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CSEE JOURNAL OF POWER AND ENERGY SYSTEMS, VOL. 2, NO. 4, DECEMBER 2016

Flexible Active Distribution System Management Considering Interaction with Transmission Networks Using Information-gap Decision Theory Tianyang Zhao, Jianhua Zhang, Member, CSEE, and Peng Wang, Senior Member, IEEE

Abstract—In this paper, a flexible management method is proposed for an active distribution system (ADS) with distributed energy resources (DERs) integrated, where DERs can provide spinning reserves to transmission networks. This method, based on the information-gap decision-making theory (IGDT) theory, could be of use to the ADS operator (ADSO) from either the opportunistic or robust perspective when reserve is called by the independent system operator (ISO). Two IGDT uncertainty models are employed to depict the characteristics of reserve uncertainty in centralized and decentralized control frameworks. The reactive power of each DER is managed by the ADSO in the immunity functions, which are reformulated as bi-level biobjective optimization problems. A hybrid multi-objective differential evolutional algorithm (MODE) is proposed to solve the optimization problems. The relationship between the uncertainty levels and robust/opportunistic limits is revealed by the Pareto fronts obtained by MODE. Effectiveness of the proposed method is demonstrated based on simulation results of a 33-bus and 123bus test system. Index Terms—Active distribution system, bi-level multiobjective optimization, information-gap decision-making theory, reserve uncertainty.

I. I NTRODUCTION ITH increasing penetration of distributed energy resources (DERs) into distribution networks, interactions between DERs and power systems are becoming active. Various methods have been proposed to realize the interactions between DERs and transmission networks, i.e., energy exchange, regulation reserves, spinning reserves and territory reserves [1]. In essence, distribution networks form the bridge between DERs and transmission networks, and it is important that they are flexible as a way to guarantee these interactions. An active distribution system (ADS) is one such effective solution for achieving flexible operations of distribution networks [2].

W

Manuscript received February 29, 2016; revised May 30, 2016 and July 31, 2016; accepted September 23, 2016. Date of publication December 30, 2016; date of current version October 20, 2016. This work was supported by the Fundamental Research Funds for the Central Universities (No. 2014XS09) and Chinese Scholarship Council of the Ministry of Education. T. Y. Zhao (corresponding author, e-mail: [email protected]) and J. H. Zhang are with State Key Laboratory of Alternate Power System with Renewable Energy Resources, North China Electric Power University, Beijing 102206, China. P. Wang is with the School of Electrical and Electronic Engineering, Nanyang Technological University, Singapore S639798. DOI: 10.17775/CSEEJPES.2016.00052

Numerous methods have been proposed to provide flexibility for ADS [2]–[4]. These methods can be categorized into deterministic [2]–[4] and non-deterministic methods [5]– [7]. The deterministic method is only applied to scenarios where outputs of sources and demands are fixed [2]–[4]. Non-deterministic methods, on the other hand, deal with uncertainties (e.g., forecasting errors of demands), and can be modeled using three approaches: probability distribution function [5], fuzzy set [6], and interval number [7]. Additionally, different decision-making methods are employed to manage the corresponding types of uncertainties. For example, the chance-constrained optimization method is applied to mitigate the adverse impacts of the stochastic outputs of renewable energy resources on ADS [5]. Similarly, the fuzzy technique is implemented to model the uncertainty and assess operation risk [6], and a two-stage robust optimization technique has been proposed to deal with the interval uncertainty of wind power output [7]. The aforementioned uncertainty models [5]–[7] only reveal the relationship between uncertainties and environmental factors, e.g., solar illumination and wind speed. However, uncertainties within ADS are also affected by interactions between DERs and transmission networks because when DERs provide reserves to transmission networks [1], their outputs need to be adjusted according to the signals obtained from the independent system operator (ISO) [8] or the DER aggregators, such as the local distribution system [9]. The signals obtained from the ISO are typically influenced by frequency derivation, area control error, short-time forecasting information [8], and control methods (e.g., fixed gain or dynamic gain in automatic generation control [10]). On the other hand, the signals obtained from aggregators are affected by the signals sent from the ISO and the aggregators’ own operation strategies, e.g., minimize the cost [9]. Other uncertainty factors include frequency derivation and area control error [11], as well as reserve utilization of DERs, which is affected by different control methods. Based on these factors, it is clear then that a novel decision making method is required to manage reserve uncertainty of DERs. Information-gap decision-making theory (IGDT) is a powerful decision tool that looks at the severe gap between predicted and actual variables of interest and then is able to model the reserve uncertainty in a more realistic fashion [12], [13]. IGDT can provide both robust and opportune strategies based on decision makers’ attitudes towards risks [12]. This makes

c 2016 CSEE 2096-0042

ZHAO et al.: FLEXIBLE ACTIVE DISTRIBUTION SYSTEM MANAGEMENT CONSIDERING INTERACTION WITH TRANSMISSION NETWORKS USING INFORMATION-GAP DECISION THEORY

IGDT more attractive than other available risk-averse decision making methods [5]–[7]. As such, IGDT has been applied in multiple uncertainty scenarios, e.g., Genco’s bidding [15], [16], restoration of distribution networks [17], and electric vehicle aggregator management [18]. Currently, existing ADS management methods are mostly focused on the selfregulation aspect of distribution networks [2]–[8]. When DERs provide reserves to transmission networks, ADS guarantees efficient interactions between DERs and transmission networks. In this paper, a novel IGDT based method is proposed for ADS to manage the reserve uncertainty. This method enables ADS to optimally utilize the reactive power provided by DERs, while enhancing the interactions between DERs and transmission networks. The main contributions of this paper can be summarized as follows: 1) Two IGDT based models are employed to depict the characteristics of reserve uncertainty, where the reserve of DERs is called by the transmission networks in centralized and decentralized control methods, respectively. 2) Two IGDT immunity functions are proposed for ADS to optimally utilize the reactive power provided by DERs from risk-averse and risk-seeking perspectives. 3) Finally, the immunity functions are reformulated as multi-objective bi-level optimization problems, which are then solved using a hybrid multi-objective differential evolutional (MODE) algorithm. The relationship between robust/opportunistic levels and performance requirements is revealed using the Pareto fronts obtained by MODE. This paper is organized as follows. An active distribution system (ADS) and its corresponding deterministic management model are presented in Section II. This model serves as the system model for IGDT. The uncertainty model and performance requirements in IGDT are depicted in Section III. The solving method for IGDT is proposed in Section IV. Simulation and conclusions are shown in Section V and Section VI, respectively.

the reactive power of DGs to improve operation efficiency of the ADS. It is also important to note that this work is carried out under a regulated power market.

DG1

DGi

DGN

Independent system operator

Distribution network Central power plants

Fig. 1.

Transmission network Reserve flow Reserve called signal Power flow Information flow within the active distribution network

Active distribution system operator

Note: All DGs are schedulable.

Framework of DERs, ADS, and transmission networks.

ADSO’s management should meet a set of constraints, e.g., power balance, voltage magnitude limitations, and reactive power adjustment capacity constraint. Based on the branch power flow for radial networks [21], the management model of the ADSO could be depicted as follows: X F = min rij lij (1) (i,j)∈E

X

s.t.

PG,i −

i∈Φj

X

X

PL,i = Pij − rij lij −

i∈Θj

II. ACTIVE D ISTRIBUTION S YSTEM M ANAGEMENT

Pjk ,

k:(j,k)∈E

∀j ∈ B X i∈Φj

QG,i −

X

(2) X

QL,i = Qij − xij lij −

i∈Θj

∀j ∈ B vj = vi − 2(rij Pij + xij Qij ) +

2 (rij

(3) +

x2ij )lij ,

∀(i, j) ∈ E (4)

Pij2 + Q2ij , ∀(i, j) ∈ E vi 2 Pij2 + Q2ij ≤ Smax,ij , ∀(i, j) ∈ E 2 Vmin,i

Qjk ,

k:(j,k)∈E

lij ≥

DERs encompass a variety of distributed technologies, i.e. internal combustion, external combustion, energy storage, renewables, and fuel cells [19]. When DERs provide reserves to transmission networks, they must meet certain grid codes, e.g., DERs’ capacities should be larger than the minimal eligible size (500 kW in CAISO [19]), and spinning reserve must be maintained for at least two hours [20]. Thus, only schedulable distributed generators (DGs), e.g., diesel generators and combined heat and power, are considered here. The integration of renewable sources and flexible loads, e.g., photovoltaic generations and electric vehicles would be considered in future work. The framework of the transmission networks for ADS and DERs is explicitly shown in Fig. 1. DGs provide spinning reserve to transmission networks and their active power is controlled by the ISO. An active distribution system operator (ADSO) acts as the system operator of the distribution networks with schedulable DGs integrated. The ADSO manages

77

≤ vi ≤

2 Vmax,i ,

∀i ∈ N

QG min,i, ≤ QG,i ≤ QG max,i , ∀i ∈ Φ

(5) (6) (7) (8)

where PG,i and QG,i are the active and reactive power output of the ith generator; PL,i and QL,i are the active and reactive power demand of the ith user; Vi is the voltage magnitude at the ith bus; vi := V2i ; rij and xij are the resistance and reactance of line (i, j); Iij is the complex current flowing from buses i to j, lij := I2ij ; Pij and Qij are the active and reactive power on line (i, j); Smax,ij is the apparent power limitation of line (i, j); Vmin,i and Vmax,i are minimum and maximum voltage magnitude limitation at the ith bus; QG min,i and QG max,i are the minimum and maximum reactive output of the ith generator; Φ and Θ are the sets of generators and users, and Φ := ΦR ∪ ΦL , where ΦL stands for eternal power system connected by ADS, Φi and Θi are the sets of generations and users connecting to the ith bus; B is the set of buses; E is the set of branches.

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CSEE JOURNAL OF POWER AND ENERGY SYSTEMS, VOL. 2, NO. 4, DECEMBER 2016

As shown in (1)–(8), the management problem is a voltage control problem. Since the voltage magnitude at each bus is treated as constraint, as shown in (7), the objective function refers to minimize real power losses, which is depicted by (1). Equations (2)–(3) represent the KCL equations, and (4) represents the KVL equations. The conic relaxation for power, voltage, and current is shown in (5). This relaxation is tight for the ADS management model [21]. Equation (6) is the thermal limitation for the distribution lines. The voltage magnitude limitation for each bus is shown in (7). The reactive power capacity of each DG is depicted by (8), which is subjective to technical characteristics of the generator and its active power output [22]. Maximum and minimal reactive power capacities of the ith DG are given as follows: q 2 , −C QG min,i = max(− SG,i − PG,i G,i PG,i ), ∀i ∈ ΦR (9) q 2 ,C (10) QG max,i = min( SG,i − PG,i G,i PG,i ), ∀i ∈ ΦR Cmin,i ≤ CG,i ≤ Cmax,i , ∀i ∈ ΦR

(11)

where SG,i is the capacity of the ith DG; CG,i is the parameter limiting reactive power of ith DG with fixed active power, and Cmin,i and Cmax,i are lower and upper boundary of CG,i . CG,i is determined by the minimum power factor PF min , which is set by the ADSO for the ith DG, where PF min := cos(tan−1 (CG,i )). To obtain the convex hull of bilinear items (CG,i PG,i ) in (9) and (10), the McCormick envelope method [23] is employed as follows: Cmin,i PG,i + CG,i PG min,i − Cmin,i PG min,i ≤ wi

(12)

Cmax,i PG,i + CG,i PG max,i − Cmax,i PG max,i ≤ wi

(13)

wi ≤ Cmax,i PG,i + CG,i PG min,i − Cmax,i PG min,i

(14)

wi ≤ Cmin,i PG,i + CG,i PG max,i − Cmin,i PG max,i

(15) A. IGTD Based Uncertainty Modeling

where wi := CG,i PG,i . Furthermore, when PG,i , PG min,i and PG max,i are equal to each other, wi = CG,i PG,i . For convenence, the ADS management model is depicted in the following compact form: F (Y ) = min cT X ( X Aeq X = beq , AX ≤ b, X ∈ Γ s.t. Ceq X + Deq Y = deq , CX + DY ≤ d

of each DG ∈ ΦR is given. In addition, the amount and direction of spinning reserves called by the ISO is difficult to be forecasted and is uncertain [11]. This uncertainty might deteriorate or benefit the operation of ADS. Thus, an IGDT based management method is proposed for the ADSO to manage the reserve uncertainty of DGs ∈ ΦR . It should be noticed that the information gap in this paper is defined as the real amount and direction of spinning reserve called by the ISO and the spinning reserve capacity cleared in the spinning reserve market. IGDT is a non-probabilistic decision-making theory, which expresses the idea that uncertainty may be either pernicious or propitious. The idea is realized by constructing two “immunity functions”: the robustness function expresses the immunity to failure, while the opportuneness function expresses the opportunity to windfall gain [12]. Both functions should be evaluated with respect to different values of uncertainty parameters under specific performance requirements. Consequently, the relationship between uncertainty levels and robust/opportunistic limits set by the ADSO can be obtained. This relationship helps the ADSO to compare different reactive power management strategies that satisfy system performance criteria as per the ADSO’s requirements or aspirations. An IGDT based decision making is specified by three component models: 1) System model; 2) uncertainty model; 3) performance requirements. The system model has been proposed in Section II. Only the uncertainty model and performance requirements are studied in this section.

