefficient of the NSC index and the expected purchased power from the upstream network relative to the electricity price in both riskneutral and risk-averse modes are shown. As expected in the risk-neutral case, the available network storage and the purchased power from the upstream network follows electricity price considering the priority of the economic benefits to the operator.
considered as a function of the energy price. As in hours with low electricity price, the power purchased from the upstream network increased, and the participation of MT resources and battery storage systems in load supply declined. In contrast, with a rise in energy prices, the power purchased from the upstream network was reduced and the network resource output power increased. In a neutral risk situation, the entire capacity of the storage systems is in line with the economic benefits considering the priority of the economic benefits to the security interests and reliability. Therefore, according to Fig. 5, the behavior of battery storage systems in a neutral risk situation follows a uniform harmony, as they are in charging mode during low energy price hours and in discharging mode during high energy price hours. 4.2. Risk-averse operation strategy In this scenario the β parameter is set to one. The results for MT 8
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online version, at https://doi.org/10.1016/j.ijepes.2019.105791.
Therefore, the battery is discharged/charged and the network storage capacity is decreased/increased with an increase/decrease in electricity prices. Likewise, the power purchased from the upstream network is decreased/increased and the participation of DERs in power generation increases/decreases with increasing/decreasing electricity prices. The negative correlation index confirms the above analysis. In risk aversion, due to the priority of risk reduction for the operator, the correlation of the network storage index to energy price decrease by 29.5%. The power purchased from the upstream network is not following energy price; rather, it is determined by customer load and operation constraints. Therefore, in a risk-averse environment, the MT resources are operated at a maximum level to reduce risks due to failure and electricity price uncertainties. The battery storage systems are operated in such a way as to reduce interruption risk and the risk of electricity price uncertainties based on load level. Table 6 shows the operation cost, risk index, and correlation between the purchased power from the upstream network and the NSC index to energy prices for different risk-taking operators according to the optimal network operation strategy. According to Table 6, total costs increased and the network risk index decreased as expected by increasing the risk-taking factor (β ). As mentioned before, the operation strategy is to decrease the purchased power from the upstream network and increase the participation of DERs in the customer load supply to reduce the risk due to price uncertainty. Hence, the purchased power from the upstream network is not following energy price, but is determined by the network load level and other network constraints by increasing the risk factor. However, the optimal strategy to reduce interruption risk is to operate MT resources at a maximum capacity and to maintain the network storage capacity at an acceptable level. Therefore, the correlation between network storage and energy price is diminished in this situation. As shown in Table 6, the correlation coefficient between the NSC index and energy cost decrease and benefits from reliability and network security are preferred to the economic benefits by increasing the operator risk-taking factor.
References [1] Chowdhury S, Crossley P. Microgrids and active distribution networks. The Institution of Engineering and Technology; 2009. [2] Farhangi H. The path of the smart grid. IEEE Power Energ Mag 2010;8. [3] Zheng C, Li W, Liang Q. An energy management strategy of hybrid energy storage systems for electric vehicle applications. IEEE Trans Sustain Energy 2018;9:1880–8. [4] Arefifar SA, Ordonez M, Mohamed YA-RI. Energy management in multi-microgrid systems—development and assessment. IEEE Trans Power Syst 2017;32:910–22. [5] Saint-Pierre A, Mancarella P. Active distribution system management: a dual-horizon scheduling framework for dso/tso interface under uncertainty. IEEE Trans Smart Grid 2017;8:2186–97. [6] Malekpour AR, Pahwa A. Stochastic networked microgrid energy management with correlated wind generators. IEEE Trans Power Syst 2017;32:3681–93. [7] Zhuang P, Liang H, Pomphrey M. Stochastic multi-timescale energy management of greenhouses with renewable energy sources. 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IEEE Trans Power Syst 2009;24:398–406. [14] Hatziargyriou N, Asano H, Iravani R, Marnay C. Microgrids. IEEE Power Energy Mag 2007;5:78–94. [15] Li J, Ma X-Y, Liu C-C, Schneider KP. Distribution system restoration with microgrids using spanning tree search. IEEE Trans Power Syst 2014;29:3021–9. [16] Xu Y, Liu C-C, Schneider KP, Tuffner FK, Ton DT. Microgrids for service restoration to critical load in a resilient distribution system. IEEE Trans Smart Grid 2018;9:426–37. [17] Liu X, Shahidehpour M, Li Z, Liu X, Cao Y, Bie Z. Microgrids for enhancing the power grid resilience in extreme conditions. IEEE Trans Smart Grid 2017;8:589–97. [18] Wang Y, Xu Y, He J, Liu C-C, Schneider KP, Hong M, et al. Coordinating multiple sources for service restoration to enhance resilience of distribution systems. IEEE Trans Smart Grid 2019:1. [19] Bajpai P, Chanda S, Srivastava AK. A novel metric to quantify and enable resilient distribution system using graph theory and choquet integral. IEEE Trans Smart Grid 2018;9:2918–29. [20] Ding T, Lin Y, Li G, Bie Z. A new model for resilient distribution systems by microgrids formation. IEEE Trans Power Syst 2017. [21] Yuan C, Illindala MS, Khalsa AS. Modified viterbi algorithm based distribution system restoration strategy for grid resiliency. IEEE Trans Power Delivery 2017;32:310–9. [22] Wang Z, Wang J, Chen C. A three-phase microgrid restoration model considering unbalanced operation of distributed generation. IEEE Trans Smart Grid 2018;9:3594–604. [23] Wang Z, Wang J. Self-healing resilient distribution systems based on sectionalization into microgrids. IEEE Trans Power Syst 2015;30:3139–49. [24] Zadsar M, Haghifam MR, Larimi SMM. Approach for self-healing resilient operation of active distribution network with microgrid. IET Gener Transm Distrib 2017;11:4633–43. [25] Mousavizadeh M, Haghifam M-R, Shariatkhah MH. A linear two-stage method for resiliency analysis in distribution systems considering renewable energy and demand response resources. Appl Energy 2018;211:443–60. [26] Baboli PT, Shahparasti M, Moghaddam MP, Haghifam MR, Mohamadian M. Energy management and operation modelling of hybrid ac–dc microgrid. IET Gener Transm Distrib 2014;8:1700–11. [27] Rockafellar1 RT, Uryasev S. Optimization of conditional value-at-risk. J Risk 2000;2:21–42. [28] Yuan H, Li F, Wei Y, Zhu J. Novel linearized power flow and linearized opf models for active distribution networks with application in distribution lmp. IEEE Trans Smart Grid 2018;9:438–48. [29] Billinton R. Reliability evaluation of power systems. Springer Science & Business Media; 2013. [30] Sebtahmadi SS, Borhan Azad H, Aghay Kaboli SH, Didarul Islam M, Mekhilef S. A pso-dq current control scheme for performance enhancement of z-source matrix converter to drive im fed by abnormal voltage. IEEE Trans Power Electron 2018;33:1666–81. 