Optimal Reactive Power Dispatch under Transmission and Distribution Coordination Based on an Accelerated Augmented Lagrangian Algorithm

Abstract

As many distributed power sources flood into the distribution network, the relationship between transmission and distribution grids in reactive power and voltage is becoming closer and closer. The traditional way of independent reactive power optimization in transmission and distribution grids is no longer appropriate. In this study, a collaborative and distributed reactive power optimization method for transmission and distribution grids based on the accelerated augmented Lagrangian (AAL) algorithm is proposed to adapt to the independence of the transmission and distribution grids in operation and management. The global reactive power optimization problem is decomposed into the transmission network subproblem and several distribution network subproblems. According to AAL, subproblems are solved in a distributed manner until the optimal global solution is finally reached after several iterations, and coordination between transmission and distribution grids is achieved only through the interaction of information on coordinating variables. For better convergence, a linearized and convergence-guaranteed optimal power flow model (OPF) with reactive power and voltage magnitude was applied to model the transmission grid optimization subproblem, while the second-order cone programming (SOCP) technique is used in the distribution network subproblems. The simulation results confirm that the method in this paper can effectively reduce network losses and achieve better economic performance, and converges better when compared to other algorithms.

1. Introduction

With the rapid development of distributed energy, distribution networks are gradually evolving into active distribution networks, and the reactive power characteristic has also changed [1]. Traditionally, the transmission and distribution network′s optimal reactive power dispatch (ORPD) is carried out separately. The lower distribution networks remain unchanged when optimizing the transmission network, and the voltage of the boundary buses are used as the reference when optimizing the distribution networks, making it prone to overvoltage problems and cascading blackouts of distributed generators when there is a large-scale influx of distributed generators into distribution networks [2]. Therefore, it is necessary to give full play to the voltage support role of the transmission network and make full use of the reactive power regulation capability of the distribution network, which helps active distribution networks to eliminate overvoltage problems and reduce network losses [3,4].

Theoretically, the centralized optimization method could be used to establish a global optimal dispatch model to realize the coordinated optimal dispatch of transmission and distribution networks [5]. However, for the existing power system, transmission and distribution networks are under the jurisdiction of different dispatching agencies, which makes it hard for centralized optimization to deal with the confidentiality of information among various stakeholders and the independence of transmission and distribution networks in the process of operation and management. On the other hand, the centralized optimization method must concentrate the data of the global system to the dispatch center for calculation. However, aggregating data from multiple distribution networks will dramatically increase the complexity and computational cost of optimization problems. Therefore, it is reasonable to use a distributed optimization method when carrying out the optimal dispatch under transmission and distribution network coordination [6].

Recently, several researches have been conducted on distributed optimization methods for coordinated transmission and distribution networks. In [7], a reactive power optimization model considering coordinated transmission and distribution networks was established and solved by the generalized Benders decomposition method. However, this approach requires the subproblems to be convex to obtain the efficient Benders cut. However, shunt capacitors (SCs) and on-load transformers (OLTCs) in distribution networks are discrete control variables, which poses a challenge to the Benders decomposition method. In [8,9], the heterogeneous decomposition (HGD) was applied to the economic scheduling problem of coordinated transmission and distribution networks. However, the HGD may lead to local optima and is unsuitable for topologically complex networks, especially when there are connections between different distribution networks [10]. As a commonly used distributed optimization algorithm in optimization problems, the alternating direction multiplier (ADMM) has been widely used in multi-region network operation problems, including distributed optimal power flow and fully distributed reactive power optimization [11,12]. In [13], ADMM was applied to the reactive power optimization problem of multiple partition coordination in active distribution networks, and the results proved that the ADMM is an effective method for dealing with collaborative optimization problems. By improving the ADMM algorithm, the accelerated augmented Lagrangian algorithm (AAL) can obtain a faster convergence speed when dealing with collaborative optimization problems [14]. In [15], the AAL was applied to the distributed restoration problem of an integrated power transmission and distribution system with distributed energy sources. In [16], a framework was established to deal with the reactive power optimization of transmission and distribution networks coordination through the method of curve fitting, but the method cannot take into account the role of DGs. In addition, methods such as multi-parameter planning and the principle of auxiliary problems were also used to optimize transmission and distribution networks coordination problems [17,18].

The optimization problem of transmission and distribution networks can be formulated by nonlinear AC-OPF, DC-OPF, or other forms, such as mixed-integer linear programming, second-order cone programming (SOCP), and semidefinite programming. In [19], the DC-OPF was used for the transmission network optimization problem, and the nonlinear AC-OPF was used for the distribution network optimization problem, thereby improving the calculation speed. However, since the variables related to reactive power are not modeled in the traditional DC-OPF, it cannot be used to deal with the optimal reactive power dispatch problem. For that issue, a novel linearized OPF model with reactive power (Q) and voltage magnitude (V) was established to achieve the better performance of the DC-OPF models [20]. By applying the novel linear approximation method to the transmission network subproblem model, the complexity of the transmission network subproblem and the global optimization problem can be significantly reduced. In addition, the effectiveness of SOCP for radial networks has been demonstrated in the related literature, and it is often used in the optimization of distribution networks [21].

To give full play to the synergy between the transmission network and the distribution network, a coordinated transmission and distribution optimal reactive power dispatch (CTD-ORPD) framework is established in this paper. The main contributions are listed as follows:

(1)

A fully distributed framework based on the AAL algorism is constructed for CTD-ORPD, through coordination between transmission and distribution grids to achieve the common goals of minimizing network losses and maximizing economic benefits, which performs better in terms of economy and safety when compared to the traditional independent optimization method of distribution grids [22].

(2)

In the proposed CTD-ORPD framework, the global reactive power optimization problem is decomposed into the transmission network subproblem and several distribution network subproblems. Based on AAL, the subproblems are solved in a distributed manner until the optimal global solution is finally reached after several iterations, and coordination between transmission and distribution grids is achieved only through the interaction of information on coordinating variables. By adopting a distributed approach, the issues of data privacy, cybersecurity, and the computing and communication of independent system operators are reasonably addressed. Additionally, the AAL-based CTD-ORPD framework proposed in this paper is proved to have better convergence performance and solving efficiency when compared to those co-optimization methods using HGD [9] or ADMM [13].

(3)

For a better convergence performance and solving efficiency of the established distributed optimization model, a novel linearized OPF and the SOCP technique are applied to the modeling of transmission network subproblem and distribution network subproblems, respectively.

The rest of this paper is organized as follows: In Section 2, the framework of the CTD-ORPD is introduced, including the separate modeling of transmission network and distribution network subproblems, as well as the novel linearized OPF and SOCP applied to the modeling of the transmission network and distribution networks, respectively. In Section 3, the distributed solving algorithm of the CTD-ORPD is detailed, and the algorithm flowchart is given. Section 3 establishes three test cases to verify the validity of the proposed CTD-ORPD and compares the performance of the proposed method in this paper with other methods. Finally, Section 4 presents our conclusions.

2. Framework of the CTD-ORPD

In this section, the CTD-ORPD framework based on AAL is introduced. In the proposed CTD-ORPD framework, the global reactive power optimization problem is decomposed into the transmission network subproblem and several distribution network subproblems. Based on AAL, the subproblems are solved in a distributed manner until the optimal global solution is finally reached after several iterations, and coordination between transmission and distribution grids is achieved through the interaction of information on coordinating variables. Coordinating variables are the variables of the points of common coupling (PCCs) between the transmission network and the distribution networks, including bus voltage magnitude and phase angle, active power output and reactive power output.

In the transmission network subproblem, control variables such as SCs, OLTCs, and reactive power output of generators need to be optimized. A novel linearized approximation method is applied to handle the nonlinearity of the power flow equations of the transmission network for better convergence [20]. In the distribution network subproblem, control variables such as SCs, OLTCs, and reactive power output of various distributed power sources need to be optimized. Additionally, SOCP method is applied to the convex modeling of the distribution network subproblems. The framework of the established CTD-ORPD is illustrated in Figure 1.