(16)

where X is the integrated representation for the decision parameters of the ADSO, [PG,i,i∈ΦL , QG,i,i∈Φ , Pij,(i,j)∈E , Qij,(i,j)∈E , lij,(i,j)∈E , vi,i∈N , CG,i,i∈ΦR , wi,i∈ΦR ]. Y is the integrated representation for active power outputs of DGs ∈ ΦR , PG,i,i∈ΦR . Γ is the rotatecone constraints (5). Moreover, problem (16) is a linear conic optimization problem (LCP) [24] and serves as the system model in IGDT. The IGDT based management is presented in Section III.

III. IGDT BASED ADS M ANAGEMENT M ODELING As the active power of each DG ∈ ΦR is controlled by the ISO and the reactive power of each DG ∈ ΦR is managed by the ADSO, the ADS management model presented in Section II can only be applied when the active power output

The DERs’ contribution to LFC has been well-described in [8]. To model the spinning reserve uncertainty more realistically, two uncertainty models are proposed to represent this uncertainty in centralized and decentralized control methods, which are two popular methods for the ISO to manage scalable DERs. The spinning reserve capacity of each DG ∈ ΦR is known before implementing the ADS management (16). Taking into account the amount of uncertainty and spinning reserves directions called by the ISO, the active power output of each DG ∈ ΦR deviates from the set-point with unknown deviation and known boundaries. This characteristic makes it suitable to apply the fractional error model [12] in the IGDT to constrain the reserve uncertainty within an expandable envelope. The two uncertainty models are shown as follows: R − RD,i PG,i − PG0,i − α U,i 2 Uc (α) = {Y : ≤ αψi , RU,i + RD,i 2 − RD,i ≤ PG,i − PG0,i ≤ RU,i , ∀i ∈ ΦR }, α ≥ 0, ψi > 0 X X Ud (α) = {Y : −α RD,i ≤ PG,i − PG0,i i∈GR

i∈GR

(17)

ZHAO et al.: FLEXIBLE ACTIVE DISTRIBUTION SYSTEM MANAGEMENT CONSIDERING INTERACTION WITH TRANSMISSION NETWORKS USING INFORMATION-GAP DECISION THEORY

≤α

X

RU,i , −RD,i ≤ PG,i − PG0,i ≤ RU,i },

i∈GR

1≥α≥0

(18)

where PG0,i is the set-point of the ith DG ∈ ΦR , RD,i and RU,i are the up spinning reserve and down spinning reserve of the ith DG, α is the index of reserve uncertainty, ψi is a parameter determining the shape of the envelope, and Uc and Ud are the functions for the spinning reserve uncertainty in centralized and decentralized LFC. The reserve uncertainty U (α) could be depicted by either Uc in (17) or Ud in (18). Remark 1: By employing the centralized model (17), the ISO can manage the active power of DGs ∈ ΦR directly. The parameter ψi , i ∈ ΦR , in (17) determines the shape of the uncertainty envelope and is predetermined by the ISO. The active power output range of DGi , i ∈ ΦR , could be depicted as follows: 1 + ψi 1 − ψi RU,i − α RD,i , PG0,i [PG0,i + α 2 2 1 − ψi 1 + ψi RU,i − α RD,i ]. (19) +α 2 2 Through properly setting ψi , the active power output of DGi , i ∈ ΦR would not fall into the following zones: 1 − ψi 1 + ψi RU,i − α RD,i ] (20) 2 2 1 + ψi 1 − ψi [PG0,i + RU,i , PG0,i + α RU,i − α RD,i ] (21) 2 2 This feature enables the stable operation of generators with special operation requirememts, e.g., forbidden operating zones of the thermal units [14]. Furthermore, if all ψi , i ∈ ΦR are set to 1, (17) could be simplified as follows: [PG0,i − RD,i , PG0,i + α

Uc (α) = {Y : −αRD,i ≤ PG,i − PG0,i ≤ −αRU,i , ∀i ∈ ΦR }, 1≥α≥0

(22)

Remark 2: By employing the decentralized model (18), the ISO should send the reserve signals to the aggregator of DGs ∈ ΦR or the ADSO. The ISO does not need to control each DG ∈ ΦR directly. The active power of each DG ∈ ΦR is controlled by the aggregator or the ADSO allowing for the total reserve requirement from ISO. B. Performance Requirements Performance requirements for making a decision are evaluated on the basis of robustness and opportuneness functions [12]. The ADSO aims to manage the reserve uncertainty shown in Section III-A. A robustness function guarantees a certain profit expectation under adverse future conditions. IGDT also examines beneficial opportunity arising from uncertainty, to obtain windfall profit. Both functions optimize uncertainty parameter as follows [12]: α = max{α : max F (Y ) ≤ πC }

(23)

β = min{β : max F (Y ) ≤ πW }

(24)

Y

Y

where πC is the critical limit, which is always to be satisfied when the real active power outputs of DGs ∈ ΦR increase the

79

real power losses, and πW is the opportunistic limit, which can be achieved when the real active power outputs of DGs ∈ ΦR decrease the real power losses. There are two outcomes of interest when using IGDT: 1) A value of robustness (25), which enables a risk-averse, conservative approach to deal with the reserve uncertainty; 2) an opportunistic value (26), which reveals the speculative, risk-seeking approach. The two immunity functions of robustness and opportuneness are shown as follows: 1) Robustness function max α

s.t.

α c T X ≤ πC max Y Y ∈ U (α) X ∈ SOL[(16)]

(25)

where SOL[(16)] represents the set of solutions for (16); 2) opportuneness function min β

s.t.

β cT X ≤ π W min Y Y ∈ U (β) X ∈ SOL[(16)].

(26)

As shown in (25) and (26), both robustness and opportuneness functions are tri-level optimization problems: the uncertainty level is determined in the upper level optimization, the active power of DGs ∈ ΦR is optimized in the middle level optimization, and the reactive power of each DG ∈ ΦR is adjusted in the lower level optimization. IV. S OLVING M ETHOD Immunity functions (25) and (26) are always formulated as a bi-level optimization problem [14], [18]. This problem is reformulated to a single stage optimization problem based on Karush-Kuhn-Tucker (KKT) conditions for optimization problems. Furthermore, the reformulated optimization problem should be solved under different performance requirements, to reveal the relationship between robust/opportunistic requirements and uncertainty level. This process might be timeconsuming. Since the middle and lower level optimization problems in (26) are a min-min optimization problem, the opportuneness function (26) could be treated as a bi-level optimization problem. The bi-level optimization problem could be solved efficiently by reformulating its lower level problem as complementaryconstraints based on KKT optimal conditions, formulating a mathematical programming with equilibrium constraints (MPEC) [23]. However, the integration of complementary constraints might result in infeasibility of this MPEC. This is owing to those inequality constraints (5) that are active when the optimal solution of (16) is found [21]. Strong duality conditions for LCP do not hold [24], and the duality gap might exist for this LCP, resulting in the infeasibility of the MPEC. What is worse, the lower level optimization problem of (22)

CSEE JOURNAL OF POWER AND ENERGY SYSTEMS, VOL. 2, NO. 4, DECEMBER 2016

is a max-min optimization problem. The prevailing method to solve the max-min problem is to reformulate the inner min optimization as a max optimization problem. Based on the same reason as stated for opportuneness function (26), the duality gap exists. As a result, immunity functions (25), (26) cannot be solved using traditional methods. A novel solution is proposed to solve the robustness and opportuneness functions as follows: 1) First, the middle optimization problems (25) and (26) are treated as equality constraints. 2) Next, the critical/opportunistic limits are treated as the second objective, resulting in two bi-objective bi-level optimization problems. 3) Then, the reformulated optimization problems are solved using a hybrid MODE, in which the classical optimization technique is deployed to solve the lower level optimization problem. A. Reformulation of IGDT Functions 1) Reformulated Robustness Function The reformulation of robustness function, (25), is based on the concept of Pareto domination, which is widely employed in evolutionary multi-objective (EMO) algorithms [25]. The Pareto domination between two solutions is defined as follows: Definition 1 [25]: A solution x is said to dominate the other solution y, if both the following conditions are true: 1) The solution x is no worse than y in all objectives. Thus, the solutions are compared based on their objective function values. 2) The solution x is strictly better than y in at least one objective. When the ADSO is risk-averse, a solution of (25) with smaller α and bigger πC is preferred. This is for the reason that: 1) As it is risk-averse, the ADSO should be informed of the smallest level of uncertainty α that it must resist, when the critical limit πC is given. The smallest uncertainty level α serves as a warning index for the ADSO to manage the reserve uncertainty. 2) For a fixed uncertainty level α, the ADSO should be notified of the maximal real losses, i.e., the critical limit πC . Fig. 2 shows a set of feasible solutions belonging to robustness function and their corresponding first non-domination front [25]. As shown in Fig. 2, solutions 1–3 are dominated by solutions 4–8, and solutions 4–8 are on the first nondomination front, i.e., not dominated by any other solution of robustness function. Thus, only solutions 4–8 would be provided to the ADSO. These solutions reveal the relationship between the critical limits and uncertainty levels for the ADSO. One of these solutions is selected according to the ADOS’s passive preference towards risk. The bi-objective bi-level optimization problem for robustness function (25) is shown as follows: min {α, −πC }

α,πC ,Y

s.t.

Y ∈ U (α) cT X = π C X ∈ SOL[(16)].

(27)

1 8 7

0.8 Uncertainty Level

80

6

2

0.6

5

1 3 4

Non-dominated front

0.4 0.2

18 are feasible solutions of (25)

0 0.02 0.04

0.06

0.08 0.1 0.12 0.14 Critical Limit (MWh)

0.16

0.18

Fig. 2. A set of feasible solutions belonging to robustness function and the corresponding first non-domination front.

As shown in (27), critical limit πC , uncertainty level α and active power outputs of DGs ∈ ΦR , Y are optimized to minimize uncertainty level and maximize critical limits simultaneously. In lower level optimization, the reactive power of DGs ∈ ΦR is determined to minimize the real power losses, as shown in (16). Furthermore, as cT X = πC in (27), the critical limit πC equals to the real power losses in the lower level optimization. 2) Reformulated Opportuneness Function When the ADSO is risk-seeking, a solution of (26) with smaller β and smaller πW is preferred. This is because 1) when the opportunistic limit πW is given, the ADSO could be notified of the minimal uncertainty level β, which the ADSO should bear in order to obtain the windfall, and 2) when the uncertainty level β is fixed, the ADSO could be informed of the opportunistic limit πW , i.e., the minimal real power losses that can be possibly obtained. A set of feasible solutions belonging to opportuneness and its corresponding first non-domination front is shown in Fig. 3. As shown in Fig. 3, solutions 1–3 are dominated by solutions 4–8, and solutions 4–8 are on the first non-domination front, i.e., not dominated by any other solution of (26). Similar to (27), solutions 4–8 reveal the relationship between the opportunistic limits and uncertainty levels. One of these solutions is selected according to the ADOS’s opportunistic preference towards risk. The bi-objective bi-level optimization problem for the opportunistic function (26) is shown as follows: min {β, πW }

β,γW ,Y

s.t.