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5. Conclusion In this paper, the decision-making model of a smart distribution system resilient operation are formulated as a two-stage optimization problem in order to create trade-off between operation cost and reduction of operational network risks, including the risk of interruption due to equipment failure, the uncertainty of renewable energy resources, customer load, and energy prices. The utility of this model is verified using the modified IEEE 33-bus network. The results recommend that the sum of operation costs and risk index are higher under the neutral-risk mode in compare with the risk-averse mode. Additionally, an operator’s strategy to reduce the interruption risk is to maintain network storage capacity in the operation period. According to the results, this is achieved by increasing the charge/discharge frequency of battery storage systems. This strategy reduces battery lifetime. However, the proposed operation strategy to reduce the interruption risk can only be used on days when the equipment outage probability is high due to operation condition or climate condition. The correlation between charging/discharging of storage systems and energy price is decreased regarding the higher priority of risk reduction for the risk-averse operator. Declaration of Competing Interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. Appendix A. Supplementary material Supplementary data associated with this article can be found, in the 9
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Electrical Power and Energy Systems journal homepage: www.elsevier.com/locate/ijepes
Two stage risk based decision making for operation of smart grid by optimal dynamic multi-microgrid☆ M. Zadsara, S.Sina. Sebtahmadia, a b
T
⁎,1
, M. Kazemib, S.M.M. Larimia, M.R. Haghifama
Faculty of Electrical and Computer Engineering, Tarbiat Modares University, Jalal Ale Ahmad, Nasr, Tehran, Iran Faculty of Electrical Engineering, University of Shahreza, Shahreza, Iran
A R T I C LE I N FO
A B S T R A C T
Keywords: Active distribution network Distributed energy resource Emergency operation Energy storage system Multi-microgrid Renewable energy resource
Resect severe economic losses caused by distribution system equipment outage have highlighted the importance of improving the system resiliency and reliability. In active distribution networks (ADNs), the distributed energy resources (DERs) managed by dynamic isolated microgrids in contingency mode provide an alternative approach to enhance the system resiliency and continue supplying critical loads after equipment outage. How to incorporate this ADNs capability into a short-term DERs scheduling is a challenging issue. In response to this challenge, in this paper, a two stage risk based decision making framework for operation of ADNs is proposed to coordinate 24-h DERs’ scheduling and outage management scheme, in a way to be immune against contingency by creating optimal dynamic multi-micrigrid, micro-turbine and energy storage island operating and load shed plan. The first stage is the normal operation condition that operation cost should be minimized considering of the uncertainties of renewable resources, the electricity price and customers loads. The second stage is the operation in contingency condition. In the second stage, the main objective is to keep the shed load at minimum level. Numerical results and sensitivity analysis from modified 33-bus IEEE network are further discussed to demonstrate the efficiency of the solution approach.
1. Introduction Active distribution networks (ADNs) is formed owing to the addition of distributed energy resources (DER) into the traditional electrical distribution systems [1]. Complicated technical constraints and economic decisions related to ADNs, requires a new management system. This management system, which is known as distributed management system (DMS). It is a hardware and software organization that is responsible for making decisions about the optimal operation of the system [2]. Proposing various methods for increasing the applicability of DMS have been widely investigated in [3–9]. An energy management strategy for hybrid energy storage system is proposed based on the Pontryagin’s minimum principle in [3]. A new energy management strategy and a novel probabilistic index for evaluating operation efficiency is proposed in [4] for performing in multi-microgrid system by various DERs. A dual-horizon rolling scheduling model for planning and operation of ADNs is proposed in [5]. In [6,7], the optimal energy
management strategies are proposed to response the renewable resources uncertainties. In energy management strategy proposed by [9], the two optimization levels are proposed, which consist of the microgrid centralized controller and local controllers and the distribution network in order to response load uncertainties. In the referred papers, the proposed methods for DMS are designed for the normal operation. Additionally, DMS should be designed in a way to be able of handling emergency situations by outage management system (OMS) application by providing a robust operation plan. Robust operation refers to the ability of the system to withstand events in an efficient manner, while ensuring the least interruption in the supply of electricity and enabling a quick recovery and restoration to the normal operation state [10]. Considering the importance of this topic, several studies have been conducted in this regard [11–13]. A fully decentralized multi-agent system is presented to handle the distribution system restoration (DSR) problem, considering incorporation of DERs in [11]. An agent-based paradigm for self-healing protection systems is introduced in [12], using the graph theory-based expert system. A new DSR procedure is
☆
Grant numbers: Iran’s National Elites Foundation – Bright Spark. Corresponding author. E-mail addresses: [email protected] (M. Zadsar), [email protected] (S.S. Sebtahmadi), [email protected] (M. Kazemi), [email protected] (S.M.M. Larimi), [email protected] (M.R. Haghifam). 1 The paper’s supervisor. ⁎
https://doi.org/10.1016/j.ijepes.2019.105791 Received 10 August 2019; Received in revised form 11 December 2019; Accepted 17 December 2019 0142-0615/ © 2019 Elsevier Ltd. All rights reserved.
