Formulations and Approximations of Branch Flow Model for General Power Networks

Zhao Yuan

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Abstract

The formulations and approximations of the branch flow model for general (radial and mesh) power networks (General-BranchFlow) are given in this paper. Using different sets of the power flow equations, six formats of the exact General-BranchFlow model are listed. The six formats are mathematically equivalent with each other. Linear approximation and second-order cone programming (SOCP) are then used to derive the six formats of the convex General-BranchFlow model. The branch ampacity constraints considering the shunt conductance and capacitance of the transmission line $Π$ -model are derived. The key foundation of deriving the ampacity constraints is the correct interpretation of the physical meaning of the transmission line $Π$ -model. An exact linear expression of the ampacity constraints of the power loss variable is derived. The applications of the General-BranchFlow model in deriving twelve formats of the exact optimal power flow (OPF) model and twelve formats of the approximate OPF model are formulated and analyzed. Using the Julia programming language, the extensive numerical investigations of all formats of the OPF models show the accuracy and computational efficiency of the General-BranchFlow model. A penalty function based approximation gap reduction method is finally proposed and numerically validated to improve the AC-feasibility of the approximate General-BranchFlow model.

Keywords

Branch flow; linear approximation; second-order cone programming; ampacity constraint; optimal power flow; radial network; mesh network

I. INTRODUCTION

EFFICIENT operations of power systems rely largely on the accurate modeling of power networks and the optimal solutions of the models [

1]. Since the first formulation in the year of 1962, the optimal power flow (OPF) model has been investigated in enormous aspects and applied in many areas including operation, planning, control and market clearing [2]-[6]. Reference [7] reports that huge economic benefits, in the scale of billions of US dollars, can be achieved for the global power industry by improving the accuracy or solution quality of the OPF model. A good summary of traditional polar power-voltage, rectangular power-voltage, and rectangular current-voltage formulations of the OPF model can be found in [7]. The large-scale integration of renewable energy resources (RERs) and the growing penetration of distributed energy resources (DERs) are pushing the power system operators to deploy the OPF model with more robust and powerful performance [8], [9]. The recent developments of OPF modeling approaches include second-order cone programming (SOCP), semi-definite programming (SDP), and polynomial optimization [10]-[12]. These approaches convexify the original nonconvex OPF model and thus are useful to find the global optimal solutions which are better than the local optimal solutions obtained from directly solving the original nonconvex OPF model. Compared with the local optimal solutions, the global optimal solutions can provide higher economic gains or engineering benefits. Extensive research efforts have been put to find or prove the conditions of the exactness of the SOCP-based OPF models [13], to ensure the rank-1 solution of the SDP-based OPF models [14] and to improve the computational efficiency of the polynomial optimization based OPF models [15]. The SOCP-based OPF models feature in better computational efficiency compared with the SDP or polynomial optimization based OPF models. This is majorly because the number of variables and constraints of the SOCP-based OPF models is less than those of the SDP or polynomial optimization based OPF models.

The dist-flow branch equations are firstly proposed in [

16] to optimally size the capacitors in distribution networks. This formulation is valid only for radial power networks since no voltage phase angle constraints are considered. The voltage phase angle constraints are necessary for mesh power networks according to the Kirchhoff’s laws for AC circuits. References [17] and [18] reformulate the dist-flow branch equations in [16] and denote the derived model as branch flow model. For radial power networks, it is proved that the branch flow model in [17] and [18] is valid if there are no upper bounds for the power loads. Though this condition of the power loads to validate the branch flow model in [17] and [18] for radial power networks is not realistic, this work has inspired a vast amount of research efforts to the branch flow model and its applications, for example, in the multi-period optimal gas-power flow (OGPF) problem and in the unbalanced three-phase distribution network context [19], [20]. References [17] and [18] also use SOCP to derive a convex relaxation of the branch flow model. It is shown in [21]-[23] that the ampacity constraint is not fully addressed in the branch flow model derived in [17] and [18]. References [21]-[23] point out that the longitude current variable used in the branch flow model in [17] and [18] is not an actual measurable current according to the physical interpretation of the transmission line

Π

-model. As an improvement, [21]-[23] formulate an exact optimal flow (OPF) model for radial power networks. Another branch flow model for general power networks (General-BranchFlow) including the voltage phase angle constraint is firstly proposed in [24]. This General-BranchFlow model is then extended, reformulated, and applied in OPF, distribution locational marginal pricing (DLMP), coordination of transmission system operator (TSO) and distribution system operator (DSO), distributed economic dispatch, and super grid coordination [25]-[29]. Reference [26] also proposes a sequential programming method to tighten the relaxation or approximation gap of the General-BranchFlow model. The research work in [30] and [31] shows the applicability of the General-BranchFlow model in the operations of voltage source converter based multi-terminal DC (VSC-MTDC) system and flexible AC transmission system (FACTS). The recent work [32] rigorously proves the relaxation property and accuracy of the reformulated convex General-BranchFlow model. This paper extends the work in [32] to formulate and validate more formats of the General-BranchFlow model, and to derive the transmission line ampacity constraint in a more accurate way.

In this paper, a comprehensive investigation of the six equivalent formats of the exact General-BranchFlow model and the six formats of the approximate General-BranchFlow model is conducted. The work in [

32] is extended to derive the ampacity constraints considering both shunt conductive and capacitive components of the transmission line

Π

-model. Taking the derived ampacity constraints into account, twelve formats of the exact OPF model and twelve formats of the approximate OPF model are formulated based on the General-BranchFlow equations. All the formats of the OPF model are implemented in Julia programming language and the JuMP optimization modeling package [33], [34]. A numerical investigation is then conducted through various IEEE test cases. To improve the accuracy of the approximate General-BranchFlow model, a penalty function based method is proposed and numerically validated. Compared with the work in [32], the following contributions of this paper are original.

1) This paper provides and investigates six formats of the exact General-BranchFlow model and six formats of the convex General-BranchFlow model. Based on these, twelve formats of the exact OPF model and twelve formats of the convex OPF model are formulated and validated. The previous work [

32] only formulates and investigates one format of the convex OPF model.

2) Although all the formats of the exact General-BranchFlow model are mathematically equivalent, their numerical performance can be different for different test cases. This means more models are provided for the power system operators to use, in case one model faces numerical difficulties or parameter non-availability problems in solving network operation problems. The numerical results in this paper confirm this point.

3) For the proposed formats of the General-BranchFlow model, they are not equivalent to each other due to the different approximations or relaxations to the different power network equations. These formats of the convex General-BranchFlow model provide more methods or techniques to find the global optimal solutions of the OPF problem. This point is validated by the numerical results of this paper.

4) This paper considers both shunt conductance and capacitance of the transmission line $Π$ -model. The previous work [

32] only considers the shunt capacitance of the transmission line

Π

-model.

5) This paper proposes and validates a penalty function based method to reduce the approximation gap of the convex OPF models, which is not mentioned in the previous work [

32].

6) More numerical examinations of all the poposed formats of the General-BranchFlow model are conducted for the base power loads and heavy power loads conditions. On the contrary, only the base power loads and light power loads for one format of convex OPF model are considered in the previous work [

32].

The rest of this paper is organized as follows. Section II presents the General-BranchFlow model including the exact mode, the approximate model, and the branch ampacity constraint. Section III formulates the OPF problem including the exact and approximate OPF models, and provides the numerical validations. Section IV proposes the approximation gap reduction of the approximate General-BranchFlow model by using the penalty function based method, and numerically validates this method. Section V concludes this paper.

II. GENERAL-BRANCHFLOW MODEL

A.　Exact Model

It is assumed that the three-phase power network is balanced and the decision variables considered are constrained in (1)-(7).

v_{n} \in (v_{n}^{m i n}, v_{n}^{m a x}) \subseteq (0.9,1.1) \forall n \in 𝒩

(1)

θ_{n} \in (θ_{n}^{m i n}, θ_{n}^{m a x}) \subseteq (0,2 π) \forall n \in 𝒩

(2)

θ_{l} = θ_{s_{l}} - θ_{r_{l}} \in (θ_{l}^{m i n}, θ_{l}^{m a x}) \subseteq (- \frac{π}{2}, \frac{π}{2}) \forall l \in ℒ

(3)

p_{n} \in (p_{n}^{m i n}, p_{n}^{m a x}) \subset R^{+} \forall n \in 𝒩

(4)

q_{n} \in (q_{n}^{m i n}, q_{n}^{m a x}) \subset R \forall n \in 𝒩

(5)

p_{d_{n}} \in (p_{d_{n}}^{m i n}, p_{d_{n}}^{m a x}) \subset R^{+} \forall n \in 𝒩

(6)

q_{d_{n}} \in (q_{d_{n}}^{m i n}, q_{d_{n}}^{m a x}) \subset R^{+} \forall n \in 𝒩

(7)

Note the subscripts s and r in all the relevant variables and parameters $θ_{s_{l}}, θ_{r_{l}}, v_{s_{l}}, v_{r_{l}}, V_{s_{l}}, p_{s_{l}}, B_{s_{l}}, G_{s_{l}}$ of this paper are not indexes but only to denote the meanings of sending-end and receiving-end. Similar reasoning holds for the subscripts d, o, which are to denote the meaning of demand and power loss in all the relevant variables $p_{d_{n}}, q_{d_{n}}, p_{o_{l}}, q_{o_{l}}$ of this paper. $p_{d_{n}}, q_{d_{n}}$ are taken as variables here to consider possible demand side responses. The demands are equal to fixed values if there is no demand side response. It is assumed that $(v_{n}^{m i n}, v_{n}^{m a x}) \subset (0.9,1.1)$ in (1) and $(θ_{l}^{m i n}, θ_{l}^{m a x}) \subset (- \frac{π}{2}, \frac{π}{2})$ in (3). These assumptions are valid in power system operations under normal conditions.

The nodal power balance equations of the General-BranchFlow model are formulated as:

p_{n} - p_{d_{n}} = \sum_{l} (A_{n l}^{+} p_{s_{l}} - A_{n l}^{-} p_{o_{l}}) + G_{n} v_{n}^{2} \forall n \in 𝒩

(8)

q_{n} - q_{d_{n}} = \sum_{l} (A_{n l}^{+} q_{s_{l}} - A_{n l}^{-} q_{o_{l}}) - B_{n} v_{n}^{2} \forall n \in 𝒩

(9)

Equation (8) represents the active power balance. Equation (9) represents the reactive power balance. $A_{n l}^{+}$ and $A_{n l}^{-}$ are defined as $A_{n l}^{+} = 1, A_{n l}^{-} = 0$ if $n$ is the sending-end of branch $l$ , and $A_{n l}^{+} = - 1, A_{n l}^{-} = - 1$ if $n$ is the receiving-end of branch $l$ . The default convention of the sending-end or receiving-end of the lines can be defined in anyway. The only difference is that, the results of the power flow variables $p_{s_{l}}, q_{s_{l}}$ from the OPF calculations are negative if the default sending-end or receiving-end is reversed. Figure 1 illustrates this issue. Note that we neglect the power loss in Fig. 1 for sake of simplicity. The default conventions are made before the OPF calculations are done. In convention 1, node $n_{1}$ is referred as the sending-end and node $n_{2}$ is referred as the receiving-end. A contrary default setting is made in convention 2. In this set-up, after the OPF calculations are done, 100 kW power flow from node $n_{1}$ to node $n_{2}$ is equivalent to -100 kW power flow from the opposite direction. So the default convention of the sending-end or the receiving-end does not affect the OPF results.

Fig. 1 Different conventions of sending-end and receiving-end of lines. (a) Convention 1. (b) Convention 2.

To relate the power variables $p_{s_{l}}, q_{s_{l}}, p_{o_{l}}, q_{o_{l}}$ with the voltage variables, the voltage drop phasor of branch $l$ is used to derive the following equations:

v_{s_{l}}^{2} - v_{r_{l}}^{2} = 2 R_{l} p_{s_{l}} + 2 X_{l} q_{s_{l}} - R_{l} p_{o_{l}} - X_{l} q_{o_{l}} \forall l \in ℒ

(10)

v_{s_{l}} v_{r_{l}} s i n θ_{l} = X_{l} p_{s_{l}} - R_{l} q_{s_{l}} \forall l \in ℒ

(11)

v_{s_{l}}^{2} - v_{s_{l}} v_{r_{l}} c o s θ_{l} = R_{l} p_{s_{l}} + X_{l} q_{s_{l}} \forall l \in ℒ

(12)

where $V_{s_{l}} = v_{s_{l}}^{2} a n d V_{r_{l}} = v_{r_{l}}^{2}$ . According to the proof of Theorem 6 in [

32], it is only necessary to use (10), (11) or (11), (12) to sufficiently express the voltage drop phasor. This reasoning is used in listing all the necessary equations of the General-BranchFlow model in Table I.

