Two-Stage Distribution Circuit Design Framework for

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Two-Stage Distribution Circuit Design Framework for High Levels of Photovoltaic Generation Suma Jothibasu, Student Member, IEEE, Anamika Dubey, Member, IEEE, and Surya Santoso, Fellow, IEEE,

Abstract—Present-day radial electric power distribution circuit faces multiple challenges due to the increased photovoltaic (PV) generation. This paper aims to examine and design modern distribution circuit topologies for accommodating high PV penetration, while addressing power quality concerns. The central hypothesis of the proposed design approach is that by decreasing the Thevenin impedance at the buses where PVs are connected, the impacts on feeder voltages due to PV generation become less pronounced thereby additional PV capacity can be integrated at the corresponding buses. A novel two-stage optimization framework is proposed. The first stage is a mixedinteger linear programming based formulation to design new optimal configuration for any given distribution circuit. The formulation allows the possibility that the circuit can be operated in either a radial or loop configuration. The second stage, a nonlinear programming based formulation, then identifies PV hosting capacity for the identified optimal configuration. The proposed two-stage framework is evaluated for IEEE 123-bus test feeder. It is demonstrated that the PV hosting capacity of the feeder can be increased by 53% by optimally adding two new distribution lines with tie-switches. Index Terms—Circuit topology, Power system planning, Distributed energy resources, Circuit reconfiguration.

N OMENCLATURE Sets: N bus Set of all buses in a distribution circuit. N Set of all nodes in a distribution circuit. Note that node is a part of a bus which refers to each phase of a bus. For instance, a three-phase bus has three nodes: nodes a, b and c. E bus Set of all three-phase and single-phase physical distribution lines between N bus E Set of all physical single-phase distribution lines, mutual coupling between the lines and shunt capacitance of the lines. For instance, a single three-phase distribution line between two buses is modeled as three single-phase physical distribution lines with six mutual coupling elements between all pairs of lines and six shunt capacitance of the lines that are lumped equally at each end of the line. ETbus ⊂ E bus , Set of all potential lines with tie-switches. Suma Jothibasu and Surya Santoso are with the Department of Electrical and Computer Engineering, University of Texas at Austin, Austin, TX 78751 USA (e-mail: [email protected]; [email protected]). Anamika Dubey is with the School of Electrical Engineering and Computer Science, Washington State University, Pullman, WA, 99163 (e-mail: [email protected]).

ET ⊂ E, Set of all single-phase lines with tie-switches. N sub Set of substation nodes in a distribution circuit. N pv Set of nodes where is PV is connected. Parameters: BT Total budget for adding new lines with tie-switches Costl Cost of laying line l, l ∈ ET Gl Conductance of line l, l ∈ E Bl Susceptance of line l, l ∈ E ZlP Magnitude of positive sequence impedance of line l, l ∈ E bus Il Rated current carrying capacity of line l, l ∈ E PiL Real power demand at node i, i ∈ N QL Reactive power demand at node i, i ∈ N i sub Pi Upper limit on real an power generation at substation node i, i ∈ N sub sub Lower limit on real power gen. at node i, i ∈ N sub Pi sub Qi Upper limit on reactive power generation at node i, i ∈ N sub sub Lower limit on reactive power generation at node i, Qi i ∈ N sub gen Iik Current injection at PV node i, i ∈ N pv sink Iik Current demand at substation node i, i ∈ N sub Functions: Pl Real power flow in line l, l ∈ E Ql Reactive power flow in line l, l ∈ E Variables: dl Binary line status variable for line l, l ∈ E dl = 1, if a line is open. dl = 0, if a line is closed. Note that for distribution lines not equipped with tie-switch, dl is always zero Vi Voltage magnitude at node i, i ∈ N θi Voltage angle at node i, i ∈ N Voltage and theta for nodes i, i ∈ N sub are known. sub Pi Real power generation at subst. node i, i ∈ N sub sub Qi Reactive power generation at subst. node i, i ∈ N sub pv Pi Real power generation at node i, i ∈ N pv Note PV injects power at unity power factor Ilk Current flow in line l, l ∈ E bus for finding Thevenin impedance at node k, k ∈ N pv I. I NTRODUCTION With the incentivized rapid decarbonization of electric power generation industry and aggressive renewable portfolio standards, an increased integration of distributed energy resources is expected largely in electric power distribution

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systems. To this regard and additionally owing to the improvements in technology performance, decrease in cost, and growing consumer interest, grid-tied photovoltaic (PV) power generation has been increasing worldwide [1], [2]. Integrating large percentages of PVs may result in several issues including voltage quality problems, increased thermal stress, and additional feeder losses. Thus there exists a maximum PV capacity, referred to as feeder’s PV hosting capacity (HC) that can be integrated in a given distribution circuit without violating any feeder’s operational limits [3]– [6]. Largely, the PV hosting capacity is limited by overvoltage problems due to excess generation and reverse power flow. Integrating more PV than the HC of the distribution circuit mandates operational changes and grid upgrades which require additional investments. Operational solutions focused on voltage control using legacy devices (capacitor banks and voltage regulators), smart inverters, and battery energy storage units have been explored to increase feeder’s PV hosting capacity [7]–[9]. Once operational solutions are exhausted, advanced planning methods are required to increase the feeder’s PV hosting capacity. This paper focuses on optimally modifying the distribution circuit topology by adding new distribution lines with tie-switches for increasing the PV hosting capacity. Distribution circuit reconfiguration approaches to minimize feeder losses [10], [11] and improve reliability [12] are well established. Recent efforts include circuit reconfiguration techniques to increase the PV hosting capacity of the distribution circuits [13]–[18]. Mathematically, the distribution network reconfiguration problem to maximize the PV hosting capacity is formulated as a mixed integer non-linear programming problem (MINLP). Unfortunately, MINLP problems are NP hard and difficult to solve for a large number of constraints [13]–[22]. For example, [13] demonstrates that the time taken for a commercial solver to solve the MINLP problem with an objective of maximizing PV penetration for a small sized distribution circuit (with 34-bus) varies between few minutes and 4 hours. Therefore, there is a need for an alternate approach for reconfiguring large distribution circuits. Heuristics based techniques such as particle swarm optimization [14], [15], genetic algorithm [16] and meta-heuristics algorithms [17] have been proposed for the distribution circuit reconfiguration problem to maximize PV hosting capacity. Since, the scale of the problem (size of the distribution circuit) greatly increases the solution space, the heuristic based approaches are not practically feasible. Other methods in power systems domain include techniques to relax the original MINLP problem into a mixed-integer linear program (MILP) using linear approximation [19]–[22] and convex relaxations techniques [18]. These methods are employed to scale the MINLP problem and solve it within reasonable amount of time. Unfortunately, linear approximations ignore the nonlinear power flow model that is critical to capture loses in three-phase unbalanced distribution feeders. Similarly convex relaxation techniques do not guarantee feasible power flow solution and do not solve for integer variables. In this paper, a tractable method is presented to solve the NP hard MINLP problem of maximizing a given feeder’s PV hosting capacity by optimal circuit design. The novelty