Y ∈ U (β) cT X = π W X ∈ SOL[(16)].

(28)

As shown in (28), opportunistic limit πW , uncertainty level β, and active power of DGs ∈ ΦR Y are optimized to minimize uncertainty level β and opportunistic limit πW . In lower level optimization, reactive power of DGs ∈ ΦR is determined to minimize the real power losses, based on model (16). As cT X = πW shown in (28), the opportunistic limit πW also equals to the real power losses in the lower level optimization.

ZHAO et al.: FLEXIBLE ACTIVE DISTRIBUTION SYSTEM MANAGEMENT CONSIDERING INTERACTION WITH TRANSMISSION NETWORKS USING INFORMATION-GAP DECISION THEORY

1

Uncertainty Level

0.8 2

0.6 0.4

Non-dominated front

1 3

4 5 6

7

8

0.2 18 are feasible solutions of (26) 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 Opportunistic Limit (MWh) Fig. 3. A set of feasible solutions belonging to opportuneness function and the corresponding first non-domination front.

B. Solving Procedure According to search techniques employed, traditional methods to solve multi-objective optimization problems could be classified into two categories: conversion to single objective optimization problem based on higher-level information and EMO algorithms [25]. Since the weight between the uncertainty levels and robustness/opportunistic limits is difficult to obtain, EMO is adopted in this paper. The frameworks of EMO algorithms are Pareto dominancebased or decomposition based [26]. A Pareto dominancebased MODE [27] is applied to solve (18) and (19). For the lower problem (16), a classical interior-point method based optimization technique is applied to solve this LCP, i.e., Gurobi [28] is employed in our work. The solving procedure of the hybrid MODE is depicted in Fig. 4. The details about initialization, mutation, recombination and selection in MODE could be found in [27]. Input Parameters Initialization

Meet the stopping criteria

Yes

Output Pareto fronts

No Mutation

Recombination Individual assessment Selection/ constraint handling Upper level optimization for (α,πC,Y) or (β,πW,Y)

Fig. 4.

Yes Y

Solving (16)

Is (16) feasible? No

Solving (29)

Lower level optimization for X

Flow chart of the hybrid MODE for bi-level bi-objective problems.

81

Since the generation of off-springs in MODE is random, the active power outputs of DGs ∈ ΦR , Y , might cause over voltage to ADS, resulting in infeasibility of the lower level optimization problem (16). To assess the violation of the lower problem corresponding to a given Y , an optimal recovery model is proposed as follows: T − V C(Y ) = min I T ε + Ieq (ε+ eq + εeq ) + − X,ε,εeq ,εeq − Aeq X = beq + ε+ eq − εeq AX ≤ b + ε s.t. X∈Γ Ceq X + Deq Y = deq , CX + DY ≤ d ε, ε+ , ε− ≥ 0 eq eq

(29)

− where ε, ε+ eq , and εeq are vectors of relaxation variables for the inequality and equality linear constraints in (16), and I and Ieq are vectors full of 1.

V. C ASE S TUDY To demonstrate the effectiveness of proposed IGDT based management method, a case study has been carried out on two test systems, including a 33-bus and 123-bus distribution network. Details of both networks are available in [7]. The simulations were implemented in MATLAB version 8.1.0.604, running on a PC with Intel Core i7-4700 MQ and 8 GB RAM. A. Case Description For the 33-bus test system, there are 4 DGs connected at buses 18, 22, 25, and 33. Additionally, the set-points of DGs are 1.0 MW, 0.5 MW, 0.5 MW and 0.5 MW, respectively. The up spinning reserve capacity of each DG is 0.5 MW, 0.3 MW, 0.5 MW and 0.5 MW, respectively. The down spinning reserve is set to 0.0 MW, 0.3 MW, 0.5 MW and 0.5 MW, respectively. The capacity of each DER is 1.5 MVA, 1 MWA, 1 MVA and 1 MVA, respectively. Moreover, three scenarios were proposed for comparison: Scenario I: Active power of each DG is controlled by the ISO through centralized control, and the reserve uncertainty follows (17), where ψi , i ∈ ΦR , is set to 1. Scenario II: Active power of each DG is controlled by the ISO through decentralized control, and the reserve uncertainty follows (18). Scenario III: The ISO sends the reserve requirements to the ADSO. The active power of DGs ∈ ΦR is controlled by the ADSO, and the reserve uncertainty follows (18). For the 123 bus test system, there are 10 DGs connected at buses 12, 25, 44, 54, 57, 64, 67, 78, 86 and 110. The setpoint, up spinning reserve, down spinning reserve capacity and capacity of each DG is set to 0.5 MW, 0.5 MW, 0.5 MW, and 1 MVA, respectively. The population size of MODE is set to 80, and the maximum iteration of MODE is set to 300. B. 33-bus Test System 1) Results Under Different Scenarios Fig. 5 shows the Pareto fronts in robustness and opportuneness functions of the three scenarios. These Pareto fronts re-

CSEE JOURNAL OF POWER AND ENERGY SYSTEMS, VOL. 2, NO. 4, DECEMBER 2016

1 Scenario I Scenario II Scenario III

Uncertainty Level

0.8 0.2 0.6

0.15 0.1

0.4

0.05 0

0.2

0.034 0.036

0.038

0.04

0 0.02 0.04 Opportuneness

0.06

0.08

0.1 0.12 0.14 Robustness

0.16

0.18

Real Power Losses (MWh) Fig. 5. system.

Critical/opportunistic limits vs. uncertainty level for 33-bus test

The extreme solutions under each scenario are given in Table I. These extreme solutions in the opportuneness or robustness functions are almost the same under different scenarios, as shown in Table I. The small gap among these extreme solutions results in the equality of minimal real power losses or maximal real power losses under all three scenarios, i.e., 0.0351 MWh and 0.1632 MWh (Fig. 5). An interesting observation of opportuneness function in Table I is that the active power outputs of DGs ∈ ΦR under all three scenarios are almost the same while the uncertainty levels differ. This is owing to the difference between the uncertainty model (17) and (18); the uncertainty level is directly imposed on each DG ∈ ΦR in (17), while the uncertainty level is imposed on DGs ∈ ΦR . The difference enables the flexible operation of aggregators when the active power of each DG ∈ ΦR is controlled by the aggregator. TABLE I C OMPARISON OF E XTREME S OLUTIONS U NDER D IFFERENT S CENARIOS Attitude Scenarios Uncertainty level PG,1 DG1 PF min PG,2 DG2 PF min PG,3 DG3 PF min PG,4 DG4 PF min

Opportuneness I II III 0.76 0.17 0.17 1.00 1.00 1.00 0.95 0.95 0.95 0.27 0.24 0.24 0.92 0.89 0.90 0.88 0.87 0.87 0.88 0.87 0.87 0.70 0.70 0.70 0.70 0.70 0.70

Robustness I II III 1.00 1.00 1.00 1.50 1.50 1.50 1.00 1.00 1.00 0.80 0.80 0.80 0.97 0.98 0.98 0.00 0.00 1.00 1.00 1.00 1.00 0.00 0.00 1.00 1.00 1.00 1.00

The relationship between critical/opportunistic limits and uncertainty levels varies in the three scenarios. In scenario I, as observed from Fig. 5, the real power losses are fixed to 0.0351 MWh when the uncertainty level is bigger than 0.7617. Accordingly, if the ADSO wants to reduce the real

power losses to 0.0351 MWh, the uncertainty level it must bear is 0.7617. Alternatively, the maximal real power losses are 0.1632 MWh while the uncertainty level reaches 1. When the ADSO sets the critical limit to 0.1632 MWh, it could bear any level of reserve uncertainty. Fig. 6(a) and Fig. 6(b) show the minimal power factor of each DG in opportuneness function and robustness function, respectively. In opportuneness function, as shown in Fig. 6(a), the power factors of DG1 and DG2 remain constant, while the power factors of DG3 and DG4 increase. This is owing to the active power outputs of DGs ∈ ΦR can also reduce the real power losses in opportuneness function. On the other hand, as shown in Fig. 6(b), the power factors of DG3 and DG4 decrease, indicating DG3 and DG4 generate more reactive power in robustness function. As the power factors of DG1 and DG2 remain almost 1, they provide less reactive power. In scenario II, the real power losses are 0.0364 MWh under zero uncertainty, which is smaller than the losses of scenario I (Fig. 5). The real power losses can be reduced to 0.0351 MWh with uncertainty level 0.1720. Moreover, under zero uncertainty level, the maximal real power losses are 0.1573 MWh, which is 2.75 times bigger than the losses obtained in scenario I under zero uncertainty level. When the ADSO sets 1

Minimal Power Factor

veal the relationships between the critical/opportunistic limits and uncertainty levels for the 33-bus test system. They verify the merit of the proposed method, providing both robust and opportunistic solutions for the ADSO.

0.9 0.8 0.7 DG1 DG2 DG3 DG4

0.6 0.5

0

0.2

0.4 0.6 Uncertainty Level (a)

0.8

1 0.8 Minimal Power Factor

82

DG1 DG2 DG3 DG4

0.6 0.4 0.2 0

0

0.2

0.4 0.6 Uncertainty Level (b)

0.8

1

Fig. 6. Minimal power factor of each DER with respect to different uncertainty levels under scenario I. (a) Minimal power factor of each DER with respect to different uncertainty levels in the opportuneness function. (b) Minimal power factor of each DER with respect to different uncertainty levels in the robustness function.

ZHAO et al.: FLEXIBLE ACTIVE DISTRIBUTION SYSTEM MANAGEMENT CONSIDERING INTERACTION WITH TRANSMISSION NETWORKS USING INFORMATION-GAP DECISION THEORY

Minimal Power Factor

1

1 0.95 Minimal Power Factor

the critical reward within (0.0364, 0.1573) MWh, there would be no feasible operation plan. The minimal power factor of each DG in the opportuneness function and robustness function under scenario II is shown in Fig. 7(a) and Fig. 7(b), respectively. Fig. 7(a) could be explained similarly to the way of Fig. 6(a). However, DG4 ’ reactive power factor is jumping, as shown in Fig. 7(b).

0.9 0.85 DG1 DG2 DG3 DG4

0.8 0.75

0.95

0.7

0.9

0.65

0

0.85 DG1 DG2 DG3 DG4

0.8 0.75

0.05

0.1 0.15 Uncertainty Level (a)

0.2

Minimal Power Factor

1 0.8

DG1 DG2 DG3 DG4

0.4 0.2

0

0.15 0.05 0.1 Uncertainty Level (b)

0.2

0.8

0.2

Fig. 7. Minimal power factor for each DER with respect to different uncertainty levels under scenario II. (a) Minimal power factor of each DER with respect to different uncertainty levels in the opportuneness function. (b) Minimal power factor of each DER with respect to different uncertainty levels in the robustness function.

