Distributed Generation Hosting Capacity Evaluation for Distribution ...

IEEE TRANSACTIONS ON SUSTAINABLE ENERGY, VOL. 7, NO. 3, JULY 2016

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Distributed Generation Hosting Capacity Evaluation for Distribution Systems Considering the Robust Optimal Operation of OLTC and SVC Shouxiang Wang, Senior Member, IEEE, Sijia Chen, Leijiao Ge, and Lei Wu, Senior Member, IEEE

Abstract—With the rapidly increasing penetration of renewable distributed generation (DG), the maximum hosting capacity (MHC) of a distribution system has become a major concern. To effectively evaluate the ability of a distribution system to accommodate DGs, this paper proposes an MHC evaluation method while considering the robust optimal operation of on load tap changers (OLTCs) and static var compensators (SVCs) in the uncertain context of DG power outputs and load consumptions. The proposed method determines the DG hosting capacities corresponding to different conservative levels. Furthermore, this paper discusses how to find out the most critical technical constraint that may limit the MHC by adjusting parameters of the proposed robust formulation. The effectiveness of the proposed method is demonstrated using a modified IEEE 33-bus distribution system. Index Terms—Distributed generation, distribution system, maximum hosting capacity, robust optimization, uncertainty.

I. I NTRODUCTION

R

ENEWABLE distributed generation (DG) technology is considered as a promising solution to energy crisis and environmental pollution [1], and many countries in the world have issued relevant policies to promote the installation of renewable DGs. More and more green sources, such as solar, wind, hydro, and methane, are being interconnected to the distribution network via DG technology. However, high penetration DG brings various impacts on distribution network planning and operation [2]–[4], including voltage rise, reverse power flow, loss increase, power quality decline, etc. Additionally, because of intermittency of renewable energy sources, the power output of renewable DG presents significant uncertainty [5]. These impacts together limit the ability of a distribution system to connect renewable DGs. Therefore, taking into account uncertainties of DG power outputs and load consumptions, the planners need an Manuscript received October 15, 2015; revised December 30, 2015; accepted February 06, 2016. Date of publication March 07, 2016; date of current version June 16, 2016. This work was supported in part by the National Natural Science Foundation of China under Grant NSFC 51361135704 and in part by the National High Technology Research and Development Program of China under Grant 2014AA052003. This work was also supported by the U.S. National Science Foundation under Grant ECCS-1254310 and Grant PFI:BIC IIP-1534035. Paper no. TSTE-00854-2015. S. Wang, S. Chen, and L. Ge are with the Key Laboratory of Smart Grid of Ministry of Education, Tianjin University, Tianjin 300072, China (e-mail: [email protected]; [email protected]; [email protected]). L. Wu is with the Department of Electrical and Computer Engineering, Clarkson University, Potsdam, NY 13699 USA (e-mail: [email protected]). Digital Object Identifier 10.1109/TSTE.2016.2529627

appropriate method for evaluating the maximum capacity of DGs that can be installed in a distribution network without technical constraint violations. Many methods have been presented to assess the DG hosting capacity of a distribution system with regard to various constraints. Reference [6]–[8] propose optimal power flow (OPF)-based methods for network DG capacity analysis, but they lack the consideration of the variation of DG power outputs and load levels. In [9], the available DG headroom is defined, and several load levels are analyzed to maximize the DG capacity. In [10], a heuristic method is applied to the optimum allocation of the maximum possible DG penetration in a distribution network, and the short-circuit level is added into technical constraints. However, the studies in [9], [10] overlook the impact of active network management (ANM). The major ANM schemes include coordinated voltage control, reactive power compensation, DG power factor control, DG curtailment, and network reconfiguration [11], which have been investigated for increasing the DG hosting capacity of a distribution system [12]–[16]. In [12], the photovoltaic inverter control schemes are incorporated in the OPF to maximize the total wind-based DG penetration on a typical U.K. distribution system. In [13], [14], a multi-period OPF-based method is proposed to evaluate the DG hosting capacity, and the ANM schemes including coordinated voltage control, power factor control, and energy curtailment are incorporated in the method for analyzing the impacts on the maximum DG hosting capacity. In [15], a multi-period OPF approach is presented to assess how the DG hosting capacity can be increased by applying static reconfiguration or dynamic reconfiguration. In [16], with the objective of maximizing the total energy exported, a dynamic OPF is developed to model energy curtailment, energy storage systems, and flexible demand. These previous works apply the OPF-based method to evaluate the maximum hosting capacity (MHC) of a distribution system via a set of scenarios, which are generated by sampling and aggregating data of coincident DG power outputs and load levels. However, DG power outputs and load levels are actually continuous uncertain variables, and the discrete data samples cannot ensure that all possible scenarios of continuous uncertain variables are taken into consideration. In [17], probability distributions of loads are derived from historical data, and a Monte Carlo-based technique is used to

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determine the maximum DG capacity for the predefined locations in a distribution network. However, it is difficult to acquire the accurate joint probability distribution of multiple uncertain variables due to inaccurate historical data and forecast data. Therefore, the stochastic optimization approach has some practical limitations. As an alternative tool to tackle with parameter uncertainty, robust optimization (RO) has gained sustained attention and development over the past dozen years [18]–[22]. Because RO does not depend on the probability distribution of the uncertain parameters, the RO framework has recently been adopted to solve power system problems including unit commitment (UC) [23]–[26], generation expansion planning [27], [28], OPF [29], demand response [30], etc. In this paper, a RO-based method is proposed to evaluate the ability of a distribution system to accommodate DGs. The main contributions of our paper are summarized as follows. 1) The RO-based method for evaluating the MHC of the distribution system is proposed, which takes into account uncertainties of DG power outputs and load levels. Specifically, uncertain DG power outputs and load levels are modeled via deterministic intervals. Since only the boundaries of uncertainty intervals need to be predefined, the proposed evaluation method presents fewer limitations compared with the stochastic optimization approaches. Moreover, the DG capacity allocation acquired by the proposed RO-based method could immunize against all changes of uncertain DG power outputs and load levels within the predefined intervals. 2) In the proposed RO formulation, a series of parameters Γ can be predefined to control the robustness of the DG capacity allocation of a distribution system. As mentioned in [11], [17], if only the worst-case scenarios are utilized to evaluate the MHC, the DG capacity allocation could be over-conservative. The RO-based method proposed in this paper can control the conservative level of the DG capacity allocation by adaptively adjusting the Γ values. 3) Voltage control and reactive power compensation, which are the most widely available ANM schemes in practice, are considered and embedded in the proposed RO formulation of our work. The tap positions of on load tap changers (OLTCs) and the reactive power outputs of static var compensators (SVCs) are modeled as control variables for improving the capability of the distribution system to accommodate DGs. The remaining part of this paper is organized as follows. In Section II, the problem formulation is presented. In Section III, the solution methodology of the proposed RO formulation is described. In Section IV and V, case studies and simulation results are shown and discussed. Finally, in Section VI, the main conclusions are summarized.