lies in the proposed two-stage formulation that can efficiently solve the original MINLP problem while incorporating the discrete decision variables along with nonlinear power flow equations for a three-phase unbalanced distribution feeder. Furthermore, the proposed framework considers the possibility of operating the feeder in loop configuration [23] and hence it caters for the following two circuit operations: 1) to maintain and operate in radial configuration and 2) to design advanced loop configurations. The central hypothesis employed in this framework is that a larger PV capacity can be integrated by decreasing the Thevenin impedance at the point of connection (PoC) [4]–[6]. Note that decreasing the Thevenin impedance at the PoC help decrease the effect of PV generation on feeder voltages and result in a lesser voltage rise and thereby allowing additional PV integration. The two-stage framework is comprised of: (1) MILP formulation to design a circuit configuration that minimizes the thevenin impedance at the PV bus (2) a non-linear optimal power flow (OPF) problem that maximizes the PV hosting capacity for the new distribution circuit configuration. The MILP problem obtains optimal locations for additional distribution lines to be integrated in a given distribution circuit, subject to budget constraints. The stage 1 MILP formulation is solved using CPLEX solver [24]. While the stage 2 non-linear OPF problem is solved using an Interior Point OPTimizer (IPOPT) [25]. The proposed framework is tested on two test feeders: a small 15-bus distribution feeder and IEEE 123-bus distribution feeder. For IEEE 123-bus feeder, it is demonstrated that the PV hosting capacity can be increased by 53% by optimally adding only two new distribution lines with tieswitches. II. G ENERALIZED F RAMEWORK FOR D ISTRIBUTION C IRCUIT D ESIGN A generalized optimization framework based on a MINLP formulation is presented in this section. The objective is to maximize the PV hosting capacity of a given distribution circuit by optimally reconfiguring it within budget constraints. A. Network Modeling of Distribution Circuits A given distribution circuit is represented as a graphical network G bus = (N bus , E bus ). Since distribution circuits are typically unbalanced, a detailed three phase modeling is carried out and the circuit is represented as a graph G = (N , E). The circuit modeling is explained with the help of a simple 3-bus circuit shown in Fig. 1(a). Reduced phase impedance matrices of the lines are also included in Fig.1(b). Further, corresponding network sets and elements are defined in Table I. It can be observed from Table I that the size of three-phase network model (G) is larger than G bus . The network (N , E) has 7 and 39 elements respectively. The elements of the set E are obtained from non-zero entries of 7 × 7 nodal admittance matrix [26]. The substation nodes (N sub ) of the distribution network are initialized as the slack bus i.e., the voltage and angle at the nodes are fixed and the power generation is limited by a large value (infinite source). The loads are modeled as constant power components and defined at loads nodes in the circuit.

3

Bus 2

Bus 1

} } (a)

} (b)

Bus 3

Fig. 1. (a) Three bus circuit model and (b) reduced phase impedance matrices L and Z L . Z12 31 TABLE I N ETWORK D EFINITIONS FOR THE T HREE B US C IRCUIT Set N bus

Size 3

E bus

2

N

7

E

39

dl , ∀l∈ET

i∈N pv

It is subjected to the following constraints, 1) Nodal Power Balance Constraints: Equations (2) and (3) represent the power balance constraints for real and reactive power respectively. The constraints ensure that the net generation equals the net demand at every node in the circuit. X X Pisub − Pl + Pl = PiL − Pipv ∀i ∈ N (2) −

l|o(l)=i

Ql +

l|e(l)=i

X

Ql = Vi Vj [Gl sin(θi − θj ) − Bl cos(θi − θj )] (1 − dl )

(5)

0.95 p.u ≤ Vi ≤ 1.05 p.u

The proposed framework aims at maximizing the PV hosting capacity of a given distribution feeder by optimally adding new distribution lines with tie-switches. It is formulated as an optimal power flow (OPF) problem subject to the following three impact criteria: 1) maintaining the feeder voltages within the specified limits, 2) ensuring that the power flow in all distribution lines are within its rated thermal capacity, and 3) satisfying the substation’s reverse power flow limits. The maximization problem is also subject to budget constraints on adding distribution lines with tie-switches. The locations of the PV systems to be integrated into the feeder and the potential locations of new distribution lines with tie-switches are assumed to be known. Decision variables for the optimization problem are the binary variables (dl ) that represent the new distribution lines that can be potentially added in the network. The framework considers the detailed three-phase modeling of the distribution circuit (G) as explained in Section II-A. The objective of the formulation is given by (1), X pv Max Pi (1)

X

(4)

3) Operational Constraints: The following operational constraints are introduced to limit the impacts due to highlevels PV penetration in the distribution circuit. Over-voltage Constraint: The voltage and angle at the substation nodes are fixed at a pre-defined value, usually 1.06 0 p.u at Phase-A. Whereas the voltage at other nodes are limited by lower and upper limits as given in (6). The box constraint on the voltage limit (6) is non-convex. Similarly the voltage angle is limited by (7).