In scenario III, the Pareto front in opportuneness function is the same to scenario II. However, the Pareto front in robustness function would be significantly improved (Fig. 5). Hence, when the ADSO is risk-averse, the best way to manage the reserve uncertainty is to let the ADSO manage the active power of DGs ∈ ΦR , corresponding to total reserve requirement from ISO. The minimal power factor of each DG in opportuneness function and robustness function under scenario III is shown in Fig. 8(a) and Fig. 8(b), respectively. In Fig. 8(b), when uncertainty level is lower than 0.7205, DG3 and DG4 provide reactive power to reduce the real power losses. However, when uncertainty level is higher than 0.7205, power factors of DG3 and DG4 decrease significantly, indicating their reactive power support reduces. Thus, the real power losses increases rapidly when uncertainty level is higher than 0.7205 (Fig. 5). According to the above observation, the IGDT based method could provide both robustness and opportuneness reactive

DG1 DG2 DG3 DG4

0.6 0.4 0.2 0

0.6

0

Minimal Power Factor

0

0.05 0.1 0.15 Uncertainty Level (a)

1

0.7 0.65

83

0

0.2

0.4 0.6 Uncertainty Level (b)

0.8

1

Fig. 8. Minimal power factor for each DER with respect to different uncertainty levels under scenario III. (a) Minimal power factor of each DER with respect to different uncertainty levels in the opportuneness function. (b) minimal power factor of each DER with respect to different uncertainty levels in the robustness function.

power management solutions under different scenarios, enabling the flexible operation of ADS. 2) Method Verification To test the effectiveness of solutions obtained by the IGDT based method, a stochastic simulation with uncertainty level 0.15 is considered under scenario I and scenario III, respectively. Based on the relationship shown in Fig. 5, the opportunistic limits are 0.0388 MWh and 0.0351 MWh, and the critical limits are 0.0488 MWh and 0.0385 MWh in scenario I and scenario III, correspondingly. 1000 times randomly simulated reserves called by ISO are tested, where the reserve uncertainty follows (17)–(18) and α is set to 0.15. The simulation results obtained in scenario I and scenario III are shown in Fig. 9 and Fig. 10, respectively. As shown in Fig. 9, in the 1000 times simulation, the ADSO could possibly reduce the real power losses to 0.0395 MWh and 0.0351 MWh, almost the same to the opportunistic limits. Furthermore, it could be concluded from Fig. 10 that the maximal real power losses would never exceed 0.0488 MWh and 0.0385 MWh, smaller than the critical limits. The ADSO could always meet the technical need by applying the reactive power management plans provided by IGDT. This demonstrates the effectiveness of the proposed method.

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250

180 160

200

120

Fequency

Fequency

140

100 80

πW

150 πC 100

60

50

40

0

20

0.04

0 0.038 0.04 0.042 0.044 0.046 0.048 0.05 0.052 Real Power Losses (MWh) (a)

0.042 0.044 0.046 0.048 Real Power Losses (MWh) (a)

700 600

400

500 πW

300 Fequency

Fequency

350

250

400

πC

300

200

200

150

100

100

0 0.0355

50 0 0.035

0.036 0.037 0.038 0.039 Real Power Losses (MWh) (b)

0.0365 0.0375 Real Power Losses (MWh) (b)

0.0385

0.04

Fig. 9. Probability density functions of minimal real power losses randomly simulated under uncertainty scenarios for β = 0.15. (a) Probability density functions of minimal real power losses under scenario I for the solution of opportuneness function. (b) Probability density functions of minimal real power losses under scenario II for the solution of opportuneness function.

Fig. 10. Probability density functions of minimal real power losses randomly simulated under uncertainty scenarios for α = 0.15. (a) Probability density functions of minimal real power losses under scenario I for the solution of robustness function. (b) Probability density functions of minimal real power losses under scenario II for the solution of robustness function.

1 0.9 0.8

The simulation results of the 123-test system under the three scenarios are shown in Fig. 11. As shown in Fig. 11, the minimal real power losses in ADS are the same among the three scenarios, i.e., 0.0034 MWh. However, the minimal uncertainty levels are 0.4684, 0.3396 and 0.3388 for scenario I, scenario II and scenario III respectively. In addition, the maximum real power losses in ADS are also the same, i.e., 0.4062 MWh. In the opportuneness part of Fig. 11, the Pareto fronts of scenario II and scenario III are the same, dominating the Pareto fronts obtained by scenario I. Furthermore, the Pareto front obtained by scenario III dominates the Pareto fronts of scenario I and scenario II in opportuneness function. The results shown above further demonstrate the effectiveness of the proposed method to manage different kinds of reserve uncertainty.

0.7

VI. C ONCLUSION In this paper, the interaction between DERs, ADS, and ISO is studied under the scenario where DERs provide spinning

Uncertainty Level

C. 123-bus Test System

0.5 0.4

0.6

0.3

0.5

0.2

0.4

0.1

0.3

00

0.2 0.1 0

0

Opportuneness

0.1

0.2 0.3 Robustness

0.01 0.02 Scenario I Scenario II Scenario III 0.4 0.5

0.03

Real Power Losses (MWh)

Fig. 11. Critical/opportunistic limits vs. uncertainty level for 123-test system.

reserve to ISO based on IGDT. The interaction is modeled through both robust and opportunistic perspectives when the ADSO tries to manage the reserve uncertainty. Two IGDT uncertainty models are deployed to depict the characteristics of the reserve uncertainty under a centralized and decentralized control framework. To obtain the relationship between

ZHAO et al.: FLEXIBLE ACTIVE DISTRIBUTION SYSTEM MANAGEMENT CONSIDERING INTERACTION WITH TRANSMISSION NETWORKS USING INFORMATION-GAP DECISION THEORY

uncertainty levels and robust/opportunistic limits, the opportuneness/robustness functions in IGDT are reformulated as bilevel, bi-objective optimization problems and solved using a hybrid MODE. The simulation results in a 33-bus and 123-bus test system demonstrate the effectiveness of the IGDT based method: 1) Results obtained by MODE could provide both risk-averse and risk-seeking reactive management solutions. 2) The relationship between robust/opportunistic levels and performance requirements could be revealed by the obtained Pareto fronts. 3) When the active power of each DG ∈ ΦR is managed by the ADSO corresponding to the total reserve requirements form ISO, the ADSO could obtain better management results. For other kinds of uncertainties, e.g., fluctuation of wind power, photovoltaic and critical load, IGDT could also provide inspiring solutions. The Minkowski-norm models [12] could be employed to depict the correlation among the uncertainty factors. The Slope-bound models [12] might provide an insightful view on the ramp reserves [29]. To measure the information gap between what the decision maker needs to know and what is known, these uncertainty models are strongly related to the system model. When renewable sources and flexible loads are integrated, the system model should be extended to consider for the technical and economic characteristics of these sources and loads. Further work would apply IGDT to manage multiple renewable sources as well as flexible loads. R EFERENCES [1] Q. Wang, C. Y. Zhang, Y. Ding, G. Xydis, J. H. Wang, and J. Østergaard, “Review of real-time electricity markets for integrating distributed energy resources and demand response,” Applied Energy, vol. 138, no. C, pp. 695–706, Jan. 2015. [2] Y. Xiang, J. Y. Liu, and Y. Liu. “Optimal active distribution system management considering aggregated plug-in electric vehicles,” Electric Power Systems Research, vol.131, pp. 105–115, Feb. 2016. [3] H. Ahmadi, J. R. Marti, and H. W. Dommel, “A framework for voltVAR optimization in distribution systems,” IEEE Transactions on Smart Grid, vol. 6, no. 3, pp. 1473–1483, May. 2015. [4] J. Barr and R. Majumder, “Integration of distributed generation in the volt/VAR management system for active distribution networks,” IEEE Transactions on Smart Grid, vol. 6, no. 2, pp. 576–586, Mar. 2015. [5] Y. J. Cao, Y. Tan, C. B. Li, and C. Rehtanz. “Chance-constrained optimization-based unbalanced optimal power flow for radial distribution networks,” IEEE Transactions on Power Delivery, vol. 28, no. 3, pp. 1855–1864, Jul. 2013. [6] S. Abapour, K. Zare, and B. Mohammadi-Ivatloo, “Evaluation of technical risks in distribution network along with distributed generation based on active management,” IET Generation, Transmission & Distribution, vol. 8, no. 4, pp. 609–618, Apr. 2014. [7] T. Ding, S. Y. Liu, W. Yuan, Z. H. Bie, and B. Zeng, “A twostage robust reactive power optimization considering uncertain wind power integration in active distribution networks,” IEEE Transactions on Sustainable Energy, vol. 7, no. 1, pp. 301–311, Jan. 2016. [8] K. Dehghanpour and S. Afsharnia, “Electrical demand side contribution to frequency control in power systems: A review on technical aspects,” Renewable and Sustainable Energy Reviews, vol. 4, pp. 1267–1276, Jan. 2015. [9] M. Doostizadeh and H. Ghasemi, “Day-ahead scheduling of an active distribution network considering energy and reserve markets,” International Transactions on Electrical Energy Systems, vol. 23, no. 7, pp. 930–945, Oct. 2013. [10] Y. Xu, F. X. Li, Z. Q. Jin, and M. H. Variani, “Dynamic gain-tuning control (DGTC) approach for AGC with effects of wind power,” IEEE Transactions on Power Systems, vol. 31, no. 5, pp. 3339–3348, Sep. 2016.

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[11] J. D. Lyon, M. H. Zhang, and K. W. Hedman, “Locational reserve disqualification for distinct scenarios,” IEEE Transactions on Power Systems, vol. 30, no. 1, pp. 357–364, Jan. 2015. [12] Y. Ben-Haim, Info-gap Decision Theory: Decisions Under Severe Uncertainty. Haifa, Israel: Academic Press, 2006. [13] D. Berleant, K. Villaverde, and O. M. Koseheleva, “Towards a more realistic representation of uncertainty: An approach motivated by infogap decision theory,” in Fuzzy Information Processing Society, 2008. NAFIPS 2008, Annual Meeting of the North America, May. 2008, pp: 1–5. [14] R. Laia, H. M. I. Pousinho, R. Mel´ıco, and V. M. F. Mendes, “Selfscheduling and bidding strategies of thermal units with stochastic emission constraints,” Energy Conversion and Management, vol. 89, pp. 975–984, Jan. 2015. [15] P. Mathuria and R. Bhakar, “GenCo’s integrated trading decision making to manage multimarket uncertainties,” IEEE Transactions on Power Systems, vol. 30, no. 3, pp. 1465–1474, May. 2015. [16] M. Kazemi, B. Mohammadi-Ivatloo, and M. Ehsan, “Risk-constrained strategic bidding of GenCos considering demand response,” IEEE Transactions on Power Systems, vol. 30, no. 1, pp. 376–384, Jan. 2015. [17] K. N. Chen, W. C. Wu, B. M. Zhang, and H. B. Sun, “Robust restoration decision-making model for distribution networks based on information gap decision theory,” IEEE Transactions on Smart Grid, vol. 6, no. 2, pp. 587–597, Mar. 2015. [18] J. Zhao, C. Wan, Z. Xu, and J. H. Wang, “Risk-based day-ahead scheduling of electric vehicle aggregator using information gap decision theory,” IEEE Transactions on Smart Grid, to be published. [19] DNV GL. (2014, Sep.). A review of distributed energy resources. DNV GL Energy, Arlington, VA. [Online]. Available: http://www.nyiso.com/ public/webdocs/media room/publications presentations/Other Reports/ Other Reports/A Review of Distributed Energy Resources September 2014.pdf. [20] Z. Zhou, T. Levin, and G. Conzelmann, “Survey of U.S. ancillary services markets,” Argonne National Laboratory, Chicago, I. L., ANL/ESD16/1, Jan. 2016. [21] M. Farivarand and S. H. Low, “Branch flow model: Relaxations and convexification – Part I,” IEEE Transactions on Power Systems, vol. 28, no. 3, pp. 2554–2564, Aug. 2013. [22] P. Cuffe, P. Smith, and A. Keane, “Capability chart for distributed reactive power resources,” IEEE Transactions on Power Systems, vol. 29, no. 1, pp. 15–22, Jan. 2014. [23] B. Colson, P. Marcotte, and G. Savard, “An overview of bilevel optimization,” Annals of Operations Research, vol. 153, no. 1, pp. 235–256, Sep. 2007. [24] A. Ben-Tal and A. Nemirovski. (2001). Lectures on modern convex optimization: analysis, algorithms, and engineering applications. Society for Industrial and Applied Mathematics, Philadelphia, US. [Online]. Available: http://www2.isye.gatech.edu/∼nemirovs/lmco run prf.pdf [25] E. K. Burke and G. Kendall, Search Methodologies. Colorado, U. S: Springer, 2014, pp. 403–449. [26] X. Qiu, J. X. Xu, K. C. Tan, and H. A. Abbass, “Adaptive crossgeneration differential evolution operators for multi-objective optimization,” IEEE Transactions on Evolutionary Computation, vol. 20, no. 2, pp. 232–244, Apr. 2016. [27] M. Varadarajan and K. S. Swarup, “Solving multi-objective optimal power flow using differential evolution,” IET Generation, Transmission & Distribution, vol. 2, no. 5, pp. 720–730, Sep. 2008. [28] Gurobi Optimization. (2016, May). Gurobi optimizer reference manual. [Online]. Available: http://www.gurobi.com [29] E. Ela, M. Milligan, and B. Kirby, “Operating reserves and variable generation,” National Renewable Energy Laboratory, Golden, CO, NREL/TP-5500-51978, Aug. 2011.