II. P ROBLEM F ORMULATION In this section, the linearized power flow equations are described and developed, and the technical limits on the total

Fig. 1. Diagram of a radial distribution system.

DG capacity are introduced. Finally, the mathematical formulation for evaluating the MHC is established. A. Distribution System Model Considering a radial distribution network as shown in Fig. 1, the complex power flow at each node i is described by the DistFlow equations [31]: ∀i ∈ N , V0 = Vsub (1 + tp · a) Pi+1 = Qi+1 = 2 Vi+1

=

Pi − ri (Pi2 + Q2i )/Vi2 − pi+1 Qi − xi (Pi2 + Q2i )/Vi2 − qi+1 Vi2 − 2(ri Pi + xi Qi ) + (ri2 + x2i )(Pi2

(1) (2) (3) +

Q2i )/Vi2 (4)

pi = qi =

pLi qiL

− pgi − qig

(5) −

qis .

(6)

where N is the set of all nodes except the root node 0, Vsub is the rated secondary voltage magnitude of the substation transformer, tp is the position of the OLTC, a is the change ratio per step, Vi is the voltage magnitude of node i, Pi + jQi is the complex power flow from node i to node i + 1, pLi + jqiL is the load consumption of node i, pgi + jqig is the complex power output of the DG at node i, qis is the reactive power output of the SVC at node i, and ri + jxi is the impedance of the line between nodes i and i + 1. In order to reduce the non-linearity of DistFlow equations, their linear versions are proposed in [31], which have been extensively justified and adopted in various distribution system optimization applications [32]–[34]. Specifically, The DistFlow equations (2)–(4) can be simplified as follows: ∀i ∈ N , Pi+1 = Pi − pi+1

(7)

Qi+1 = Qi − qi+1 Vi+1 = Vi − (ri Pi + xi Qi )/Vsub .

(8) (9)

In turn, the branch power flow and the bus voltage magnitude can be expressed by the voltage magnitude of the root node and the active/reactive injection of any other node. As shown in Fig. 1, the radial distribution network has (|N | + 1) nodes and the node 0 represents the root node. Thus, the network has |N | independent nodes and |N | branches. The “path” of node i refers to the set of branches along the route from the root node to the node i, which is unique in the radial distribution network. By introducing path-branch incidence matrix [Tij ], the topology of the radial distribution network can be easily described. If branch j is on path i, Tij = 1; otherwise, Tij = 0. Here, we define that matrix [Bij ] is the transpose

WANG et al.: DISTRIBUTED GENERATION HOSTING CAPACITY EVALUATION FOR DISTRIBUTION SYSTEMS

of matrix [Tij ]. Based on Kirchhoff’s Current Law, (7), and (8), branch active and reactive powers can be reformulated as: ∀i ∈ N ,  Pi = Bij pj (10) j∈N

Qi =



Bij qj .

(11)

j∈N

That is, branch power flows are formulated based on active/reactive power injections of independent nodes. Next, the node-voltage mathematical expression will be deduced. By assigning different values to variable i of (9), a set of voltage equations can be derived. After replacing intermediate variables, namely, other node voltages, the node-voltage mathematical equation can be reformulated as follows: ∀i ∈ N , Vi = V0 −

1 Vsub



(pj Rij + qj Xij ).

(12)

j∈N

where Rij is the sum of resistance for all overlapping branches between the two paths (i.e., one is from the root node to node i, and the other is from the root node to node j.), and Xij is the sum of reactance of all overlapping branches between the two.

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currently work at fixed power factor and are not dispatchable. On the other hand, static/dynamic network reconfiguration itself is a complex optimization problem. Our research mainly focuses on how to deal with uncertainty and variability of DG outputs and load levels, rather than involves all types of ANM schemes. Therefore, DG power factor control, DG power curtailment, and network reconfiguration are not considered in our present research.

C. Technical Constraints The following technical constraints are taken into consideration when determining the maximum DG capacity: 1) Bus Voltage Magnitude Constraint: Vi,min ≤ Vi ≤ Vi,max ,

∀i ∈ N.

where Vi,min and Vi,max are the lower and the upper voltage limits at bus i. 2) Thermal Capacity Constraint: The complex power flow across a transformer or a line is constrained by its maximum apparent power. 2 2 ≤ Ssub,max p2sub + qsub

(14)

2 , ∀i ∈ N . p2i + qi2 ≤ Si,max

B. Considerations About ANM Schemes ANM schemes include coordinated voltage control, reactive power compensation, DG power factor control, DG power curtailment, network reconfiguration, etc. Voltage control and reactive power compensation equipment is widely used in traditional distribution systems, and has an influence on the MHC. Therefore, coordinated voltage regulation and reactive power compensation are taken into consideration when evaluating the MHC. In our research, OLTC and SVC are selected to represent coordinated voltage control and reactive power compensation respectively. Many traditional distribution systems utilize shunt capacitor banks (SCBs) for reactive power compensation. Compared with SCBs, SVCs are usually more expensive. However, the rapid development of power electronics technology brings potential to significantly reduce the cost of SVCs. On the other hand, high penetration DG may bring a series of technical problems, such as reverse power flow and voltage rise. Different from SCBs, SVCs can not only compensate capacitive reactive power [35], but also give out inductive reactive power to relieve the issue of voltage rise [36]. In addition, the action of SVCs is faster than SCBs [34], and they are capable of responding quickly to support voltages during the transient state after a contingency. Therefore, in this paper we consider SVC to compensate reactive power. In regard to DG power factor control and DG power curtailment, they largely depend on specific commercial arrangements and regulatory incentives, considering the fact that many DGs are owned by consumers rather than distribution system operators (DNOs). Therefore, most of renewable energy based DGs

(13)

(15)

where psub and qsub are active and reactive powers across the substation transformer, Ssub,max is the allowed maximum apparent power of the transformer, and Si,max is the allowed maximum apparent power of line i. Because the feasible region restricted by (14) or (15) is the interior of a circle, a polygonal inner-approximation method (given in Appendix A) is employed to linearize the thermal capacity constraint. Hence, a range of linear equations can be derived to replace constraints (14)–(15). αc psub + βc qsub + δc Ssub,max ≤ 0,

∀c ∈ {1, 2, . . . , 12}. (16)

αc pi + βc qi + δc Si,max ≤ 0, ∀c ∈ {1, 2, . . . , 12}, ∀i ∈ N . (17) where αc , βc , and δc are coefficients of the set of the linearized constraints. 3) Upstream Power Grid Constraint: The distribution network is connected to the upstream power grid by the substation transformer. In order to ensure the security of the upstream grid, the exchanged active/reactive power is limited by: psub,min ≤ psub ≤ psub,max qsub,min ≤ qsub ≤ qsub,max .