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Nomenclature
β ηiES γs κ i, s , t
Sets
i, j, s, t DsF , T ΩB , ΩS , ΩMT , ΩES ΩMaster , MT
Index of buses, scenarios, time Fault period in scenario s, Horizon time Set of network’s buses, uncertainty scenarios Set of micro-turbine buses, energy storage buses Set of buses connected to master micro-turbine
κiRamp
Risk- aversion coefficient Charging/discharging efficiency of battery storage in bus i Probability of scenario s Customer damage weight factor in scenario s in bus i at time t Ramp rate limit of the micro-turbine located at bus i [kW/ min]
Variables Voltage angle difference between bus i and bus j in scenario s in time t The auxiliary variable used in CVaR calculation η PiD, s,,shed / QiD, s,,shed Active/reactive load curtailment in bus i in scenario s t t at time t [kW/kVar] dch Pich Charge/discharging power of the battery storage located , t / Pi, t at bus i at time t [kW] PiG, t / QiG, t Active/Reactive power generated by the micro-turbine located at bus i at time t [kW] PijP, s, t / QijQ, s, t Active/reactive power of the line connecting bus i to bus j in scenarios at time t [kW/kVar] Psex, t / Qsex, t Active/Reactive power transacted with the upstream network in scenario s at time t [kW/kVar] Ss The auxiliary variable used for CVaR calculation in scenario s SOCi, t State of charge of the battery storage located at bus i at time t Voltage magnitude of bus i in scenario s in time t [kV] Vi, s, t Charging/discharging state of battery storage in bus i in ψich ,t scenario s at time t υij, s, t Binary variable indicating the connection status of the line connecting bus i to bus j in scenario s at time t
δij, s, t
Parameters
aiG , biG , ciG Micro-turbine cost function coefficients Customer damage cost in scenario s at bus i at time t CDFi, s, t [$/kWh] PijP, max / QijQ, max Maximum active/reactive power capacity of the line connecting bus i to bus j [kW/kVar] Pich, max Maximum charge rate of battery storage in bus i [kW] Pidch, max Maximum discharge rate of battery storage in bus i [kW] Predicted active power output of renewable energy rePiR, s, t sources in bus i in scenario s at time i [kW] PiD, s, t / QiD, s, t Predicted active/reactive load demand in bus i in scenario s at time t [kW] PiMT , max / QiMT , max Maximum active/reactive power capacity of the micro-turbine located at bus i [kW/kVar] ri, j / x i, j Line resistance/reactance connecting bus i to j P sub, max / Q sub, max Maximum active/reactive power capacity of substation [kW/kVar] SOCimin Minimum energy limit of battery storage in bus i SOCimax Maximum energy limit of battery storage in bus i Vimin/ Vimax Minimum/Maximum voltage limit in bus i [kV] V Set Voltage magnitude of slack bus i [kV] WiES Capacity of battery storage in bus i [kWh] α Confidence factor of CVaR method
However, in the first category, only the set of switches status are evaluated considering DERs’ generation level as fixed parameters; and, in the second category, the DERs’ optimum generation level is determined assuming a predetermined multi-MG configuration plan. This paper covers the gap of the literature by proposing a model handling both of the decision variable sets consist of DERs scheduling and switches status in contingency mode, simultaneously. This point of view has been also investigated in [24,25]. In [24,25] the real-time operation model is proposed, in which the post-contingency operation strategy (consist of DERs’ scheduling and MG formation) is determined based on post-contingency conditions in order to reduce load shed and improve system’s resiliency. Thus, in [24,25], only an outage scenario is investigated. However, in this paper, the 24-h scheduling plan has been determined in order to reduce system interruption risks due to several outage scenarios. Therefore, the coordinating adaptive dynamic mutiMG and DERs’ islanding operation in the proposed model improves system’s efficiency and reliability. The proposed model compromises between operation cost and operation risk in ADNs. In order to achieve a resilient adaptive distribution network, different items are performed including dynamic Multi-MG configuration, DGs inslanded operation mode, load shedding plan and energy storage systems. One of the advantages of using storage systems is that some of the operation risks are covered in normal and contingency modes. In order to reduce the interruption risks in distribution systems, storage systems should be fully charged to be ready to feed the islanded loads in the contingency condition [25]. However, in the normal operation condition, the storage system is charged and discharged periodically to minimize the operation cost of purchasing energy from the upstream network [3]. These operation strategies are against each other and a balanced strategy is required to cover both of the operational cost and
developed based on graph models and a knapsack problem formulation using the dispersed generation availability in [13]. The above-mentioned researches do not consider some advantage of ADNs’ options such as scheduling of distributed generation (DG) and energy storage system resources in island operation mode. According to the IEEE standard 1547.4, dividing ADNs into multiMGs and island operating is an effective method to improve system reliability during the contingency condition [14]. The idea of optimal switching with the purpose of MG creation under emergency condition is investigated in [15–23]. Several studies which is conducted in this area can be categorized in two groups. The first group of researches have proposed methods for switching and forming of multi-MGs during contingency condition with different objectives energy not supplied and non-served load [16–21], as well as the number of switching [15]. In this category, scheduling of DERs are not involved in the planning problem and it is assumed that their optimal level of generation are predetermined. Moreover, the only decision