Tianyang Zhao received the B.Sc. and M.Sc. degree in automation of electric power systems from North China Electric Power University, Beijing, China in 2011 and 2013, respectively. Currently, he is pursuing his Ph.D. degree in automation of electric power systems in North China Electric Power University, Beijing, China. His research interests include power system operation optimization and game theory.

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Jianhua Zhang (M’98) received the B.Sc. and M.Sc. degrees in electrical engineering from North China Electric Power University, Baoding, China in 1982 and 1984, respectively. Currently, he is working as a Professor in the Department of Electrical and Electronic Engineering, North China Electric Power University. He has been the IET Fellow since 2005, and is also a member of the PES Committee of China National “973 Project.”

Peng Wang (M’00–SM’11) received the B.Sc. degree in electronic engineering from Xi’an Jiaotong University, Xi’an, China, in 1978, the M.Sc. degree from Taiyuan University of Technology, Taiyuan, China, in 1987, and the M.Sc. and Ph.D. degrees in electrical engineering from the University of Saskatchewan, Saskatoon, SK, Canada, in 1995 and 1998, respectively. Currently, he is a Professor of the School of Electrical and Electronic Engineering at Nanyang Technological University, Singapore.

CSEE JOURNAL OF POWER AND ENERGY SYSTEMS, VOL. 2, NO. 4, DECEMBER 2016

Flexible Active Distribution System Management Considering Interaction with Transmission Networks Using Information-gap Decision Theory Tianyang Zhao, Jianhua Zhang, Member, CSEE, and Peng Wang, Senior Member, IEEE

Abstract—In this paper, a flexible management method is proposed for an active distribution system (ADS) with distributed energy resources (DERs) integrated, where DERs can provide spinning reserves to transmission networks. This method, based on the information-gap decision-making theory (IGDT) theory, could be of use to the ADS operator (ADSO) from either the opportunistic or robust perspective when reserve is called by the independent system operator (ISO). Two IGDT uncertainty models are employed to depict the characteristics of reserve uncertainty in centralized and decentralized control frameworks. The reactive power of each DER is managed by the ADSO in the immunity functions, which are reformulated as bi-level biobjective optimization problems. A hybrid multi-objective differential evolutional algorithm (MODE) is proposed to solve the optimization problems. The relationship between the uncertainty levels and robust/opportunistic limits is revealed by the Pareto fronts obtained by MODE. Effectiveness of the proposed method is demonstrated based on simulation results of a 33-bus and 123bus test system. Index Terms—Active distribution system, bi-level multiobjective optimization, information-gap decision-making theory, reserve uncertainty.

I. I NTRODUCTION ITH increasing penetration of distributed energy resources (DERs) into distribution networks, interactions between DERs and power systems are becoming active. Various methods have been proposed to realize the interactions between DERs and transmission networks, i.e., energy exchange, regulation reserves, spinning reserves and territory reserves [1]. In essence, distribution networks form the bridge between DERs and transmission networks, and it is important that they are flexible as a way to guarantee these interactions. An active distribution system (ADS) is one such effective solution for achieving flexible operations of distribution networks [2].

W

Manuscript received February 29, 2016; revised May 30, 2016 and July 31, 2016; accepted September 23, 2016. Date of publication December 30, 2016; date of current version October 20, 2016. This work was supported by the Fundamental Research Funds for the Central Universities (No. 2014XS09) and Chinese Scholarship Council of the Ministry of Education. T. Y. Zhao (corresponding author, e-mail: [email protected]) and J. H. Zhang are with State Key Laboratory of Alternate Power System with Renewable Energy Resources, North China Electric Power University, Beijing 102206, China. P. Wang is with the School of Electrical and Electronic Engineering, Nanyang Technological University, Singapore S639798. DOI: 10.17775/CSEEJPES.2016.00052

Numerous methods have been proposed to provide flexibility for ADS [2]–[4]. These methods can be categorized into deterministic [2]–[4] and non-deterministic methods [5]– [7]. The deterministic method is only applied to scenarios where outputs of sources and demands are fixed [2]–[4]. Non-deterministic methods, on the other hand, deal with uncertainties (e.g., forecasting errors of demands), and can be modeled using three approaches: probability distribution function [5], fuzzy set [6], and interval number [7]. Additionally, different decision-making methods are employed to manage the corresponding types of uncertainties. For example, the chance-constrained optimization method is applied to mitigate the adverse impacts of the stochastic outputs of renewable energy resources on ADS [5]. Similarly, the fuzzy technique is implemented to model the uncertainty and assess operation risk [6], and a two-stage robust optimization technique has been proposed to deal with the interval uncertainty of wind power output [7]. The aforementioned uncertainty models [5]–[7] only reveal the relationship between uncertainties and environmental factors, e.g., solar illumination and wind speed. However, uncertainties within ADS are also affected by interactions between DERs and transmission networks because when DERs provide reserves to transmission networks [1], their outputs need to be adjusted according to the signals obtained from the independent system operator (ISO) [8] or the DER aggregators, such as the local distribution system [9]. The signals obtained from the ISO are typically influenced by frequency derivation, area control error, short-time forecasting information [8], and control methods (e.g., fixed gain or dynamic gain in automatic generation control [10]). On the other hand, the signals obtained from aggregators are affected by the signals sent from the ISO and the aggregators’ own operation strategies, e.g., minimize the cost [9]. Other uncertainty factors include frequency derivation and area control error [11], as well as reserve utilization of DERs, which is affected by different control methods. Based on these factors, it is clear then that a novel decision making method is required to manage reserve uncertainty of DERs. Information-gap decision-making theory (IGDT) is a powerful decision tool that looks at the severe gap between predicted and actual variables of interest and then is able to model the reserve uncertainty in a more realistic fashion [12], [13]. IGDT can provide both robust and opportune strategies based on decision makers’ attitudes towards risks [12]. This makes

c 2016 CSEE 2096-0042

ZHAO et al.: FLEXIBLE ACTIVE DISTRIBUTION SYSTEM MANAGEMENT CONSIDERING INTERACTION WITH TRANSMISSION NETWORKS USING INFORMATION-GAP DECISION THEORY

IGDT more attractive than other available risk-averse decision making methods [5]–[7]. As such, IGDT has been applied in multiple uncertainty scenarios, e.g., Genco’s bidding [15], [16], restoration of distribution networks [17], and electric vehicle aggregator management [18]. Currently, existing ADS management methods are mostly focused on the selfregulation aspect of distribution networks [2]–[8]. When DERs provide reserves to transmission networks, ADS guarantees efficient interactions between DERs and transmission networks. In this paper, a novel IGDT based method is proposed for ADS to manage the reserve uncertainty. This method enables ADS to optimally utilize the reactive power provided by DERs, while enhancing the interactions between DERs and transmission networks. The main contributions of this paper can be summarized as follows: 1) Two IGDT based models are employed to depict the characteristics of reserve uncertainty, where the reserve of DERs is called by the transmission networks in centralized and decentralized control methods, respectively. 2) Two IGDT immunity functions are proposed for ADS to optimally utilize the reactive power provided by DERs from risk-averse and risk-seeking perspectives. 3) Finally, the immunity functions are reformulated as multi-objective bi-level optimization problems, which are then solved using a hybrid multi-objective differential evolutional (MODE) algorithm. The relationship between robust/opportunistic levels and performance requirements is revealed using the Pareto fronts obtained by MODE. This paper is organized as follows. An active distribution system (ADS) and its corresponding deterministic management model are presented in Section II. This model serves as the system model for IGDT. The uncertainty model and performance requirements in IGDT are depicted in Section III. The solving method for IGDT is proposed in Section IV. Simulation and conclusions are shown in Section V and Section VI, respectively.

the reactive power of DGs to improve operation efficiency of the ADS. It is also important to note that this work is carried out under a regulated power market.

DG1

DGi

DGN

Independent system operator

Distribution network Central power plants

Fig. 1.

Transmission network Reserve flow Reserve called signal Power flow Information flow within the active distribution network

Active distribution system operator

Note: All DGs are schedulable.

Framework of DERs, ADS, and transmission networks.

ADSO’s management should meet a set of constraints, e.g., power balance, voltage magnitude limitations, and reactive power adjustment capacity constraint. Based on the branch power flow for radial networks [21], the management model of the ADSO could be depicted as follows: X F = min rij lij (1) (i,j)∈E

X

s.t.

PG,i −

i∈Φj

X

X

PL,i = Pij − rij lij −

i∈Θj

II. ACTIVE D ISTRIBUTION S YSTEM M ANAGEMENT

Pjk ,

k:(j,k)∈E

∀j ∈ B X i∈Φj

QG,i −

X

(2) X

QL,i = Qij − xij lij −

i∈Θj

∀j ∈ B vj = vi − 2(rij Pij + xij Qij ) +

2 (rij

(3) +

x2ij )lij ,

∀(i, j) ∈ E (4)

Pij2 + Q2ij , ∀(i, j) ∈ E vi 2 Pij2 + Q2ij ≤ Smax,ij , ∀(i, j) ∈ E 2 Vmin,i

Qjk ,

k:(j,k)∈E

lij ≥

DERs encompass a variety of distributed technologies, i.e. internal combustion, external combustion, energy storage, renewables, and fuel cells [19]. When DERs provide reserves to transmission networks, they must meet certain grid codes, e.g., DERs’ capacities should be larger than the minimal eligible size (500 kW in CAISO [19]), and spinning reserve must be maintained for at least two hours [20]. Thus, only schedulable distributed generators (DGs), e.g., diesel generators and combined heat and power, are considered here. The integration of renewable sources and flexible loads, e.g., photovoltaic generations and electric vehicles would be considered in future work. The framework of the transmission networks for ADS and DERs is explicitly shown in Fig. 1. DGs provide spinning reserve to transmission networks and their active power is controlled by the ISO. An active distribution system operator (ADSO) acts as the system operator of the distribution networks with schedulable DGs integrated. The ADSO manages

77

≤ vi ≤

2 Vmax,i ,

∀i ∈ N

QG min,i, ≤ QG,i ≤ QG max,i , ∀i ∈ Φ

(5) (6) (7) (8)

where PG,i and QG,i are the active and reactive power output of the ith generator; PL,i and QL,i are the active and reactive power demand of the ith user; Vi is the voltage magnitude at the ith bus; vi := V2i ; rij and xij are the resistance and reactance of line (i, j); Iij is the complex current flowing from buses i to j, lij := I2ij ; Pij and Qij are the active and reactive power on line (i, j); Smax,ij is the apparent power limitation of line (i, j); Vmin,i and Vmax,i are minimum and maximum voltage magnitude limitation at the ith bus; QG min,i and QG max,i are the minimum and maximum reactive output of the ith generator; Φ and Θ are the sets of generators and users, and Φ := ΦR ∪ ΦL , where ΦL stands for eternal power system connected by ADS, Φi and Θi are the sets of generations and users connecting to the ith bus; B is the set of buses; E is the set of branches.