Set element details Buses 1, 2 and 3 One three-phase line between bus 1 and 2 and one single-phase line between bus 1 and 3 Three nodes each in buses 1 and 2, and one node in bus 3 Non-zero entries in 7x7 nodal admittance matrix includes physical single-phase distribution lines, mutual coupling elements between the lines, and shunt capacitance elements.

l|o(l)=i

Pl = Vi Vj [Gl cos(θi − θj ) + Bl sin(θi − θj )] (1 − dl ) ∀ l ∈ E, i = o(l), j = e(l)

B. Problem Formulation

Qsub i

2) Power Flow Constraints: The power flow constraints are non-linear as they are dependent on admittance, node voltage and voltage angle (θ). The equations for real and reactive power flow are given in (4) and (5), respectively. The binary variable dl is included in the constraint to account for nonoperational state of the line. If a distribution line is open, (1 − dl ) is equal to zero which enforces the power flow to be equal to zero. Note that the power flow constraints (4) and (5) are non-linear and have integer variables as well.

pv Ql = QL i − Qi

∀i ∈ N

(3)

l|e(l)=i

where, o(l) and e(l) represents the origin and destination nodes of l, ∀l ∈ E.

−π ≤ θi ≤ π

∀i ∈ N

(6)

∀i ∈ N

(7)

Thermal Power Flow Constraint: The magnitude of the current flow (Il ) is limited by its specified current carrying capacity (I l ) as presented in (8). The equation also has the term (1 − dl ), to account for line’s non-operational state. Il ≤ I l (1 − dl ) ∀l ∈ E

(8)

where, Il = |(Vi 6 θi − Vj 6 θj )(Gl + jBl )| Reverse Power Flow Constraint: Power generation at the substation nodes are limited by the upper and lower limits as given by (9) and (10). The lower limit for real power generation is negative to allow a reverse power flow at the substation, that may occur when there is excess PV generation in the distribution circuit. sub

∀i ∈ N sub

(9)

sub Qi

sub

(10)

P sub ≤ Pisub ≤ P i i Qsub i

≤

Qsub i

≤

∀i ∈ N

4) Budget Constraints: Given a budget BT and costs of laying new lines with tie-switches, constraint (11) is introduced to realize the best circuit configuration within the budget. X (1 − dl ) Costl ≤ BT ∀l ∈ ET (11) l

5) Network Connectivity Constraints: Adding a new distribution line may result in radial, loop or mesh configurations. Depending on the disposition to operate a distribution circuit, two frameworks are proposed: 1) radial topology is maintained while adding new distribution lines and 2) feeder is operated in non-radial topologies. A non-radial operation requires no additional connectivity constraints. However, the following necessary conditions must be satisfied to ensure a radial operation for the planned circuit topology [27]:

4

•

•

Condition 1 — Total number of distribution lines that are operational in a circuit should be equal to one less than the total number of buses N bus Condition 2 — The network should be connected.

Both conditions are essential to maintain radiality in a given circuit and they are ensured in this formulation. Condition 1 is enforced by (12). X

(1 − dl ) = n(N bus ) − 1

∀l ∈ E bus

(12)

l

Condition 2 requires that the network be connected i.e., all nodes have a path to the substation. It is enforced by the following constraints:

X l|o(l)=i

fl +

X

Fi = 1

∀i ∈ / N sub

(13)

f l = Fi

sub

(14)

∀l ∈ E bus

(15)

∀i ∈ /N

Algorithm 1 Optimal Design of Distribution System Topology Require: Existing radial circuit G = (N , E) with known PV locations (N pv ) and locations of potential distribution lines with tie-switch (ET ). The tie-line budget (BT ) is given as well. 1: if The circuit can only be operated in a radial configuration then 2: Stage 1 - Radial: Use the MILP framework from Section III-B to optimally add new distribution lines with tie-switches while satisfying the two necessary conditions for radiality. 3: else 4: Stage 1 - Non-Radial: Use the MILP framework from Section III-B to optimally add new distribution lines with tie-switches 5: end if 6: Define the graph Gnew = (N , Enew ), where, Enew represents the existing distribution lines and new additional distribution lines. 7: Stage 2: Solve the OPF defined in Section III-C to find the maximum PV HC at N pv nodes, while satisfying the overvoltage limit and other operational constraints pv 8: return The graph Gnew = (N , Enew ), and Pi . The graph Gnew represents the optimal circuit topology and Pipv ∀i ∈ N pv represents maximum PV HC of the circuit.

l|e(l)=i

|fl | ≤ n(N bus ) (1 − dl )

where Fi is fictitious load that can be fed only by the substation and fl is the fictitious power flow. Fi is initialized to be equal to 1 in (13) at all bus except the substation in the circuit. Power balance constraint for fictitious power flow is defined in (14). Finally, the fictitious power flow is defined as the product of binary line status variable (dl ) and upper limit on power flow similar to (8). The constraints (13) to (15) ensure a path between the substation and all bus in the designed distribution circuit. Thus a MINLP based optimal power flow framework (1)(15) is derived to maximize PV hosting capacity in a given distribution circuit. The decision variable is an integer line switching variable dl .