Table I Exact General-BranchFlow Model Expressed in Different Sets of Equations

Format	Set of equations
1	${(1) - (7)}, {(8) - (11)}, {(13), (14)}$
2	${(1) - (7)}, {(8) - (11)}, {(13)}, {(15)}$
3	${(1) - (7)}, {(8) - (11)}, {(14), (15)}$
4	${(1) - (7)}, {(8), (9)}, {(11), (12)}, {(13), (14)}$
5	${(1) - (7)}, {(8), (9)}, {(11), (12)}, {(13)}, {(15)}$
6	${(1) - (7)}, {(8), (9)}, {(11), (12)}, {(14), (15)}$

The active power loss and reactive power loss $p_{o_{l}}, q_{o_{l}}$ are expressed as:

p_{o_{l}} = \frac{p_{s_{l}}^{2} + q_{s_{l}}^{2}}{v_{s_{l}}^{2}} R_{l} \forall l \in ℒ

(13)

q_{o_{l}} = \frac{p_{s_{l}}^{2} + q_{s_{l}}^{2}}{v_{s_{l}}^{2}} X_{l} \forall l \in ℒ

(14)

A linear relationship between $p_{o_{l}}$ and $q_{o_{l}}$ exists:

p_{o_{l}} X_{l} = q_{o_{l}} R_{l} \forall l \in ℒ

(15)

For mesh power networks, the sum of the phase angles of the voltage drop phasors along each closed network loop $𝒞$ should satisfy the following cyclic constraint:

\sum_{l \in 𝒞} θ_{l} = 0 m o d 2 π

(16)

As proved and explained in [

32], this constraint is implicitly satisfied if

θ_{l}

is expressed explicitly using

θ_{s_{l}}, θ_{r_{l}}

in constraint (3). So the constraint (16) is not required in the General-BranchFlow model of this paper.

For radial power networks, there are no closed loops. Using $θ_{s_{l}}, θ_{r_{l}}$ in constraint (3) does not enforce any cyclic constraint which is not necessary for radial power networks.

The exact General-BranchFlow model expressed by selecting different sets of power flow equations is summarized in Table I of this section. The six formats are mathematically equivalent. The exact General-BranchFlow model is valid for both radial and mesh power networks.

B. Approximate Model

Using $V_{n} = v_{n}^{2}$ to replace the voltage magnitude variables, the voltage magnitude bounds (1) can be replaced by:

V_{n} \in (V_{n}^{m i n}, V_{n}^{m a x}) \subseteq (0.81,1.21) \forall n \in 𝒩

(17)

Equations (8)-(10) are linearized to [

32]:

p_{n} - p_{d_{n}} = \sum_{l} (A_{n l}^{+} p_{s_{l}} - A_{n l}^{-} p_{o_{l}}) + G_{n} V_{n} \forall n \in 𝒩

(18)

q_{n} - q_{d_{n}} = \sum_{l} (A_{n l}^{+} q_{s_{l}} - A_{n l}^{-} q_{o_{l}}) - B_{n} V_{n} \forall n \in 𝒩

(19)

V_{s_{l}} - V_{r_{l}} = 2 R_{l} p_{s_{l}} + 2 X_{l} q_{s_{l}} - R_{l} p_{o_{l}} - X_{l} q_{o_{l}} \forall l \in ℒ

(20)

Note the solutions of the original voltage variable $v_{n}$ can be obtained by $v_{n} = \sqrt[]{V_{n}}$ after solving the approximate General-BranchFlow model.

Equation (11) can be linearized to [

32]:

θ_{l} = X_{l} p_{s_{l}} - R_{l} q_{s_{l}} \forall l \in ℒ

(21)

It is proposed to linearize (12)-(22) in this paper as:

\frac{V_{s_{l}} - V_{r_{l}}}{2} = R_{l} p_{s_{l}} + X_{l} q_{s_{l}} \forall l \in ℒ

(22)

This linearization is based on the approximation $v_{s_{l}} v_{r_{l}} c o s θ_{l} \approx (V_{s_{l}} + V_{r_{l}}) / 2$ because $v_{s_{l}} v_{r_{l}} \approx (V_{s_{l}} + V_{r_{l}}) / 2$ for $v_{n} \in (0.9,1.1)$ and $θ_{l} \approx 0$ are valid in power system operations under normal conditions.

Using rotated second-order cone, (13) and (14) can be approximated to:

p_{o_{l}} \geq \frac{p_{s_{l}}^{2} + q_{s_{l}}^{2}}{V_{s_{l}}} R_{l} \forall l \in ℒ

(23)

q_{o_{l}} \geq \frac{p_{s_{l}}^{2} + q_{s_{l}}^{2}}{V_{s_{l}}} X_{l} \forall l \in ℒ

(24)

Note that (20)-(24) are convex since they are rotated cones.

The approximate General-BranchFlow model expressed by selecting different sets of the linearized or convexified power flow equations is summarized in Table II of this section. The six formats are not mathematically equivalent because of the linearizations and approximations of different power flow equations. They are approximate to each other. The approximate General-BranchFlow model is valid for both radial and mesh power networks.

Table II Approximate General-BranchFlow Model Expressed by Selecting Different Sets of Equations

Format	Set of equations
1	${(2) - (7)}, {(17) - (21)}, {(23), (24)}$
2	${(2) - (7)}, {(17) - (21)}, {(15)}, {(23)}$
3	${(2) - (7)}, {(17) - (21)}, {(15)}, {(24)}$
4	${(2) - (7)}, {(17) - (19)}, {(21), (22)}, {(23), (24)}$
5	${(2) - (7)}, {(17) - (19)}, {(15)}, {(21), (22)}, {(23)}$
6	${(2) - (7)}, {(17) - (19)}, {(15)}, {(21), (22)}, {(24)}$

C. Branch Ampacity Constraint

The ampacity constraint of the transmission or distribution line is a very important constraint to avoid over-loading of the corresponding transmission or distribution line. It is important to emphasize the significance of the ampacity constraint considering that several big black-outs such as the Northeast blackout of 2003 in the United States and Canada were caused by the over-loading of transmission lines [

35]. This paper extends the previous work [32] to consider both shunt conductance and capacitance of the transmission line

Π

-model in deriving the ampacity constraint. Though the shunt conductance is normally very small, it is more accurate to quantify the ampacity constraint by considering this element. The key point here is a correct interpretation of the physical meaning of the transmission line

Π

-model, i.e., the line-to-ground shunt capacitance and conductance are actually distributed and there is a difference between the actual measurable power flow variables

{\tilde{p}}_{s_{l}}, {\tilde{q}}_{s_{l}}

and the power flow variables

p_{s_{l}}, q_{s_{l}}

used in the General-BranchFlow equations. The details about this difference are analyzed in [32]. The transmission line

Π

-model used in this paper and many other papers in the literature is a lumped representation of the actual line with distributed parameters (resistance, reactance, conductance, and capacitance) by nature. This representation is generally valid for transmission lines above 80 km and below 250 km [1]. Equivalent transmission line

Π

-model for other line lengths might be derived. Deriving the exact ampacity constraint of the transmission line enables the accurate consideration of its loadability.

From the transmission line $Π$ -model shown in Fig. 2, the following equations can be derived:

{\tilde{p}}_{s_{l}} = p_{s_{l}} + p_{g s_{l}}

(25)

{\tilde{q}}_{s_{l}} = q_{s_{l}} - q_{b s_{l}}

(26)

p_{g s_{l}} = V_{s_{l}} G_{s_{l}}

(27)

q_{b s_{l}} = V_{s_{l}} B_{s_{l}}

(28)

Fig. 2 Transmission line $Π$ -model considering shunt capacitance and conductance.

The branch ampacity constraint is derived as:

{‖{\tilde{i}}_{s_{l}}‖}^{2} = \frac{{\tilde{p}}_{s_{l}}^{2} + {\tilde{q}}_{s_{l}}^{2}}{V_{s_{l}}} \leq {\tilde{K}}_{l}

(29)

The amapcity bound of the transmission line ${\tilde{K}}_{l}$ is normally provided by the transmission line manufacturer. ${\tilde{q}}_{s_{l}}, {\tilde{p}}_{s_{l}}, {\tilde{i}}_{s_{l}}$ can be measured from the sending-end of the transmission line. But $q_{s_{l}}, p_{s_{l}}, i_{s_{l}}$ can not be measured because they are physically distributed across the transmission line. From (25)-(29), the gap $Δ^{2} I$ between the current amplitude square ${‖i_{s_{l}}‖}^{2}$ and the measurable current amplitude square ${‖{\tilde{i}}_{s_{l}}‖}^{2}$ in (29) can be formulated as:

\begin{array}{l} Δ^{2} I = {‖i_{s_{l}}‖}^{2} - {‖{\tilde{i}}_{s_{l}}‖}^{2} = \frac{- q_{b s_{l}}^{2} + 2 q_{s_{l}} q_{b s_{l}} - p_{g s_{l}}^{2} - 2 p_{s_{l}} p_{g s_{l}}}{V_{s_{l}}} = \\ \frac{- V_{s_{l}}^{2} B_{s_{l}}^{2} + 2 q_{s_{l}} V_{s_{l}} B_{s_{l}}}{V_{s_{l}}} + \frac{- V_{s_{l}}^{2} G_{s_{l}}^{2} - 2 p_{s_{l}} V_{s_{l}} G_{s_{l}}}{V_{s_{l}}} = \\ - V_{s_{l}} B_{s_{l}}^{2} + 2 q_{s_{l}} B_{s_{l}} - V_{s_{l}} G_{s_{l}}^{2} - 2 p_{s_{l}} G_{s_{l}} \end{array}

(30)

The branch ampacity constraint (29) is equivalent to:

i_{s_{l}}^{2} = \frac{p_{s_{l}}^{2} + q_{s_{l}}^{2}}{V_{s_{l}}} \leq K_{l} = {\tilde{K}}_{l} + Δ^{2} I

(31)

The upper bounds of active power loss and reactive power loss $K_{o l}^{p}, K_{o l}^{q}$ can be quantified as:

\begin{array}{l} K_{o_{l}}^{p} = K_{l} R_{l} = ({\tilde{K}}_{l} + Δ^{2} I) R_{l} = \\ ({\tilde{K}}_{l} - V_{s_{l}} B_{s_{l}}^{2} + 2 q_{s_{l}} B_{s_{l}} - V_{s_{l}} G_{s_{l}}^{2} - 2 p_{s_{l}} G_{s_{l}}) R_{l} \end{array}

(32)

\begin{array}{l} K_{o_{l}}^{q} = K_{l} X_{l} = ({\tilde{K}}_{l} + Δ^{2} I) X_{l} = \\ ({\tilde{K}}_{l} - V_{s_{l}} B_{s_{l}}^{2} + 2 q_{s_{l}} B_{s_{l}} - V_{s_{l}} G_{s_{l}}^{2} - 2 p_{s_{l}} G_{s_{l}}) X_{l} \end{array}

(33)

Since $G_{s_{l}} a n d B_{s_{l}}$ are constants, (32) and (33) are linear (the final expressions). The expressions of $K_{o_{l}}^{p} a n d K_{o_{l}}^{q}$ from (32) and (33) are used in the exact and approximate General-BranchFlow models. The approximate General-BranchFlow model is still convex. In this way, any approximation on the branch ampacity constraint is avoided. This ampacity constraint can be used to constrain the power loss variable which equivalently constrains the capacity of the transmission lines expressed as:

p_{o_{l}} \leq K_{o_{l}}^{p}

(34)

q_{o_{l}} \leq K_{o_{l}}^{q}

(35)

III. OPF Problem

The OPF problem is a fundamental mathematical optimization model used widely in power system operations. Power system operators solve the OPF problem to make optimal decisions in the control room. The objective of the decision making can be to minimize the economic generation cost (economic dispatch), to minimize the power loss or maximize the security margin, etc. Any decisions in operating the power network must take into account physical laws of power flow and various operational constraints. In this paper, $Ω \subseteq {p_{n}, q_{n}, p_{d_{n}}, q_{d_{n}}, v_{n}, V_{n}, θ_{l}, p_{s_{l}}, q_{s_{l}}, p_{o_{l}}, q_{o_{l}}}$ is used to represent the set of decision variables in expressing the OPF model. For ease of the comparison with the proposed OPF model in this paper, the original OPF formulation in the polar format is re-stated as the objective function (36) subject to constraints (1), (2), (4)-(7), and (37)-(39).

m i n f = f (Ω)

(36)

p_{n} - p_{d_{n}} = v_{n} \sum_{n^{'} \in 𝒩_{n}} v_{n^{'}} (G_{n n^{'}} c o s θ_{n n^{'}} + B_{n n^{'}} s i n θ_{n n^{'}}) \forall n \in 𝒩

(37)

q_{n} - q_{d_{n}} = v_{n} \sum_{n^{'} \in 𝒩_{n}} v_{n^{'}} (G_{n n^{'}} s i n θ_{n n^{'}} - B_{n n^{'}} c o s θ_{n n^{'}}) \forall n \in 𝒩

(38)

θ_{n n^{'}} \in (θ_{n n^{'}}^{m i n}, θ_{n n^{'}}^{m a x}) \forall n, n^{'} \in 𝒩

(39)

where $n^{'} \in 𝒩$ is the alias of $n$ . Equation (37) is the active power balance equation; (38) is the reactive power balance equation; and $θ_{n n^{'}} = θ_{n} - θ_{n^{'}}$ .