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As shown in (1)–(8), the management problem is a voltage control problem. Since the voltage magnitude at each bus is treated as constraint, as shown in (7), the objective function refers to minimize real power losses, which is depicted by (1). Equations (2)–(3) represent the KCL equations, and (4) represents the KVL equations. The conic relaxation for power, voltage, and current is shown in (5). This relaxation is tight for the ADS management model [21]. Equation (6) is the thermal limitation for the distribution lines. The voltage magnitude limitation for each bus is shown in (7). The reactive power capacity of each DG is depicted by (8), which is subjective to technical characteristics of the generator and its active power output [22]. Maximum and minimal reactive power capacities of the ith DG are given as follows: q 2 , −C QG min,i = max(− SG,i − PG,i G,i PG,i ), ∀i ∈ ΦR (9) q 2 ,C (10) QG max,i = min( SG,i − PG,i G,i PG,i ), ∀i ∈ ΦR Cmin,i ≤ CG,i ≤ Cmax,i , ∀i ∈ ΦR

(11)

where SG,i is the capacity of the ith DG; CG,i is the parameter limiting reactive power of ith DG with fixed active power, and Cmin,i and Cmax,i are lower and upper boundary of CG,i . CG,i is determined by the minimum power factor PF min , which is set by the ADSO for the ith DG, where PF min := cos(tan−1 (CG,i )). To obtain the convex hull of bilinear items (CG,i PG,i ) in (9) and (10), the McCormick envelope method [23] is employed as follows: Cmin,i PG,i + CG,i PG min,i − Cmin,i PG min,i ≤ wi

(12)

Cmax,i PG,i + CG,i PG max,i − Cmax,i PG max,i ≤ wi

(13)

wi ≤ Cmax,i PG,i + CG,i PG min,i − Cmax,i PG min,i

(14)

wi ≤ Cmin,i PG,i + CG,i PG max,i − Cmin,i PG max,i

(15) A. IGTD Based Uncertainty Modeling

where wi := CG,i PG,i . Furthermore, when PG,i , PG min,i and PG max,i are equal to each other, wi = CG,i PG,i . For convenence, the ADS management model is depicted in the following compact form: F (Y ) = min cT X ( X Aeq X = beq , AX ≤ b, X ∈ Γ s.t. Ceq X + Deq Y = deq , CX + DY ≤ d

of each DG ∈ ΦR is given. In addition, the amount and direction of spinning reserves called by the ISO is difficult to be forecasted and is uncertain [11]. This uncertainty might deteriorate or benefit the operation of ADS. Thus, an IGDT based management method is proposed for the ADSO to manage the reserve uncertainty of DGs ∈ ΦR . It should be noticed that the information gap in this paper is defined as the real amount and direction of spinning reserve called by the ISO and the spinning reserve capacity cleared in the spinning reserve market. IGDT is a non-probabilistic decision-making theory, which expresses the idea that uncertainty may be either pernicious or propitious. The idea is realized by constructing two “immunity functions”: the robustness function expresses the immunity to failure, while the opportuneness function expresses the opportunity to windfall gain [12]. Both functions should be evaluated with respect to different values of uncertainty parameters under specific performance requirements. Consequently, the relationship between uncertainty levels and robust/opportunistic limits set by the ADSO can be obtained. This relationship helps the ADSO to compare different reactive power management strategies that satisfy system performance criteria as per the ADSO’s requirements or aspirations. An IGDT based decision making is specified by three component models: 1) System model; 2) uncertainty model; 3) performance requirements. The system model has been proposed in Section II. Only the uncertainty model and performance requirements are studied in this section.

(16)

where X is the integrated representation for the decision parameters of the ADSO, [PG,i,i∈ΦL , QG,i,i∈Φ , Pij,(i,j)∈E , Qij,(i,j)∈E , lij,(i,j)∈E , vi,i∈N , CG,i,i∈ΦR , wi,i∈ΦR ]. Y is the integrated representation for active power outputs of DGs ∈ ΦR , PG,i,i∈ΦR . Γ is the rotatecone constraints (5). Moreover, problem (16) is a linear conic optimization problem (LCP) [24] and serves as the system model in IGDT. The IGDT based management is presented in Section III.

III. IGDT BASED ADS M ANAGEMENT M ODELING As the active power of each DG ∈ ΦR is controlled by the ISO and the reactive power of each DG ∈ ΦR is managed by the ADSO, the ADS management model presented in Section II can only be applied when the active power output

The DERs’ contribution to LFC has been well-described in [8]. To model the spinning reserve uncertainty more realistically, two uncertainty models are proposed to represent this uncertainty in centralized and decentralized control methods, which are two popular methods for the ISO to manage scalable DERs. The spinning reserve capacity of each DG ∈ ΦR is known before implementing the ADS management (16). Taking into account the amount of uncertainty and spinning reserves directions called by the ISO, the active power output of each DG ∈ ΦR deviates from the set-point with unknown deviation and known boundaries. This characteristic makes it suitable to apply the fractional error model [12] in the IGDT to constrain the reserve uncertainty within an expandable envelope. The two uncertainty models are shown as follows: R − RD,i PG,i − PG0,i − α U,i 2 Uc (α) = {Y : ≤ αψi , RU,i + RD,i 2 − RD,i ≤ PG,i − PG0,i ≤ RU,i , ∀i ∈ ΦR }, α ≥ 0, ψi > 0 X X Ud (α) = {Y : −α RD,i ≤ PG,i − PG0,i i∈GR

i∈GR

(17)

ZHAO et al.: FLEXIBLE ACTIVE DISTRIBUTION SYSTEM MANAGEMENT CONSIDERING INTERACTION WITH TRANSMISSION NETWORKS USING INFORMATION-GAP DECISION THEORY

≤α

X

RU,i , −RD,i ≤ PG,i − PG0,i ≤ RU,i },

i∈GR

1≥α≥0

(18)

where PG0,i is the set-point of the ith DG ∈ ΦR , RD,i and RU,i are the up spinning reserve and down spinning reserve of the ith DG, α is the index of reserve uncertainty, ψi is a parameter determining the shape of the envelope, and Uc and Ud are the functions for the spinning reserve uncertainty in centralized and decentralized LFC. The reserve uncertainty U (α) could be depicted by either Uc in (17) or Ud in (18). Remark 1: By employing the centralized model (17), the ISO can manage the active power of DGs ∈ ΦR directly. The parameter ψi , i ∈ ΦR , in (17) determines the shape of the uncertainty envelope and is predetermined by the ISO. The active power output range of DGi , i ∈ ΦR , could be depicted as follows: 1 + ψi 1 − ψi RU,i − α RD,i , PG0,i [PG0,i + α 2 2 1 − ψi 1 + ψi RU,i − α RD,i ]. (19) +α 2 2 Through properly setting ψi , the active power output of DGi , i ∈ ΦR would not fall into the following zones: 1 − ψi 1 + ψi RU,i − α RD,i ] (20) 2 2 1 + ψi 1 − ψi [PG0,i + RU,i , PG0,i + α RU,i − α RD,i ] (21) 2 2 This feature enables the stable operation of generators with special operation requirememts, e.g., forbidden operating zones of the thermal units [14]. Furthermore, if all ψi , i ∈ ΦR are set to 1, (17) could be simplified as follows: [PG0,i − RD,i , PG0,i + α

Uc (α) = {Y : −αRD,i ≤ PG,i − PG0,i ≤ −αRU,i , ∀i ∈ ΦR }, 1≥α≥0

(22)

Remark 2: By employing the decentralized model (18), the ISO should send the reserve signals to the aggregator of DGs ∈ ΦR or the ADSO. The ISO does not need to control each DG ∈ ΦR directly. The active power of each DG ∈ ΦR is controlled by the aggregator or the ADSO allowing for the total reserve requirement from ISO. B. Performance Requirements Performance requirements for making a decision are evaluated on the basis of robustness and opportuneness functions [12]. The ADSO aims to manage the reserve uncertainty shown in Section III-A. A robustness function guarantees a certain profit expectation under adverse future conditions. IGDT also examines beneficial opportunity arising from uncertainty, to obtain windfall profit. Both functions optimize uncertainty parameter as follows [12]: α = max{α : max F (Y ) ≤ πC }

(23)

β = min{β : max F (Y ) ≤ πW }

(24)

Y

Y

where πC is the critical limit, which is always to be satisfied when the real active power outputs of DGs ∈ ΦR increase the

79

real power losses, and πW is the opportunistic limit, which can be achieved when the real active power outputs of DGs ∈ ΦR decrease the real power losses. There are two outcomes of interest when using IGDT: 1) A value of robustness (25), which enables a risk-averse, conservative approach to deal with the reserve uncertainty; 2) an opportunistic value (26), which reveals the speculative, risk-seeking approach. The two immunity functions of robustness and opportuneness are shown as follows: 1) Robustness function max α

s.t.

α c T X ≤ πC max Y Y ∈ U (α) X ∈ SOL[(16)]

(25)

where SOL[(16)] represents the set of solutions for (16); 2) opportuneness function min β

s.t.

β cT X ≤ π W min Y Y ∈ U (β) X ∈ SOL[(16)].

(26)

As shown in (25) and (26), both robustness and opportuneness functions are tri-level optimization problems: the uncertainty level is determined in the upper level optimization, the active power of DGs ∈ ΦR is optimized in the middle level optimization, and the reactive power of each DG ∈ ΦR is adjusted in the lower level optimization. IV. S OLVING M ETHOD Immunity functions (25) and (26) are always formulated as a bi-level optimization problem [14], [18]. This problem is reformulated to a single stage optimization problem based on Karush-Kuhn-Tucker (KKT) conditions for optimization problems. Furthermore, the reformulated optimization problem should be solved under different performance requirements, to reveal the relationship between robust/opportunistic requirements and uncertainty level. This process might be timeconsuming. Since the middle and lower level optimization problems in (26) are a min-min optimization problem, the opportuneness function (26) could be treated as a bi-level optimization problem. The bi-level optimization problem could be solved efficiently by reformulating its lower level problem as complementaryconstraints based on KKT optimal conditions, formulating a mathematical programming with equilibrium constraints (MPEC) [23]. However, the integration of complementary constraints might result in infeasibility of this MPEC. This is owing to those inequality constraints (5) that are active when the optimal solution of (16) is found [21]. Strong duality conditions for LCP do not hold [24], and the duality gap might exist for this LCP, resulting in the infeasibility of the MPEC. What is worse, the lower level optimization problem of (22)

CSEE JOURNAL OF POWER AND ENERGY SYSTEMS, VOL. 2, NO. 4, DECEMBER 2016

is a max-min optimization problem. The prevailing method to solve the max-min problem is to reformulate the inner min optimization as a max optimization problem. Based on the same reason as stated for opportuneness function (26), the duality gap exists. As a result, immunity functions (25), (26) cannot be solved using traditional methods. A novel solution is proposed to solve the robustness and opportuneness functions as follows: 1) First, the middle optimization problems (25) and (26) are treated as equality constraints. 2) Next, the critical/opportunistic limits are treated as the second objective, resulting in two bi-objective bi-level optimization problems. 3) Then, the reformulated optimization problems are solved using a hybrid MODE, in which the classical optimization technique is deployed to solve the lower level optimization problem. A. Reformulation of IGDT Functions 1) Reformulated Robustness Function The reformulation of robustness function, (25), is based on the concept of Pareto domination, which is widely employed in evolutionary multi-objective (EMO) algorithms [25]. The Pareto domination between two solutions is defined as follows: Definition 1 [25]: A solution x is said to dominate the other solution y, if both the following conditions are true: 1) The solution x is no worse than y in all objectives. Thus, the solutions are compared based on their objective function values. 2) The solution x is strictly better than y in at least one objective. When the ADSO is risk-averse, a solution of (25) with smaller α and bigger πC is preferred. This is for the reason that: 1) As it is risk-averse, the ADSO should be informed of the smallest level of uncertainty α that it must resist, when the critical limit πC is given. The smallest uncertainty level α serves as a warning index for the ADSO to manage the reserve uncertainty. 2) For a fixed uncertainty level α, the ADSO should be notified of the maximal real losses, i.e., the critical limit πC . Fig. 2 shows a set of feasible solutions belonging to robustness function and their corresponding first non-domination front [25]. As shown in Fig. 2, solutions 1–3 are dominated by solutions 4–8, and solutions 4–8 are on the first nondomination front, i.e., not dominated by any other solution of robustness function. Thus, only solutions 4–8 would be provided to the ADSO. These solutions reveal the relationship between the critical limits and uncertainty levels for the ADSO. One of these solutions is selected according to the ADOS’s passive preference towards risk. The bi-objective bi-level optimization problem for robustness function (25) is shown as follows: min {α, −πC }

α,πC ,Y

s.t.