III. P ROPOSED F RAMEWORK The above posed optimization formulation in (1)-(15) is a non-convex MINLP problem. Solving the formulation for optimality is difficult and intractable for a large-scale distribution system [13]–[22]. Various heuristic based approaches [13]– [17] and linearization techniques [18] have been employed to solve this MINLP-based circuit reconfiguration problem for increasing PV hosting capacity. All the above methods [13]– [18] are applied from a mathematical standpoint. This paper utilizes a fundamental principle to decompose the original MINLP problem into a MILP and a NLP problem that are relatively easier to solve using the commercially available solvers. The first-stage solves a MILP problem to obtain the optimal locations for additional distribution lines and the second-stage solves a non-linear OPF to obtain the maximum PV capacity that can be integrated while satisfying power flow and other operational constraints in the distribution circuit. The proposed formulation is detailed in Algorithm 1. The rationale for the problem decomposition and proposed algorithm are explained in detail in the following sub-sections.

A. Maximum PV Penetration and Thevenin Impedance The central hypothesis used for the two-stage decomposition of the original MINLP problem is validated in this section. We obtain an analytical derivation on feeder’s maximum PV capacity and its sensitivity to Thevenin impedance at the PoC [4], [5]. A simplified feeder model as shown in Fig. 2 that is comprised of two buses with an equivalent Thevenin impedance, Z Th 6 θz = Rth + jX th , at the PoC is used for the analysis. The feeder load and PV generation are modeled as constant power models. The voltage at the PV bus, V pv 6 δ, is given by: ∗  S6 φ pv 6 6 V (Z th 6 θz ) δ = V1 0 + (16) V pv 6 δ where, V1 6 0◦ is the source voltage and the net power injected at the PoC is S 6 φ = P + jQ = (P pv − P L ) + j(Qpv − QL ).

Vpv /δ

V1

Rth+jXth th

Z /θz

Ppv+jQpv

PL+jQL Load

PV

Fig. 2. Simplified feeder model with PV.

Assuming that δ is negligible in distribution circuits, (16) can be simplified to (17) in rectangular coordinates. Furthermore the maximum net real power injected at the PoC (P ) to result in the maximum voltage at the bus is derived in (18), P Rth + QX th V pv pv pv V (V − V1 ) P = th Z (cos θz + tan φ sin θz ) V pv ≈ V1 +

pv

(17) (18)

Let the source voltage (V1 ) be 16 0◦ and V be 1.05 p.u [28]. Assuming that the power factor is 0.95 and X/R ratio is 2 at

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the PoC, the maximum power that can be injected at the bus is P = 0.07/Z th . It can be observed that the P is inversely proportional to the equivalent Thevenin impedance at the PoC. This concludes that a larger PV capacity can be integrated at the buses with lower Thevenin impedance without resulting in any overvoltage limit violations. Based on this observation, we postulate that PV injection at any given bus can be maximized by appropriately redesigning and reconfiguring the distribution circuit such that the Thevenin impedances at the corresponding bus is minimized. B. Stage 1: MILP for Optimal Locations for Additional Distribution Lines with Tie-switches The first stage of the proposed approach identifies an optimal circuit configuration for any given distribution circuit. In this framework, the distribution circuit is represented as a positive sequence equivalent (N bus , E bus ) network. This simplification is carried out because typically the Thevenin impedance at a bus does not vary significantly between the phases. The sequence impedance (Zs ) of a three-phase distribution line is calculated using the following expression [29], [30], Zs = A−1 ZlL A

(19)

where, ZlL is the 3×3 reduced phase impedance matrix for the three phase line l as explained in Section II-A and A is the 3×3 symmetrical component transformation matrix. Magnitude of positive sequence impedance (ZlP ) is then obtained from the sequence impedance (Zs ). The objective of the framework is to minimize the unweighted sum of the positive-sequence equivalent Thevenin impedance (Zkth ) at each individual PV bus k and it is given by (20). Weights can be introduced to prioritize the PV buses. The higher the value of the weights, the greater is its priority in the circuit. X Min Zkth (20) bus dl ∀l∈ET

k∈N pv

The decision variables are the binary line status variables dl as given in (1). Note that in Stage-1, l ∈ ETbus . It is because the distribution circuit is reduced to the positive-sequence equivalent network. Similarly, N pv and N sub indicate the set of PV buses and substation buses, respectively. In this study, the distribution circuit is supplied by a single substation and so N sub is a singleton set. Thevenin impedance (Z th ) is calculated for each PV location separately. For this reason the system variables including the voltage and currents are defined a total of n(N pv ) number of times. Note that k is the index for the PV bus under consideration in the following formulation (21)-(28). The Thevenin impedance at a bus of interest (k) is calculated by the following initialization [26], [29], [30] : • Shorting the substation - Initializing the voltage equal to zero and introducing a 1 A current demand, as given by (21) and (22) respectively. • Injecting 1 A current at the bus of interest as defined in the equation (23).