(18) (19)

where psub,min and psub,max are the lower and the upper bounds of the exchanged active power, and qsub,min and

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qsub,max are the lower and the upper bounds of the exchanged reactive power. 4) Tap Position Constraint of OLTC: tpmin ≤ tp ≤ tpmax ,

tp ∈ int .

(20)

where tpmin and tpmax are the lower and the upper limits on tap position of OLTC. 5) Reactive Power Constraint of SVC: s s qj,min ≤ qjs ≤ qj,max , s qj,min

∀j ∈ S.

(21)

s qj,max

where and are the lower and the upper limits on SVC at node j, and S is the set of nodes at which SVCs have been installed. For simplicity, the technical constraint of short-circuit level is not taken into account in this paper. However, they can be included in the proposed model based on the methodology proposed in [37].



(αc B1j + βc B1j tan ϕLj )˜ pLj −

j∈N

−

βc B1j qjs

j∈S



(αc B1j + βc B1j tan ϕgj )˜ pgj + δc Ssub,max

j∈M

 − (αc B1j + βc B1j tan ϕG ˜ j pG j )w j ≤ 0, ∀c∈{1, 2, . . . , 12} j∈G

(24)   (αc Bij +βc Bij tan ϕLj )˜ pLj − (αc Bij +βc Bij tan ϕgj )˜ pgj j∈N

−

j∈M



(αc Bij + βc Bij tan ϕG ˜ j pG j )w j

j∈G

−



βc Bij qjs + δc Si,max ≤ 0, ∀c ∈ {1, 2, . . . , 12}, ∀i ∈ N

j∈S

(25) psub,min ≤

   B1j p˜Lj − B1j p˜gj − B1j w ˜ j pG j ≤ psub,max

j∈N

D. Mathematical Formulation The objective of the optimization problem is to evaluate the maximum DG capacity that can be accommodated by the distribution system. In order to ensure the security of the distribution system, different load and DG output levels need to be taken into consideration. In this paper, all loads and DGs are simulated to have constant power factors. In this paper, M is defined as the set of nodes at which DGs have been installed, and G is the set of the candidate nodes at which DGs could be installed. The uncertain active power output is modeled as a random variable p˜gj (j ∈ M ) which takes values in the interval [¯ pgj − pˆgj , p¯gj + pˆgj ]. Note that, p¯gj is the midpoint of the interval [¯ pgj − pˆgj , p¯gj + pˆgj ], and pˆgj is the undulation amplitude with respect to p¯gj . In the same way, the interval [¯ pLj − pˆLj , p¯Lj + pˆLj ] represents the range of the uncertain active load level p˜Lj (j ∈ N ). In addition, pG j is defined as the capacity of DG which will be installed at node j(j ∈ G), and the capac˜ j ∈ [w ¯j − w ˆj , w ¯j + w ˆj ]) is the uncertain ratio ity factor w ˜ j (w . It is important to highlight that of the active power output to pG j G w ˜j pj represents the actual active power output after the DG is installed at node j(j ∈ G). Based on the aforementioned definitions, the MHC evaluation problem can be formulated as follows:  max pG (22) j



j∈M

j∈G

(26) qsub,min ≤



B1j p˜Lj tan ϕLj −

j∈N

−





B1j qjs ≤ qsub,max

(27)

j∈S

p¯gj − pˆgj ≤ p˜gj ≤ p¯gj + pˆgj , p¯Lj − pˆLj ≤ p˜Lj ≤ p¯Lj + pˆLj , w ¯j − w ˆj ≤ w ˜j ≤ w ¯j + w ˆj , pG j ≥ 0,

B1j p˜gj tan ϕgj

j∈M

G B1j w ˜ j pG j tan ϕj −

j∈G



∀j ∈ G.

∀j ∈ M

(28)

∀j ∈ N

(29)

∀j ∈ G

(30) (31)

Where ϕ is load and DG’s power factor angle. In the above formulation, bus voltage constrains can be obtained as (23) by substituting (1), (5), (6), and (12) into (13). Thermal capacity constraints (24), (25) can be derived from (5), (6), (10), (11), (16) and (17). Similarly, upstream power grid constraints can be reformulated as (26), (27). In this paper, the optimization problem of evaluating the MHC is formulated to maximize the total capacity of DGs, which will be installed at the predefined locations. In this optimization problem, tap positions of OLTCs and reactive power outputs of SVCs are modeled as control variables, rather than regarded as fixed values. This is because voltage j∈G regulation and reactive power compensation could affect the total DG capacity that can be hosted by the distribution netsubject to (20), (21), work, and in turn needs to be integrated in the optimization  problem. 2 G Vi,min Vsub ≤ Vsub (1 + tp · a)+ w ˜ j pG j (Rij + Xij tan ϕj ) The formulation (22)–(31) presents an optimization problem j∈G ˜j . It should be noted with uncertain parameters p˜gj , p˜Lj and w  g  g s qj Xij + p˜j (Rij + Xij tan ϕj ) + that uncertain parameters (load and generation) are not decision j∈S j∈M variables. The MHC is required to guarantee that the techni L L p˜j (Rij + Xij tan ϕj ) ≤ Vi,max Vsub , ∀i ∈ N (23) cal constraints will not be violated when load levels and DG − power outputs vary within the predefined uncertainty interval. j∈N

WANG et al.: DISTRIBUTED GENERATION HOSTING CAPACITY EVALUATION FOR DISTRIBUTION SYSTEMS

However, existing deterministic methods and tools cannot be used to directly solve the optimization problem with uncertain parameters. In the next section, the solution methodology of this problem will be elaborated.

max

⎧ ⎨ ⎩

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  G G  w ˆj Rij + Xij tan ϕG j pj zub,ij

j∈G



+

  g pˆgj Rij + Xij tan ϕgj  zub,ij

j∈M

III. S OLUTION M ETHODOLOGY In this section, the RO formulation for evaluating the MHC is proposed. The RO method proposed by Bertsimas and Sim [22] is employed, and a set of Γ(Γ ∈ int) is introduced to adjust the degree of the solution conservativeness. Consequently, we can assess the potential capacity of the distribution network for accommodating DGs with the presence of various uncertainties. In the following, the establishment and transformation of the robust counterpart are discussed.