2.1. Linear Approximation of Power Flow Equations

A novel linear approximation of power flow equations to construct the linearized OPF model of the transmission subproblem is detailed in this part, which is the foundation for formulating the proposed CTD-ORPD model.

Taking advantage of the power flow equations in the polar coordination, the expression for power flows on branch i-j are as follows:

$$P_{ij}=\left(v_i^2-v_i{v}_j\cos {\theta }_{ij}\right)g_{ij}-v_i{v}_j{b}_{ij}\sin {\theta }_{ij}$$

(1)

$$Q_{ij}=-\left(v_i^2-v_i{v}_j\cos {\theta }_{ij}\right)b_{ij}-v_i{v}_j{g}_{ij}\sin {\theta }_{ij}$$

(2)

where $P_{ij}$ and $Q_{ij}$ are the active and reactive power passing on branch i-j, respectively; $v_i$ and $v_j$ are the voltage magnitudes of bus i and bus j, respectively; and $g_{ij}$ and $b_{ij}$ are conductance and susceptance of branch i-j. Given the initial values of voltage magnitudes and phase angles $(v_0,{\theta }_0)$, the trigonometric function in the above formula can be linearized and approximated by the first-order Taylor series as follows:

$$\sin {\theta }_{ij}\approx s_{ij}^1{\theta }_{ij}+s_{ij}^0$$

(3)

$$\cos {\theta }_{ij}\approx c_{ij}^1{\theta }_{ij}+c_{ij}^0$$

(4)

$$s_{ij}^1=\cos {\theta }_{ij,0},s_{ij}^0=\sin {\theta }_{ij,0}-{\theta }_{ij,0}\cos {\theta }_{ij,0}$$

(5)

$$c_{ij}^1=-\sin {\theta }_{ij,0},c_{ij}^0=\cos {\theta }_{ij,0}+{\theta }_{ij,0}\sin {\theta }_{ij,0}$$

(6)

The initial value of the voltage phase angle is very close to the final value after iteration due to the quasi-linear relationship between the active power and the voltage phase angle in the power system [23]. Therefore, Equations (3) and (4) have a good accuracy. Substituting Equations (3) and (4) into Equations (1) and (2), we can obtain:

$$P_{ij}=v_i^2{g}_{ij}-v_i{v}_j\left(g_{ij}{c}_{ij}^0+b_{ij}{s}_{ij}^0\right)-v_i{v}_j{\theta }_{ij}\left(g_{ij}{c}_{ij}^1+b_{ij}{s}_{ij}^1\right)$$

(7)

$$Q_{ij}=-v_i^2{b}_{ij}+v_i{v}_j\left(-g_{ij}{s}_{ij}^0+b_{ij}{c}_{ij}^0\right)-v_i{v}_j{\theta }_{ij}\left(g_{ij}{s}_{ij}^1-b_{ij}{c}_{ij}^1\right)$$

(8)

Considering $v_i{v}_j$ as a whole, use the first-order Taylor series expansion to decouple the variable $v_i{v}_j$ from the variable ${\theta }_{ij}$ in the expression $v_i{v}_j{\theta }_{ij}$:

$$v_i{v}_j{\theta }_{ij}\approx v_{i,0}{v}_{j,0}{\theta }_{ij}+\left(v_i{v}_j-v_{i,0}{v}_{j,0}\right){\theta }_{ij,0}$$

(9)

By substituting Equation (9) into Equations (7) and (8), the following expressions can be obtained:

$$P_{ij}=g_{ij}{v}_i^2-g_{ij}^{\mathrm{P}}{v}_i{v}_j-b_{ij}^{\mathrm{P}}\left({\theta }_{ij}-{\theta }_{ij,0}\right)$$

(10)

$$Q_{ij}=-b_{ij}{v}_i^2+b_{ij}^{\mathrm{Q}}{v}_i{v}_j-g_{ij}^{\mathrm{Q}}\left({\theta }_{ij}-{\theta }_{ij,0}\right)$$

(11)

where $g_{ij}^{\mathrm{P}}$, $b_{ij}^{\mathrm{P}}$, $g_{ij}^{\mathrm{Q}}$, and $b_{ij}^{\mathrm{Q}}$ are the active equivalent conductance, active equivalent susceptance, reactive equivalent conductance, and reactive equivalent susceptance of line i-j, respectively, and their expressions are as follows:

$$g_{ij}^{\mathrm{P}}=\left(g_{ij}{c}_{ij}^0+b_{ij}{s}_{ij}^0\right)+\left(g_{ij}{c}_{ij}^1+b_{ij}{s}_{ij}^1\right){\theta }_{ij,0}$$

(12)

$$b_{ij}^{\mathrm{P}}=\left(g_{ij}{c}_{ij}^1+b_{ij}{s}_{ij}^1\right)v_{i,0}{v}_{j,0}$$

(13)

$$g_{ij}^{\mathrm{Q}}=\left(g_{ij}{s}_{ij}^1-b_{ij}{c}_{ij}^1\right)v_{i,0}{v}_{j,0}$$

(14)

$$b_{ij}^{\mathrm{Q}}=\left(-g_{ij}{s}_{ij}^0+b_{ij}{c}_{ij}^0\right)-\left(g_{ij}{s}_{ij}^1-b_{ij}{c}_{ij}^1\right){\theta }_{ij,0}$$

(15)

There is still a nonlinear variable $v_i{v}_j$ in Equations (10) and (11), respectively. In order to eliminate the nonlinearity of the model, the following mathematical transformations are performed on $v_i{v}_j$:

$$v_i{v}_j=\frac{1}{2}\left[v_i^2+v_j^2-{\left(v_i-v_j\right)}^2\right]=\frac{v_i^2+v_j^2}{2}-\frac{v_{ij}^2}{2}$$

(16)

The expression $\left(v_i^2+v_j^2\right)/2$ is linear when considering $v^2$ as an independent variable. In practical power systems, the values of $v_{ij}^2$ are very small and often negligible. Thus, $\left(v_i^2+v_j^2\right)/2$ can be used as an approximation of $v_i{v}_j$, the approximation error would be small and not affected by the initial value of the voltage magnitude [24]. For keeping the error as low as possible, the following approximation is made to $v_{ij}^2$ in Equation (16) based on a Taylor series expansion:

$$v_{ij}^2\approx 2v_{ij,0}{v}_{ij}-v_{ij,0}^2\approx 2v_{ij,0}{v}_{ij}\frac{v_i+v_j}{v_{i,0}+v_{j,0}}-v_{ij,0}^2=2\frac{v_{i,0}-v_{j,0}}{v_{i,0}+v_{j,0}}\left(v_i^2-v_j^2\right)-v_{ij,0}^2=v_{ij,\mathrm{L}}^{\mathrm{s}}$$

(17)

where $v_{ij,\mathrm{L}}^{\mathrm{s}}$ is an approximation of $v_{ij}^2$. Equation (17) are linear when considered $v^2$ as an independent variable. Since $v_{ij}^2$ is non-negativity, the following non-negativity constraint should be applied:

$$v_{ij,L}^s+{\epsilon }_{ij}\ge 0,{\epsilon }_{ij}\ge 0$$

(18)

By substituting Equations (16) and (17) into Equations (10) and (11), the linearized approximation of the power flow equation is obtained:

$$P_{ij}^{\mathrm{L}}=g_{ij}{v}_i^2-g_{ij}^{\mathrm{P}}\frac{v_i^2+v_j^2}{2}-b_{ij}^{\mathrm{P}}\left({\theta }_{ij}-{\theta }_{ij,0}\right)+g_{ij}^{\mathrm{P}}\frac{v_{ij,\mathrm{L}}^{\mathrm{s}}}{2}$$

(19)

$$Q_{ij}^{\mathrm{L}}=-b_{ij}{v}_i^2+b_{ij}^{\mathrm{Q}}\frac{v_i^2+v_j^2}{2}-g_{ij}^{\mathrm{Q}}\left({\theta }_{ij}-{\theta }_{ij,0}\right)-b_{ij}^{\mathrm{Q}}\frac{v_{ij,\mathrm{L}}^{\mathrm{s}}}{2}$$

(20)

In Equations (19) and (20), the active and reactive components of the power flow are both linear functions when considering $v^2$ as an independent variable.