Y ∈ U (α) cT X = π C X ∈ SOL[(16)].

(27)

1 8 7

0.8 Uncertainty Level

80

6

2

0.6

5

1 3 4

Non-dominated front

0.4 0.2

18 are feasible solutions of (25)

0 0.02 0.04

0.06

0.08 0.1 0.12 0.14 Critical Limit (MWh)

0.16

0.18

Fig. 2. A set of feasible solutions belonging to robustness function and the corresponding first non-domination front.

As shown in (27), critical limit πC , uncertainty level α and active power outputs of DGs ∈ ΦR , Y are optimized to minimize uncertainty level and maximize critical limits simultaneously. In lower level optimization, the reactive power of DGs ∈ ΦR is determined to minimize the real power losses, as shown in (16). Furthermore, as cT X = πC in (27), the critical limit πC equals to the real power losses in the lower level optimization. 2) Reformulated Opportuneness Function When the ADSO is risk-seeking, a solution of (26) with smaller β and smaller πW is preferred. This is because 1) when the opportunistic limit πW is given, the ADSO could be notified of the minimal uncertainty level β, which the ADSO should bear in order to obtain the windfall, and 2) when the uncertainty level β is fixed, the ADSO could be informed of the opportunistic limit πW , i.e., the minimal real power losses that can be possibly obtained. A set of feasible solutions belonging to opportuneness and its corresponding first non-domination front is shown in Fig. 3. As shown in Fig. 3, solutions 1–3 are dominated by solutions 4–8, and solutions 4–8 are on the first non-domination front, i.e., not dominated by any other solution of (26). Similar to (27), solutions 4–8 reveal the relationship between the opportunistic limits and uncertainty levels. One of these solutions is selected according to the ADOS’s opportunistic preference towards risk. The bi-objective bi-level optimization problem for the opportunistic function (26) is shown as follows: min {β, πW }

β,γW ,Y

s.t.

Y ∈ U (β) cT X = π W X ∈ SOL[(16)].

(28)

As shown in (28), opportunistic limit πW , uncertainty level β, and active power of DGs ∈ ΦR Y are optimized to minimize uncertainty level β and opportunistic limit πW . In lower level optimization, reactive power of DGs ∈ ΦR is determined to minimize the real power losses, based on model (16). As cT X = πW shown in (28), the opportunistic limit πW also equals to the real power losses in the lower level optimization.

ZHAO et al.: FLEXIBLE ACTIVE DISTRIBUTION SYSTEM MANAGEMENT CONSIDERING INTERACTION WITH TRANSMISSION NETWORKS USING INFORMATION-GAP DECISION THEORY

1

Uncertainty Level

0.8 2

0.6 0.4

Non-dominated front

1 3

4 5 6

7

8

0.2 18 are feasible solutions of (26) 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 Opportunistic Limit (MWh) Fig. 3. A set of feasible solutions belonging to opportuneness function and the corresponding first non-domination front.

B. Solving Procedure According to search techniques employed, traditional methods to solve multi-objective optimization problems could be classified into two categories: conversion to single objective optimization problem based on higher-level information and EMO algorithms [25]. Since the weight between the uncertainty levels and robustness/opportunistic limits is difficult to obtain, EMO is adopted in this paper. The frameworks of EMO algorithms are Pareto dominancebased or decomposition based [26]. A Pareto dominancebased MODE [27] is applied to solve (18) and (19). For the lower problem (16), a classical interior-point method based optimization technique is applied to solve this LCP, i.e., Gurobi [28] is employed in our work. The solving procedure of the hybrid MODE is depicted in Fig. 4. The details about initialization, mutation, recombination and selection in MODE could be found in [27]. Input Parameters Initialization

Meet the stopping criteria

Yes

Output Pareto fronts

No Mutation

Recombination Individual assessment Selection/ constraint handling Upper level optimization for (α,πC,Y) or (β,πW,Y)

Fig. 4.

Yes Y

Solving (16)

Is (16) feasible? No

Solving (29)

Lower level optimization for X

Flow chart of the hybrid MODE for bi-level bi-objective problems.

81

Since the generation of off-springs in MODE is random, the active power outputs of DGs ∈ ΦR , Y , might cause over voltage to ADS, resulting in infeasibility of the lower level optimization problem (16). To assess the violation of the lower problem corresponding to a given Y , an optimal recovery model is proposed as follows: T − V C(Y ) = min I T ε + Ieq (ε+ eq + εeq ) + − X,ε,εeq ,εeq − Aeq X = beq + ε+ eq − εeq AX ≤ b + ε s.t. X∈Γ Ceq X + Deq Y = deq , CX + DY ≤ d ε, ε+ , ε− ≥ 0 eq eq

(29)

− where ε, ε+ eq , and εeq are vectors of relaxation variables for the inequality and equality linear constraints in (16), and I and Ieq are vectors full of 1.

V. C ASE S TUDY To demonstrate the effectiveness of proposed IGDT based management method, a case study has been carried out on two test systems, including a 33-bus and 123-bus distribution network. Details of both networks are available in [7]. The simulations were implemented in MATLAB version 8.1.0.604, running on a PC with Intel Core i7-4700 MQ and 8 GB RAM. A. Case Description For the 33-bus test system, there are 4 DGs connected at buses 18, 22, 25, and 33. Additionally, the set-points of DGs are 1.0 MW, 0.5 MW, 0.5 MW and 0.5 MW, respectively. The up spinning reserve capacity of each DG is 0.5 MW, 0.3 MW, 0.5 MW and 0.5 MW, respectively. The down spinning reserve is set to 0.0 MW, 0.3 MW, 0.5 MW and 0.5 MW, respectively. The capacity of each DER is 1.5 MVA, 1 MWA, 1 MVA and 1 MVA, respectively. Moreover, three scenarios were proposed for comparison: Scenario I: Active power of each DG is controlled by the ISO through centralized control, and the reserve uncertainty follows (17), where ψi , i ∈ ΦR , is set to 1. Scenario II: Active power of each DG is controlled by the ISO through decentralized control, and the reserve uncertainty follows (18). Scenario III: The ISO sends the reserve requirements to the ADSO. The active power of DGs ∈ ΦR is controlled by the ADSO, and the reserve uncertainty follows (18). For the 123 bus test system, there are 10 DGs connected at buses 12, 25, 44, 54, 57, 64, 67, 78, 86 and 110. The setpoint, up spinning reserve, down spinning reserve capacity and capacity of each DG is set to 0.5 MW, 0.5 MW, 0.5 MW, and 1 MVA, respectively. The population size of MODE is set to 80, and the maximum iteration of MODE is set to 300. B. 33-bus Test System 1) Results Under Different Scenarios Fig. 5 shows the Pareto fronts in robustness and opportuneness functions of the three scenarios. These Pareto fronts re-

CSEE JOURNAL OF POWER AND ENERGY SYSTEMS, VOL. 2, NO. 4, DECEMBER 2016

1 Scenario I Scenario II Scenario III

Uncertainty Level

0.8 0.2 0.6

0.15 0.1

0.4

0.05 0

0.2

0.034 0.036

0.038

0.04

0 0.02 0.04 Opportuneness

0.06

0.08

0.1 0.12 0.14 Robustness

0.16

0.18

Real Power Losses (MWh) Fig. 5. system.

Critical/opportunistic limits vs. uncertainty level for 33-bus test

The extreme solutions under each scenario are given in Table I. These extreme solutions in the opportuneness or robustness functions are almost the same under different scenarios, as shown in Table I. The small gap among these extreme solutions results in the equality of minimal real power losses or maximal real power losses under all three scenarios, i.e., 0.0351 MWh and 0.1632 MWh (Fig. 5). An interesting observation of opportuneness function in Table I is that the active power outputs of DGs ∈ ΦR under all three scenarios are almost the same while the uncertainty levels differ. This is owing to the difference between the uncertainty model (17) and (18); the uncertainty level is directly imposed on each DG ∈ ΦR in (17), while the uncertainty level is imposed on DGs ∈ ΦR . The difference enables the flexible operation of aggregators when the active power of each DG ∈ ΦR is controlled by the aggregator. TABLE I C OMPARISON OF E XTREME S OLUTIONS U NDER D IFFERENT S CENARIOS Attitude Scenarios Uncertainty level PG,1 DG1 PF min PG,2 DG2 PF min PG,3 DG3 PF min PG,4 DG4 PF min

Opportuneness I II III 0.76 0.17 0.17 1.00 1.00 1.00 0.95 0.95 0.95 0.27 0.24 0.24 0.92 0.89 0.90 0.88 0.87 0.87 0.88 0.87 0.87 0.70 0.70 0.70 0.70 0.70 0.70

Robustness I II III 1.00 1.00 1.00 1.50 1.50 1.50 1.00 1.00 1.00 0.80 0.80 0.80 0.97 0.98 0.98 0.00 0.00 1.00 1.00 1.00 1.00 0.00 0.00 1.00 1.00 1.00 1.00

The relationship between critical/opportunistic limits and uncertainty levels varies in the three scenarios. In scenario I, as observed from Fig. 5, the real power losses are fixed to 0.0351 MWh when the uncertainty level is bigger than 0.7617. Accordingly, if the ADSO wants to reduce the real

power losses to 0.0351 MWh, the uncertainty level it must bear is 0.7617. Alternatively, the maximal real power losses are 0.1632 MWh while the uncertainty level reaches 1. When the ADSO sets the critical limit to 0.1632 MWh, it could bear any level of reserve uncertainty. Fig. 6(a) and Fig. 6(b) show the minimal power factor of each DG in opportuneness function and robustness function, respectively. In opportuneness function, as shown in Fig. 6(a), the power factors of DG1 and DG2 remain constant, while the power factors of DG3 and DG4 increase. This is owing to the active power outputs of DGs ∈ ΦR can also reduce the real power losses in opportuneness function. On the other hand, as shown in Fig. 6(b), the power factors of DG3 and DG4 decrease, indicating DG3 and DG4 generate more reactive power in robustness function. As the power factors of DG1 and DG2 remain almost 1, they provide less reactive power. In scenario II, the real power losses are 0.0364 MWh under zero uncertainty, which is smaller than the losses of scenario I (Fig. 5). The real power losses can be reduced to 0.0351 MWh with uncertainty level 0.1720. Moreover, under zero uncertainty level, the maximal real power losses are 0.1573 MWh, which is 2.75 times bigger than the losses obtained in scenario I under zero uncertainty level. When the ADSO sets 1

Minimal Power Factor

veal the relationships between the critical/opportunistic limits and uncertainty levels for the 33-bus test system. They verify the merit of the proposed method, providing both robust and opportunistic solutions for the ADSO.