The resulting voltage magnitude at the bus (k) is nonetheless its Thevenin impedance by Ohm’s law and it is given by (24). ∀i ∈ N sub

(21)

=1

∀i ∈ N

sub

(22)

=1

∀k ∈ N pv

(23)

Vkk

pv

(24)

Vik = 0 sink Iik gen Ikk Zkth =

∀k ∈ N

Current flow in the circuit is determined by flow balance constraints defined in (25) which is similar to (2) to enforce the net current flow at a bus is zero. The flow constraint at a bus i corresponding to the PV bus k is given by, X X gen sink Iik − Ilk + Ilk = Iik ∀i ∈ N bus (25) l|o(l)=i

l|e(l)=i

The constraint (26) for current flow is defined in a similar way as (4). This relates the current flow in a line to the physical characteristics of the circuit, i.e positive sequence line impedance (ZlP ) and the source and destination bus voltages, Vi and Vj . Ilk −

(Vik − Vjk ) (1 − dl ) = 0 ∀ l = (i, j) ∈ E bus ZlP

(26)

Similar to (4), the binary variable dl is included in the above constraint to account for a non-operational state of the line. This however makes the constraint non-linear due to the product of two variables Vik (continuous variable) and dl (integer variable). The constraint is linearized by the big M method as defined in (27) - (28). (Vik − Vjk ) Ilk − ≤ M (dl ) (27) ZlP (Vik − Vjk ) Ilk − ≥ −M (dl ) (28) ZlP where, M is a large positive real number. When a line is nonoperational, i.e., when dl is equal to ‘1’, the power flow in the line is unconstrained due to large value of M . However when the line is operational, i.e., when d is equal to ‘0’, the inequalities in the constraints (27) and (28) reduce to an equality constraint similar to (26). The above constraints (20) - (28) along with the budget constraint (11) form the Stage 1 framework for determining the optimal non-radial configurations with minimum overall Thevenin impedance at the PV buses. Next, for the optimal radial configurations, the two necessary conditions for radiality are included in the framework. The first condition is ensured by including the constraint (12) in the framework. Condition 2 requires that the network be connected i.e., all buses have a path to the substation. It is enforced by assuming that all buses have fictitious load demand and a fictitious source at the substation similar to the constraints (14) - (15). Thus every bus is linked with the substation bus, forming a connected graph. Stage 1 framework is formulated as a MILP optimization for determining the new graph Gnew constrained by the given budget. The objective is to minimize the overall Thevenin impedance at each individual PV buses. The proposed MILP problem is solved using Pyomo optimization in Python [31] and CPLEX solver [24].

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C. Stage 2: Maximize PV Hosting Capacity The second stage programming determines the maximum PV capacity that can be integrated in the newly designed Gnew , considering operational conditions such as over-voltage, thermal limit, and reverse power flow at the substation. The objective for Stage 2 optimization framework is given by (29) X pv Max Pi (29) i∈N pv

It is similar to (1), except that the binary variable dl are eliminated in Stage 1. The objective (29) along with the constraints (2)-(10) form the Stage 2 framework. Without the integer variables, Stage 2 framework is a NLP problem. In this paper we have solved the resulting problem using Pyomo modeling language in Python and interior-point based opensource optimization solver IPOPT [25]. The maximum PV HC in this study are calculated at the bus-level and the feeder-level of the distribution circuit. They are defined as the following. • Nodal PV Hosting Capacity (HC) - The largest PV size at a bus assuming that no other PV source is injecting power in the distribution circuit. • Feeder PV Hosting Capacity (HC) - All the PVs in the set N pv are simultaneously injecting power in the distribution circuit. The PV HC values are calculated for the optimal distribution circuit configuration designed in Stage 1. Conservative conditions such as (1) the PV is generating its rated output at unity power factor and (2) the load is at its minimum are considered for the analysis. Typically, the minimum load of distribution circuit is between 30%-40% of peak load during the peak time PV generation period (10am to 2pm). Since the yearly load demand is not known, the minimum load is assumed to be 30% of peak load for this analysis. If the circuit’s load demand profile is known, then the specific loading condition can be easily incorporated in the proposed formulation for the specified distribution circuit.

in addition to determining the new lines, the framework also identifies existing lines that need to be opened. A. 15-Bus Distribution Circuit A small 15-bus radial distribution circuit at 12.47 kV as shown in Fig. 3(a) is considered as the base-case radial topology for this study. The voltage level of the feeder is stepped down to 4.16 kV at the substation. It has 14 threephase distribution lines (refer Fig. 3 (a)) and 8 possible locations for deploying new lines with tie-switches. All the three phase distribution lines in the circuit are assumed to have the same characteristics and running for 0.5 miles in length with an X/R ratio of 2. The circuit serves three loads of 360 kW each at buses 9, 13 and 14. Potential PV locations are buses 9 and 13 (indicated as star in Fig. 3). The nodal and feeder PV hosting capacity of the base-case circuit are determined. Definitions for the same are specified in Section III-C. The nodal PV HC of the circuit at bus 9 and 13 are 1.75 and 1.29 MW respectively. The feeder PV HC of the base-case circuit is calculated to be 2.92 MW, which is 1.69 MW at bus 9 and 1.23 MW at bus 13 respectively.

Substation

Substation

Substation

1

2

3

1

2

3

1

2

3

4

5

6

4

5

6

4

5

6

7

8

7

8

9

7

8

9

10

11

12

10

11

12

10

11

12

14

15

14

15

14

15

13

9

13

(a)

13

(c)

(b) Feeder PV HC 2.9 MW

2.9 MW

3.3 MW

2.9 MW

4.2 MW

IV. S IMULATION R ESULTS The effectiveness of the proposed two-stage framework is evaluated on a small and a medium sized distribution circuit. Stage 1 determines new distribution circuit configuration Gnew within a given budget (BT ). For the purpose of this study, BT refers to the physical number of new distribution lines that are added in a circuit. Also, the budget constraint (11) is modified to an equality constraint (30) to evaluate the effect of adding new distribution lines on net Thevenin impedance and the PV hosting capacity of the circuit. X (1 − dl ) = BT ∀l ∈ ET (30) l

The above constraint can be modified if the actual cost of adding a new distribution line at a particular location of a given length is known. The results of the analysis for the two test feeders are discussed in the following sections for both radial and nonradial configurations. For maintaining the radial configuration,