The following are the expressions of the nodal injection power equations:

$$P_i={\displaystyle \sum }_{j=1}^{\mathrm{N}}\left(v_i{v}_j{G}_{ij}\cos {\theta }_{ij}+v_i{v}_j{B}_{ij}\sin {\theta }_{ij}\right)$$

(21)

$$Q_i=-{\displaystyle \sum }_{j=1}^{\mathrm{N}}\left(v_i{v}_j{B}_{ij}\cos {\theta }_{ij}-v_i{v}_j{G}_{ij}\sin {\theta }_{ij}\right)$$

(22)

where $G_{ij}$ and $B_{ij}$ are the real and imaginary parts of the node admittance matrix, respectively. Equations (21) and (22) can be converted into the following forms:

$$P_i={\displaystyle \sum }_{(i,j)}{P}_{ij}+\left({\displaystyle \sum }_{j=1}^{\mathrm{N}}{G}_{ij}\right)v_i^2$$

(23)

$$Q_i={\displaystyle \sum }_{(i,j)}{Q}_{ij}+\left({\displaystyle \sum }_{j=1}^N-B_{ij}\right)v_i^2$$

(24)

Finally, the linearized power flow equation constraints are obtained, which are composed of Equations (19), (20), (23), and (24). They are all linear functions when $v^2$ is considered as an independent variable.

2.2. Transmission Network Subproblem Model

In this part, the linearized OPF model of the transmission network subproblem is established based on the above linear approximation of power flow equations.

In traditional optimal reactive power dispatch, minimizing network losses is often applied as the objective function, where network losses are functions of the voltage magnitude and the voltage phase angle [25]. However, minimizing network losses without considering generator costs may be in conflict with economic principles. Thus, minimizing active power injections at the root bus is also usually used as the objective function of the optimal reactive power dispatch problem [26]. In this paper, minimizing the operating cost was adopted as the objective function of ORPD of the transmission network subproblem, and its expression is as follows:

$$\min F_{\mathrm{T}}={\displaystyle \sum }_{g\in G}C(P_g)$$

(25)

2.2.1. The Cost of Coal Consumption

According to the characteristics of coal consumption, a convex quadratic function is used as the generation cost of generator g [27]. By default, the unit commitment is already achieved, so the binary variables indicating the on/off status of the unit are not included in the equation.

$$C(P_g)=\left(c_{g,2}^{\mathrm{P}}{P}_g^2+c_{g,1}^{\mathrm{P}}{P}_g+c_{g,0}^{\mathrm{P}}\right)$$

(26)

where $c_{g,2}^{\mathrm{P}}$, $c_{g,1}^{\mathrm{P}}$, and $c_{g,0}^{\mathrm{P}}$ are the quadratic term coefficient, primary term coefficient, and constant term of the quadratic cost curve of the generator g, respectively. $P_g$ denotes the active power output of the generator g.

For better convergence, a segmented linearization technique is applied to the linearization of the generator cost function. Figure 2 shows the linearized cost curve of generator g. We equally divided $[P_g^{\min },P_g^{\max }]$, the power output range of generator g, into m_g segments, and the length of each segment is denoted as $P{I}_{g,1}$, $P{I}_{g,2}$, …, $P{I}_{g,m_g}$; the slope is denoted as $K{I}_{g,1}$, $K{I}_{g,2}$, …, $K{I}_{g,m_g}$, respectively. Since $C(P_g)$ is a convex function, then we have $K{I}_{g,1}<K{I}_{g,2}<\dots <K{I}_{g,m_g}$ and the power of the later segments can only be used after the power of the previous segments has been used to the upper limit. Denoting the power at the segmented points as $P_g^{\min }$, $P_{g,1}^{}$, $P_{g,2}^{}$, …, $P_{g,m_g-1}^{}$, $P_g^{\max }$, then the linearized operating cost and power output can be expressed as follows:

$$C(P_g)=C(P_g^{\min })+{\displaystyle \sum _{k=1}^{m_g}P{I}_{g,k}\times K{I}_{g,k}}$$

(27)

$$P_g=P_g^{\min }+{\displaystyle \sum _{k=1}^{m_g}P{I}_{g,k}}$$

(28)

$$0⩽P{I}_{g,1}⩽P_{g,1}-P_g^{\min },0⩽P{I}_{g,2}⩽P_{g,2}-P_{g,1},\dots ,0⩽P{I}_{g,m_g}⩽P_g^{\max }-P_{g,m_g-1}$$

(29)

When the initial values of the voltage magnitude and phase angle $(v_0,{\theta }_0)$ are given, according to the description in Section 2.1, the constraints of the transmission network subproblem are constructed as follows:

2.2.2. Nodal Injection Power Balancing Constraints

For all the nodes, the following constraints need to be satisfied:

$$P_i^k={\displaystyle \sum }_{g\in i}{P}_g-P_{i,d}={\displaystyle \sum }_{(i,j)}{P}_{ij}^k+\left({\displaystyle \sum }_{j=1}^N{G}_{ij}\right)V_i$$

(30)

$$Q_i^k={\displaystyle \sum }_{g\in i}{Q}_g-Q_{i,d}={\displaystyle \sum }_{(i,j)}{Q}_{ij}^k+\left({\displaystyle \sum }_{j=1}^N-B_{ij}\right)V_i+Q_{i,\mathrm{sc}}$$

(31)

where $P_{i,d}$ and $Q_{i,d}$ are the active and reactive loads at bus i; $G_{ij}$ and $B_{ij}$ are the real and imaginary parts of the node admittance matrix, respectively; $Q_{i,\mathrm{sc}}^{}$ is the reactive power output of SCs at bus i.; $V_i=v_i^2$ represents the square of voltage magnitude at node i, and it is used as an independent variable in this paper.

2.2.3. Branch Power Balancing Constraints

Equations (19) and (20) are the branch power balancing constraints for those branches without OLTCs. For those branches with OLTCs, the branch power balancing constraints are as follows:

$$P_{ij}^k=g_{ij}{V}_{i,\mathrm{OLTC}}^{}-g_{ij}^{\mathrm{P}}\frac{V_{i,\mathrm{OLTC}}^{}+V_j^{}}{2}-b_{ij}^{\mathrm{P}}\left({\theta }_{ij}-{\theta }_{ij,0}\right)+g_{ij}^{\mathrm{P}}\frac{V_{ij,\mathrm{L}}^{\mathrm{OLTC}}}{2}$$

(32)

$$Q_{ij}^k=-b_{ij}{V}_{i,\mathrm{OLTC}}^{}+b_{ij}^{\mathrm{Q}}\frac{V_{i,\mathrm{OLTC}}^{}+V_j^{}}{2}-g_{ij}^{\mathrm{Q}}\left({\theta }_{ij}-{\theta }_{ij,0}\right)-b_{ij}^{\mathrm{Q}}\frac{V_{ij,\mathrm{L}}^{\mathrm{OLTC}}}{2}$$

(33)

$$V_{ij,\mathrm{L}}^{\mathrm{OLTC}}=2\frac{v_{i,\mathrm{OLTC},0}-v_{j,0}}{v_{i,\mathrm{OLTC},0}+v_{j,0}}\left(V_{i,\mathrm{OLTC}}^{}-V_j^2\right)-{\left(V_{i,\mathrm{OLTC},0}-V_{j,0}\right)}^2$$

(34)

2.2.4. Constraints of OLTCs

Denote $K\in \left\{K_0,K_1,K_2,\dots ,K_n\right\}$ as the optional status of OLTCs. When the operation status of OLTCs is $K_m$, the following constraint should be satisfied:

$$V_i^{}=K_{}^2{V}_{i,\mathrm{OLTC}}^{}=K_0^2{V}_{i,\mathrm{OLTC}}^{}+{\displaystyle \sum _{m=1}^t(K_m^2-K_{m-1}^2)}{V}_{i,\mathrm{OLTC}}^{}$$