In the proposed MHC evaluation optimization model, uncertain parameters all appear in the constraints. Because the process of the robust counterpart establishment is similar for individual constraints, we use the voltage upper limit constraints as an example. According to the notion of the RO method [22], the constraint (Vi≤ Vi,max , i ∈ N ) can be formulated as follows:   2 G Vsub (1 + tp · a) + qjs Xij + w ¯ j pG j (Rij + Xij tan ϕj ) +



j∈S

p¯gj (Rij + Xij tan ϕgj ) −

j∈M



+

∀j ∈ G

(36)

g zub,ij L zub,ij

≤ 1,

∀j ∈ M

(37)

≤ 1,

∀j ∈ N.

(38)

pG j ≥ 0, ∀j ∈ G.

(33)

where Γv,i is a predefined parameter to adjust the conservativeness of the robust solution. By changing Γv, i, the maximum function in (32) could take different values. For the original optimization problem, this is equivalent to taking into consideration different load levels and DG power outputs. The larger the Γv, i is, the more conservative DG capacity would be. It should be noted that (32) includes the maximum function, which may not be effectively solved. Therefore, the robust counterpart needs to be transformed properly.

  G  ˆj Rij + Xij tan ϕG zub,i + hG ub,ij ≥ w j pj , ∀j ∈ G   g g g zub,i + hub,ij ≥ pˆj Rij + Xij tan ϕj , ∀j ∈ M   zub,i + hLub,ij ≥ pˆLj Rij + Xij tan ϕLj  , ∀j ∈ N hG ub,ij hgub,ij hLub,ij

First of all, the maximum function in (32) is replaced by the following equivalent form: ∀i ∈ N ,

(40) (41) (42)

≥ 0, ∀j ∈ G

(43)

≥ 0, ∀j ∈ M

(44)

≥ 0, ∀j ∈ N

(45)

zub,i ≥ 0.

(46)

g L where hG ub,ij , hub,ij , hub,ij and zub,i are dual variables of constraints (35)–(38), respectively. Substituting (39) to (32), the robust counterpart composed of (22) and (32) can be recast in terms of a max-min optimization problem:  pG (47) max j j∈G

subject to 2 (1 + tp · a) + Vsub

+

 j∈M

B. Transformation of the Robust Counterpart

j∈N

subject to

∀i ∈ N (32)

j∈M

(39)



j∈Nv,i

(35)

j∈N

j∈G

p¯Lj (Rij + Xij tan ϕLj )

≤ Vi,max Vsub ,

j∈M

L zub,ij ≤ Γv,i

g G L , zub,ij , and zub,ij are auxiliary variables. where zub,ij By applying the duality theory [38], the equivalent maximum function can be recast as follows: ∀i ∈ N , ⎫ ⎧ ⎬ ⎨   g  hG hub,ij + hLub,ij min Γv,i zub,i + ub,ij + ⎭ ⎩

j∈Mv,i

  pˆLj Rij + Xij tan ϕLj 



g zub,ij +

G ≤ 1, 0 ≤ zub,ij

0≤

 + max {Gv,i ∪Mv,i ∪Nv,i |Gv,i ⊆G,Mv,i ⊆M,Nv,i ⊆N,|Gv,i|+|Mv,i |+|Nv,i|=Γv,i }    G  g   w ˆj Rij + Xij tan ϕG pˆj Rij + Xij tan ϕgj  j pj + 



G zub,ij +

j∈G

j∈N

j∈Gv,i

⎭

subject to

j∈G



  L pˆLj Rij + Xij tan ϕLj  zub,ij

j∈N

0≤ A. Establishment of the Robust Counterpart



+

(34) ⎫ ⎬

+ min



qjs Xij +

j∈S

p¯gj (Rij ⎧ ⎨ ⎩

+



G w ¯ j pG j (Rij + Xij tan ϕj )

j∈G

Xij tan ϕgj )

−



p¯Lj (Rij + Xij tan ϕLj )

j∈N

Γv,i zub,i +

≤ Vi,max Vsub ,



hG ub,ij +

j∈G

∀i ∈ N.

 j∈M

hgub,ij +

 j∈N

hLub,ij

⎫ ⎬ ⎭

(48)

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Moreover, reference [22, Theorem 1] proves that the optimization model composed of (47) and (48) is equivalent to the following linear programming model:  pG (49) max j j∈G

subject to

  2 G (1 + tp · a) + qjs Xij + w ¯ j pG Vsub j (Rij + Xij tan ϕj )+ 

j∈S

Fig. 2. Diagram of the modified IEEE 33-bus distribution network.

j∈G

p¯gj (Rij + Xij tan ϕgj ) −

j∈M



p¯Lj (Rij + Xij tan ϕLj )+

TABLE I C APACITY A LLOCATION (MW) OF E ACH PV (Γ = 39)