By deploying one format of the exact General-BranchFlow model in Table I or the approximate General-BranchFlow model in Table II as the constraints of the OPF problem, different formats of the exact OPF problem or the approximate OPF problem can be formulated. For the branch ampacity constraint, either constraints (32)-(34) or (33)-(35) described in the Section II of this paper can be used. Twelve formats of the exact OPF model are listed in Table III. They are mathematically equivalent.

Table III Formats of Exact OPF Model

Format	Exact OPF Model
1	$\underset{Ω}{a r g m i n} f (Ω) : = {Ω \in {(1) - (7)}, {(8) - (11)}, {(13), (14)}, {(32)}, {(34)}}$
2	$\underset{Ω}{a r g m i n} f (Ω) : = {Ω \in {(1) - (7)}, {(8) - (11)}, {(13)}, {(15)}, {(32)}, {(34)}}$
3	$\underset{Ω}{a r g m i n} f (Ω) : = {Ω \in {(1) - (7)}, {(8) - (11)}, {(14), (15)}, {(32)}, {(34)}}$
4	$\begin{array}{l} \underset{Ω}{a r g m i n} f (Ω) : = {Ω \in {(1) - (7)}, {(8), (9)}, {(11), (12)}, {(13), (14)}, \\ {(32)}, {(34)}} \end{array}$
5	$\begin{array}{l} \underset{Ω}{a r g m i n} f (Ω) : = {Ω \in {(1) - (7)}, {(8), (9)}, {(11), (12)}, {(13)}, {(15)}, \\ {(32)}, {(34)}} \end{array}$
6	$\begin{array}{l} \underset{Ω}{a r g m i n} f (Ω) : = {Ω \in {(1) - (7)}, {(8), (9)}, {(11), (12)}, {(14), (15)}, \\ {(32)}, {(34)}} \end{array}$
7	$\underset{Ω}{a r g m i n} f (Ω) : = {Ω \in {(1) - (7)}, {(8) - (11)}, {(13), (14)}, {(33)}, {(35)}}$
8	$\underset{Ω}{a r g m i n} f (Ω) : = {Ω \in {(1) - (7)}, {(8) - (11)}, {(13)}, {(15)}, {(33)}, {(35)}$
9	$\underset{Ω}{a r g m i n} f (Ω) : = {Ω \in {(1) - (7)}, {(8) - (11)}, {(14), (15)}, {(33)}, {(35)}}$
10	$\begin{array}{l} \underset{Ω}{a r g m i n} f (Ω) : = {Ω \in {(1) - (7)}, {(8), (9)}, {(11), (12)}, {(13), (14)}, \\ {(33)}, {(35)}} \end{array}$
11	$\begin{array}{l} \underset{Ω}{a r g m i n} f (Ω) : = {Ω \in {(1) - (7)}, {(8), (9)}, {(11), (12)}, {(13)}, {(15)}, \\ {(33)}, {(35)}} \end{array}$
12	$\begin{array}{l} \underset{Ω}{a r g m i n} f (Ω) : = {Ω \in {(1) - (7)}, {(8), (9)}, {(11), (12)}, {(14), (15)}, \\ {(33)}, {(35)}} \end{array}$

To improve the AC-feasibility of the approximate General-BranchFlow model, the following conic constraint (40) is proven to be a necessary condition to recover the AC-feasible solution [

32]:

V_{s_{l}} V_{r_{l}} s i n^{2} (θ_{l}^{m a x}) \geq θ_{l}^{2} \forall l \in ℒ

(40)

This constraint is included in the approximate OPF model based on the approximate General-BranchFlow equations. Since this constraint is conic, it is convex. Twelve formats of the approximate OPF model are listed in Table IV. They are not mathematically equivalent but are approximate to each other.

Table IV Formats of Approximate OPF Model

Format	Approximate OPF model
1	$\begin{array}{l} \underset{Ω}{a r g m i n} f (Ω) : = {Ω \in {(2) - (7)}, {(17) - (21)}, {(23), (24)}, {(32)}, \\ {(34)}, {(40)}} \end{array}$
2	$\begin{array}{l} \underset{Ω}{a r g m i n} f (Ω) : = {Ω \in {(2) - (7)}, {(17) - (21)}, {(15)}, {(23)}, {(32)}, \\ {(34)}, {(40)}} \end{array}$
3	$\begin{array}{l} \underset{Ω}{a r g m i n} f (Ω) : = {Ω \in {(2) - (7)}, {(17) - (21)}, {(15)}, {(24)}, {(32)}, \\ {(34)}, {(40)}} \end{array}$
4	$\begin{array}{l} \underset{Ω}{a r g m i n} f (Ω) : = {Ω \in {(2) - (7)}, {(17) - (19)}, {(21), (22)}, {(23), (24)}, \\ {(32)}, {(34)}, {(40)}} \end{array}$
5	$\begin{array}{l} \underset{Ω}{a r g m i n} f (Ω) : = {Ω \in {(2) - (7)}, {(17) - (19)}, {(15)}, {(21), (22)}, {(23)}, \\ {(32)}, {(34)}, {(40)}} \end{array}$
6	$\begin{array}{l} \underset{Ω}{a r g m i n} f (Ω) : = {Ω \in {(2) - (7)}, {(17) - (19)}, {(15)}, {(21), (22)}, {(24)}, \\ {(32)}, {(34)}, {(40)}} \end{array}$
7	$\begin{array}{l} \underset{Ω}{a r g m i n} f (Ω) : = {Ω \in {(2) - (7)}, {(17) - (21)}, {(23), (24)}, {(33)}, \\ {(35)}, {(40)}} \end{array}$
8	$\begin{array}{l} \underset{Ω}{a r g m i n} f (Ω) : = {Ω \in {(2) - (7)}, {(17) - (21)}, {(15)}, {(23)}, {(33)}, \\ {(35)}, {(40)}} \end{array}$
9	$\begin{array}{l} \underset{Ω}{a r g m i n} f (Ω) : = {Ω \in {(2) - (7)}, {(17) - (21)}, {(15)}, {(24)}, {(33)}, \\ {(35)}, {(40)}} \end{array}$
10	$\begin{array}{l} \underset{Ω}{a r g m i n} f (Ω) : = {Ω \in {(2) - (7)}, {(17) - (19)}, {(21), (22)}, {(23), (24)}, \\ {(33)}, {(35)}, {(40)}} \end{array}$
11	$\begin{array}{l} \underset{Ω}{a r g m i n} f (Ω) : = {Ω \in {(2) - (7)}, {(17) - (19)}, {(15)}, {(21), (22)}, {(23)}, \\ {(33)}, {(35)}, {(40)}} \end{array}$
12	$\begin{array}{l} \underset{Ω}{a r g m i n} f (Ω) : = {Ω \in {(2) - (7)}, {(17) - (19)}, {(15)}, {(21), (22)}, {(24)}, \\ {(33)}, {(35)}, {(40)}} \end{array}$

For the approximate OPF model, the approximation gaps of active power loss $g a p_{l}^{p o}$ and reactive power loss $g a p_{l}^{q o}$ are defined as:

g a p_{l}^{p o} : = |p_{o_{l}} - \frac{p_{s_{l}}^{2} + q_{s_{l}}^{2}}{V_{s_{l}}} R_{l}| \forall l \in ℒ

(41)

g a p_{l}^{q o} : = |q_{o_{l}} - \frac{p_{s_{l}}^{2} + q_{s_{l}}^{2}}{V_{s_{l}}} X_{l}| \forall l \in ℒ

(42)

The corresponding maximum approximation gaps (of active and reactive power losses) are defined as:

g a p^{p o, m a x} : = m a x \{g a p_{l}^{p o}, \forall l \in ℒ\}

(43)

g a p^{q o, m a x} : = m a x \{g a p_{l}^{q o}, \forall l \in ℒ\}

(44)

These approximation gaps are useful to quantify the AC-feasibility of the solutions from the approximate OPF model. A fully AC-feasible solution of the approximate OPF model means that $g a p^{p o, m a x} = g a p^{q o, m a x} = 0$ . When $g a p^{p o, m a x} \neq 0$ or $g a p^{q o, m a x} \neq 0$ , smaller values of $g a p^{p o, m a x}, g a p^{q o, m a x}$ mean better solution quality in terms of AC-feasibility.

A typical example of the OPF problem to minimize a quadratic power generation cost function as the objective function $f$ is formulated in (45). This formulation is also used in the OPF formulations in MATPOWER, which is a benchmark OPF software package based on MATLAB.

f (p_{n}) = \sum_{n} (α_{n} p_{n}^{2} + β_{n} p_{n} + γ_{n})

(45)

where $α_{n}, β_{n}, γ_{n} \geq 0$ . Numerical investigations of the OPF model using (45) as the objective function are conducted in this section.

All the formats of the OPF model listed in Table III and Table IV are implemented in the Julia programming language and the JuMP optimization modeling package. The codes are running on the 64-bit Windows 10 operating system. A personal computer with Intel-i7 3 GHz CPU and 32 GB RAM is deployed. The IPOPT solver is used to solve the exact OPF model and the approximate OPF model in Julia [

36]. Some test cases of the approximate OPF model are solved by the CPLEX solver. The power network data (including the cost parameters

α_{n}, β_{n}, γ_{n}

) from MATPOWER is used here [37]. The evaluated power networks include case9, IEEE14, case30, IEEE57, case89pegase, IEEE118, IEEE300, ACTIVSg200, and ACTIVSg500 [38]-[41]. The OPF solutions from MATPOWER are used as the benchmark. Note that even the test cases used in this paper are mesh power networks, while there are many radial branches in these power networks. For example, in Fig. 3, one sub-network in radial topology is shown for the IEEE300 test case. These radial branches can be regarded as sub-networks in radial topology. In other words, the numerical results in this paper validate the OPF models for both radial and mesh power networks.

Fig. 3 Network topology of IEEE300 test case.

A.　Base Power Loads

The performance of the proposed OPF models for the base power loads is examined in this subsection. The base power loads are equal to the original power loads according to the test case data in MATPOWER.

The objective solutions of the exact OPF model are listed in Table V. All the proposed twelve formats of the exact OPF model have the same or very close objective solutions compared with MATPOWER. These results show that all the proposed formats of the exact OPF model are accurate. The computational CPU time of the exact OPF model are listed in Table VI. Note that the computational time for the proposed twelve formats of the OPF model includes both optimization model construction time in JuMP and the solver time in IPOPT.