Fig. 3. 15-bus feeder configurations and the corresponding feeder voltage displayed as box plot (a) Base-case topology with box plot when feeder HC (2.9 MW) is interconnected (b) New radial topology with voltage box plot when base-case feeder HC - 2.9 MW (orange plot) and the new feeder HC - 3.3 MW (blue plot) are interconnected (c) New non-radial topology with voltage box plot when base-case feeder HC - 2.9 MW (orange plot) and the new . feeder HC - 4.2 MW (blue plot) are interconnected

The HC of the circuit is increased using the proposed two-stage framework. For the base-case topology, the positive sequence Thevenin impedances at buses 9 and 13 are 1.41 Ω and 2.07 Ω, respectively. The Stage 1 MILP problem designs a new circuit configuration within a given budget constraint such that the Thevenin impedances at the PV buses are further decreased. New radial and non-radial configurations for BT = 2 are shown in Figs. 3(b) and (c). New lines are highlighted in red and the existing lines that are removed are indicated as green dotted lines. Modified bus impedances for each of the configurations are reported in Table II. For radial configuration, it can be observed that the impedance

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TABLE III R ESULTS FOR D ISTRIBUTION C IRCUIT D ESIGN P ROBLEM FOR 15-N ODE T EST F EEDER

BT 0 1 2 3

New Ties Added 4−7 4−7 2−5 4−7 2−5 11 − 14

Radial Old Lines Opened 4−5 4−5 7−8 4−5 7−8 11 − 12

Feeder HC 2.93 3.39 3.36 3.36

Non-radial New Ties Feeder Added HC − 2.93 8−9 3.50 8−9 4.27 4−7 8−9 4−7 4.34 13 − 14

Thevenin Impedance (Ω)

Feeder PV HC

3

2

1

6

4

2

3

0

4

4

Feeder PV HC

Thevenin Impedance (Ω)

1

Thevenin Impedance

0

Nodal PV HC

0

3

2

2

1

0

0

1

2 3 Tie-line Budget (BT)

0

4

Fig. 4. The maximum nodal and feeder PV HC for 15-bus feeder in new (a) Radial and (b) Non-radial configurations for tie-line budgets 0 to 4. The nodal PV HC and Thevenin impedance are represented as stacked bar graph. While Feeder PV HC is illustrated as a line graph.

B. 123-bus Distribution Circuit In this section, the proposed algorithm is tested on a medium scale IEEE test feeder at 4.16 kV. The sets N bus , E bus has 123 buses and 122 distribution lines and the sets N , E has 256 nodes and 1974 elements. The peak load demand of the circuit is 3.49 + j1.91 MVA. The line, transformer and load data are defined in [32] for simulation studies. The circuit has 5 normally open tie-switches. Apart from the original 5 lines, ten other locations are identified for deploying new distribution lines with tie-switches. The new potential tie-switch locations are highlighted in red color in Fig. 5.

32

250

29

33

30

251 49

28

31

26

47

23

44

64

22

66

40 38

36

39

62

58

7 149

9

1

8

12

150

152

13

4

5

6

69 74

73

57 61

52

53

54

55

610

56

72

17

76

16

95

93

91

89

84

81

88

90

92

85

79

77 78 80

96

34

75

67

94

15 3

71 70

68

59 10

99

97 160

60

14

2

98

197

450

100

101

37 11

103

102

63

41

35 18 135

19

20

105

114

451

104

106

65

113

107

109 108

43

42 21

300

151

45

27

350 111 110 112

51

50 46

25 48

24

Details on line modifications are reported for various budget constraints in Table III. Fig. 4 shows stacked bar chart for change in the Thevenin impedance and corresponding change in nodal PV HC. It also shows change in feeder HC of the circuit. For a radial configuration, the Thevenin impedance does not vary for BT > 1, whereas the impedance of the non-radial configuration steadily decreases for increasing BT . The corresponding increase in nodal and feeder PV HC of the circuit are also presented in Fig. 4. The feeder HC for radial configuration of 15-bus circuit is 3.3 MW which is 15% more than base-case topology. It is achieved by adding one additional line. Further, if the circuit can be operated in loop/mesh configuration, the feeder PV HC is 4.3 MW, by adding two additional lines, which is 45% than that of the base-case.

Bus 13

2

8

(b)

Thevenin Impedance

Nodal PV HC

4

Bus 9

Thevenin Impedance

(a)

6

Nodal PV HC

at bus 9 does not change, whereas the impedance at bus 13 is decreased by 33%. Following the same trend, the nodal PV HC at the buses also increase by 0% and 35% (from 1.29 MW to 1.75 MW), respectively. Similarly for the nonradial configuration, the nodal PV HC at the buses 9 and 13 increase by 113.7% and 60%, respectively and there is a similar percentage decrease in Thevenin impedances at the two buses. The results corroborate that the increase in maximum PV size at a given bus corresponds to the decrease in the Thevenin impedance at the respective bus. Next the feeder PV HC capacity for the new configurations are determined. Fig. 3 shows box plots for voltage profile across the feeder topology when 2.9 MW (feeder HC of basecase circuit) is interconnected in all three configurations. It can be observed that the mean voltage of the feeder is lower in the new radial and non-radial configurations compared to that of the base-case. This allows for additional PV integration in the topologies. The box plots for voltage profile across the feeder when interconnected with calculated feeder PV HC of 3.3 MW and 4.2 MW respectively in the new configurations are also shown in Fig. 3.

Nodal PV HC (MW)

Base Radial Non-radial

Bus 13 Z th (Ω) Nodal HC 2.07 1.29 MW 1.41 1.75 MW 1.22 2.07 MW

Nodal PV HC (MW)

Topology

Bus 9 Z th (Ω) Nodal HC 1.41 1.75 MW 1.41 1.78 MW 0.66 3.74 MW

4

8

TABLE II T HEVENIN I MPEDANCE AND N ODAL PV HC OF 15 B US FOR BT = 2

86 83

87 82

195

Fig. 5. 123 bus distribution circuit topologies with potential new tie-line locations (highlighted in red). Stars indicate new PV locations

The following three-phase buses are considered as the potential PV integration locations: 114, 83, 94, 66, and 151.