j∈N

  g  hub,ij + hLub,ij ≤Vi,max Vsub , ∀i∈N. Γv,i zub,i + hG ub,ij + j∈G

j∈M

j∈N

(50) According to the aforementioned process, the complete robust counterpart of the original optimization model (22)–(31) can be obtained similarly. Thus, the RO formulation for evaluating the MHC will be transformed into a mixed-integer linear programming (MILP) problem, which can be solved by commercial software. The completed MILP formulation of the RO counterpart is given in Appendix B. IV. C ASE S TUDY This section profoundly investigates the performance of the proposed RO-based method in terms of evaluating the MHC of a modified IEEE 33-bus distribution system. First, the validity of the proposed RO-based method is demonstrated by comparing with the deterministic method. Then, the MHC of the distribution network is evaluated corresponding to different Γ values, and the probability of constraint violation is summarized by using numerical simulations. Finally, by adjusting Γ, we point out the bottlenecks that may limit the MHC. All numerical experiments are implemented using LINGO 14.0 at Intel Quad Core 2.66 GHz with 2 GB memory. A. Test System and Constraints The proposed RO-based method is tested on the widely used IEEE 33-bus distribution system [31]. As shown in Fig. 2, the benchmark system is modified to take into account the presence of the OLTC and SVCs (S1 and S2), which may affect the capability of the distribution system for accepting power injections from DGs. It should be noted that the OLTC and SVCs are three-phase devices. In the numerical experiments, all the loads and DGs are assumed to have constant factors. The peak load is 3.715 MW and 2.3 MVar, and the minimum load is 20% of the peak. Seven potential sites (i.e., G1 to G7) are identified for connecting DGs. In this paper, solar photovoltaic (PV) units are chosen as an example to validate the proposed methodology. According to the recommendation of the IEEE 1547 Standard [39], PV units operate at a unity power factor in our experiments. The maximum power output of each PV unit is its installed capacity, and the minimum power output is zero. The OLTC on the 6–MVA transformer has 17 tap positions, and can regulate the second voltage within a ±10% range. The

TABLE II T HE S TATE OF OLTC AND SVC (Γ = 39)

reactive power output of each SVC can continuously vary from -100 kVar to 300 kVar for each phase. The lower and the upper voltage limits are set to 0.95 p.u. and 1.05 p.u. at all nodes, and the thermal limit of all lines is set to 6.6 MVA (which corresponds to a current of 300 A). Finally, up to 6 MW of active and 3 MVar of reactive power can be exchanged with the upstream power grid through the substation transformer.

B. Effectiveness of the Proposed RO-Based Method As shown in Fig. 2, considering 32 load consumptions and 7 PV units that will be installed at 7 predefined sites (G1 to G7), the total number of uncertain power injections is 39. As a result, the evaluator can select the value of Γ in the interval [0,39]. In this section, it is assumed that all the parameters Γ corresponding to different constraints take the same value. Aimed at illustrating the effectiveness of the proposed RO-based method, we need to assess the capability of the distribution network to accept PVs in the most conservative condition (Γ = 39). Table I presents the capacity allocation of each PV when Γ of each constraint takes the value of 39. The MHC of the distribution network is 6.508 MW. Together with the MHC, the proposed RO approach also derives optimal values of other key control variables as shown in Table II. According to the notion of the proposed RO-based method, the PV capacity allocation as shown in Table I will not jeopardize the security of the distribution system in Fig. 2, even if the worst scenarios happen. Generally speaking, the worst scenarios usually correspond to the maximum PV output at the minimum load level (named as Scenario A in the following for short), and no PV output at the maximum load level (named as Scenario B in the following for short). In order to demonstrate the robustness of the proposed evaluation method directly, the exact power calculations [40] are implemented for Scenarios A

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TABLE III T HE P OWER E XCHANGES W ITH THE S UPPLY G RID

Fig. 3. Voltage profiles in Scenarios A and B. (a) The result of the exact power flow calculation. (b) The result of the linearized power flow calculation.

violated in Scenario B. In order to account for the constraint violation, the linearized power flow calculations [31] are also implemented, and corresponding voltage profiles are shown in Fig. 3(b). It can be seen from Fig. 3(b) voltage constraints are not violated. Therefore, the error of voltage profiles is caused by the difference between the two power flow models. The average relative error of voltage profiles between the exact and the linearized power flow equations is 0.24%, which is acceptable from the engineer point of view. In order to avoid such violations caused by the approximations, distribution system operators (DSOs) can implement the exact and the linearized power flow calculations with respect to different load levels; then, the average relative error between the exact and the linearized power flow equations can be obtained; at last, DSOs can tighten constraints accordingly corresponding to the relative error before assessing the DG hosting capacity of the distribution system. In addition, it can be seen from Fig. 3(b) the voltage magnitudes of bus 22 and 25 are equal to 1.05 p.u. in scenario A, and the voltage magnitudes of bus 13 and 30 are equal to 0.95 p.u.. Therefore, both scenario A (the overall load is minimal and the generation is also maximum) and scenario B (the overall load is maximum and the generation is also minimal) are the worstcase scenarios. All loads are binding at the (lower or upper) interval limit in the worst-case scenarios. In conclusion, the RO solution immunizes against the worst cases, and the effectiveness of the proposed RO method can be verified in the most conservative condition.

Fig. 4. Apparent powers of individual branches in Scenarios A and B.

C. Selection of the Parameter Γ and B. Based on the power flow results, we can check whether certain constraints are satisfied. Before the power flow analysis, it is assumed that PV units have been installed in the modified IEEE 33-bus distribution network in accordance with the capacity allocation in Table I. Note that the tap position of the OLTC and the reactive compensation of the SVCs are set as those values in Table II for both Scenario A and Scenario B during the power flow analysis. This is because the tap position of the OLTC and the reactive power of the SVCs corresponding to ANM schemes are optimized to handle indeterminate DG powers and uncertain loads in a robust way, i.e., being capable of handling all the scenarios sampled from indeterminate DG powers and uncertain loads. Scenarios A and B are just two extreme cases of countless scenarios. The results of the power flow calculations are presented in Figs. 3(a), 4, and Table III. In this paper, each branch of the distribution network is named after the number of the bus at its end. It can be seen in Figs. 3(a), 4, and Table III that only voltage constraints at some buses (such as bus 12) are slightly

Before evaluating the MHC of the distribution network by using the proposed RO-based method, DNOs firstly need to predefine the Γ values. According to the idea of RO, a higher Γ value means being more conservative, while a lower Γ value may result in a larger allowable PV installation capacity. Quantitative analysis is presented in the following to explore the relationship between the Γ value and the MHC. The MHC values corresponding to different Γ values are shown in Fig. 5. It can be observed in Fig. 5 that the curve is steep when Γ < 9, and becomes more gentle when Γ > 9. Specifically, the MHC value with Γ = 0 increases by 152.9% compared with the case of Γ = 39, while Γ = 9 only causes a 21.3% growth compared with the case of Γ = 39. It should be noted that Γ = 0 means that the DNOs will evaluate the DG hosting capacity with the most optimistic attitude. In other words, when the DNOs set Γ = 0, they believe that DG power and load values that may cause the constraint violation will not happen. In conclusion, the MHC value will not increase significantly until the parameter Γ is reduced to a certain extent. As a result, the DNOs can adjust the parameter Γ to improve the

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Fig. 5. The MHC values with different Γ values. Fig. 7. PV power outputs.