Table V Objective Solutions of Exact OPF Model for Base Power Loads

Format	Objective solution ($)
Format	case9	IEEE14	case30	IEEE57	case89pegase	IEEE118	ACTIVSg200	IEEE300	ACTIVSg500
1	5296.69	8081.61	576.89	41738.11	5819.69	129660.63	27557.57	719732.11	71817.42
2	5314.02	8078.80	573.94	41698.64	5814.88	129695.30	27557.57	719377.57	71817.42
3	5296.69	8081.61	576.89	41738.11	5819.69	129660.63	27557.57	719732.11	71817.42
4	5296.69	8081.54	576.89	41737.93	5819.51	129660.54	27557.57	719732.22	71817.42
5	5296.68	8080.11	576.21	41696.02	5831.56	129930.62	27557.57	719396.40	71817.42
6	5296.69	8081.54	576.89	41737.93	5819.51	129660.54	27557.57	719731.22	71817.42
7	5296.69	8081.61	576.89	41738.11	5819.69	129660.63	27557.57	719732.11	71817.42
8	5296.68	8078.80	577.77	41698.64	5849.61	129695.30	27557.57	719377.57	71817.42
9	5296.69	8081.61	576.89	41738.11	5819.69	129660.63	27557.57	719732.11	71817.42
10	5296.69	8081.54	576.89	41737.93	5819.51	129660.54	27557.57	719731.22	71817.42
11	5309.00	8080.12	578.38	41696.02	5824.01	129930.62	27557.57	719396.40	71817.42
12	5296.69	8081.54	576.89	41737.93	5819.51	129660.54	27557.57	719731.22	71817.42
MATPOWER	5296.69	8081.53	576.89	41737.79	5819.81	129660.70	27557.57	719725.11	72578.30

Table VI Computational Time of Exact OPF Model for Base Power Loads

Format	Computational time (s)
Format	case9	IEEE14	case30	IEEE57	case89pegase	IEEE118	ACTIVSg200	IEEE300	ACTIVSg500
1	0.03	0.02	0.05	0.08	0.33	0.27	0.63	0.95	1.67
2	0.16	0.03	0.08	0.14	5.20	1.36	0.59	2.33	1.64
3	0.05	0.02	0.06	0.06	0.33	0.27	0.83	0.88	1.64
4	0.03	0.02	0.06	0.08	0.27	0.25	0.66	0.98	1.77
5	0.33	0.03	0.31	0.13	3.56	0.81	0.86	1.36	1.72
6	0.03	0.02	0.06	0.06	0.36	0.24	0.86	0.95	1.70
7	0.05	0.03	0.09	0.08	0.22	0.25	0.59	0.95	2.66
8	0.94	0.03	0.52	0.13	2.80	1.36	0.56	2.12	5.02
9	0.05	0.03	0.06	0.08	0.20	0.25	0.55	0.91	2.94
10	0.05	0.02	0.06	0.06	0.22	0.25	0.61	1.02	2.77
11	1.61	0.03	5.06	0.11	3.41	0.83	0.55	1.34	3.94
12	0.05	0.03	0.06	0.06	0.22	0.27	0.56	0.95	3.33
MATPOWER	1.20	1.67	2.14	1.88	2.50	2.02	2.09	2.13	2.78

With the increase of the network scale, the required computational CPU time increases. This is reasonable since the number of model variables and constraints increases. It can also be observed that, in some test cases, the computational time of one specific format of the OPF model is very different from the other formats. This is majorly because of the different formulations or constraints in different formats of the OPF model. These results also demonstrate the advantages of providing more formats of the OPF models to give power system operators more options in modeling and solving network operation problems. In other words, if one format of OPF models faces the numerical inefficiency problem for a specific power network, another format can be used.

The objective solutions of the approximate OPF model are listed in Table VII. The proposed twelve formats of the approximate OPF model have slightly different objective solutions due to the approximations of different constraints and the different selection of the constraints. The computational time of the approximate OPF model is listed in Table VIII. Note that the computational time for the proposed twelve formats of the approximate OPF model includes both optimization model construction time in JuMP and the solver time in IPOPT. Similarly, with the increase of the network scale, the required computational CPU time increases. This is reasonable since the numbers of model variables and constraints increase. Compared with the computational time of the exact OPF model, when the IPOPT solver converges, the required computational time of the approximate OPF model is more or less the same. The non-uniformity of the computational time listed in Table VIII is majorly because of the different network parameters in the test cases. For example, in the test case of IEEE300, some transmission lines have negative reactance values which is the major cause of the non-uniformity of the computational time.

Table VII Objective Solutions of Approximate OPF Model for Base Power Loads

Format	Objective solution ($)
Format	case9	IEEE14	case30	IEEE57	case89pegase	IEEE118	ACTIVSg200	IEEE300	ACTIVSg500
1	5296.69	8080.44	575.33	41734.92	5814.56	129618.44	27553.67	719596.69	71891.92
2	5296.69	8080.64	576.51	41728.82	5819.04	129625.35	27557.57	719548.84	71893.41
3	5296.69	8081.55	576.85	41735.91	5819.05	129626.18	27557.57	719699.91	71893.41
4	5296.49	8081.97	575.68	41726.41	5813.30	129606.68	27553.70	719710.59	71930.49
5	5315.48	8081.43	576.28	41719.61	5816.87	129614.02	27557.58	719456.87	71931.89
6	5315.49	8081.97	576.29	41726.41	5816.87	129615.15	27557.58	719824.73	71931.89
7	5296.69	8080.44	576.66	41734.92	5815.25	129618.44	27553.79	719596.69	71891.99
8	5296.69	8080.64	576.51	41728.82	5819.04	129625.35	27557.57	719548.84	71893.42
9	5296.69	8081.55	576.85	41735.91	5819.05	129626.18	27557.57	719699.91	71893.42
10	5315.49	8081.97	576.15	41726.41	5813.30	129606.68	27553.70	719710.59	71930.49
11	5315.48	8081.43	576.28	41719.61	5816.87	129614.02	27557.58	719456.87	71931.89
12	5315.49	8081.97	576.29	41726.41	5816.87	129615.15	27557.58	719824.73	71931.89

Table VIII Computational Time of Approximate OPF Model for Base Power Loads

Format	Computational time (s)
Format	case9	IEEE14	case30	IEEE57	case89pegase	IEEE118	ACTIVSg200	IEEE300	ACTIVSg500
1	0.03	0.05	0.09	0.16	0.56	0.48	0.77	1.49	2.83
2	0.05	0.05	0.16	0.31	1.08	0.80	0.64	4.06	2.27
3	0.03	0.03	0.08	0.11	0.41	0.33	0.62	1.38	2.13
4	0.03	0.05	0.13	0.16	1.00	0.44	0.78	1.56	2.89
5	0.06	0.06	0.09	0.28	0.41	0.94	0.72	5.67	2.25
6	0.03	0.03	0.06	0.14	0.41	0.38	0.69	1.75	2.19
7	0.03	0.05	0.09	0.16	0.53	0.45	0.72	1.77	3.17
8	0.05	0.05	0.08	0.28	0.83	0.80	0.69	4.00	4.47
9	0.03	0.03	0.08	0.11	0.37	0.33	0.66	1.49	3.31
10	0.03	0.05	0.13	0.14	0.52	0.42	0.78	1.75	2.69
11	0.06	0.05	0.05	0.26	1.00	0.91	0.69	5.42	2.34
12	0.05	0.03	0.09	0.11	0.41	0.34	0.67	1.88	2.25

The results of the maximum approximation gaps of active power loss of the approximate OPF model are listed in Table IX. Most values are very small or negligible. The results of the maximum approximation gaps of reactive power loss of the approximate OPF model are listed in Table X. Most of these values are larger compared with those of the maximum approximation gaps of active power loss listed in Table IX. Comparing the results of objective solutions in Table V and Table VII, most of the objective solutions of the convex OPF models are lower than the corresponding objective solutions of the original nonconvex OPF models. This means the convex OPF models can find the lower bounds of the objective solution in most cases. When the relaxation (approximation) gap is zero, the solutions from the convex OPF models are global optimal for the corresponding test cases. For example, for the test cases of ACTIVSg200, the relaxation (approximation) gaps of most formats of the convex OPF models are almost equal to 0 (below $10^{- 9}$ ). In other words, the solutions of these test cases from the convex OPF models are globally optimal. For the test cases with non-zero or large relaxation (approximation) gaps, a method to reduce the approximation gaps is proposed and validated in the next section.

Table IX The Maximum Approximation Gap of Active Power Loss of Approximate OPF Model for Base Power Loads

Format	The maximum approximation gap
Format	case9	IEEE14	case30	IEEE57	case89pegase	IEEE118	ACTIVSg200	IEEE300	ACTIVSg500
1	0.00×10⁰	0.00×10⁰	0.00×10⁰	0.00×10⁰	2.53×10^-11	0.00×10⁰	2.01×10^-12	5.15×10^-14	7.11×10^-14
2	7.70×10^-14	1.29×10^-10	5.39×10^-15	8.00×10^-11	1.85×10^-2	2.54×10^-12	3.49×10^-12	2.84×10^-13	2.95×10^-15
3	0.00×10⁰	0.00×10⁰	0.00×10⁰	0.00×10⁰	1.85×10^-2	0.00×10⁰	3.49×10^-12	3.39×10^-14	2.95×10^-15
4	0.00×10⁰	0.00×10⁰	0.00×10⁰	0.00×10⁰	2.53×10^-11	0.00×10⁰	3.42×10^-13	5.60×10^-14	7.06×10^-14
5	3.02×10^-13	1.30×10^-10	8.23×10^-15	7.85×10^-11	1.61×10^-2	2.95×10^-12	2.09×10^-12	4.38×10^-3	3.42×10^-14
6	0.00×10⁰	0.00×10⁰	0.00×10⁰	0.00×10⁰	1.61×10^-2	0.00×10⁰	2.09×10^-12	4.32×10^-3	3.42×10^-14
7	0.00×10⁰	0.00×10⁰	0.00×10⁰	0.00×10⁰	2.53×10^-11	0.00×10⁰	2.01×10^-12	5.15×10^-14	7.11×10^-14
8	7.06×10^-11	1.29×10^-10	2.14×10^-8	8.00×10^-11	1.85×10^-2	2.54×10^-12	3.49×10^-12	2.84×10^-13	2.95×10^-15
9	0.00×10⁰	0.00×10⁰	0.00×10⁰	0.00×10⁰	1.85×10^-2	0.00×10⁰	3.49×10^-12	3.39×10^-14	2.95×10^-15
10	0.00×10⁰	0.00×10⁰	0.00×10⁰	0.00×10⁰	2.53×10^-11	0.00×10⁰	2.01×10^-12	5.60×10^-14	7.06×10^-14
11	7.06×10^-11	1.30×10^-10	4.11×10^-9	7.85×10^-11	1.61×10^-2	2.95×10^-12	2.09×10^-12	4.38×10^-3	3.42×10^-14
12	0.00×10⁰	0.00×10⁰	0.00×10⁰	0.00×10⁰	1.61×10^-2	0.00×10⁰	2.09×10^-12	4.32×10^-3	3.42×10^-14

Table X The Maximum Approximation Gap of Reactive Power Loss of Approximate OPF Model for Base Power Loads

Format	The maximum approximation gap
Format	case9	IEEE14	case30	IEEE57	case89pegase	IEEE118	ACTIVSg200	IEEE300	ACTIVSg500
1	3.32×10^-1	2.43×10^-1	3.08×10^-1	1.25×10^-1	3.47×10⁰	6.02×10⁰	4.84×10^-1	3.13×10⁰	2.98×10^-1
2	2.40×10^-1	4.63×10^-10	6.14×10^-2	1.14×10^-1	1.47×10⁰	3.99×10⁰	2.55×10^-10	1.00×10⁰	3.35×10^-13
3	3.32×10^-1	4.07×10^-10	8.49×10^-2	8.30×10^-2	1.47×10⁰	4.23×10⁰	2.55×10^-10	8.14×10^-1	3.35×10^-13
4	2.07×10⁰	2.13×10^-1	2.61×10^-1	4.75×10^-1	2.15×10⁰	7.42×10⁰	3.20×10^-1	2.27×10⁴	1.01×10^-1
5	1.71×10⁰	8.65×10^-2	1.75×10^-1	1.47×10^-1	1.27×10⁰	2.96×10⁰	1.52×10^-10	1.00×10⁵	3.88×10^-12
6	2.12×10⁰	9.58×10^-2	2.18×10^-1	1.44×10^-1	1.27×10⁰	3.45×10⁰	1.52×10^-10	1.05×10⁴	3.88×10^-12
7	2.41×10^-1	2.43×10^-1	9.89×10^-2	1.25×10^-1	1.53×10⁰	6.02×10⁰	3.69×10^-1	3.13×10⁰	1.52×10^-1
8	1.41×10^-1	4.63×10^-10	1.30×10^-2	1.14×10^-1	1.44×10⁰	3.99×10⁰	2.55×10^-10	1.00×10⁰	3.35×10^-13
9	2.41×10^-1	4.07×10^-10	2.63×10^-2	8.30×10^-2	1.47×10⁰	4.23×10⁰	2.55×10^-10	8.14×10^-1	3.35×10^-13
10	1.85×10^-1	2.13×10^-1	9.69×10^-2	4.75×10^-1	2.15×10⁰	7.42×10⁰	5.42×10^-1	2.27×10⁴	1.53×10^-1
11	1.01×10^-1	8.65×10^-2	3.28×10^-2	1.47×10^-1	1.27×10⁰	2.96×10⁰	1.52×10^-10	1.00×10⁵	3.88×10^-12
12	1.85×10^-1	9.58×10^-2	9.78×10^-2	1.44×10^-1	1.27×10⁰	3.45×10⁰	1.52×10^-10	1.05×10⁴	3.88×10^-12

B.　Heavy Power Loads

The performance of the proposed OPF models for the heavy power loads is examined in this subsection. The heavy power loads are obtained by increasing the original power loads at all the nodes of the test case data from MATPOWER. The heavy power loads range from 110% to 200% (with a gradual increment of 10%) of the base power loads. Since the IPOPT solver can find all the optimal solutions in heavy power loads for the case9 and IEEE118 in MATPOWER, we list all results for the proposed formats of the OPF models for case9 and IEEE118.