(35)

However, $K_t^2{V}_{i,\mathrm{OLTC}}^{}$ in the above equation contains a nonlinear term, which is detrimental to the solution of the model. A set of 0–1 variables $\left\{z_1,z_2,\dots ,z_{\mathrm{n}}\right\}$ is introduced to linearize it.

$$V_i=K_0^2{V}_{i,\mathrm{OLTC}}^{}+{\displaystyle \sum _{m=1}^{\mathrm{n}}\Delta }{V}_m$$

(36)

$$0\le \Delta V_m\le z_m{V}_{\max }\left(K_m^2-K_{m-1}^2\right)$$

(37)

$$\Delta V_m^{}\le \left(K_m^2-K_{m-1}^2\right)V_{i,\mathrm{OLTC}}^{}$$

(38)

$$\Delta V_m^{}\ge \left(K_m^2-K_{m-1}^2\right)V_{i,\mathrm{OLTC}}^{}-\left(1-z_m\right)V_{\max }\left(K_m^2-K_{m-1}^2\right)$$

(39)

$$z_m\ge z_{m+1},m=1,2,\dots ,\mathrm{n}-1$$

(40)

where $\Delta V_m$ stands for the term $(K_m^2-K_{m-1}^2)V_{i,\mathrm{OLTC}}^{}$, replacing $K$ as the decision variable about OLTCS in the optimization process. When $z_m=0$, then $\Delta V_m^{}=0$ according to Equation (37); when $z_m=1$, then $\Delta V_m^{}=\left(K_m^2-K_{m-1}^2\right)V_{i,\mathrm{OLTC}}^{}$ according to Equations (38) and (39). According to Equation (40), when z_t − z_{t − 1} = 1, then the values of the 1st to the tth term of the set $\left\{z_1,z_2,\dots ,z_{\mathrm{n}}\right\}$, then the values of the 1st to the tth term of the set are 1, while all the remaining values are 0. Substituting all the values into Equation (36), we can see that the set of constraints Equations (36)–(40) are equivalent to the original constraints Equation (35).

2.2.5. Constraints of SCs

The set $\{B_{i,1},B_{i,2},\dots ,B_{i,\mathrm{n}}\}$ is used to represent the conductance of SCs available at bus i. Assuming that the capacitors are put in sequentially, i.e., $B_{i,t}$ is put in before $B_{i,t+1}$, then the reactive power output of SCs can be expressed as follows:

$$Q_{i,\mathrm{sc}}^{}=V_i{\displaystyle \sum _{m=1}^t{B}_{i,m}}={\displaystyle \sum _{m=1}^t\left(V_i{B}_{i,m}\right)}$$

(41)

Similar to the modeling process of OLTCs, a set of 0–1 variables $\{x_1,x_2,\dots ,x_{\mathrm{n}}\}$ is introduced to represent the operational state of SCs, then the constraint Equation (41) can be equivalently represented by the following set of linear constraints:

$$Q_{i,\mathrm{sc}}^{}={\displaystyle \sum _{m=1}^{\mathrm{n}}{Q}_{i,m}^{\mathrm{sc}}}$$

(42)

$$0\le Q_{i,m}^{\mathrm{sc}}\le x_m{V}_{\max }{B}_{i,m}$$

(43)

$$Q_{i,m}^{\mathrm{sc}}\le V_i{B}_{i,m}$$

(44)

$$Q_{i,m}^{\mathrm{sc}}\ge V_i{B}_{i,m}-\left(1-x_m\right)V_{i,\max }^{}{B}_{i,m}$$

(45)

$$x_t\ge x_{t+1},t=1,2,\dots ,\mathrm{n}-1$$

(46)

where $Q_{i,m}^{\mathrm{sc}}$ stands for the term $V_i{B}_{i,m}$.

2.2.6. Operational Constraints

For the maintenance of the system in a normal working condition, the following constraints need to be satisfied:

$$P_g^{\min }\le P_g\le P_g^{\max }$$

(47)

$$Q_g^{\min }\le Q_g\le Q_g^{\max }$$

(48)

$$v_{i,\min }^2\le V_i^{}\le v_{i,\max }^2$$

(49)

In addition, the linearized form of line capacity constraint can be referred to [20] and will not be repeated in this paper.

In the above transmission network ORPD subproblem, all constraints are linear when considering $V=v^2$ as an independent variable. Since discrete variables are included, the above-mentioned ORPD subproblem is a mixed-integer programming problem with linear constraints.

2.3. Distribution Network Subproblem Model

In this part, the distribution network subproblem model is established based on the SOCP method. In the ORPD of distribution network subproblems, in addition to the traditional OLTCs and SCs, the variables that need to be optimized also include Static Var Compensators (SVCs) and various controllable DGs. Since there are generally no conventional generators in the distribution network, using the minimum network loss as the objective function of the distribution network subproblems is in line with the economic principle. The objective function of the ORPD of the distribution network DSO_k is as follows:

$$\min F_{\mathrm{D},k}=\sum _{(i,j)}{r}_{ij,k}^{}{L}_{ij,k}^{}$$

(50)

where $L_{\mathrm{ij},\mathrm{k}}^{}$ represents the square of the current on branch i-j. The constraints of the DSO_k subproblem are as follows:

2.3.1. Power Flow Equation Constraints

For a typical radial distribution network, the power flow constraint can be expressed as follows:

$${\left(P_{ij,k}^{}\right)}^2+{\left(Q_{ij,k}^{}\right)}^2=L_{ij,k}{u}_{i,k}$$

(51)

$${\displaystyle \sum }_{i\in u(j)}\left(P_{ij,k}-L_{ij,k}{r}_{ij,k}\right)+P_{\mathrm{G}j,k}={\displaystyle \sum }_{t\in v(j)}\left(P_{jt,k}\right)+P_{\mathrm{D}j,k}$$

(52)

$${\displaystyle \sum }_{i\in u(j)}\left(Q_{ij,k}-L_{ij,k}{x}_{ij,k}\right)+Q_{\mathrm{G}j,k}={\displaystyle \sum }_{t\in v(j)}\left(Q_{jt,k}\right)+Q_{\mathrm{D}j,k}$$

(53)

$$u_{j,k}=u_{i,k}-2\left(r_{ij,k}^{}{P}_{ij,k}^{}+x_{ij,k}^{}{Q}_{ij,k}^{}\right)+\left({\left(r_{ij,k}^{}\right)}^2+{\left(x_{ij,k}^{}\right)}^2\right)L_{ij,k}^{}$$

(54)

In the above distribution network subproblem, Equation (51) contains quadratic terms, and SOCP is usually applied to transform it into the following form:

$${\Vert \begin{array}{c}2P_{ij,k}^{}\\ 2Q_{ij,k}^{}\\ L_{ij,k}^{}-u_{i,k}\end{array}\Vert }_2\le L_{ij,k}^{}+u_{i,k}$$

(55)

2.3.2. Security Constraints

For a typical radial distribution network, the power flow constraint can be expressed as follows:

$$L_{ij,k,\min }^{}\le L_{ij,k}^{}\le L_{ij,k,\max }^{}$$

(56)

$$u_{i,k,\min }\le u_{i,k}\le u_{i,k,\max }$$

(57)

2.3.3. Constraints of DGs

There are a large number of wind turbines, photovoltaics (PVs), and micro gas turbines (GTs) in distribution networks. They are generally connected to the grid through inverters, have good reactive power potentials, and participate in the distribution network′s reactive power optimization process. Depending on the types of DGs and their inverter control modes, they have different reactive power regulation characteristics.