Fig. 8. The probability of constraint violation with different Γ values. Fig. 6. Load profiles.

MHC of the distribution system. However, it should be noted that the larger the parameter Γ is, the worse the robustness of the optimal solution will be. Consequently, the larger Γ value will increase the risk of technical constraint violation. In the proposed RO-based evaluation method, multiple load and generation values may correspond to the solution of RO, because they act as given uncertain parameters rather than decision variables in the RO. In fact, one solution of RO can handle multiple load and generation values, i.e. being able to maximize the MHC without constraint violation under a large number of rather than only one possible load and generation values. Note that load and generation uncertainty sets that the solution of RO can handle get “smaller” with the decrease of Γ, but always contain a large number of scenarios. Quantitative evaluation on the violation risk under different Γ values is presented as follows. It is worth noting that in the proposed RO approach, the tap position of OLTCs and power outputs of SVCs are optimized with respect to different load and PV power output variations within the predefined interval. That is, they are considered as uncertainty-independent control variables. However, the tap position of OLTCs and power outputs of SVCs may be flexibly adjusted according to the operator’s instructions for responding to variations of load and PV outputs. Therefore, when evaluating the risk for decreasing the Γ value, the adjustment of OLTCs and SVCs are taken into consideration. One year data of load and PV power is utilized to test the probability of constraint violation. Figs. 6 and 7 show the 1year profiles for load and PV generation for all nodes. The load of each node at each hour can be calculated by multiplying the load profile shown in Fig. 6 by the peak load level of the IEEE 33-bus system. In a similar way, the PV power generation at the predefined nodes in each hour can be obtained.

In order to calculate the probability of constraint violation, the evaluation procedure is composed of the following steps: a) Read load and PV power data for each node at h (h represents the simulation time). b) Run an optimal power flow (check if the optimal solution can be found by changing the tap position of OLTCs and the power outputs of SVCs, without violating technical constraints). c) Let h = h + 1. d) If h < 8760, repeat steps a)-d); otherwise stop. After the above steps have been completed successfully, we can figure out the probability that the constraints are violated. As shown in Fig. 8, lower Γ values results in higher risk levels. Therefore, according to Figs. 5 and 8, the DNOs can choose a proper Γ value as a tradeoff between the MHC and the risk based on the active adjustment ability of the system. If enough data on load and PV power is available, the DNOs can evaluate the risk with different Γ values and choose the appropriate Γ value; otherwise, the DNOs can select the most conservative scheme by predefining the maximum Γ value. D. Analysis of Technical Constraints The MHC of a distribution system is limited by a set of technical constraints, but their influencing degrees are different. If the DNOs want to increase the MHC, they need to find out the most critical technical bottlenecks. In this section, it is assumed that parameters Γ for all the voltage constraints take the same value, which is represented by Γv . In a similar way, Γs represents parameters Γ for all thermal capacity constraints, and Γpq represents parameters Γ for the upstream power grid constraints. In turn, the following simulation is performed: for Γv , Γs , and Γpq , make two of them take the maximum value, and

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embedded in the RO framework for improving the DG hosting capacity. Other ANM schemes, such as shunt capacitor compensation, DG power factor control, DG power curtailment, and network reconfiguration, also present significant impacts on the MHC [12]–[16]. It is necessary to discuss whether the method proposed in this paper can be extended to other ANM schemes. 1) Shunt Capacitor Compensation: It is assumed that a bank of capacitors is installed at node j. Thus, the reactive power output of the capacitor bank (CB) at node j can be represented as follows:  sj,k Qj,k,CB (51) qj,CB = k

Fig. 9. The MHC values with different Γ values.

calculate the MHC when another Γ varies from 0 to 39. The simulation result is shown in Fig. 9(a). It is evidently observed from Fig. 9(a) that no matter how large the Γ value is, the values of the MHC lie in the interval [6.5 MW, 6.8 MW]. In another word, if the Γ value of only one set of technical constraints decreases, the MHC will not significantly increase. Therefore, we can conclude that the three sets of technical constraints limit the MHC almost evenly. In order to demonstrate the availability of the above method, another simulation is implemented, which increases the boundaries of thermal capacity constraints and upstream power grid constraints by 2 times. The results are presented in Fig. 9(b). It can be seen from Fig. 9(b) that the lower Γv value is, the higher the MHC is, while the MHC stays the same no matter what value is set for Γs and Γpq . Therefore, the bus voltage magnitude constraints are the most critical factor of the MHC in this case. From the above simulations, it can be concluded that the proposed RO approach can help DNOs identify the most critical factor that may impact the MHC. According to the most critical factor, the DNOs can take corresponding measures to improve the ability of the distribution system for effectively accommodating more DGs. V. D ISCUSSION A. Extended Application In this paper, a RO-based method is proposed to assess the MHC of a distribution system for accommodating DGs, and this method mainly focuses on how to cope with uncertainty and variation of DG power outputs and load levels. In addition, as two of the most widely used ANM schemes in practice, coordinated voltage control and reactive power compensation are

where sj,k,CB is a binary indicator of switch status of the kth capacitor in the CB at node j, Qj,k,CB is the size of the kth capacitor in the CB at node j, and qj,CB is the reactive power output of CB at node j. Compared with the SVC model, binary variables are introduced in the model of shunt capacitor. Because the RO approach employed in this paper [21] can address data uncertainty for discrete optimization, shunt capacitor compensation can be incorporated into the method proposed in this paper. 2) DG Power Factor Control: DG power factor control can be implemented by adjusting reactive power output of the DG inverter. The DG reactive power limit can be represented in terms of the power factor (μ) limit as follows [12]: ∀j ∈ M ∪ G,

pgj 2

(pgj ) + (qjg )

2

≥ μmin

(52)

where μmin is the lower limit of DG power factor. In turn, (52) can be transformed into the following linear constraint:

pgj 1 − μ2min pgj 1 − μ2min g − ≤ qj ≤ . (53) μmin μmin The DG power factor control is modeled by the linear constraint (53), so it can be embedded into the RO framework in this paper [22]. 3) DG Power Curtailment: Curtailment of DG power output can be represented by following constraints:  g  g pj,t,curt τ t ≤ γ p˜j τ t , ∀j ∈ M (54) t∈T



t∈T

t pG j,t,curt τ

≤γ

t∈T



t w ˜ j pG j τ ,

∀j ∈ G.