The objective solutions of the exact OPF model for heavy power loads of case9 are listed in Table XI. The computational time of the exact OPF model for heavy power loads of case9 is listed in Table XII. The objective solutions of the exact OPF model for heavy power loads of IEEE118 are listed in Table XIII. The computational time of the exact OPF model for heavy power loads of IEEE118 is listed in Table XIV. The objective solutions of the approximate OPF model for heavy power loads of case9 are listed in Table XV. The computational time of the exact OPF model for heavy power loads of case9 is listed in Table XVI. The maximum approximation gaps of the active power loss of the approximate OPF model for the heavy power loads of case9 are listed in Table XVII. The maximum approximation gaps of the reactive power loss of the approximate OPF model for the heavy power loads of case9 are listed in Table XVIII. The objective solutions of the approximate OPF model for heavy power loads of IEEE118 are listed in Table XIX. The computational time of the exact OPF model for heavy power loads of IEEE118 is listed in Table XX. The maximum approximation gaps of the active power loss of the approximate OPF model for the heavy power loads of IEEE118 are listed in Table XXI. The maximum approximation gaps of the reactive power loss of the approximate OPF model for the heavy power loads of IEEE118 are listed in Table XXII. These results show that the proposed OPF models based on the General-BranchFlow equations in this paper are robust in terms of the power loads, which can force the decision variables or line capacities to the limits. In most cases, the computational efficiency of the proposed OPF models executed in Julia outperforms MATPOWER.

Table XI Objective Solutions of Exact OPF Model for Heavy Power Loads of Case9

Format	Objective solution ($)
Format	110%	120%	130%	140%	150%	160%	170%	180%	190%	200%
1	6114.27	7006.02	7972.52	9014.71	10133.71	11330.27	12605.50	13960.63	15397.09	16916.52
2	6114.03	7004.97	7982.45	9013.78	10127.29	11384.69	12583.93	13938.51	15452.18	16970.22
3	6114.27	7006.02	7972.52	9014.71	10133.71	11330.27	12605.50	13960.63	15397.09	16916.52
4	6114.03	7006.02	7972.52	9014.71	10133.71	11330.27	12605.50	13960.63	15397.09	16916.52
5	6114.03	7005.23	7975.05	9011.40	10144.66	11323.91	12616.04	14018.66	15422.55	16870.09
6	6114.27	7006.02	7972.52	9014.71	10133.71	11330.27	12605.50	13960.63	15397.09	16916.52
7	6114.27	7006.02	7972.52	9014.71	10133.71	11330.27	12605.50	13960.63	15397.09	16916.71
8	6115.38	7026.93	8002.58	9014.50	10131.20	11326.63	12608.77	14538.75	15411.56	16941.55
9	6114.27	7006.02	7972.52	9014.71	10133.71	11330.27	12605.50	13960.63	15397.09	16916.71
10	6114.27	7006.02	7972.52	9014.71	10133.71	11330.27	12605.50	13960.63	15397.09	16916.71
11	6129.82	7008.02	7976.51	9016.57	10142.61	11328.90	12625.99	13969.64	15400.50	16903.12
12	6114.27	7006.02	7972.52	9014.71	10133.71	11330.27	12605.50	13960.63	15397.09	16916.71
MATPOWER	6114.27	7006.02	7972.52	9014.71	10133.71	11330.27	12605.50	13960.63	15412.54	17004.28

Table XII Computational Time of Exact OPF Model for Heavy Power Loads of Case9

Format	Computational time (s)
Format	110%	120%	130%	140%	150%	160%	170%	180%	190%	200%
1	0.05	0.05	0.05	0.03	0.03	0.03	0.05	0.03	0.08	0.05
2	0.17	0.23	1.06	0.53	0.25	0.08	0.19	0.31	0.08	0.08
3	0.05	0.03	0.03	0.03	0.03	0.03	0.05	0.03	0.08	0.05
4	0.06	0.03	0.05	0.08	0.03	0.03	0.05	0.03	0.19	0.03
5	0.38	0.89	0.09	0.13	1.34	0.09	0.11	0.03	0.44	0.38
6	0.06	0.03	0.05	0.08	0.03	0.03	0.06	0.08	0.06	0.03
7	0.06	0.03	0.05	0.14	0.06	0.03	0.03	0.14	0.06	0.03
8	0.22	3.00	0.66	0.08	0.67	0.05	0.34	0.34	2.09	1.44
9	0.06	0.05	0.05	0.06	0.06	0.05	0.03	0.24	0.05	0.03
10	0.05	0.08	0.06	0.09	0.05	0.03	0.03	0.09	0.08	0.06
11	0.16	0.27	0.47	0.27	0.42	1.45	0.31	0.33	0.88	0.27
12	0.06	0.08	0.08	0.13	0.05	0.03	0.03	0.06	0.08	0.05
MATPOWER	1.03	1.05	1.14	1.30	1.31	1.25	1.28	1.30	1.40	1.25

Table XIII Objective Solutions of Exact OPF Model for Heavy Power Loads of IEEE118

Format	Objective solution ($)
Format	110%	120%	130%	140%	150%	160%	170%	180%	190%	200%
1	146584.41	163717.23	181026.22	198505.38	216185.52	234071.02	252203.09	270661.72	289547.38	309049.00
2	146629.57	163767.22	181076.43	198565.32	216389.18	234301.10	252464.21	270747.87	289855.94	309115.65
3	146584.41	163717.23	181026.22	198505.38	216185.52	234071.02	252203.09	270661.72	289547.37	309049.00
4	146584.34	163717.15	181026.15	198505.31	216185.44	234070.99	252203.09	270661.72	289547.49	309049.23
5	146674.46	163811.03	181123.74	198611.39	216447.91	234459.16	270777.81	271214.68	290280.28	309101.80
6	146584.34	163717.15	181026.15	198505.31	216185.44	234070.99	252203.09	270661.76	289547.49	309049.23
7	146584.41	163717.23	181026.22	198505.38	216185.52	234071.02	252203.09	270661.72	289547.38	309049.00
8	146629.57	163767.22	1810076.43	198565.32	216389.18	234301.10	252464.21	270747.87	289855.94	309115.65
9	146584.41	163717.23	181026.22	198505.38	216185.52	234071.02	252203.09	270661.72	289547.38	309049.00
10	146584.34	163717.15	181026.15	198505.31	216185.52	234070.99	252203.09	270661.76	289547.49	309049.23
11	146674.46	163811.03	181123.74	198611.39	216447.91	234459.16	252629.32	271214.68	290280.28	309101.80
12	146584.34	163717.15	181026.15	198505.31	216185.44	234070.99	252203.09	270661.76	289547.49	309049.23
MATPOWER	146584.49	163717.30	181026.29	198505.42	216185.54	234070.66	252202.15	270659.85	289544.15	309043.83

Table XIV Computational Time of Exact OPF Model for Heavy Power Loads of IEEE118

Format	Computational time (s)
Format	110%	120%	130%	140%	150%	160%	170%	180%	190%	200%
1	0.16	0.22	0.19	0.20	0.39	0.25	0.22	0.20	0.23	0.22
2	0.42	0.50	0.47	0.72	0.45	0.47	0.78	0.66	0.61	0.63
3	0.19	0.17	0.17	0.20	0.36	0.28	0.27	0.22	0.22	0.24
4	0.17	0.16	0.20	0.25	0.38	0.30	0.22	0.22	0.22	0.24
5	0.47	0.38	0.50	0.83	0.84	0.50	0.70	0.69	0.97	0.89
6	0.16	0.16	0.19	0.23	0.39	0.23	0.25	0.22	0.22	0.25
7	0.16	0.16	0.16	0.24	0.38	0.25	0.23	0.24	0.22	0.25
8	0.41	0.53	0.45	0.64	0.44	0.48	0.77	0.67	0.58	0.63
9	0.19	0.14	0.16	0.19	0.36	0.28	0.20	0.20	0.20	0.22
10	0.20	0.19	0.20	0.27	0.44	0.31	0.25	0.24	0.25	0.25
11	0.47	0.38	0.48	0.80	0.89	0.49	0.67	0.70	0.98	0.83
12	0.16	0.16	0.16	0.22	0.39	0.23	0.22	0.22	0.22	0.23
MATPOWER	1.77	1.90	1.77	2.09	1.98	1.84	1.89	1.98	1.97	2.14

Table XV Objective Solutions of Approximate OPF Model for Heavy Power Loads of Case9

Format	Objective solution ($)
Format	110%	120%	130%	140%	150%	160%	170%	180%	190%	200%
1	6114.28	7006.04	7972.55	9014.76	10133.79	11330.38	12605.65	13960.84	15397.39	16916.93
2	6114.14	7005.57	7971.77	9013.35	10131.27	11326.73	12600.42	13953.45	15387.04	16902.57
3	6114.28	7006.04	7972.55	9014.76	10133.79	11330.38	12605.65	13960.84	15397.39	16916.93
4	6113.90	7005.22	7971.39	9013.32	10131.76	11327.56	12601.78	13955.60	15390.36	16907.56
5	6113.81	7004.94	7970.87	9012.09	10129.56	11324.30	12597.02	13948.75	15380.63	16893.90
6	6113.90	7005.22	7971.39	9013.32	10131.76	11327.56	12601.78	13955.60	15390.36	16907.56
7	6114.28	7006.04	7972.55	9014.76	10133.79	11330.38	12605.65	13960.84	15397.39	16917.12
8	6114.14	7005.57	7971.77	9013.35	10131.27	11326.73	12600.42	13953.45	15387.04	16902.57
9	6114.28	7006.04	7972.55	9014.76	10133.79	11330.38	12605.65	13960.84	15397.39	16917.12
10	6113.90	7005.22	7971.39	9013.32	10131.76	11327.56	12601.78	13955.60	15390.36	16907.56
11	6113.81	7004.94	7970.87	9012.09	10129.56	11324.30	12607.02	13948.75	15380.63	16893.90
12	6113.90	7005.22	7971.39	9013.32	10131.76	11327.56	12601.78	13955.60	15390.36	16907.56

Table XVI Computational Time of Approximate OPF Model for Heavy Power Loads of Case9

Format	Computational time (s)
Format	110%	120%	130%	140%	150%	160%	170%	180%	190%	200%
1	0.03	0.05	0.03	0.03	0.03	0.05	0.05	0.03	0.03	0.03
2	0.03	0.05	0.03	0.05	0.05	0.03	0.05	0.03	0.03	0.03
3	0.03	0.03	0.05	0.05	0.05	0.03	0.05	0.06	0.03	0.03
4	0.05	0.05	0.05	0.05	0.03	0.06	0.05	0.03	0.03	0.03
5	0.05	0.05	0.05	0.05	0.05	0.05	0.05	0.03	0.05	0.05
6	0.03	0.05	0.03	0.05	0.03	0.05	0.05	0.06	0.05	0.03
7	0.03	0.05	0.03	0.03	0.03	0.06	0.03	0.05	0.03	0.03
8	0.03	0.05	0.05	0.05	0.05	0.03	0.03	0.03	0.05	0.03
9	0.03	0.03	0.03	0.05	0.03	0.05	0.05	0.06	0.03	0.05
10	0.03	0.03	0.03	0.03	0.03	0.06	0.03	0.03	0.03	0.05
11	0.05	0.03	0.05	0.05	0.05	0.05	0.03	0.03	0.03	0.03
12	0.03	0.03	0.03	0.05	0.03	0.05	0.03	0.03	0.03	0.03