8

TABLE IV R ESULTS FOR D ISTRIBUTION C IRCUIT D ESIGN P ROBLEM FOR IEEE 123 B US T EST F EEDER

0 1 2 3

4

5

Old Lines Opened 93 − 94 76 − 86 62 − 63 72 − 76 62 − 63 76 − 77 76 − 86 62 − 63 67 − 72 105 − 108 76 − 86 60 − 62 81 − 82 105 − 108 70 − 71

30 20

(MW) 94 4.2 8.9

at Buses: 66 151 2.7 5 2.7 5

4.4

8.9

8.9

5

13.2

3.8

6.3

8.9

8.9

5

13.2

3.2

6.2

8.9

8.9

5

13.2

3.2

6.1

8.8

8.9

4.5

13.2

Bus 83 Bus 151

Bus 94

2 1

0

1

2

3

4

Nodal PV HC

Thevenin Impedance

20

Thevenin Impedance

Nodal PV HC

Nodal PV HC (MW)

(b)

30

Feeder PV HC

4 3 2 1

10 0

0

5

50 40

4 3

Feeder PV HC

10 0

Feeder HC (MW) 8.6 11.9

3.9

Bus 114 Bus 66

Thevenin Impedance

40

Nodal PV HC

(a)

Nodal PV HC (MW)

50

Radial Nodal HC 114 83 4.1 4.6 4.1 4.6

0

1

2 3 4 Tie-line Budget (BT)

5

Thevenin Inpedance (Ω)

New Ties Added 54 − 94 54 − 94 66 − 35 54 − 94 66 − 35 86 − 82 54 − 94 66 − 35 86 − 82 64 − 108 54 − 94 66 − 35 86 − 82 64 − 108 71 − 100

Thevenin Inpedance (Ω)

BT

0

Fig. 6. The maximum PV size in 123 Bus Feeder in new (a) Radial and (b) Non-radial configurations for tie-line budgets 0 to 5. Nodal PV HC and Thevenin impedance as stacked bars.

Both nodal and feeder PV hosting capacity of the circuit is determined. Note that the analysis is carried out based on a conservative assumption that the PV output is at its maximum (equal to its rated capacity) and the load demand is minimum (30% of peak load). For the base-case topology, the nodal PV HC at the buses 114, 83, 94, 66, and 151 are calculated to be 4.1, 4.6, 4.2, 2.7, and 5 MW, respectively. The circuit’s PV HC is further increased by optimally modifying the topology within the budget constraint. The Stage 1 framework designs new radial and non-radial configurations to decrease the net Z th at the PV buses. Details on line modification and the corresponding change in

New Ties Added 54 − 94 54 − 94 66 − 35 54 − 94 66 − 35 151 − 300 54 − 94 66 − 35 151 − 300 86 − 82 54 − 94 66 − 35 151 − 300 86 − 82 450 − 114

Non-radial Nodal HC (MW) at Buses: 114 83 94 66 151 4.1 4.6 4.2 2.7 5 4.2 5.2 8.9 2.7 5

Feeder HC (MW) 8.6 11.9

4.1

5.1

9

8.9

5

13.2

5.1

5.4

11.7

10

7.7

13.2

5.1

7.8

11.7

9.9

7.7

13.2

6.6

7.9

11.7

9.9

7.8

13.2

impedance and nodal PV HC for varying tie-line budgets are provided in Fig. 6 and Table IV. Similar to the 15 bus circuit, it can be observed that the Thevenin impedance of radial configurations does not decrease beyond BT > 2, whereas the impedance steadily decreases for increasing tie-line additions in the non-radial configurations. The increase in the nodal PV HC in the circuit is also proportional to the corresponding change in the circuit impedance. For BT = 2 in a radial configuration, the nodal PV HC at buses 83, 94 and 66 are increased by 36.9%, 119%, and 230% respectively. However, there is no further increase in PV capacity for budget constraint greater than two. For BT = 5 in non-radial configurations, the nodal PV capacities at the five PV buses 114, 83, 94, 66, and 151 are 60%, 71.7%, 178%, 266%, and 56% more than the respective HC for base-case topology. Results for feeder PV HC are also included in Fig. 6 and Table IV. The base-case feeder HC is calculated to be 8.6 MW. The HC for both the configurations are increased to 13.2 MW for BT = 2, which is a 53% increase over that of basecase topology. Interestingly, the increase in feeder PV HC is not appreciable in both radial and non-radial configurations for BT > 2. It is because there is not sufficient voltage headroom for more PV integration. Fig. 7 (a) show base-case feeder voltage profile without any PV capacity integrated in the circuit. By integrating the calculated PV HC (8.6 MW), it can be observed from Fig. 7 (b) that the maximum voltage reaches 1.05 p.u at the feeder end, while voltage along majority of the feeder is below the ANSI limit. With the proposed framework, the topology is modified such that more PV can be integrated in the circuit while still maintaining the overall feeder voltage below 1.05 p.u (refer Fig. 7 (c) and (d)) and retaining the radial configuration. Since there is no voltage headroom, the PV HC cannot be increased further with the available tie-line options. Note that the voltage profiles for the non-radial configurations are also similar to the profiles shown in Fig. 7. The PV distribution for both radial and non-radial configurations for BT = 5 is given in Table V. Buses 94 and 66 have the highest PV size in the optimal solution. The corresponding