(55)

t∈T

where τ t is the time duration of the period t, and γ is a curtailment factor to quantify the amount of curtailed energy. Curtailment variables pgj,t,curt and pG j,t,curt need to be limited by the following constraints: pgj,t,curt ≤ p˜gj , pG j,t,curt

≤

∀j ∈ M, ∀t ∈ T

w ˜ j pG j ,

∀j ∈ G, ∀t ∈ T.

(56) (57)

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TABLE IV R ANG OF C OMPUTATIONAL T IME ( S )

The aforementioned model of DG power curtailment is linear and meets the requirement of the RO framework [22]. It should be noted that the DG power curtailment inherently requires analyzing multiple periods. Using RO framework to tackle the multi-period issues involves extending variables and constraints used in the paper to time-series variables, applying them at each time-step and also considering the interaction among different time-steps. The RO framework employed in this paper has been applied to UC [24], [41], which is a multi-period optimization problem. Therefore, DG power curtailment has the potential to be incorporated in the method proposed in this paper. 4) Network Reconfiguration: In general, network reconfiguration can be classified into static reconfiguration and dynamic reconfiguration. The static reconfiguration approach has been investigated via MILP models [42], [43]. Specifically, in [44], constraints of the static reconfiguration are linearized by means of adopting and simplifying the DistFlow equations [31]. Therefore, static reconfiguration can be integrated into the method proposed in this paper. When distribution system status is transferred from one to another, dynamic reconfiguration limits the number of switching actions (Δd), which can be modeled as follows:  |dij,t − dij,t−1 | ≤ Δd, ∀t ∈ T. (58) ij∈Ω

where Ω represents the subset of lines with remotely controlled switches, and dij,t is a binary indicator for the switch status of a line at time t. Constraint (58) can be linearized according to the transformation of the constraint that limits the number of switch actions for capacitors [34]. In addition, dynamic network reconfiguration requires multi-period analysis. Similar with the discussion on DG power curtailment, dynamic reconfiguration has the potential to be embedded into the RO framework employed in this paper. B. Computational Performance Table IV shows computational times of the cases in section IV-A, which are all less than 15 s with different Γ values. It should be noted that the computational time includes the time that was spent on passing input data and building solution report. Besides, it is necessary to analyze the general computational burden of the method proposed in this paper, especially when evaluating the MHC of large distribution systems with the consideration of multiple periods. Taking into account uncertainties of DG power and load, an MILP formulation (22)–(31) with the uncertain parameters is proposed in this paper to assess the DG hosting capacity of a distribution system. By means of appropriate transformation, the RO counterpart of the original MILP formulation is still

an MILP problem, even when evaluating a large distribution system for multiple periods. The time complexity of a linear programming (LP) problem is polynomially dependent on the number of continuous variables. When dealing with an MILP problem, a solution tree is established and an LP is solved at each node of the tree. It should be noted that the time complexity of an MILP may be exponentially dependent on the number of integer variables [45]. If the formulation has nz integer variables at each time step, nz T integer variables will be embedded in the multi-period (T ) problem. For an MILP problem with moderate number of integer variables, it usually can be solved by state-of-the-art MILP solvers in reasonable computational time. VI. C ONCLUSION This paper presents a RO-based method for evaluating the ability of a distribution system to accommodate DGs, which uses distribution-free bounded intervals to model uncertainty of DG power outputs and load levels. The proposed method ensures that the DG capacity allocation immunizes against the inherent uncertainty of DGs and loads. In addition, voltage control and reactive power compensation are incorporated into the proposed RO-based formulation for maximizing the DG hosting capacity of a distribution system. Moreover, the planner can control the robustness of the DG capacity allocation by flexibly adjusting the parameter Γ of the proposed RO-based formulation. Simulation results on a modified IEEE 33-bus distribution system verify the effectiveness of the proposed RO-based method by revealing that all the technical constraints hold at the presence of uncertainties in the most conservative case. Moreover, quantitative analysis is given to evaluate the violation risk of MHC decisions under different robustness levels, which offers a practical method to optimally determine Γ. Finally, two specific cases are presented to illustrate the proposed RO-based method for identifying the most critical technical bottlenecks that may limit the MHC of the distribution system. A PPENDIX A The feasible region limited by constraint (14) is the interior of the circle, whose radius equals to Ssub,max . As shown in Fig. 10, a regular polygon with 12 edges is used to linearize the quadratic inequality corresponding to the thermal capacity constraint of the substation transformer. Because vertices of the regular polygon are all located at the circumference, the coordinates of the regular polygon as well as the coefficients of constraints (16), (17) can be obtained. In addition, the values of the coefficients are listed in Table V. A PPENDIX B In this section, the equivalent robust counterpart of the formulation (22)–(31) is fully described as follow: ∀i ∈ N ,  pG (59) max j j∈G

WANG et al.: DISTRIBUTED GENERATION HOSTING CAPACITY EVALUATION FOR DISTRIBUTION SYSTEMS

−



(αc B1j + βc B1j tan ϕG ¯ j pG j )w j + zsub Γsub +

j∈G



1121

hG sub,j +

j∈G



 hgsub,j + hLsub,j ≤ 0, ∀c ∈ {1, 2, . . . , 12}

j∈M

j∈N

(68)   G G  ˆj pj , ∀j ∈ G ≥ αc B1j + βc B1j tan ϕj w zsub + (69)   g g g  zsub + hsub,j ≥ αc B1j + βc B1j tan ϕj pˆj , ∀j ∈ M (70)   zsub + hLsub,j ≥ αc B1j + βc B1j tan ϕLj  pˆLj , ∀j ∈ N (71)   (αc Bij + βc Bij tan ϕLj )¯ pLj − βc Bij qjs hG sub,j

Fig. 10. Diagram of a polygonal inner-approximation method.

j∈N

TABLE V C OEFFICIENTS OF THE L INEARIZED T HERMAL C APACITY C ONSTRAINTS

−

j∈S



(αc Bij +

βc Bij tan ϕgj )¯ pgj

+ Δc Si,max

j∈M

−



(αc Bij + βc Bij tan ϕG ¯ j pG j )w j + zc,i Γs,i +

j∈G



hG c,ij +

j∈G

zc,i +



hgc,ij +

j∈M



hLc,ij ≤ 0, ∀c ∈ {1, 2, . . . , 12}

j∈N

∀c ∈ {1, 2, . . . , 12}, ∀j ∈ G   zc,i + hg ≥ αc Bij + βc Bij tan ϕg  pˆg ,

(73)