The inverter of the micro GTs generally adopts the PQ decoupling control method, and the reactive power regulation capacity is determined by the active power output and the total capacity of the inverter:

$$Q_{\mathrm{DG},\max }=\sqrt{S_{\max }^2-P_{\mathrm{DG}}^2}$$

(58)

$$Q_{\mathrm{DG},\min }=-Q_{\mathrm{DG},\max }$$

(59)

where $Q_{\mathrm{DG},\max }$ and $Q_{\mathrm{DG},\min }$ represent the upper and lower limits of the reactive power regulation capability of the micro gas turbine; $S_{\max }^{}$ is the maximum apparent power of the inverter; and $P_{\mathrm{DG}}^{}$ is the active power output of the micro gas turbine.

For Doubly fed Induction Generators (DFIGs), the reactive power regulation capability is mainly determined by the reactive power capacity on the stator side and the total capacity of the inverter:

$$Q_{\mathrm{DG},\max }=Q_{\mathrm{s},\max }-Q_{\mathrm{c},\min }$$

(60)

$$Q_{\mathrm{DG},\min }=Q_{\mathrm{s},\min }-Q_{\mathrm{c},\max }$$

(61)

For PVs, the grid-connected inverters mainly have two control modes: voltage control and current control. Due to the limitations of the voltage control mood, the photovoltaic inverter usually adopts the current control mood. In the reactive power optimization process, the reactive power output of photovoltaics is jointly determined by the nodal voltage, active power output, and current constraints:

$$Q_{\mathrm{DG},\max }=\sqrt{V_{\mathrm{DG}}^2{I}_{\mathrm{DG},\max }^2-P_{\mathrm{DG}}^2}$$

(62)

$$Q_{\mathrm{DG},\min }=0$$

(63)

2.3.4. Other Constraints

The modeling of the OLTCs and SCs in the distribution networks is similar to that in the transmission network and will not be repeated here.

3. Solving Algorithm of the CTD-ORPD

This section describes the details of the algorithm AAL for solving the proposed CTD-ORPD framework.

The AAL algorithm is a novel distributed algorithm that combines the advantages of the Diagonal Quadratic Approximation (DQA) method and the ADMM method. It uses a local augmented Lagrangian function similar to that of DQA, while discarding its inner loop step. Compared to ADMM, the updated rules for original variables and dual variables are improved. The original variables are updated one more time compared to ADMM on the basis of the solution of the local augmented Lagrangian function, which is then used for the update of the dual variables. In addition, a fully distributed rule is used for the update of dual variables in the AAL algorithm [14].

Based on AAL, the global ORPD problem is decomposed into the TSO subproblem and several DSO_k subproblems, and the subproblems are solved in a distributed manner until the optimal global solution is finally reached after several iterations. Coordination between the TSO subproblem and DSO_k subproblems is achieved only through the interaction of information on coordinating variables, which consists of the variables of PCCs, including bus voltage magnitude and phase angle, active power output and reactive power output, and denoted as $x_{\mathrm{CT}}$ and $x_{\mathrm{CD}}$ in this paper. The following is the definition of the coordinating variables:

$$x_{\mathrm{CT}}={\displaystyle \underset{k=1}{\overset{\mathrm{d}}{\cup }}{x}_{\mathrm{CT},k}^{}},\hspace{1em}{x}_{\mathrm{CT},k}^{}=\left[\begin{array}{cccc}{P}_{\mathrm{CT},k}^{}& Q_{\mathrm{CT},k}^{}& V_{\mathrm{CT},k}^{}& {\delta }_{\mathrm{CT},k}^{}\end{array}\right]$$

(64)

$$x_{\mathrm{CD}}={\displaystyle \underset{k=1}{\overset{\mathrm{d}}{\cup }}{x}_{\mathrm{CD},k}^{}},\hspace{1em}{x}_{\mathrm{CD},k}^{}=[\begin{array}{llll}{P}_{\mathrm{CD},k}^{}\hfill & Q_{\mathrm{CD},k}^{}\hfill & V_{\mathrm{CD},k}^{}\hfill & {\delta }_{\mathrm{CD},k}^{}]\hfill \end{array}$$

(65)

The CTD-ORPD problem is an optimization problem defined by the objective functions and related constraints of the TSO and DSO_k subproblems, which can be expressed as follows:

$$\underset{x}{\min }{F}_{\mathrm{T}}\left(x_{\mathrm{T}},x_{\mathrm{CT}}\right)+{\displaystyle \sum }_{k=1}^{\mathrm{d}}{F}_{\mathrm{D},k}\left(x_{\mathrm{D},\mathrm{k}},x_{\mathrm{CD},k}\right)$$

(66)

$$s.t.\hspace{1em}{g}_{\mathrm{T}}\left(x_{\mathrm{T}},x_{\mathrm{CT}}\right)\le 0,h_{\mathrm{T}}\left(x_{\mathrm{T}},x_{\mathrm{CT}}\right)=0$$

(67)

$$g_{\mathrm{D},k}\left(x_{\mathrm{CD},k}^{},x_{\mathrm{D},k}^{}\right)\le 0,h_{\mathrm{D},k}\left(x_{\mathrm{CD},k}^{},x_{\mathrm{D},k}^{}\right)=0\hspace{1em}k=1,2,\dots \mathrm{d}$$

(68)

$$x_{\mathrm{CT}}-x_{\mathrm{CD}}=0$$

(69)

In Equation (66), $F_{\mathrm{T}}$ and $F_{\mathrm{D},k}^{}$ are the objective functions of the TSO and DSO_k subproblems established in the previous section, respectively. Equations (67) and (68) represent the inequality constraints and equality constraints of TSO and DSO_k (k = 1,2,…,d), respectively. $x_{\mathrm{T}}$ and $x_{\mathrm{D},k}^{}$ are local variables in the respective systems of TSO and DSO_k (k = 1,2,…,d). Equation (69) demonstrates the consistency constraint of the coordinating variables.

In the proposed distributed method, the CTD-ORPD problem Equation (66) is decomposed into (d + 1) subproblems and coordinated optimization is performed. According to AAL, the decomposed subproblems have the following expressions:

$$\min {\mathsf{\Lambda }}_{\mathrm{TSO}}\left(x_{\mathrm{T}}^{},x_{\mathrm{CT}}^{},\lambda \right)=\min F_{\mathrm{T}}\left(x_{\mathrm{T}}^{},x_{\mathrm{CT}}^{}\right)+\sum _{k=1}^{\mathrm{d}}\left[{\lambda }_k\left(x_{\mathrm{CT},k}^{}-{\tilde{x}}_{\mathrm{CD},k}^{}\right)+\frac{\rho }{2}{\Vert x_{\mathrm{CT},k}^{}-{\tilde{x}}_{\mathrm{CD},k}^{}\Vert }^2\right]$$

(70)

$$\begin{array}{l}\min {\mathsf{\Lambda }}_{\mathrm{DSO},k}^{}\left(x_{\mathrm{D},k}^{},x_{\mathrm{CD},k}^{},{\lambda }_k\right)=\min F_{\mathrm{D},k}\left(x_{\mathrm{D},k}^{},x_{\mathrm{CD},k}^{}\right)+\\ \left[{\lambda }_k\left({\tilde{x}}_{\mathrm{CT},k}^{}-x_{\mathrm{CD},k}^{}\right)+\frac{\rho }{2}{\Vert {\tilde{x}}_{\mathrm{CT},k}^{}-x_{\mathrm{CD},k}^{}\Vert }^2\right], k=1,2,\dots ,\mathrm{d}\end{array}$$

(71)

${\mathsf{\Lambda }}_{\mathrm{TSO}}$/${\mathsf{\Lambda }}_{\mathrm{DSO},k}^{}$ are the local augmented Lagrangian functions of the TSO/DSO_k subproblems, respectively. They are, respectively, composed of the corresponding objective function, the penalty term of the consistency constraint, and a quadratic term that improves the convergence speed of the Lagrangian function. $\lambda$ is the Lagrange multiplier associated with coordinating variables, and ρ > 0 is the penalty term coefficient. The expression of $\lambda$ is as follows:

$$\lambda ={\displaystyle \underset{k=1}{\overset{\mathrm{d}}{\cup }}{\lambda }_k}, {\lambda }_k=\left[\begin{array}{llll}{\lambda }_{P,k}^{}\hfill & {\lambda }_{Q,k}^{}\hfill & {\lambda }_{V,k}^{}\hfill & {\lambda }_{\delta ,k}^{}\hfill \end{array}\right]$$