Table XVII The Maximum Approximation Gap of Active Power Loss of Approximate OPF Model for Heavy Power Loads of Case9

Format	The maximum approximation gap
Format	110%	120%	130%	140%	150%	160%	170%	180%	190%	200%
1	0.00×10⁰	0.00×10⁰	0.00×10⁰	0.00×10⁰	0.00×10⁰	0.00×10⁰	0.00×10⁰	0.00×10⁰	0.00×10⁰	0.00×10⁰
2	1.87×10^-15	2.11×10^-13	1.47×10^-13	3.57×10^-13	5.27×10^-13	5.47×10^-15	2.43×10^-13	1.85×10^-13	1.21×10^-13	1.69×10^-13
3	0.00×10⁰	0.00×10⁰	0.00×10⁰	0.00×10⁰	0.00×10⁰	0.00×10⁰	0.00×10⁰	0.00×10⁰	0.00×10⁰	0.00×10⁰
4	0.00×10⁰	0.00×10⁰	0.00×10⁰	0.00×10⁰	0.00×10⁰	0.00×10⁰	0.00×10⁰	0.00×10⁰	0.00×10⁰	0.00×10⁰
5	1.70×10^-14	1.88×10^-13	6.11×10^-13	4.11×10^-13	3.33×10^-13	2.62×10^-13	1.93×10^-13	3.14×10^-15	1.27×10^-15	7.81×10^-15
6	0.00×10⁰	0.00×10⁰	0.00×10⁰	0.00×10⁰	0.00×10⁰	0.00×10⁰	0.00×10⁰	0.00×10⁰	0.00×10⁰	0.00×10⁰
7	0.00×10⁰	0.00×10⁰	0.00×10⁰	0.00×10⁰	0.00×10⁰	0.00×10⁰	0.00×10⁰	0.00×10⁰	0.00×10⁰	0.00×10⁰
8	7.09×10^-11	7.14×10^-11	7.19×10^-11	7.30×10^-11	7.31×10^-11	7.39×10^-11	7.47×10^-11	7.56×10^-11	7.65×10^-11	7.75×10^-11
9	0.00×10⁰	0.00×10⁰	0.00×10⁰	0.00×10⁰	0.00×10⁰	0.00×10⁰	0.00×10⁰	0.00×10⁰	0.00×10⁰	0.00×10⁰
10	0.00×10⁰	0.00×10⁰	0.00×10⁰	0.00×10⁰	0.00×10⁰	0.00×10⁰	0.00×10⁰	0.00×10⁰	0.00×10⁰	0.00×10⁰
11	7.09×10^-11	7.18×10^-11	7.18×10^-11	7.23×10^-11	7.30×10^-11	7.37×10^-11	7.45×10^-11	7.53×10^-11	7.62×10^-11	7.71×10^-11
12	0.00×10⁰	0.00×10⁰	0.00×10⁰	0.00×10⁰	0.00×10⁰	0.00×10⁰	0.00×10⁰	0.00×10⁰	0.00×10⁰	0.00×10⁰

Table XIX Objective Solution of Approximate OPF Model for Heavy Power Loads of IEEE118

Format	Objective solution ($)
Format	110%	120%	130%	140%	150%	160%	170%	180%	190%	200%
1	146544.37	163678.09	180986.97	198463.56	216139.58	234018.65	252135.44	270575.68	289444.00	308925.65
2	146550.23	163683.12	180991.60	198467.97	216143.89	234022.81	252138.26	270569.30	289423.11	308888.97
3	146551.11	163684.06	180992.61	198469.05	216145.06	234024.47	252141.43	270580.54	289448.93	308931.22
4	146531.65	163664.05	180971.26	198444.66	216115.93	233989.29	252097.65	270522.21	289373.44	308833.42
5	146538.89	163671.16	180978.35	198451.76	216123.05	233996.20	252103.35	270522.29	289357.32	308797.67
6	146540.11	163672.48	180979.76	198453.28	216124.68	233998.00	252107.05	270533.23	289384.36	308843.26
7	146544.37	163678.09	180986.97	198463.56	216139.58	234018.65	252135.44	270575.68	289444.00	308925.65
8	146550.23	163683.12	180991.60	198467.97	216143.89	234022.81	252138.26	270569.30	289423.11	308888.97
9	146551.11	163684.06	180992.61	198469.05	216145.06	234024.47	252141.43	270580.54	289448.93	308931.22
10	146531.65	163664.05	180971.26	198444.66	216115.93	233989.29	252097.65	270522.21	289373.44	308833.42
11	146538.89	163671.16	180978.35	198451.76	216123.05	233996.20	252103.35	270522.29	289357.32	308797.67
12	146540.11	163672.48	180979.76	198453.28	216124.68	233998.00	252107.05	270533.23	289384.36	308843.26

Table XX Computational Time of Approximate OPF Model for Heavy Power Loads of IEEE118

Format	Computational time (s)
Format	110%	120%	130%	140%	150%	160%	170%	180%	190%	200%
1	0.53	0.80	0.66	0.52	0.50	0.52	0.52	0.50	0.50	0.64
2	0.91	0.88	1.00	0.78	0.92	0.78	0.76	0.78	0.95	0.86
3	0.66	0.42	0.56	0.33	0.36	0.41	0.34	0.36	0.36	0.41
4	0.44	0.52	0.53	0.45	0.44	0.42	0.44	0.45	0.58	0.55
5	1.00	0.81	0.84	0.86	0.92	0.98	0.80	0.83	0.72	0.84
6	0.39	0.44	0.45	0.34	0.36	0.38	0.36	0.41	0.41	0.45
7	0.50	0.83	0.64	0.50	0.50	0.50	0.50	0.50	0.50	0.62
8	0.89	0.91	1.01	0.78	0.97	0.77	0.81	0.75	0.97	0.81
9	0.67	0.45	0.56	0.36	0.39	0.36	0.34	0.36	0.36	0.39
10	0.45	0.58	0.50	0.47	0.52	0.44	0.44	0.45	0.52	0.56
11	0.97	0.84	0.78	0.88	0.94	0.95	0.80	0.78	0.77	0.84
12	0.39	0.44	0.42	0.36	0.36	0.36	0.36	0.41	0.39	0.48

Table XXI The Maximum Approximation Gap of Active Power Loss of Approximate OPF Model for Heavy Power Loads of IEEE118

Format	The maximum approximation gap
Format	110%	120%	130%	140%	150%	160%	170%	180%	190%	200%
1	0.00×10⁰	0.00×10⁰	0.00×10⁰	0.00×10⁰	0.00×10⁰	0.00×10⁰	0.00×10⁰	0.00×10⁰	0.00×10⁰	0.00×10⁰
2	2.56×10^-10	2.57×10^-12	2.58×10^-12	2.58×10^-12	2.58×10^-12	2.58×10^-12	2.57×10^-12	2.57×10^-12	2.56×10^-12	2.57×10^-12
3	0.00×10⁰	0.00×10⁰	0.00×10⁰	0.00×10⁰	0.00×10⁰	0.00×10⁰	0.00×10⁰	0.00×10⁰	0.00×10⁰	0.00×10⁰
4	0.00×10⁰	0.00×10⁰	0.00×10⁰	0.00×10⁰	0.00×10⁰	0.00×10⁰	0.00×10⁰	0.00×10⁰	0.00×10⁰	0.00×10⁰
5	2.95×10^-12	2.55×10^-12	2.56×10^-12	2.56×10^-12	2.57×10^-12	2.57×10^-12	2.58×10^-12	2.57×10^-12	2.54×10^-12	2.54×10^-12
6	0.00×10⁰	0.00×10⁰	0.00×10⁰	0.00×10⁰	0.00×10⁰	0.00×10⁰	0.00×10⁰	0.00×10⁰	0.00×10⁰	0.00×10⁰
7	0.00×10⁰	0.00×10⁰	0.00×10⁰	0.00×10⁰	0.00×10⁰	0.00×10⁰	0.00×10⁰	0.00×10⁰	0.00×10⁰	0.00×10⁰
8	2.56×10^-12	2.57×10^-12	2.58×10^-12	2.58×10^-12	2.58×10^-12	2.58×10^-12	2.57×10^-12	2.57×10^-12	2.56×10^-12	2.57×10^-12
9	0.00×10⁰	0.00×10⁰	0.00×10⁰	0.00×10⁰	0.00×10⁰	0.00×10⁰	0.00×10⁰	0.00×10⁰	0.00×10⁰	0.00×10⁰
10	0.00×10⁰	0.00×10⁰	0.00×10⁰	0.00×10⁰	0.00×10⁰	0.00×10⁰	0.00×10⁰	0.00×10⁰	0.00×10⁰	0.00×10⁰
11	2.95×10^-12	2.55×10^-12	2.56×10^-12	2.56×10^-12	2.57×10^-12	2.57×10^-12	2.58×10^-12	2.57×10^-12	2.54×10^-12	2.54×10^-12
12	0.00×10⁰	0.00×10⁰	0.00×10⁰	0.00×10⁰	0.00×10⁰	0.00×10⁰	0.00×10⁰	0.00×10⁰	0.00×10⁰	0.00×10⁰

Table XXII The Maximum Approximation Gap of Reactive Power Loss of Approximate OPF Model for Heavy Power Loads of IEEE118

Format	The maximum approximation gap (×10⁰)
Format	110%	120%	130%	140%	150%	160%	170%	180%	190%	200%
1	6.07	6.09	6.09	6.05	5.98	5.73	5.56	5.00	4.59	4.08
2	3.99	4.00	4.02	4.03	4.05	4.03	4.03	4.01	3.97	3.92
3	4.24	4.25	4.25	7.40	2.29	4.28	4.28	4.26	4.21	4.16
4	7.42	7.42	7.41	2.76	7.39	7.39	7.38	7.38	7.38	7.37
5	2.92	2.88	2.82	2.76	2.69	2.63	2.54	2.50	2.69	2.66
6	3.42	3.37	3.32	3.26	3.19	3.13	3.05	3.12	3.01	2.88
7	6.07	6.09	6.09	6.05	5.99	5.73	5.55	5.00	4.59	4.08
8	3.99	4.00	4.02	4.03	4.05	4.04	4.03	4.01	3.97	3.92
9	4.24	4.25	4.26	4.28	4.29	4.28	4.28	4.26	4.21	4.16
10	7.42	7.42	7.41	7.40	7.39	7.39	7.38	7.38	7.38	7.37
11	2.92	2.88	2.82	2.76	2.69	2.63	2.54	2.50	2.69	2.66
12	3.42	3.37	3.32	3.26	3.19	3.13	3.05	3.12	3.01	2.88

Table XVIII TheMaximum Approximation Gap of Reactive Power Loss of Approximate OPF Model for Heavy Power Loads of Case9

Format	The maximum approximation gap
Format	110%	120%	130%	140%	150%	160%	170%	180%	190%	200%
1	2.54×10^-1	1.57×10^-1	5.82×10^-2	3.28×10^-10	1.60×10^-10	2.69×10^-11	7.64×10^-11	1.57×10^-10	2.21×10^-10	2.72×10^-10
2	1.84×10^-1	1.21×10^-1	4.41×10^-2	6.27×10^-10	7.08×10^-10	6.71×10^-10	6.68×10^-10	6.70×10^-10	6.73×10^-10	6.77×10^-10
3	2.55×10^-1	1.57×10^-1	5.84×10^-2	1.88×10⁰	1.82×10⁰	2.68×10^-11	7.65×10^-11	1.57×10^-10	2.21×10^-10	2.72×10^-10
4	2.04×10⁰	2.00×10⁰	1.95×10⁰	1.50×10⁰	1.81×10⁰	1.74×10⁰	1.66×10⁰	1.58×10⁰	1.50×10⁰	1.41×10⁰
5	1.65×10⁰	1.60×10⁰	1.54×10⁰	1.88×10⁰	1.44×10⁰	1.37×10⁰	1.29×10⁰	1.22×10⁰	1.14×10⁰	1.06×10⁰
6	1.99×10⁰	2.00×10⁰	1.95×10⁰	3.29×10^-10	1.82×10⁰	1.74×10⁰	1.67×10⁰	1.58×10⁰	1.49×10⁰	1.41×10⁰
7	2.04×10^-1	1.40×10^-1	5.56×10^-2	3.29×10^-10	1.60×10^-10	2.72×10^-11	7.62×10^-11	1.57×10^-10	2.21×10^-10	2.90×10^-10
8	1.16×10^-1	8.08×10^-2	2.90×10^-2	6.18×10^-10	6.95×10^-10	6.67×10^-10	6.33×10^-10	6.65×10^-10	6.68×10^-10	6.72×10^-10
9	2.04×10^-1	1.39×10^-1	5.50×10^-2	3.30×10^-10	1.60×10^-10	2.71×10^-11	7.62×10^-11	1.57×10^-10	2.21×10^-10	2.89×10^-10
10	1.71×10^-1	1.56×10^-1	1.40×10^-1	1.22×10^-1	1.04×10^-1	8.41×10^-2	6.41×10^-2	4.33×10^-2	2.19×10^-2	2.06×10^-10
11	8.28×10^-2	6.24×10^-2	3.95×10^-2	1.44×10^-2	6.75×10^-10	6.62×10^-10	6.31×10^-10	6.62×10^-10	6.64×10^-10	6.68×10^-10
12	1.71×10^-1	1.56×10^-1	1.40×10^-1	1.22×10^-1	1.04×10^-1	8.43×10^-2	6.48×10^-2	4.26×10^-2	2.18×10^-2	2.08×10^-10

IV. APPROXIMATIOn GAP REDUCTION

To reduce the approximation gap of the reactive power loss for the approximate OPF model, a penalty function based method which adds an penalty term $ξ \sum_{l} q_{o l}$ to the original objective function of the OPF model is proposed as:

f' = f (p_{n}) + ξ \sum_{l} q_{o_{l}}

(46)

where the penalty coefficient $ξ > 0$ . In this paper, $ξ = 0.3$ is used.