∀c ∈ {1, 2, . . . , 12}, ∀j ∈ M   zc,i + hLc,ij ≥ αc Bij + βc Bij tan ϕLj  pˆLj

(74)

∀c ∈ {1, 2, . . . , 12}, ∀j ∈ N    B1j p¯Lj − B1j p¯gj − B1j w ¯ j pG j + zup Γup

(75)

c,ij

subject to (20), (21), (31),   2 G Vsub (1 + tp · a) + qjs Xij + w ¯ j pG j (Rij + Xij tan ϕj ) +



j∈S

j∈G

p¯gj (Rij + Xij tan ϕgj ) −

j∈M



p¯Lj (Rij + Xij tan ϕLj )

j∈N

  g  hub,ij + hLub,ij ≤ Vi,max Vsub + Γv,i zub,i + hG ub,ij + j∈G

j∈M

j∈N

(60)

  G  ˆj Rij + Xij tan ϕG ∀j ∈ G (61) zub,i + hG ub,ij ≥ w j pj ,   g g g (62) zub,i + hub,ij ≥ pˆj Rij + Xij tan ϕj , ∀j ∈ M   (63) zub,i + hLub,ij ≥ pˆLj Rij + Xij tan ϕLj  , ∀j ∈ N   2 s G − Vsub (1 + tp · a) − qj Xij − w ¯j pj (Rij + Xij tan ϕG j ) −



j∈S

p¯gj (Rij + Xij tan ϕgj ) +

j∈M

j∈G



p¯Lj (Rij + Xij tan ϕLj )

j∈N

  g  hlb,ij + hLlb,ij ≤ −Vi,min Vsub + zlb,i Γv,i + hG lb,ij + j∈G

zlb,i + zlb,i +

hG lb,ij hglb,ij hLlb,ij

zlb,i +  (αc B1j j∈N

−



j∈M

j∈M

j∈N

  G  ≥w ˆj Rij + Xij tan ϕG ∀j ∈ G j pj ,   g g ≥ pˆj Rij + Xij tan ϕj , ∀j ∈ M   ≥ pˆLj Rij + Xij tan ϕLj  , ∀j ∈ N  + βc B1j tan ϕLj )¯ pLj − βc B1j qjs j∈S

(αc B1j + βc B1j tan ϕgj )¯ pgj + δc Ssub,max

(64) (65) (66) (67)

(72)

   ˆ j pG ≥ αc Bij + βc Bij tan ϕG j w j ,

hG c,ij

j

j∈N

+



j∈M

hG up,j +

j∈G



j∈G

hgup,j

+

j∈M

zup +

≥



hLup,j ≤ psub,max

(76)

j∈N

ˆ j pG zup + hG up,j ≥ B1j w j , hgup,j hLup,j

j

B1j pˆgj , B1j pˆLj ,

∀j ∈ G

(77)

∀j ∈ M

(78)

≥ ∀j ∈ N (79) zup +    g L G − B1j p¯j + B1j p¯j + B1j w ¯j pj + zlp Γlp j∈N

+



j∈M

hG lp,j

+

j∈G



j∈G

hglp,j

+

j∈M

zlp +

≥

hLlp,j ≤ −psub,min

(80)

j∈N

ˆ j pG zlp + hG lp,j ≥ B1j w j , hglp,j hLlp,j



B1j pˆgj , B1j pˆLj ,

∀j ∈ G

(81)

∀j ∈ M

(82)

≥ ∀j ∈ N (83) zlp +    g g L L G B1j p¯j tan ϕj − B1j p¯j tan ϕj − B1j w ¯j pj tan ϕG j j∈N

−



j∈M

B1j qjs

j∈S

≤ qsub,max zuq + zuq + zuq +

hG uq,j hguq,j hLuq,j

+ zuq Γuq +

 j∈G

j∈G

hG uq,j

+



hguq,j +

j∈M

  G  ≥ B1j w ˆj tan ϕG ∀j ∈ G j pj ,   g g ≥ B1j pˆj tan ϕj , ∀j ∈ M   ≥ B1j pˆLj tan ϕLj  , ∀j ∈ N



hLuq,j

j∈N

(84) (85) (86) (87)

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IEEE TRANSACTIONS ON SUSTAINABLE ENERGY, VOL. 7, NO. 3, JULY 2016

−

 j∈N

+



+

j∈G

B1j p¯gj tan ϕgj

j∈M

B1j w ¯ j pG j

j∈G





B1j p¯Lj tan ϕLj +

hG lq,j +

tan ϕG j

 j∈M

+



B1j qjs + zlq Γlq

j∈S

hglq,j

+



hLlq,j ≤ −qsub,min

(88)

j∈N

  G  ˆj tan ϕG ∀j ∈ G zlq + hG lq,j ≥ B1j w j pj ,   g g g zlq + hlq,j ≥ B1j pˆj tan ϕj , ∀j ∈ M   zlq + hLlq,j ≥ B1j pˆLj tan ϕLj  , ∀j ∈ N

(91)

z≥0 h ≥ 0.

(92) (93)

(89) (90)

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Shouxiang Wang (SM’12) received the B.S. and M.S. degrees from Shandong University of Technology, Jinan, China, in 1995 and 1998, respectively, and the Ph.D. degree from Tianjin University, Tianjin, China, in 2001, all in electrical engineering. He is currently a Professor with the School of Electrical Engineering and Automation, Tianjin University. His research interests include distributed generation, microgrid, and smart distribution system.

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Sijia Chen received the B.S. degree in electrical engineering from Tianjin University, Tianjin, China, in 2011. He is currently pursuing the Ph.D. degree at the School of Electrical Engineering and Automation, Tianjin University. His research interests include volt/var control, system modeling, and robust optimization in power systems.

Leijiao Ge is currently pursuing the Ph.D. degree at the School of Electrical Engineering and Automation, Tianjin University, Tianjin, China. His research interests include cloud computing in power system analysis.

Lei Wu (SM’13) received the B.S. degree in electrical engineering and the M.S. degree in systems engineering from Xi’an Jiaotong University, Xi’an, China, in 2001 and 2004, respectively, and the Ph.D. degree in electrical engineering from Illinois Institute of Technology, Chicago, IL, USA, in 2008. Currently, he is an Associate Professor with the Department of Electrical and Computer Engineering, Clarkson University, Potsdam, NY, USA. His research interests include power systems optimization and economics.