(72)

where ${\tilde{x}}_{\mathrm{CT},k}^{}$ and ${\tilde{x}}_{\mathrm{CD},k}^{}$ denote the reference values of the DSO_k and TSO coordination variables, respectively, which are derived from the last optimization results of the TSO and DSO_k subproblems. When optimizing the TSO subproblem, the coordinating variables of the DSO_k subproblem ${\tilde{x}}_{\mathrm{CD},k}^{}$ serve as the benchmark values of $x_{\mathrm{CT},k}^{}$ in the TSO subproblem. Similarly, ${\tilde{x}}_{\mathrm{CT},k}^{}$ would be served as the benchmark values of $x_{\mathrm{CD},k}^{}$ when optimizing the DSO_k subproblem. Figure 3 illustrates the overall pictorial view of the interactions between TSO and DSOs in the proposed distributed framework, where the solid line connections between TSOs and DSOs represent connections through transformers and the dotted line connections indicate information interactions.

In each iteration, the transmission network subproblem is solved first, and then the coordinating variables are transferred to each distribution network subproblem. After the distribution network receives the coordinating variables, the subproblems are solved in parallel, and then the updated coordinating variables are transmitted to the transmission network. After the transmission grid receives data from all distribution grids, feasibility constraints and optimality constraints are checked. The optimal solution is obtained, and the optimization process ends if these two constraints are satisfied. Otherwise, the next iteration begins. Due to the parallelism of the subproblems, the time consumed by each iteration process depends on the transmission network optimization time and the distribution network with the longest optimization time. The detailed iterative steps of the algorithm are as follows.

1.

Set t = 1, initialize ${\lambda }^t$, $\rho$ > 0, $\tau$∈(0,0.5), initial local variables $x_{\mathrm{T}}^0$/$x_{\mathrm{D},k}^0$, initial coordinating variables $x_{\mathrm{CT}}^0$/$x_{\mathrm{CD},k}^0$, and ${\tilde{x}}_{\mathrm{CD},k}^{}=x_{\mathrm{CD},k}^0$.

2.

Optimization of the TSO subproblem: Optimize the problem Equation (70) at fixed $\lambda$ and ${\tilde{x}}_{\mathrm{CD},k}^{}$, and obtain the local/coordinating variables ${\widehat{x}}_{\mathrm{CT}}^t$/${\widehat{x}}_{\mathrm{T}}^t$. After the following update steps are completed, transfer ${\tilde{x}}_{\mathrm{CT},k}^{}$ to DSO_k (k = 1,2,…,d).

$$x_{\mathrm{T}}^t=x_{\mathrm{T}}^{t-1}+\tau ({\widehat{x}}_{\mathrm{T}}^t-x_{\mathrm{T}}^{t-1})$$

(73)

$${\tilde{x}}_{\mathrm{CT}}^{}=x_{\mathrm{CT}}^t=x_{\mathrm{CT}}^{t-1}+\tau ({\widehat{x}}_{\mathrm{CT}}^t-x_{\mathrm{CT}}^{t-1})$$

(74)

3.

Optimization of the DSO_k subproblem: Optimize the problem Equation (71) at fixed $\lambda$ and ${\tilde{x}}_{\mathrm{CT},k}^{}$, and obtain the local/coordinating variables ${\widehat{x}}_{\mathrm{CD}}^t$/${\widehat{x}}_{\mathrm{D}}^t$. After the following update steps are completed, then transfer ${\tilde{x}}_{\mathrm{CD},k}^{}$ to TSO.

$$x_{\mathrm{D},k}^t=x_{\mathrm{D},k}^{t-1}+\tau ({\widehat{x}}_{\mathrm{D}}^t-x_{\mathrm{D},k}^{t-1})$$

(75)

$${\tilde{x}}_{\mathrm{CD},k}^{}=x_{\mathrm{CD},k}^t=x_{\mathrm{CD},k}^{t-1}+\tau ({\widehat{x}}_{\mathrm{CD}}^t-x_{\mathrm{CD},k}^{t-1})$$

(76)

4.

When the TSO receives all the data from DSO_k, iterations enter the next step, or else hold and wait.

5.

Use Equation (77) to check whether the feasibility constraint and optimality constraint are satisfied. If both are satisfied, then the optimization process ends; otherwise, update the dual variables according to Equation (78), set t = t + 1, and return to step 1.

$$x_{\mathrm{CT}}^t-{\tilde{x}}_{\mathrm{CD}}^{}<\psi , x_e^t-{\widehat{x}}_e^t<{\gamma }_e, F_e^t-F_e^{t-1}<{\xi }_e, e\in \left\{\mathrm{TSO},{\mathrm{DSO}}_1,\dots ,{\mathrm{DSO}}_{\mathrm{d}}\right\}$$

(77)

$${\lambda }^{t+1}={\lambda }^t+\rho \tau (x_{\mathrm{CT}}^t-{\tilde{x}}_{\mathrm{CD}}^{})$$

(78)

In Equation (77), $\psi$, ${\gamma }_e$, and ${\xi }_e$ together determine the accuracy of the algorithm, all of which took the value of 10⁻² in this paper.

Based on the concept of independent system operators, the program only needs to exchange a small amount of boundary information, which protects the information privacy and independent decision-making of system operators at all levels. Each distribution network subproblem can be optimized in parallel, which is beneficial to improving computational efficiency.

4. Analysis of the Simulation Case

4.1. Introduction of the Simulation Case

In order to prove the effectiveness of the proposed distributed CTD-ORPD framework, three test cases were constructed based on the IEEE-30 node system (transmission network) and the IEEE-33 node system (distribution network) for simulation calculation.

(1)

Case 1: Connect an IEEE-33 node test system at node No.26 of the IEEE-30 node test system, which is denoted by TSO and D26 in the following text.

(2)

Case 2: Connect an IEEE-33 node test system at the 7th, 19th, and 26th nodes of the IEEE-30 node test system, respectively, which are represented by TSO, D7, D19, and D26 in the following text.

(3)

Case 3: Connect an IEEE-33 node test system at the 7th, 17th, 19th, 26th, and 29th nodes of the IEEE-30 node test system, respectively, which are represented by TSO, D7, D17, D19, D26, and D29 in the following text.

In the three test cases, the transmission network and the distribution network were connected through 132/12.66 kv transformers. The high voltage side bus of the transformer was treated as the PCC. The active and reactive power exchanged between transmission and distribution networks through PCC, as well as the magnitude and phase angle of PCC, were used as the coordinating variables in the process of transmission and distribution co-optimization. The power reference value of both the distribution network and the transmission network was set to 100 MVA.

In the IEEE-30 node system, the OLTCs were set to be adjustable within 0.95~1.05, and the adjustment step was 0.01.

In the IEEE-33 node systems, nodes 2, 4, 6, 10, 13, 24, and 28 were connected to DGs, and the detailed parameters of the DGs are shown in

Table 1. Buses and parameters of the DGs.

DGs	Buses	Control Modes	Parameters/p.u.
GT	4	PQ decoupling	P = 0.002, Smax = 0.005
DFIG	2, 28	Constant voltage control	P = 0.002, Qs,max = 0.003, Qs,min = −0.003, Qc,max = 0.005, Qc,min = −0.005
PV	6, 13	Current control	P = 0.002, Imax = 0.005

.

Since the accuracy of the linearization of the power flow equation in the transmission network is related to the quality of the initial value $(v_0,{\theta }_0)$, the latest result of the integrated power flow calculation was used as the value of $(v_0,{\theta }_0)$.

Other parameters in the transmission and distribution grids, such as load data, impedance data, and unit cost coefficients, were derived from MATPOWER (v.7.1, Cornell University’s Charles H. Dyson School, U.S.).