A.　Base Power Loads

The performance of the proposed penalty function based approximation gap reduction method to tighten the approximation gap in the base power loads is shown in Tables XXIII and XXIV. It can be observed that the approximation gaps of the reactive power loss are much smaller than the values listed in Table X. For the maximum approximation gaps of the active power loss, the values are more or less the same compared with the results listed in Table IX. Since the maximum approximation gaps of the active power loss are very small, it is not necessary to reduce them.

B.　Heavy Power Loads

The performance of the proposed penalty function based approximation gap reduction method to tighten the approximation gap in the heavy power loads for case9 is shown in Tables XXV and XXVI. The performance of the proposed method to tighten the approximation gap in the heavy power loads for IEEE118 is shown in Tables XXVII and XXVIII. It can be observed that the approximation gaps of the reactive power loss are much smaller than the values listed in Tables XVIII and XVII. These results show that the proposed method is also useful for the power networks in heavy power loads.

Table XXV Reduced Maximum Approximation Gap of Active Power Loss of Approximate OPF Model for Heavy Power Loads of Case9

Format	Reduced maximum approximation gap
Format	110%	120%	130%	140%	150%	160%	170%	180%	190%	200%
1	0.00×10⁰	0.00×10⁰	0.00×10⁰	0.00×10⁰	0.00×10⁰	0.00×10⁰	0.00×10⁰	0.00×10⁰	0.00×10⁰	0.00×10⁰
2	4.18×10^-14	8.69×10^-14	7.43×10^-15	6.67×10^-14	2.96×10^-13	1.21×10^-13	3.65×10^-13	1.59×10^-14	1.05×10^-13	1.48×10^-13
3	0.00×10⁰	0.00×10⁰	0.00×10⁰	0.00×10⁰	0.00×10⁰	0.00×10⁰	0.00×10⁰	0.00×10⁰	0.00×10⁰	0.00×10⁰
4	0.00×10⁰	0.00×10⁰	0.00×10⁰	0.00×10⁰	0.00×10⁰	0.00×10⁰	0.00×10⁰	0.00×10⁰	0.00×10⁰	0.00×10⁰
5	3.84×10^-14	5.19×10^-14	4.69×10^-14	1.84×10^-14	4.65×10^-14	1.69×10^-14	8.38×10^-14	1.57×10^-13	1.56×10^-13	8.63×10^-15
6	0.00×10⁰	0.00×10⁰	0.00×10⁰	0.00×10⁰	0.00×10⁰	0.00×10⁰	0.00×10⁰	0.00×10⁰	0.00×10⁰	0.00×10⁰
7	0.00×10⁰	0.00×10⁰	0.00×10⁰	0.00×10⁰	0.00×10⁰	0.00×10⁰	0.00×10⁰	0.00×10⁰	0.00×10⁰	0.00×10⁰
8	7.11×10^-11	7.14×10^-11	7.20×10^-11	7.25×10^-11	7.32×10^-11	7.40×10^-11	7.48×10^-11	7.56×10^-11	7.65×10^-11	7.75×10^-11
9	0.00×10⁰	0.00×10⁰	0.00×10⁰	0.00×10⁰	0.00×10⁰	0.00×10⁰	0.00×10⁰	0.00×10⁰	0.00×10⁰	0.00×10⁰
10	0.00×10⁰	0.00×10⁰	0.00×10⁰	0.00×10⁰	0.00×10⁰	0.00×10⁰	0.00×10⁰	0.00×10⁰	0.00×10⁰	0.00×10⁰
11	7.10×10^-11	7.19×10^-11	7.19×10^-11	7.24×10^-11	7.30×10^-11	7.38×10^-11	7.45×10^-11	7.54×10^-11	7.65×10^-11	7.71×10^-11
12	0.00×10⁰	0.00×10⁰	0.00×10⁰	0.00×10⁰	0.00×10⁰	0.00×10⁰	0.00×10⁰	0.00×10⁰	0.00×10⁰	0.00×10⁰

Table XXVI Reduced Maximum Approximation Gap of Reactive Power Loss of Approximate OPF Model for Heavy Power Loads of Case9

Format	Reduced maximum approximation gap (×10^-10)
Format	110%	120%	130%	140%	150%	160%	170%	180%	190%	200%
1	3.97	4.00	4.03	4.09	4.12	4.15	4.17	4.21	4.24	4.28
2	6.03	7.06	6.66	6.48	6.99	6.71	6.69	6.70	6.73	6.78
3	3.97	4.00	4.03	4.09	4.12	4.14	4.17	4.21	4.24	4.28
4	3.99	4.04	4.01	4.16	4.19	4.22	4.26	4.30	4.34	4.38
5	6.03	8.26	6.78	6.54	6.23	6.66	6.65	6.66	6.70	6.56
6	3.99	4.04	4.01	4.16	4.19	4.22	4.26	4.30	4.34	4.38
7	3.97	4.00	4.03	4.09	4.12	4.15	4.17	4.21	4.24	4.29
8	6.21	6.05	6.62	6.43	6.99	6.67	6.64	6.66	6.69	6.73
9	3.97	4.00	4.03	4.09	4.12	4.15	4.17	4.21	4.24	4.29
10	3.99	4.04	4.01	4.16	4.19	4.22	4.26	4.30	4.34	4.38
11	6.28	8.26	6.74	6.50	6.76	6.61	6.60	6.62	6.48	6.69
12	3.99	4.04	4.01	4.16	4.19	4.22	4.26	4.30	4.34	4.38

Table XXIII Reduced Maximum Approximation Gap of Active Power Loss of Approximate OPF Model for Base Power Loads

Format	Reduced maximum approximation gap
Format	case9	IEEE14	case30	IEEE57	case89pegase	IEEE118	ACTIVSg200	IEEE300	ACTIVSg500
1	0.00×10⁰	0.00×10⁰	0.00×10⁰	0.00×10⁰	9.41×10^-10	0.00×10⁰	1.96×10^-12	3.70×10^-7	7.21×10^-14
2	3.68×10^-15	1.29×10^-10	6.60×10^-15	8.12×10^-11	2.83×10^-9	2.72×10^-12	7.77×10^-13	3.99×10^-8	5.71×10^-13
3	0.00×10⁰	0.00×10⁰	0.00×10⁰	0.00×10⁰	6.12×10^-12	0.00×10⁰	7.77×10^-13	7.34×10^-8	5.71×10^-13
4	0.00×10⁰	0.00×10⁰	0.00×10⁰	0.00×10⁰	3.11×10^-8	0.00×10⁰	1.96×10^-12	4.55×10^-7	7.17×10^-14
5	1.09×10^-14	1.30×10^-10	3.55×10^-15	7.19×10^-11	2.44×10^-11	2.72×10^-12	1.37×10^-12	6.97×10^-8	5.66×10^-13
6	0.00×10⁰	0.00×10⁰	0.00×10⁰	0.00×10⁰	6.15×10^-12	0.00×10⁰	7.80×10^-13	1.61×10^-3	5.66×10^-13
7	0.00×10⁰	0.00×10⁰	0.00×10⁰	0.00×10⁰	3.11×10^-8	0.00×10⁰	1.96×10^-12	3.70×10^-7	7.21×10^-14
8	7.08×10^-11	1.29×10^-10	1.04×10^-9	8.13×10^-11	1.16×10^-7	2.72×10^-12	1.37×10^-12	3.99×10^-8	5.71×10^-13
9	0.00×10⁰	0.00×10⁰	0.00×10⁰	0.00×10⁰	6.12×10^-12	0.00×10⁰	7.77×10^-13	7.34×10^-8	5.71×10^-13
10	0.00×10⁰	0.00×10⁰	0.00×10⁰	0.00×10⁰	1.24×10^-11	0.00×10⁰	1.96×10^-12	4.55×10^-7	7.17×10^-14
11	7.07×10^-11	1.30×10^-10	2.84×10^-9	7.19×10^-11	2.44×10^-11	2.72×10^-12	7.80×10^-13	6.97×10^-8	5.66×10^-13
12	0.00×10⁰	0.00×10⁰	0.00×10⁰	0.00×10⁰	6.15×10^-12	0.00×10⁰	7.80×10^-13	1.61×10^-3	5.66×10^-13

Table XXIV Reduced Maximum Approximation Gap of Reactive Power Loss of Approximate OPF Model for Base Power Loads

Format	Reduced maximum approximation gap
Format	case9	IEEE14	case30	IEEE57	case89pegase	IEEE118	ACTIVSg200	IEEE300	ACTIVSg500
1	3.94×10^-10	3.62×10^-10	1.04×10^-10	7.21×10^-11	8.50×10^-11	4.38×10^-11	7.05×10^-11	4.75×10^-2	1.95×10^-10
2	6.64×10^-10	4.74×10^-10	1.76×10^-10	8.00×10^-12	4.56×10^-8	3.43×10^-10	9.84×10^-12	3.09×10^-2	9.38×10^-12
3	3.95×10^-10	4.07×10^-10	1.76×10^-10	8.94×10^-11	9.61×10^-11	3.25×10^-11	9.84×10^-12	7.53×10^-7	9.38×10^-12
4	3.94×10^-10	3.56×10^-10	1.05×10^-10	3.09×10^-2	1.86×10^-7	4.39×10^-11	7.06×10^-11	8.96×10^-2	1.84×10^-10
5	5.74×10^-10	4.42×10^-10	1.77×10^-10	5.00×10^-2	2.10×10^-9	2.02×10^-9	2.07×10^-11	3.04×10^-2	9.44×10^-12
6	3.94×10^-10	4.11×10^-10	1.77×10^-10	3.09×10^-2	9.58×10^-11	3.24×10^-11	9.87×10^-12	4.60×10^-2	9.44×10^-12
7	3.94×10^-10	3.62×10^-10	1.04×10^-10	7.21×10^-11	1.13×10^-7	4.38×10^-11	7.05×10^-11	4.75×10^-2	1.95×10^-10
8	6.60×10^-10	4.74×10^-10	4.03×10^-9	8.00×10^-12	8.36×10^-7	3.43×10^-10	2.07×10^-11	3.09×10^-2	9.38×10^-12
9	3.95×10^-10	4.07×10^-10	1.76×10^-10	8.94×10^-11	9.61×10^-11	3.25×10^-11	9.84×10^-12	7.53×10^-7	9.38×10^-12
10	3.94×10^-10	3.56×10^-10	1.05×10^-10	3.09×10^-2	3.31×10^-11	4.39×10^-11	7.06×10^-11	8.96×10^-2	1.84×10^-10
11	5.72×10^-10	4.42×10^-10	1.14×10^-8	5.00×10^-2	2.10×10^-9	2.02×10^-9	9.87×10^-12	3.04×10^-2	9.44×10^-12
12	3.94×10^-10	4.11×10^-10	1.77×10^-10	3.09×10^-2	9.62×10^-11	3.24×10^-11	9.87×10^-12	4.60×10^-2	9.44×10^-12

Table XXVII Reduced Maximum Approximation Gap of Active Power Loss of Approximate OPF Model for Heavy Power Loads of IEEE118