variable of this category is the status of switches. The second category [22,23], the multi-MG configuration are predetermined and the planning problem is solved from the perspective of DERs’ optimum generation level. Across the first category, this group concerns about the scheduling of DERs and switches status are not a decision variable here. The simplifying assumptions made in these categories are could affect the optimally of their results. For example, the predetermined configuration assumed in the second category, is not necessarily the optimal one at the time of fault occurrence. The optimal configuration can be changed depending on the fault location, DERs generation level, the renewable resources available power and etc. Furthermore, the optimal value for both of the decision variable sets, i.e., the set of switches status and the set of generation levels of DERs, should be simultaneously evaluated. 2
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2.1. Stage 1: Normal operation scheduling
operational risk. This paper provides an operation model of energy storage systems with the aim of compromising the operational risk of a ADN versus operating costs considering the ADNs’ capabilities including self-healing. Thus, considering the capability of DERs’ islanding operation, especially energy storage system, by multi-MG formation in contingency mode in the evaluating the optimal 24-h operation plan is the most important contribution of this paper. Also, in this paper a new viewpoint of energy storage systems and their applications in smart distribution systems is presented. The energy storage systems are used as reserve resources in contingency mode modeled by the second stage. Considering this point of view, a minimum level of state of charge (SOC) can be evaluated as the required reserve capacity of the distribution systems. This paper proposes a framework, in which the minimum SOC of energy storage systems can be calculated. The main contributions of the paper are highlighted bellow:
In this operation condition, a 24-h scheduling plan is determined. The decision variables at this level are the scheduling of MT generation units, charging/discharging of battery storage systems and the energy level transacted with the bulk power system. The objective of this optimization problem is to minimize the operation cost of the AND in a 24-h period. Various sources of uncertainty can affect the scheduling problem of normal condition, such as, the generation of renewable resources, the electricity prices and the demand level. In this section, the uncertainty of the mentioned resources are modeled through scenario tree method [26]. The conditional value at risk (CVaR) is used to measure and control the risk level of the decision making process. According to the above-mentioned explanation, the optimization problem of the normal condition can be formulated as follow. 2.1.1. Objective function The objective function has been formulated as follow,
1. A two-stage resilient operational framework of ADNs that considers compromising between operation cost and operational risk due to equipment outage and renewable resources uncertainties is proposed. 2. For calculating the operational risk, renewable resources uncertainties and equipment outage has been modeled by scenario tree method and sequential Monte Carlo method, respectively. In the contingency mode the second stage is proposed, where the optimal Multi-MG structure is calculated by new graph-based algorithm in order to reduce load shed cost. 3. Compared with models presented in the literature, the proposed model lead system operator to schedule DERs in 24-h horizon by taking into account ADNs’ capabilities such as adaptive control and protection and dynamic multi-MG operation in contingency mode. 4. In the proposed model, coordinating the islanding operation of DERs, especially the energy storage system, and an adaptive dynamic Multi-MG in contingency mode improves the system’s efficiency and reliability. 5. This paper by proposing a risk-based decision making platform has investigated that how neutral-risk and risk-averse operation strategy could overshadow the scheduling of DERs, especially energy storage systems, under uncertainties of equipment outages, renewable resources, loads, and electricity price. 6. This paper proposed a framework, in which the minimum SOC of energy storage systems can be calculated as the required reserve capacity of the distribution systems in order to achieve an adaptive resilient operation plan.
⎛ min (1 − β ) ⎜ ∑ ⎝t∈T +
∑ s ∈ Ωs
ai + bi PiG, t + ci (PiG, t )2
∑ i ∈ ΩMT
⎛ γs ⎜ ∑ ⎝t∈T
CiShed , s, t +
∑ i ∈ ΩB
∑ t∈T
⎛ 1 +β ⎜ 1 − α ⎝
∑ s ∈ Ωs
⎞⎞ Csex, t ⎟ ⎟ ⎠⎠
⎞ γs Ss − η⎟ ⎠
(1)
D,shed CiShed , s, t = κ i,s,t CDFi,s P i,s,t
(2)
Csex, t
(3)
=
πs, t Psex, t
The objective function (1) consists of two parts, the operation cost and therisk index measured by CVaR The operation cost is made up of three terms; (1) The operation cost of MTs, (2) The cost or benefit of transacting power with the upstream network and, (3) The load interruption cost. The interruption cost,that are presented in (2), is evaluated from the second stage optimization, and then it will be sent back to the first stage as an input parameter. The power exchange cost or benefit are presented in (3). 2.1.2. Risk model constraints Considering the probability density function of exchange cost, interruption cost, and α as a parameter indicating the right tail probability of that function, CVaR α is defined as the expected value in the worst 100(1 − α )% of the probability density function. The CVaR term has been considered as a linear formulation by introducing additional variables and constraints [27] are presented in (4) and (5).
The paper is organized into five sections; Section 2introduces the problem formulation; Section 3describes the proposed solution methodology, Section 4presents the numerical results, and Section 5concludes and closes this paper.