The program in this paper was established in MATLAB R2019b (v.9.7, MathWorks, U.S.) and ran on a computer with Intel(R) Core(TM) i7-10850H CPU @ 2.70GHz CPU and 32 GB memory.

4.2. Comparison with Other Methods

To verify the superiority of the transmission and distribution coordination method proposed in the paper over the traditional independent method and centralized method, the centralized method, the independent method and the method proposed in the paper were used for simulation calculations for three test cases. “Centralized method” means that the distribution network is regarded as a new branch connected to the PCC node in the transmission network, and the centralized optimization calculation is performed by the control center. “Independent method” means that there is no information interaction between the transmission network and the distribution network, and independent optimization is carried out according to the boundary power flow information.

Table 2. Comparison of the results of solving test case 1 by different methods.

		Independent Method	Proposed Method	Centralized Method
System operating costs/thousand USD		61.6128	61.3055	61.3074
Network losses/MW	D26	0.0816	0.0756	0.0769
	TSO	5.1020	5.0728	5.0756
	Total	5.1836	5.1484	5.1525
Boundary voltage magnitudes (phase angles/(°))		1.0066(0)	1.0379 (−10.38)	1.0381 (−10.38)

,

Table 3. Comparison of the results of solving test case 2 by different methods.

		Independent Method	Proposed Method	Centralized Method
System operating costs/thousand USD		61.6134	61.2250	61.2265
Network losses /MW	D7	0.0812	0.0769	0.0774
	D19	0.0796	0.0781	0.0794
	D26	0.0826	0.0759	0.0765
	TSO	5.0836	5.0307	5.0324
	Total	5.3270	5.2642	5.2657
Boundary voltage magnitudes (phase angles/(°))	D7	1.0031(0)	1.0192 (−6.52)	1.0190 (−6.52)
	D19	1.0155(0)	1.0287 (−9.83)	1.0284 (−9.83)
	D26	0.9985(0)	1.0305 (−10.46)	1.0300 (−10.46)

and

Table 4. Comparison of the results of solving test case 3 by different methods.

		Independent Method	Proposed Method	Centralized Method
System operating costs/thousand USD		61.6150	61.1229	61.1265
Network losses /MW	D7	0.0818	0.0779	0.0801
	D17	0.0750	0.0718	0.0729
	D19	0.0796	0.0791	0.0795
	D26	0.0903	0.0821	0.0826
	D29	0.0801	0.077	0.0776
	TSO	5.0801	5.0109	5.0134
	Total	5.4869	5.3988	5.4061
Boundary voltage magnitudes (phase angles/(°))	D7	1.0033(0)	1.0193 (−6.54)	1.0190 (−6.52)
	D17	1.0276 (0)	1.0433 (−8.64)	1.0429 (−8.64)
	D19	1.0144(0)	1.0296 (−9.82)	1.0284 (−9.83)
	D26	0.9901(0)	1.0309 (−10.50)	1.0300 (−10.46)
	D29	1.0066 (0)	1.0234 (−6.10)	1.0230 (−6.10)

shows the results of solving the three test cases by different methods, and the boundary voltage magnitudes are in the form of per-unit values. It can be seen from the tables that in the three calculation cases, the network losses and system operating costs of the independent optimization method are significantly higher than those of the centralized optimization method. However, the results obtained by the method proposed in this paper are very close to that of the centralized optimization method. This is due to the fact that the initial value $(v_0,{\theta }_0)$ used in the linearization of the power flow equations of the transmission network adopts the result of the last integrated power flow calculation. This proves that our proposed method can obtain a solution with high precision when the initial value $(v_0,{\theta }_0)$ has high accuracy. This also demonstrates that it is feasible and effective to realize the ORPD problem of the global system through the cooperative calculation between the transmission and distribution networks.

In test case 1, the method proposed in the paper converges after three iterations, the total duration is 4.3 s, and the total duration of centralized optimization is 1.890 s. The total duration of the algorithm in this paper is 2.3 times that of centralized optimization. In test case 2, the method proposed in this paper converges to the optimal solution after four iterations. The whole process takes 5.1 s, and the total duration of centralized optimization is 7.269 s. At this time, the total duration of the method proposed in this paper is only 0.7 times that of centralized optimization. As for the situation in test case 3, the method proposed in this paper converges to the optimal solution after four iterations. The whole process takes 5.8 s, and the total duration for centralized optimization is 21.326 s. Eventually, the method proposed in this paper is obviously less time-consuming than centralized optimization, and the total duration of the algorithm in this paper is only 0.23 times that for centralized optimization. It can be inferred that, when the number of distribution networks connected to the transmission grid is further increased, the time consumption of centralized optimization increases sharply, while the total time consumption of the method proposed in this paper would be basically stable. This is because the distribution network optimization subproblems are solved in parallel in the proposed distributed computing method. In this case, the total duration is only related to the number of iterations, the time consumption of the transmission network subproblem, and the most time-consuming distribution network subproblem, but has little to do with the number of connected distribution networks. At the same time, the linearization approximation on the transmission subproblem greatly reduces the complexity and time consumption of the transmission subproblem, so the total time consumption of the method proposed in this paper is relatively low.

Figure 4 and Figure 5 show the voltage distribution of TSO and D26 using different methods to solve test case 1, respectively. (The results of test case 2 and test case 3 are similar.)

Table 5. Power output of generators in TSO with different methods on test case 1.

	Independent Method		Proposed Method		Centralized Method
	P (MW)	Q (MVAr)	P (MW)	Q (MVAr)	P (MW)	Q (MVAr)
G1	41.52	−5.13	41.31	−5.14	41.31	−5.14
G2	55.37	2.53	55.09	2.49	55.09	2.49
G3	16.13	35.21	16.05	35.22	16.05	35.22
G4	22.7	33.59	22.58	33.35	22.58	33.35
G5	16.13	6.87	16.05	6.63	16.05	6.63
G6	39.62	31.95	39.44	30.85	39.44	30.85
Total	191.47	105.02	190.52	103.4	190.52	103.4

and

Table 6. Reactive power output of DGs in D26 with different methods on test case 1.

	Independent Method	Proposed Method	Centralized Method
DFIG-2/MVar	0.1264	0.3712	0.3717
GT-4/MVar	0.2012	0.3873	0.3869
PV-6/MVar	0.2367	0.3426	0.3430
PV-13/MVar	0.3246	0.3651	0.3654
DFIG-28/MVar	0.4000	0.4000	0.4000

present the optimization results of controllable resources in TSO and D26, respectively, while

Table 7. Power transmitted from TSO to D26 with different methods on test case 1.

	Independent Method	Proposed Method	Centralized Method
Active Power/MW	2.9310	2.9266	2.9253
Reactive Power/MVar	1.4407	0.8632	0.8628

shows the power exchange between TSO and D26. The result indicates that the proposed method makes fuller use of the reactive power regulation potential of the DGs in the distribution network. On the one hand, it improves the voltage distribution and reduces the network losses in D26. On the other hand, the demand of D26 for remote reactive resources in TSO is reduced, thus further improving the voltage distribution and reducing the network losses in TSO. As can be seen from the figures, the nodal voltage distributions obtained by the proposed method are very close to that of the centralized optimization method. For the transmission grid, the local voltage distribution is improved by making full use of the reactive power regulation capability of the distribution grid in the transmission and distribution coordination optimization process. In either the proposed method or the centralized optimization method, the voltage magnitudes of the PCC node of the transmission network, i.e., node 26 and its nearby nodes, are significantly improved when compared to that of the independent optimization case. In addition, Appendix A shows the detailed branch power flow of TSO, which demonstrates the influence of coordinated transmission and distribution ORPD on the branch power flow of TSO.

4.3. Comparison of Different Distributed Algorithms

In order to test the superiority of the proposed method in relation to the other distributed methods, ADMM [13], G-MSS [9], ATC [28], AAL (without the linearized approximation method to model the TSO subproblem), and the proposed method in this paper were applied to test case 1, and the results are shown in

Table 8. Comparison of the results of different distributed algorithms to solve test case 1.