Format	Reduced maximum approximation gap
Format	110%	120%	130%	140%	150%	160%	170%	180%	190%	200%
1	0.00×10⁰	0.00×10⁰	0.00×10⁰	0.00×10⁰	0.00×10⁰	0.00×10⁰	0.00×10⁰	0.00×10⁰	0.00×10⁰	0.00×10⁰
2	2.73×10^-12	2.73×10^-12	2.73×10^-12	2.74×10^-12	2.75×10^-12	2.76×10^-12	3.12×10^-12	2.78×10^-12	2.80×10^-12	2.80×10^-12
3	0.00×10⁰	0.00×10⁰	0.00×10⁰	0.00×10⁰	0.00×10⁰	0.00×10⁰	0.00×10⁰	0.00×10⁰	0.00×10⁰	0.00×10⁰
4	0.00×10⁰	0.00×10⁰	0.00×10⁰	0.00×10⁰	0.00×10⁰	0.00×10⁰	0.00×10⁰	0.00×10⁰	0.00×10⁰	0.00×10⁰
5	2.72×10^-12	1.85×10^-11	2.73×10^-12	2.74×10^-12	2.75×10^-12	2.76×10^-12	2.76×10^-12	2.75×10^-12	2.74×10^-12	2.71×10^-12
6	0.00×10⁰	0.00×10⁰	0.00×10⁰	0.00×10⁰	0.00×10⁰	0.00×10⁰	0.00×10⁰	0.00×10⁰	0.00×10⁰	0.00×10⁰
7	0.00×10⁰	0.00×10⁰	0.00×10⁰	0.00×10⁰	0.00×10⁰	0.00×10⁰	0.00×10⁰	0.00×10⁰	0.00×10⁰	0.00×10⁰
8	2.73×10^-12	2.73×10^-12	2.73×10^-12	2.74×10^-12	2.75×10^-12	2.76×10^-12	3.12×10^-12	2.78×10^-12	2.80×10^-12	2.80×10^-12
9	0.00×10⁰	0.00×10⁰	0.00×10⁰	0.00×10⁰	0.00×10⁰	0.00×10⁰	0.00×10⁰	0.00×10⁰	0.00×10⁰	0.00×10⁰
10	0.00×10⁰	0.00×10⁰	0.00×10⁰	0.00×10⁰	0.00×10⁰	0.00×10⁰	0.00×10⁰	0.00×10⁰	0.00×10⁰	0.00×10⁰
11	2.72×10^-12	1.85×10^-11	2.73×10^-12	2.74×10^-12	2.75×10^-12	2.76×10^-12	2.76×10^-12	2.75×10^-12	2.74×10^-12	2.71×10^-12
12	0.00×10⁰	0.00×10⁰	0.00×10⁰	0.00×10⁰	0.00×10⁰	0.00×10⁰	0.00×10⁰	0.00×10⁰	0.00×10⁰	0.00×10⁰

Table XXVIII

;

Reduced Maximum Approximation Gap of Reactive Power Loss of Approximate OPF Model for Heavy Power Loads of IEEE118

Format	Reduced maximum approximation gap
Format	110%	120%	130%	140%	150%	160%	170%	180%	190%	200%
1	4.39×10^-11	4.39×10^-11	4.39×10^-11	4.39×10^-11	4.39×10^-11	4.39×10^-11	4.39×10^-11	4.39×10^-11	1.47×10^-10	3.11×10^-10
2	3.25×10^-11	3.25×10^-11	1.04×10^-8	1.11×10^-9	6.88×10^-9	4.66×10^-8	3.72×10^-11	1.05×10^-7	3.64×10^-8	5.65×10^-2
3	3.71×10^-11	3.71×10^-11	3.27×10^-11	3.28×10^-11	3.29×10^-11	3.30×10^-11	3.30×10^-11	3.32×10^-11	1.48×10^-10	2.96×10^-10
4	4.40×10^-11	4.40×10^-11	4.40×10^-11	4.40×10^-11	4.40×10^-11	4.40×10^-11	1.47×10^-10	5.10×10^-9	2.52×10^-1	5.25×10^-1
5	3.24×10^-11	1.65×10^-11	3.25×10^-11	5.17×10^-9	3.27×10^-11	1.38×10^-11	5.42×10^-9	3.28×10^-11	3.37×10^-1	1.02×10⁰
6	3.70×10^-11	3.25×10^-11	3.26×10^-11	3.27×10^-11	3.27×10^-11	3.28×10^-11	1.42×10^-10	5.10×10^-9	4.61×10^-1	1.01×10⁰
7	4.39×10^-11	4.39×10^-11	4.39×10^-11	4.39×10^-11	4.39×10^-11	4.39×10^-11	4.39×10^-11	4.39×10^-11	1.47×10^-10	3.11×10^-10
8	4.25×10^-11	3.25×10^-11	1.04×10^-8	1.11×10^-9	6.88×10^-9	4.66×10^-8	3.72×10^-11	1.05×10^-7	3.64×10^-8	5.65×10^-2
9	3.71×10^-11	3.71×10^-11	3.27×10^-11	3.28×10^-11	3.29×10^-11	3.30×10^-11	3.30×10^-11	3.32×10^-11	1.48×10^-10	2.96×10^-10
10	4.40×10^-11	4.40×10^-11	4.40×10^-11	4.40×10^-11	4.40×10^-11	4.40×10^-11	1.47×10^-10	5.10×10^-9	2.52×10^-1	5.25×10^-1
11	3.24×10^-11	1.65×10^-11	3.25×10^-11	5.17×10^-9	3.27×10^-11	1.38×10^-11	5.42×10^-9	3.28×10^-11	3.37×10^-1	1.02×10⁰
12	3.70×10^-11	3.25×10^-11	3.26×10^-11	3.27×10^-11	3.27×10^-11	3.28×10^-11	1.42×10^-10	4.10×10^-9	4.61×10^-1	1.01×10⁰

V. CONCLUSION

This paper formulates six formats of the exact General-BranchFlow model and six formats of the approximate General-BranchFlow model. All the formats are valid for both radial and mesh power networks. By taking into account the shunt conductive components of the transmission line $Π$ -model, a linear amapacity constraint is derived. Based on the formulated General-BranchFlow model and the ampacity constraint, twelve formats of the exact OPF model and twelve formats of the approximate OPF model are proposed. Providing different formats of the OPF models gives the power system operators more options in solving the network operation problems when using one format may face numerical difficulties or parameter non-availability problems. The reason of proposing the approximate OPF model is to give a convex formulation which advantages in finding the globally optimal solution using available optimization solvers.

The numerical investigations using various IEEE test cases for different power loads prove the accuracy and computational efficiency of all the twelve formats of the exact and approximate OPF models. For the twelve formats of the approximate OPF model, non-negligible reactive power loss approximation gaps exist for some test cases if the IPOPT solver is used. To reduce the approximation gaps, this paper proposes to use the penalty function in the objective function of the OPF model. The numerical results show that the approximation gaps of reactive power loss are largely reduced. In this way, the AC-feasibility of the approximate General-BranchFlow model is enhanced.

NOMENCLATURE

Symbol	——	Definition
A.	——	Sets
$𝒩, ℒ$	——	Sets of nodes (or buses) and lines (or branches)
$𝒞, Ω$	——	Sets of cycles (or closed loops) and decision variables
$𝒩_{n}$	——	Set of nodes with lines directly connected to node $n$
B.	——	Indices
$n (n^{'}), l$	——	Indices of nodes and lines
C.	——	Variables
$θ_{n}$	——	Phase-to-ground voltage phase angle at node $n$
$θ_{l}$	——	Phase angle difference between the sending-end and the receiving-end voltage of branch $l$
$θ_{s_{l}}$ , $θ_{r_{l}}$	——	Sending-end and receiving-end voltage phase angles of branch l
$θ_{n n^{'}}$	——	Phase angle difference between nodes $n$ and $n^{'}$
$f$	——	Original objective function
$f^{'}$	——	Modified objective function
$K_{o_{l}}^{p}, K_{o_{l}}^{q}$	——	Equivalent ampacity constraints for active and reactive power losses
$p_{n}, q_{n}$	——	Active and reactive power generations at node $n$
$p_{d_{n}}, q_{d_{n}}$	——	Active and reactive power demands at node $n$
$p_{g s_{l}}, q_{b s_{l}}$	——	Sending-end shunt conductance active power and capacitance reactive power
$p_{g r_{l}}, q_{b r_{l}}$	——	Receiving-end shunt conductance active power and capacitance reactive power
$p_{s_{l}}, q_{s_{l}}$	——	Non-measurable sending-end active and reactive power flows for branch $l$
${\tilde{p}}_{s_{l}}, {\tilde{q}}_{s_{l}}$	——	Measurable sending-end active and reactive power flows for branch $l$
$p_{r_{l}}, q_{r_{l}}$	——	Non-measurable receiving-end active and reactive power injections of branch $l$
${\tilde{p}}_{r_{l}}, {\tilde{q}}_{r_{l}}$	——	Measurable receiving-end active and reactive power injections of branch $l$
$v_{n}, V_{n}$	——	Phase-to-ground voltage magnitude and voltage square at node $n$
$v_{s_{l}}, v_{r_{l}}$	——	Sending-end and receiving-end phase-to-ground voltages of branch $l$
$V_{s_{l}}, V_{r_{l}}$	——	Sending-end and receiving-end phase-to-ground voltage squares of branch $l$
D.	——	Parameters
$α_{n}, β_{n}, γ_{n}$	——	Cost parameters of active power generation
$θ_{n}^{m i n}, θ_{n}^{m a x}$	——	Lower and upper bounds of $θ_{n}$
$θ_{l}^{m i n}, θ_{l}^{m a x}$	——	Lower and upper bounds of $θ_{l}$
$θ_{n n^{'}}^{m i n}, θ_{n n^{'}}^{m a x}$	——	Lower and upper bounds of $θ_{n n^{'}}$
$ξ$	——	Penalty coefficient
$A_{n l}^{+}, A_{n l}^{-}$	——	Node-to-line incidence matrices
$B_{s_{l}}, B_{r_{l}}$	——	Sending-end and receiving-end shunt susceptances of branch $l$
$G_{n}, B_{n}$	——	Shunt conductance and susceptance of node $n$
$G_{n n^{'}}, B_{n n^{'}}$	——	Real and imaginary parts of the element in the bus admittance matrix corresponding to the $n$ ^th row and ${n'}^{t h}$ column
$G_{s_{l}}$	——	Sending-end shunt conductance of branch l
${\tilde{K}}_{l}, K_{l}$	——	Actual and approximate ampacities of branch $l$
$p_{d_{n}}, q_{d_{n}}$	——	Active and reactive power loads of node $n$
$p_{n}^{m i n}, p_{n}^{m a x}$	——	Lower and upper bounds of $p_{n}$
$q_{n}^{m i n}, q_{n}^{m a x}$	——	Lower and upper bounds of $q_{n}$
$v_{n}^{m i n}, v_{n}^{m a x}$	——	Lower and upper bounds of $v_{n}$
$X_{l}, R_{l}$	——	Longitudinal reactance and resistance of branch $l$ modelled as a passive $Π$ -model

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Introduction

Editorial Board

For Author

Call For Papers

APC

Sponsor & Publisher

Formulations and Approximations of Branch Flow Model for General Power Networks PDF

Abstract

Keywords

I. INTRODUCTION

II. GENERAL-BRANCHFLOW MODEL

A.　Exact Model

B. Approximate Model

C. Branch Ampacity Constraint

III. OPF Problem

A.　Base Power Loads

B.　Heavy Power Loads

IV. APPROXIMATIOn GAP REDUCTION

A.　Base Power Loads

B.　Heavy Power Loads

V. CONCLUSION

NOMENCLATURE

REFERENCES

Home

Introduction

Editorial Board

For Author

Call For Papers

APC

Sponsor & Publisher

Formulations and Approximations of Branch Flow Model for General Power Networks PDF

Abstract

Keywords

I. INTRODUCTION

II. GENERAL-BRANCHFLOW MODEL

A. Exact Model

B. Approximate Model

C. Branch Ampacity Constraint

III. OPF Problem

A. Base Power Loads

B. Heavy Power Loads

IV. APPROXIMATIOn GAP REDUCTION

A. Base Power Loads

B. Heavy Power Loads

V. CONCLUSION

NOMENCLATURE

REFERENCES

A.　Exact Model

A.　Base Power Loads

B.　Heavy Power Loads

A.　Base Power Loads

B.　Heavy Power Loads