0.9 0.8 0.7 DG1 DG2 DG3 DG4

0.6 0.5

0

0.2

0.4 0.6 Uncertainty Level (a)

0.8

1 0.8 Minimal Power Factor

82

DG1 DG2 DG3 DG4

0.6 0.4 0.2 0

0

0.2

0.4 0.6 Uncertainty Level (b)

0.8

1

Fig. 6. Minimal power factor of each DER with respect to different uncertainty levels under scenario I. (a) Minimal power factor of each DER with respect to different uncertainty levels in the opportuneness function. (b) Minimal power factor of each DER with respect to different uncertainty levels in the robustness function.

ZHAO et al.: FLEXIBLE ACTIVE DISTRIBUTION SYSTEM MANAGEMENT CONSIDERING INTERACTION WITH TRANSMISSION NETWORKS USING INFORMATION-GAP DECISION THEORY

Minimal Power Factor

1

1 0.95 Minimal Power Factor

the critical reward within (0.0364, 0.1573) MWh, there would be no feasible operation plan. The minimal power factor of each DG in the opportuneness function and robustness function under scenario II is shown in Fig. 7(a) and Fig. 7(b), respectively. Fig. 7(a) could be explained similarly to the way of Fig. 6(a). However, DG4 ’ reactive power factor is jumping, as shown in Fig. 7(b).

0.9 0.85 DG1 DG2 DG3 DG4

0.8 0.75

0.95

0.7

0.9

0.65

0

0.85 DG1 DG2 DG3 DG4

0.8 0.75

0.05

0.1 0.15 Uncertainty Level (a)

0.2

Minimal Power Factor

1 0.8

DG1 DG2 DG3 DG4

0.4 0.2

0

0.15 0.05 0.1 Uncertainty Level (b)

0.2

0.8

0.2

Fig. 7. Minimal power factor for each DER with respect to different uncertainty levels under scenario II. (a) Minimal power factor of each DER with respect to different uncertainty levels in the opportuneness function. (b) Minimal power factor of each DER with respect to different uncertainty levels in the robustness function.

In scenario III, the Pareto front in opportuneness function is the same to scenario II. However, the Pareto front in robustness function would be significantly improved (Fig. 5). Hence, when the ADSO is risk-averse, the best way to manage the reserve uncertainty is to let the ADSO manage the active power of DGs ∈ ΦR , corresponding to total reserve requirement from ISO. The minimal power factor of each DG in opportuneness function and robustness function under scenario III is shown in Fig. 8(a) and Fig. 8(b), respectively. In Fig. 8(b), when uncertainty level is lower than 0.7205, DG3 and DG4 provide reactive power to reduce the real power losses. However, when uncertainty level is higher than 0.7205, power factors of DG3 and DG4 decrease significantly, indicating their reactive power support reduces. Thus, the real power losses increases rapidly when uncertainty level is higher than 0.7205 (Fig. 5). According to the above observation, the IGDT based method could provide both robustness and opportuneness reactive

DG1 DG2 DG3 DG4

0.6 0.4 0.2 0

0.6

0

Minimal Power Factor

0

0.05 0.1 0.15 Uncertainty Level (a)

1

0.7 0.65

83

0

0.2

0.4 0.6 Uncertainty Level (b)

0.8

1

Fig. 8. Minimal power factor for each DER with respect to different uncertainty levels under scenario III. (a) Minimal power factor of each DER with respect to different uncertainty levels in the opportuneness function. (b) minimal power factor of each DER with respect to different uncertainty levels in the robustness function.

power management solutions under different scenarios, enabling the flexible operation of ADS. 2) Method Verification To test the effectiveness of solutions obtained by the IGDT based method, a stochastic simulation with uncertainty level 0.15 is considered under scenario I and scenario III, respectively. Based on the relationship shown in Fig. 5, the opportunistic limits are 0.0388 MWh and 0.0351 MWh, and the critical limits are 0.0488 MWh and 0.0385 MWh in scenario I and scenario III, correspondingly. 1000 times randomly simulated reserves called by ISO are tested, where the reserve uncertainty follows (17)–(18) and α is set to 0.15. The simulation results obtained in scenario I and scenario III are shown in Fig. 9 and Fig. 10, respectively. As shown in Fig. 9, in the 1000 times simulation, the ADSO could possibly reduce the real power losses to 0.0395 MWh and 0.0351 MWh, almost the same to the opportunistic limits. Furthermore, it could be concluded from Fig. 10 that the maximal real power losses would never exceed 0.0488 MWh and 0.0385 MWh, smaller than the critical limits. The ADSO could always meet the technical need by applying the reactive power management plans provided by IGDT. This demonstrates the effectiveness of the proposed method.

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250

180 160

200

120

Fequency

Fequency

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100 80

πW

150 πC 100

60

50

40

0

20

0.04

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0.042 0.044 0.046 0.048 Real Power Losses (MWh) (a)

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400

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0.036 0.037 0.038 0.039 Real Power Losses (MWh) (b)

0.0365 0.0375 Real Power Losses (MWh) (b)

0.0385

0.04

Fig. 9. Probability density functions of minimal real power losses randomly simulated under uncertainty scenarios for β = 0.15. (a) Probability density functions of minimal real power losses under scenario I for the solution of opportuneness function. (b) Probability density functions of minimal real power losses under scenario II for the solution of opportuneness function.

Fig. 10. Probability density functions of minimal real power losses randomly simulated under uncertainty scenarios for α = 0.15. (a) Probability density functions of minimal real power losses under scenario I for the solution of robustness function. (b) Probability density functions of minimal real power losses under scenario II for the solution of robustness function.

1 0.9 0.8

The simulation results of the 123-test system under the three scenarios are shown in Fig. 11. As shown in Fig. 11, the minimal real power losses in ADS are the same among the three scenarios, i.e., 0.0034 MWh. However, the minimal uncertainty levels are 0.4684, 0.3396 and 0.3388 for scenario I, scenario II and scenario III respectively. In addition, the maximum real power losses in ADS are also the same, i.e., 0.4062 MWh. In the opportuneness part of Fig. 11, the Pareto fronts of scenario II and scenario III are the same, dominating the Pareto fronts obtained by scenario I. Furthermore, the Pareto front obtained by scenario III dominates the Pareto fronts of scenario I and scenario II in opportuneness function. The results shown above further demonstrate the effectiveness of the proposed method to manage different kinds of reserve uncertainty.

0.7

VI. C ONCLUSION In this paper, the interaction between DERs, ADS, and ISO is studied under the scenario where DERs provide spinning

Uncertainty Level

C. 123-bus Test System

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0.03

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Fig. 11. Critical/opportunistic limits vs. uncertainty level for 123-test system.

reserve to ISO based on IGDT. The interaction is modeled through both robust and opportunistic perspectives when the ADSO tries to manage the reserve uncertainty. Two IGDT uncertainty models are deployed to depict the characteristics of the reserve uncertainty under a centralized and decentralized control framework. To obtain the relationship between

ZHAO et al.: FLEXIBLE ACTIVE DISTRIBUTION SYSTEM MANAGEMENT CONSIDERING INTERACTION WITH TRANSMISSION NETWORKS USING INFORMATION-GAP DECISION THEORY

uncertainty levels and robust/opportunistic limits, the opportuneness/robustness functions in IGDT are reformulated as bilevel, bi-objective optimization problems and solved using a hybrid MODE. The simulation results in a 33-bus and 123-bus test system demonstrate the effectiveness of the IGDT based method: 1) Results obtained by MODE could provide both risk-averse and risk-seeking reactive management solutions. 2) The relationship between robust/opportunistic levels and performance requirements could be revealed by the obtained Pareto fronts. 3) When the active power of each DG ∈ ΦR is managed by the ADSO corresponding to the total reserve requirements form ISO, the ADSO could obtain better management results. For other kinds of uncertainties, e.g., fluctuation of wind power, photovoltaic and critical load, IGDT could also provide inspiring solutions. The Minkowski-norm models [12] could be employed to depict the correlation among the uncertainty factors. The Slope-bound models [12] might provide an insightful view on the ramp reserves [29]. To measure the information gap between what the decision maker needs to know and what is known, these uncertainty models are strongly related to the system model. When renewable sources and flexible loads are integrated, the system model should be extended to consider for the technical and economic characteristics of these sources and loads. Further work would apply IGDT to manage multiple renewable sources as well as flexible loads. R EFERENCES [1] Q. Wang, C. Y. Zhang, Y. Ding, G. Xydis, J. H. Wang, and J. Østergaard, “Review of real-time electricity markets for integrating distributed energy resources and demand response,” Applied Energy, vol. 138, no. C, pp. 695–706, Jan. 2015. [2] Y. Xiang, J. Y. Liu, and Y. Liu. “Optimal active distribution system management considering aggregated plug-in electric vehicles,” Electric Power Systems Research, vol.131, pp. 105–115, Feb. 2016. [3] H. Ahmadi, J. R. Marti, and H. W. Dommel, “A framework for voltVAR optimization in distribution systems,” IEEE Transactions on Smart Grid, vol. 6, no. 3, pp. 1473–1483, May. 2015. [4] J. Barr and R. Majumder, “Integration of distributed generation in the volt/VAR management system for active distribution networks,” IEEE Transactions on Smart Grid, vol. 6, no. 2, pp. 576–586, Mar. 2015. [5] Y. J. Cao, Y. Tan, C. B. Li, and C. Rehtanz. “Chance-constrained optimization-based unbalanced optimal power flow for radial distribution networks,” IEEE Transactions on Power Delivery, vol. 28, no. 3, pp. 1855–1864, Jul. 2013. [6] S. Abapour, K. Zare, and B. Mohammadi-Ivatloo, “Evaluation of technical risks in distribution network along with distributed generation based on active management,” IET Generation, Transmission & Distribution, vol. 8, no. 4, pp. 609–618, Apr. 2014. [7] T. Ding, S. Y. Liu, W. Yuan, Z. H. Bie, and B. Zeng, “A twostage robust reactive power optimization considering uncertain wind power integration in active distribution networks,” IEEE Transactions on Sustainable Energy, vol. 7, no. 1, pp. 301–311, Jan. 2016. [8] K. Dehghanpour and S. Afsharnia, “Electrical demand side contribution to frequency control in power systems: A review on technical aspects,” Renewable and Sustainable Energy Reviews, vol. 4, pp. 1267–1276, Jan. 2015. [9] M. Doostizadeh and H. Ghasemi, “Day-ahead scheduling of an active distribution network considering energy and reserve markets,” International Transactions on Electrical Energy Systems, vol. 23, no. 7, pp. 930–945, Oct. 2013. [10] Y. Xu, F. X. Li, Z. Q. Jin, and M. H. Variani, “Dynamic gain-tuning control (DGTC) approach for AGC with effects of wind power,” IEEE Transactions on Power Systems, vol. 31, no. 5, pp. 3339–3348, Sep. 2016.

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Tianyang Zhao received the B.Sc. and M.Sc. degree in automation of electric power systems from North China Electric Power University, Beijing, China in 2011 and 2013, respectively. Currently, he is pursuing his Ph.D. degree in automation of electric power systems in North China Electric Power University, Beijing, China. His research interests include power system operation optimization and game theory.

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Jianhua Zhang (M’98) received the B.Sc. and M.Sc. degrees in electrical engineering from North China Electric Power University, Baoding, China in 1982 and 1984, respectively. Currently, he is working as a Professor in the Department of Electrical and Electronic Engineering, North China Electric Power University. He has been the IET Fellow since 2005, and is also a member of the PES Committee of China National “973 Project.”

Peng Wang (M’00–SM’11) received the B.Sc. degree in electronic engineering from Xi’an Jiaotong University, Xi’an, China, in 1978, the M.Sc. degree from Taiyuan University of Technology, Taiyuan, China, in 1987, and the M.Sc. and Ph.D. degrees in electrical engineering from the University of Saskatchewan, Saskatoon, SK, Canada, in 1995 and 1998, respectively. Currently, he is a Professor of the School of Electrical and Electronic Engineering at Nanyang Technological University, Singapore.