⎛ ⎜∑ ⎝t∈T
2. Problem formulation
∑
CiShed , s, t +
i ∈ ΩB
∑ t∈T
⎞ Csex, t ⎟ − η ⩽ Ss ∀ s ∈ ΩS ⎠
Ss ⩾ 0 ∀ s ∈ ΩS
In this paper, a framework for short-term scheduling of smart grids is provided, in a way to be immune against outages of distribution elements. This purpose can be achieved by making decision considering two different conditions. The first one is the normal operation condition in which, the operation cost should be minimized. The second one is the operation during contingency condition. In the second condition, the main objective is to keep the shed load at minimum level. Decisions made in each conditions, i.e., normal and contingency conditions, can affect the other one. Thus, the decision making frameworks for each condition should be designed in a way to reach a compromise. In this section, the decision making formulation for each condition is provided.
(4) (5)
2.1.3. Upstream power exchange constraints The limit of energy transmission from upstream network is presented in (6) and (7).
− P Sub,max ⩽ Psex, t ⩽ P Sub,max − Q Sub,max ⩽ Qsex, t ⩽ Q Sub,max
∀ s ∈ ΩS , t ∈ T ∀ s ∈ ΩS , t ∈ T
(6) (7)
2.1.4. Energy storage constraints The battery storage system operation constraints are provided by (8)–(11). In this paper, the time interval (Δt ) for determining the 24-h scheduling plan is set to one hour. 3
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M. Zadsar, et al. ch ch,max 0 ⩽ Pich ∀ i ∈ ΩES , t ∈ T , t ⩽ ψi, t Pi
(8)
ch dch,max 0 ⩽ Pidch ∀ i ∈ ΩES , t ∈ T , t ⩽ (1 − ψi, t ) Pi
(9)
SOCi, t = SOCi, t + Pich ,t
SOCimin
⩽ SOCi, t ⩽
2.1.9. Load shed constraints Constraints (25) and (26) guarantees that bus voltage magnitude is within predefined range.
0.9 ⩽ Vi, s, t ⩽ 1.1
ηiES Δt
− Pidch ,t
WiES
SOCimax ,t
Δt
∀ i ∈ ΩES , t ∈ T
ηiES WiES
∀ i∈
ΩES ,
t∈T
(10)
Vi, s, t = V
(11)
In the previous subsection, the operation in normal condition is discussed. It is seen that one of the most important objective’s items is the amount of shed load, which can be evaluated from the second stage optimization problem. The input of the second stage is the optimum operation determined by the first stage. Moreover, the second stage optimization problem, uses the output of the first stage optimization problem, i.e., the generation level of MTs, the energy transactions with the upstream network and the battery storage systems schedules, as the base point of its analysis. Thus, switching for creating the optimal adaptive multi-MG configuration is calculated in stage 2 based on operation condition in stage 1. In order to model the outage conditions in this stage, the probabilistic nature of elements failure should be involved in the optimization problem. Another source of uncertainty can be introduced as the failure of elements, including their failure time and their repair duration. In this way, the uncertain parameters of this problem can be categorized into two groups. The first group is the uncertainty associated with the wind and solar generations, energy prices, and customer loads. These sources of uncertainties are modeled in the first level of optimization problem using scenario tree method [24]. The second group is the uncertainty of elements failure, which should be modeled in the second level optimization problem. The nature of this group of uncertainty is different from the uncertainties of the first group, i.e., uncertainties of the first level. The elements failure include uncertainties of the location, occurrence time, and clearance time (repair or replace time) of the failure. Therefore, sequential Monte Carlo methods presented in [29] have been used to model the uncertainty of this group. An illustrative example for the generated scenarios according to (27) and (28) presented in [29] that models failure of n elements is depicted in Fig. 1.
(12)
QijQ, s, t = −αijL (δi, s, t − δj, s, t ) + βijL (Vi, s, t − V j, s, t )
−
PijP,max
⩽
PijP, s, t
⩽
PijP,max
∀ ij ∈ ΩB , s ∈ ΩS , t ∈ T
− QijQ,max ⩽ QijQ, s, t ⩽ QijQ,max ∀ ij ∈ ΩB , s ∈ ΩS , t ∈ T In this model αijL and calculate as αlL = rl/(rl2
βijL
(13) (14) (15)
are distribution line characteristic that are
+ xl2) and βlL = xl /(rl2 + xl2) ,respectively [28].
2.1.6. Power balance in each bus Constraints of the active and reactive power balance in each bus are presented in (16)–(19). dch PiG, t + PiR, s, t + Pich , t − Pi, t −
∑
υij, s, t PijP, s, t
j ∈ ΩB
=PiD, s, t − PiD, s,,shed ∀ ij ∈ ΩB , s ∈ ΩS , t ∈ T t
QiG, t −
∑
(16)
υij, s, t QijQ, s, t = QiD, s, t − QiD, s,,shed t
j ∈ ΩB
∀ ij ∈ ΩB , s ∈ ΩS , t ∈ T Psex, t −
∑
(17)
υ1j, s, t P1Pj, s, t = P1,Ds, t − P1,Ds,,shed t
j ∈ ΩB
∀ j ∈ ΩB , s ∈ ΩS , t ∈ T Qsex, t −
∑
(18)
υ1j, s, t Q1Qj, s, t = Q1,Ds, t − Q1,Ds,,shed t
j ∈ ΩB
∀ j ∈ ΩB , s ∈ ΩS , t ∈ T
(19)
∀ i ∈ ΩMT , t ∈ T
QiMT ,min ⩽ QiG, t ⩽ QiMT ,max
∀ i ∈ ΩMT , t ∈ T
|PiG, t − PiG, t − 1| ⩽ κiRamp PiMT ,max
∀ i ∈ ΩMT , t ∈ T
1 TTF = − ln (U1) λ
(27)
1 TTR = − ln (U2) μ
(28)
At this level of optimization, different types of corrective action can be used to restore the system with the minimum level of load shedding. These actions in a smart distribution system can be listed as: (1) optimal switching for dynamic multi-MG structure creation in accordance with the fault location, (2) using the stored energy in the energy storage
2.1.7. MTs’ constraints The MT generation operation constraints including their maximum active and reactive power capacity and their ramp rate limits are provided by (20)–(22), respectively. In constraint (20), the minimum limit of MTs output (PiMT,min ) is assumed set to be zero.