9

Feeder PV HC = 8.6 MW

(a) Base-case

(b) Base-case with feeder PV HC

Feeder PV HC = 11.9 MW

Feeder PV HC = 13.2 MW

(c) With feeder PV HC for BT = 1

(d) With feeder PV HC for BT = 2

Fig. 7. 123 bus feeder voltage profile for various radial configurations and PV penetration (a) Base-case topology without PV (b) Base-case integrated with PV feeder HC of 8.6 MW (c) New feeder topology for BT = 1 with PV HC of 11.9 MW (d) New feeder topology for BT = 2 with PV HC of 13.2 MW Voltage (pu)

4500

Bus 151

Substation PV bus

4000

of enumeration is 122 CBT , and so the time taken is only about a second (0.02 minute). In stage 2, optimal power flow is carried out for the unbalanced distribution circuit. The 123-bus circuit has 256 nodes and the set E has 1974 elements. Also, it is a nonconvex and non-linear problem and yet the optimal solution is converged in few seconds. Note that no starting point is provided or warm-start is carried out in this study.

1.045 1.04

3500

1.035

Bus 114

3000

1.03

2500 Bus 66

1.025

2000 1.02

1500

Bus 94

1.015

1000

1.01

500

1.005

Bus 83 0

1

0

1000

2000

3000

4000

5000

6000

Fig. 8. Voltage contour plot of 123 bus radial circuit for BT = 5 with PV. TABLE V 123 B US F EEDER PV H OSTING C APACITY FOR BT = 5

XXX X Bus Topology XXX X 114 Radial Non-radial

0.05 0.12

PV Distribution (MW) 83 94 66 151 0.06 7.86 5.1 0.05 0.11 7.6 5 0.41

Total 13.15 13.24

voltage contour plot for radial configuration of the circuit is shown in Fig. 8. It can be observed that the voltage profile downstream from buses 94 and 66 is close to 1.05 p.u., which restricts additional PV integration in the circuit. Note that the PV HC is not limited by the other operational constraints such as the thermal limits of the line and reverse power flow constraint. Total three phase reverse power flow at the substation for BT = 5 case in radial configuration is calculated to be 11.4 MW. C. Computational Complexity Computational time taken by the proposed two stage algorithm to solve the MINLP problem in a personal computer with 3.4 GHz quad-core CPU and 8 GB of RAM is presented in Table VI. For a given budget BT , Stage 1 determines new radial topology by adding new tie-lines and simultaneously finding the same of number of existing lines that can be opened, such that radiality is maintained by (12). The total number of possible enumerations for the IEEE test feeder with 123 bus circuit with 122 distribution lines, and 15 new distribution lines with tie-switches is given by 122 CBT ×15 CBT . The time take by the Stage 1 framework for BT = 4 is about 17 minutes. Whereas, for non-radial configurations, the number

TABLE VI C OMPUTATIONAL T IME FOR 123 B US D ISTRIBUTION C IRCUIT

BT 0 1 2 3 4

Radial (minutes) Stage 1 Stage 2 0.02 0.03 0.03 0.04 0.11 0.04 1.86 0.04 17.24 0.04

Non-Radial (minutes) Stage 1 Stage 2 0.03 0.03 0.02 0.04 0.02 0.04 0.02 0.04 0.02 0.04

V. C ONCLUSION In this paper, new distribution feeder topologies are designed, while satisfying the given tie-line budget, to accommodate increased PV penetration. The design algorithm is based on a novel two-stage approach that decomposes the initial MINLP distribution network reconfiguration problem into MILP and NLP problems. Based on the disposition to operate a distribution circuit, the framework can design either radial or non-radial configurations for the circuit as well. The algorithm has been demonstrated to be effective in significantly increasing the PV HC of test feeders. The PV hosting capacity of IEEE 123-bus test feeder is shown to be increased by 53% by optimally placing 2 new distribution lines with tie-switches. R EFERENCES [1] C. Philibert, P. Frankl, C. Tam, Y. Abdelilah, H. Bahar, Q. Marchais, S. Mueller, U. Remme, M. Waldron, and H. Wiesner, “Technology roadmap: solar photovoltaic energy,” International Energy Agency: Paris, France, 2014. [2] S. Santoso, H. W Beaty, Standard Handbook for Electrical Engineers, 17th Ed, McGraw-Hill, 2018, sec. 10.4. [3] J. Smith, “Stochastic analysis to determine feeder hosting capacity for distributed solar PV,” Electric Power Research Inst., Palo Alto, CA, Tech. Rep, vol. 1026640, 2012.

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Suma Jothibasu (S’10) received her B.E. degree in Electrical and Electronics Engineering from Anna University, India, in 2011, M.S in Electrical Engineering from Indian Institute of Technology Madras, India, in 2014 and Ph.D. in Electrical and Computer Engineering from the University of Texas at Austin, in 2018. Her research interests include distribution system analysis and planning for high renewable energy integration, power electronic applications in power systems, and power quality.

Anamika Dubey (M’16) received the M.S.E and Ph.D. degrees in Electrical and Computer Engineering from the University of Texas at Austin in 2012 and 2015, respectively. Currently, she is an Assistant Professor in the School of Electrical Engineering and Computer Science at Washington State University, Pullman. Her research focus is on the analysis, operation, and planning of the modern power distribution systems for enhanced service quality and grid resilience. At WSU, her lab focuses on developing new planning and operational tools for the current and future power distribution systems that help in effective integration of distributed energy resources and responsive loads.

Surya Santoso (F’15) is Professor of Electrical and Computer Engineering at the University of Texas at Austin. His research interests include power quality, power systems, and renewable energy integration in transmission and distribution systems. He is coauthor of Electrical Power Systems Quality (3rd edition), sole author of Fundamentals of Electric Power Quality, and editor of Handbook of Electric Power Calculations (4th edition) and Standard Handbook for Electrical Engineers (17th edition). He is an IEEE Fellow.