0 ⩽ PiG, t ⩽ PiMT ,max
(20) (21) (22)
2.1.8. Load shed constraints Load shedding constraint are presented in (23) and (24).
0 ⩽ PiD, s,,shed ⩽ PiD, s, t ∀ i ∈ ΩB , s ∈ ΩS , t ∈ T t
(23)
0 ⩽ QiD, s,,shed ⩽ QiD, s, t ∀ i ∈ ΩB , s ∈ ΩS , t ∈ T t
(24)
(26)
2.2. Stage 2: Operation during contingency conditions
PijP, s, t = βijL (δi, s, t − δj, s, t ) + αijL (Vi, s, t − V j, s, t )
∀ ij ∈ ΩB , s ∈ ΩS , t ∈ T
δi, s, t = 0
∀ i = 1, s ∈ ΩS , t ∈ T
2.1.5. Lines flow constraints Line flow equations of the distribution system are imposed to the problem by (12)–(15).
∀ ij ∈ ΩB , s ∈ ΩS , t ∈ T
set
(25)
Fig. 1. Uncertainty model of equipment failure. 4
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upstream network (30), load flow equations, and operation constraints including the thermal capacity constraints of the network lines (31), the load shedding limits (32), and the network bus voltage limits (34).
systems. Considering these corrective actions, the optimization problem of the second level can be formulated as follows.
min
∑ ∑
CiShed , s, t
t ∈ DsF i ∈ ΩB
(29) 3. Metaheuristic-based solution methodology
s.t.
(6) − (7)
∀ t ∈ DsF
(12) − (17)
∀ ij ∈ ΩB , t ∈ DsF
(31)
(18) − (19)
∀ i = 1, j ∈ ΩB , t ∈ DsF
(32)
(23) − (25)
∀ i ∈ ΩB , t ∈ DsF
(33)
(26)
∀ i = {1, Ωmaster , MT }, t ∈ DsF
The formulation presented in this paper is nonlinear and nonconvex due to the complex nature of the problem. The two-stage structure of the optimization problem and the existence of complex binary constraints associated with radial configuration of the network and multi-MG creation constraints, make it difficult to use mathematical approaches to solve this problem. In addition, at this level of complexity, heuristic methods with their high ability to handle the complicated binary constraints can result in a better solution. Among the heuristic methods, particle swarm optimization (PSO) method is used in this paper, because of simple concept, easy implementation, robustness to control parameters, and computational efficiency when compared with mathematical algorithm and other heuristic optimization techniques [30]. The proposed framework for optimization is presented in Fig. 2. According to the proposed algorithm, variable set in stage 1 in a physical 2-dimensional search space (variable and horizon time) are generated using the PSO method for all particles. For each particle, PSO optimization procedure is performed on normal network structure in case of normal condition. This part of optimization process solves the
(30)
(34)
This objective function (29) is the interruption cost of the customer loads based on the time varying customer damage function during the contingency mode. It should be noted that the optimization problem of the second stage is performed for a specific scenario. In a specific scenario, all the stochastic variables of the problem, including the renewable resources generation, energy prices, customers’ load, outage location, outage time occurrence, and the repair time, are determined. Therefore, the index of the scenario used in the equation of this section is merely for a specific scenario. The constraints in the second-stage optimization problem include the power exchange limit with the
Fig. 2. Optimization algorithm and proposed method framework. 5
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optimization problem are: (1) the operation scheduling of MT and energy storage systems obtained in the first stage, (2) the stochastic data of uncertain parameters includes the renewable resources generation and the faults scenario data, i.e., the fault locations, their occurrence and repair times. The main outputs of the second stage optimization problem are: (1) the optimal switching for muli-MG forming, and (2) load shedding plan. υij, s, t variables in ((16)–(19)indicate the connection status of the line connecting bus i to bus j in scenario window s at time t (connect = 1, disconnect = 0). Thus, at each time of each scenario window where the equipment outage has not occurred, these variables shall be considered based on the basic network’s configuration under normal operation conditions. With the outage occurring in any hour of the scenario window, the these variables are controlled in stage 2 in order to create the optimal multi-MG formation. The second stage optimization problem has a complex structure due to the existence of complicated binary constraints associated with Multi-MG forming through optimal switching actions. In addition, the ac power flow constraints, radial structure constraints, the presence of MT with frequency control capability in each MG, increase the complexity of the second stage optimization problem. Due to this level of complexities, the second stage optimization problem is solved by a full search method. In this paper, the spanning forest search algorithm presented in [24] is used to form MGs through switching actions. This method ensures that the MGs structure in the post-contingency period is radial and there is at least one MT in each MG with capability of voltage and frequency control. In this method, all possible switching states are generated to create feasible multi-MG structures. The operating constraints are checked using ac load flow, and the objective function of the second stage is calculated in each structure. The best structure of multi-MG is chosen as an optimal operation strategy in contingency mode. Certainly, the run time of the second stage will be increased by increasing the number of controllable switches in the network. In the second stage, the method presented in [15] is used to simplify the network graph and limit possible responses in order to improve the algorithm’s efficiency. After assessing all scenarios, the objective function of the first stage can be calculated. Then, all particles’ position and velocity, i.e., the first layer problem variables, are updated in the PSO optimization algorithm [30]. This process continues until the convergence criteria are met.
Table 1 MTs’ data. Bus
15 18 19 25 29
aG
bG
cG
($)
($/kW)
($/kW2)
P MT , max (kW)
P MT , min (kW)
Factor
0.025 0.039 0.023 0.010 0.010
0.071 0.055 0.062 0.075 0.061
0.00005 0.00006 0.00005 0.00006 0.00005
400 375 50 350 350
0 0 0 0 0
0.95 0.95 0.95 0.95 0.95
Power
Table 2 ESs’ data. Bus
5 14 20 33
WES (kWh)
P ch, max (kW)
P dch, max (kW)
Initial
200 150 150 200
100 75 75 100
100 75 75 100
0.95 0.3 0.5 0.1
SOC
Fig. 3. Electricity price.
4. Numerical results The modified IEEE 33-bus network that was presented in [24], is used for the numerical studies. This network includes 33 load points, 37 lines, and 5 lines with normally open switches for maneuvering on the network [31]. The MTs and ESs data are presented in Table 1 and Table 2, respectively. Also, the electricity price is shown in Fig. 3. All test data are available online at https://www.dropbox.com/s/ o08911dagpo5qlq/InputData.xlsx?dl=0. The failure of distribution lines are considered in this paper. As noted before, wind and solar generation, customer load, and electricity price uncertainties are considered by generating scenarios based on the discrete probability distribution function. According to the five states for each of this stochastic variables, a total of 625 uncertain scenarios are generated. Therefore, the number of uncertain scenarios has been reduced to 30 in order to reduce the computational time while maintaining accuracy using the probability distance scenario reduction method [32] in this study. Meanwhile, in addition to the uncertain scenarios, the Monte Carlo simulation scenarios for the modeling of the network equipment failure greatly increase the time of the calculations. In these circumstances, the number of scenarios should be selected to reduce the computational time while maintaining the accuracy of calculations. Taking into account 100,000 scenarios (days) for modeling network failure, the variance of the objective function in a 50times calculation is less than 1, which is reasonable. According to the proposed model, the coefficient β in the objective function models the
Fig. 4. MT sources and expected purchased power from upstream network in risk-neutral scenario. Table 3 values of various parameters of the objective function
MT operation cost Expected exchange cost Expected interruption cost Total cost Risk index
Neutral-risk
Risk-averse
2144.6 3349.70 6.29 5500.6 2340.1
3116.04 2516.83 5.11 5639.2 1839.6
first stage formulated by (1)–(26). In scenarios involving failures, the outage management scheme is implemented in contingency mode. In fact, this part of optimization process solves the second stage optimization problem, in which the shed load is minimized by taking remedial actions explained in Section 2.2. The inputs of the second stage
6
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Fig. 5. ESs SOC and charging/discharging in risk-neutral scenario.
operator risk-taking. Therefore, the operator strategy is investigated by changing this parameter. The program of network operation in two modes of neutral risk (β = 0) and risk aversion (β = 1) is presented and analyzed. The results are then presented for different scenarios. 4.1. Neutral risk operation strategy In this scenario the β parameter is set to zero. The results for MT resources and the expected value of purchasing electricity from the upstream network, optimal operation of the ES systems, and the state of charge for each ES system from this operating scenarios is shown in Figs. 3 and 4, respectively. According to Table 3, the sum of operation costs and risk index in the neutral-risk scenario are 5500.6 $ US and 2340.1 $ US, respectively. Based on Fig. 4, in the case of neutral risk, the amount of power purchased from the upstream network can be
Fig. 6. MT sources and expected purchased power from upstream network in risk-averse scenario.
Fig. 7. ESs SOC and charging/discharging in risk-averse mode. 7
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resources and the expected value of purchasing electricity from the upstream network, optimal operation of the ES systems, and the state of charge for each ES system from this operating scenarios is shown in Fig. 6 and Fig. 7, respectively. According to Table 3, the total operation costs and risk index in risk-averse case, are 5639.2 $ US and 1839.6 $ US, which shows a 2.5% increase and 21.39% decrease. In contrast the risk-natural scenario, a risk-averse operator prefers to provide security benefits and reduce network risks. Due to the high interruption costs of customers, the operation strategy has led to a reduced network interruption. The MT resources are operated at their maximum power as are shown in Fig. 6 due to the probability of failure occurrence at any time of day as well as taking into account the ramp rate of MTs, in most hours of the day. Furthermore, reducing the power purchased from the upstream network and increasing the participation of DERs in power generation presents the operator with a lower electricity price risk.
Fig. 8. Network storage capacity index.
Table 4 the charging/discharging of ES systems to electricity price correlation coefficient ES Unit
Neutral-risk
Risk-averse
Comparison (%)
ES5 ES14 ES20 ES33
0.305 0.3218 0.396 0.2304
0.1476 0.1383 0.207 0.2012
−51.6 −57 −47.7 −12.7
4.3. Sensitivity analysis and comparison Exploring the operation strategy of battery storage systems provides an impressive result. The optimal operation strategy is expected to reduce the network interruption risk, keeping the SOC at a minimum level for storage systems. Hence, the operation scheduling is performed in such a way that the storage systems SOC does not get lower than the minimum SOC. However, according to Fig. 7, the optimal strategy for reducing interruption risk is not to maintain the minimum SOC for storage units. In contrast, the operator strategy to reduce interruption risk is to maintain the stored energy for the network based on the results of the proposed model. The battery charging/discharging scheduling is carried out in such a way as to preserve the amount of stored energy in the network while increasing the level of storage systems generation and share of DERs so that the operator is less affected by energy price risk. To illustrate this, the index is calculated as the Network Storage Capacity (NSC) which, shown as follow,
Table 5 the NSC index and the expected value of purchased power to electricity price correlation coefficient
NSC Index Power Exchange
Neutral-risk
Risk-averse
Comparison (%)
−0.4237 −0.438
−0.29832 0.5985
−29.5 -
NSCt =
Table 6 Risk index and correlation of purchased power from the upstream network and NSC index to electricity prices
β
Total Cost ($)
Risk Index
Power Exchange and Electricity Price Correlation
0 0.25 0.50 0.75 1
5500.6 5603.43 5619.23 5636.16 5639.2
2340.1 1856.9 1844.3 1839.62 1839.6
−0.438 0.5732 0.5894 0.5954 0.5958
NSC Index and Electricity Price Correlation −0.4237 −0.3004 −0.2992 −0.2986 −0.2983
∑ES ∈ ΩES SOCES, t WES ∑ES ∈ ΩES WES
(35)
This index represents the amount of available storage capacity in each hour relative to the total storage capacity of the system. The index value is shown for both risk-neutral and risk-averse operators in Fig. 8. As can be seen in the case of neutral risk, the value of the index is a function of energy price, therefore, during high energy price hours, the network storage decreases, and it increases during low energy price hours. In contrast, the index is at a high level most of the time in risk-averse mode. It is noteworthy that, the index decreases in the last hours of operation in Fig. 8. As it is expected, failure probability decreases during the last hours of operation; moreover, increases in energy prices during the last hours of operation caused the operator to use the full capacity of the network during the final hours of the operating period. To numerically evaluate the subjects discussed in this section, the correlation coefficient method is used. Table 4 shows a comparison of the correlation coefficient of the battery charging/discharging scheduling relative to the electricity price in both risk-neutral and riskaverse modes. As can be seen, the correlation index is positive in both cases. This suggests an increase in the level of resource generation with an increase in energy prices in both cases. Further, the correlation index in risk-averse mode for ES5, ES14, ES20, and ES33 sources are 51.5%, 57.7%, 47.7%, and 12.7% smaller, respectively, compared to riskneutral mode. This is more evident in Table 5, where the correlation coefficient of the NSC index and the expected purchased power from the upstream network relative to the electricity price in both riskneutral and risk-averse modes are shown. As expected in the risk-neutral case, the available network storage and the purchased power from the upstream network follows electricity price considering the priority of the economic benefits to the operator.
considered as a function of the energy price. As in hours with low electricity price, the power purchased from the upstream network increased, and the participation of MT resources and battery storage systems in load supply declined. In contrast, with a rise in energy prices, the power purchased from the upstream network was reduced and the network resource output power increased. In a neutral risk situation, the entire capacity of the storage systems is in line with the economic benefits considering the priority of the economic benefits to the security interests and reliability. Therefore, according to Fig. 5, the behavior of battery storage systems in a neutral risk situation follows a uniform harmony, as they are in charging mode during low energy price hours and in discharging mode during high energy price hours. 4.2. Risk-averse operation strategy In this scenario the β parameter is set to one. The results for MT 8
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online version, at https://doi.org/10.1016/j.ijepes.2019.105791.
Therefore, the battery is discharged/charged and the network storage capacity is decreased/increased with an increase/decrease in electricity prices. Likewise, the power purchased from the upstream network is decreased/increased and the participation of DERs in power generation increases/decreases with increasing/decreasing electricity prices. The negative correlation index confirms the above analysis. In risk aversion, due to the priority of risk reduction for the operator, the correlation of the network storage index to energy price decrease by 29.5%. The power purchased from the upstream network is not following energy price; rather, it is determined by customer load and operation constraints. Therefore, in a risk-averse environment, the MT resources are operated at a maximum level to reduce risks due to failure and electricity price uncertainties. The battery storage systems are operated in such a way as to reduce interruption risk and the risk of electricity price uncertainties based on load level. Table 6 shows the operation cost, risk index, and correlation between the purchased power from the upstream network and the NSC index to energy prices for different risk-taking operators according to the optimal network operation strategy. According to Table 6, total costs increased and the network risk index decreased as expected by increasing the risk-taking factor (β ). As mentioned before, the operation strategy is to decrease the purchased power from the upstream network and increase the participation of DERs in the customer load supply to reduce the risk due to price uncertainty. Hence, the purchased power from the upstream network is not following energy price, but is determined by the network load level and other network constraints by increasing the risk factor. However, the optimal strategy to reduce interruption risk is to operate MT resources at a maximum capacity and to maintain the network storage capacity at an acceptable level. Therefore, the correlation between network storage and energy price is diminished in this situation. As shown in Table 6, the correlation coefficient between the NSC index and energy cost decrease and benefits from reliability and network security are preferred to the economic benefits by increasing the operator risk-taking factor.
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5. Conclusion In this paper, the decision-making model of a smart distribution system resilient operation are formulated as a two-stage optimization problem in order to create trade-off between operation cost and reduction of operational network risks, including the risk of interruption due to equipment failure, the uncertainty of renewable energy resources, customer load, and energy prices. The utility of this model is verified using the modified IEEE 33-bus network. The results recommend that the sum of operation costs and risk index are higher under the neutral-risk mode in compare with the risk-averse mode. Additionally, an operator’s strategy to reduce the interruption risk is to maintain network storage capacity in the operation period. According to the results, this is achieved by increasing the charge/discharge frequency of battery storage systems. This strategy reduces battery lifetime. However, the proposed operation strategy to reduce the interruption risk can only be used on days when the equipment outage probability is high due to operation condition or climate condition. The correlation between charging/discharging of storage systems and energy price is decreased regarding the higher priority of risk reduction for the risk-averse operator. Declaration of Competing Interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. Appendix A. Supplementary material Supplementary data associated with this article can be found, in the 9