Distributed Methods	System Operating Costs/thousand USD	Iterations	Total Duration/s
ADMM	61.3034	5	9.3
G-MSS	61.3022	6	11.6
ATC	61.3046	5	10.2
AAL	61.3058	5	7.1
The proposed method	61.3055	3	4.3

.

It can be seen from Table 8 that the operating costs of the systems optimized by various methods are not much different, and all the results are fairly close to those of the centralized optimization method, which demonstrates the effectiveness of various distributed methods for solving transmission and distribution synergy problems. However, compared with the other methods, the iterations and the total duration of the proposed method are significantly reduced. This means that the accuracies of various distributed methods are almost the same, but the method proposed in this paper has a higher efficiency. On the one hand, the AAL algorithm, which is the basis of the proposed method in the paper, combines the advantages of other algorithms to obtain improved convergence performance and solution efficiency. On the other hand, the linear approximation is applied to the power flow equations of the TSO subproblem, which greatly reduces the complexity of the optimization model, thereby improving the calculation speed and convergence performance. With the simultaneous application of the novel linearized approximation method and AAL algorithms, the proposed method achieves a significant improvement in convergence performance and solution efficiency compared to other methods.

The above results are obtained with a convergence accuracy of 10⁻². Figure 6 and Figure 7 illustrate the system operating costs and iterations of the various methods for different convergence accuracies. As can be seen, the proposed method consistently achieves better optimization results as well as convergence performance compared to other distributed algorithms when the convergence accuracy varies. At the same time, as the convergence accuracy increases, the optimization results of various methods gradually converge and approach the results of centralized optimization, but the number of required iterations also increases. As the convergence accuracy increases from 10⁻² to 10⁻³, the quality of the optimization results of various algorithms improves significantly, while the number of iterations increases only slightly. When the convergence accuracy is increased from 10⁻⁴ to 10⁻⁵, the quality of the optimization results of various methods improves just a little, while the number of required iterations increases a lot. We can thus see that the distributed methods can obtain the same results as the centralized optimization when the convergence accuracy is high enough, however, at the expense of the convergence performance. In practice, a reasonable compromise between optimization accuracy and convergence performance should be made before the computation process. It can also be seen that, when compared to the case of using only the AAL algorithm for distributed computation without using the novel linear approximation method to model the TSO subproblem, the proposed method in the paper has a better convergence performance at all convergence accuracies, which indicates that using both methods can effectively improve the convergence performance of the transmission and distribution collaborative framework. Furthermore, the inconsistency in the total operating cost between the two is caused when the nonlinear network loss term is linearized according to the linearized approximation method.

4.4. Effect of the Linearized Approximation on the TSO Subproblem

As can be seen from the analysis of the above arithmetic cases, the nonlinear optimization solver used by MATPOWER converges in all arithmetic scenarios. However, the linearized approximation method used in the paper for the TSO subproblem has the following unique advantages:

• All formulas are convex functions, which theoretically guarantee convergence;
• The local optimum corresponds to the global optimum;
• The linearized OPF model of the transmission grid subproblem leads to a significant reduction in the complexity of the global optimization model, especially considering the weight of the TSO time consumption in the global optimization process.

However, the accuracy of the linearized approximation of the flow equations depends on the accuracy of the initial value $(v_0,{\theta }_0)$, and in the previous analysis, the results of the latest integrated flow calculation are used as the initial value $(v_0,{\theta }_0)$. The above analysis shows that this approach causes little error in the optimization process when the loads do not vary much. To further verify the feasibility of the linearized approximation method adopted in the paper, load variation coefficients ${\alpha }_{\mathrm{P}}$ and ${\alpha }_{\mathrm{Q}}$ are introduced to analyze the errors caused by the linearization of the power flow equations when the loads vary. The node loads after changes are as follows:

$$P_{i,\mathrm{d}}^{\prime }=P_{i,\mathrm{d}}\times \left(1+{\alpha }_{\mathrm{P}}\times \frac{2i-\mathrm{N}}{\mathrm{N}}\right),\hspace{1em}{Q}_{i,\mathrm{d}}^{\prime }=Q_{i,\mathrm{d}}\times \left(1+{\alpha }_{\mathrm{Q}}\times \frac{2i-\mathrm{N}}{\mathrm{N}}\right)$$

(79)

where $P_{i,\mathrm{d}}^{\prime }/Q_{i,\mathrm{d}}^{\prime }$ represents the changed load at bus i, and N is the number of transmission grid buses. When calculating the errors, the calculated results after the load change are used as the base values, and the errors at different load variation coefficients are shown in

Table 9. Errors of the linearized approximation method at different load variation coefficients.

${\mathit{\alpha }}_{\mathbf{P}}$	${\mathit{\alpha }}_{\mathbf{Q}}$	Errors in Total Costs	Maximum Error of Qij (p.u.)
20%	20%	0.078%	0.27
30%	20%	0.30%	0.22
40%	20%	0.47%	0.26
20%	40%	0.10%	0.27
30%	40%	0.31%	0.26
40%	40%	0.57%	0.27

. As can be seen from the table, the errors in operating costs as the objective function of the optimization problem increase when the absolute values of ${\alpha }_{\mathrm{P}}$ and ${\alpha }_{\mathrm{Q}}$ increase, because the errors due to the linearization of the network losses become larger in that case. It can also be seen that the operating costs errors of the system and the modeling accuracy of Q are satisfactory for different load variation coefficients. This shows that the method in the paper of directly adopting the results of the latest integrated power flow calculation as the initial values $(v_0,{\theta }_0)$ and linearizing the transmission grid power flow equations is feasible and effective when the accuracy requirement is not particularly high. Alternatively, when a higher accuracy is required, the accuracy of the linearized approximation method can be further promoted by using the warm-start iterative method described in [20], where only one more iteration is required to obtain a solution significantly close to the exact modeling case.

5. Conclusions

This paper established a distributed framework of the coordinated transmission and distribution networks optimal reactive power dispatch (CTD-ORPD) based on the AAL method. The global reactive power optimization problem of transmission and distribution grids was decomposed into a TSO subproblem and several DSO_k subproblems. Each of the subproblems ran in a distributed manner and iterated until the optimal global solution is finally reached. The transmission grid and distribution grid subproblems achieved fully distributed cooperative optimal reactive power control only through the information interaction of the coordinating variables. The main conclusions are as follows:

(1)

Under the background of a large influx of new energy in the distribution networks, the coordinated reactive power optimization of transmission and distribution networks can better allocate the reactive power resources of the system than the independent optimization method. On the one hand, it gives full play to the voltage supportability of the transmission network to the distribution networks, and also makes full use of the reactive power regulation ability of the distribution networks. Thereby minimizing system network losses and maximizing economic benefits;

(2)

Under the proposed framework, each subproblem is solved in a distributed manner, which protects the privacy of each independent subject and greatly reduces the complexity of the problem. The distribution network subproblems run in parallel; when there are many distribution networks connected to the transmission network, the complexity of the problem only depends on the distribution network subproblem with the longest solution time, and has little to do with the number of distribution networks. Thus, compared with the centralized optimization method, the solution speed of the optimization problem is greatly improved. Compared with the ADMM method, the AAL method updates the variables twice in each iteration, and has advantages in convergence performance and solving efficiency;

(3)

The transmission subproblem plays a decisive role in the complexity of the entire model and the total time consumption. By applying a linear approximation to the power flow equations of the transmission subproblem, the complexity of the optimization problem and the total time consumption are both significantly reduced.

The AAL algorithm and the novel linearized approximation method improve the convergence performance and solving efficiency of the cooperative transmission and distribution reactive power optimization problem in terms of the distributed computation method and the model building method, respectively, to achieve more stable convergence and faster solution. Considering the demand for regulating the voltage problems in the distribution networks in the background of the influx of DERs, the proposed CTD-ORPD framework in this paper is expected to play a good role in the short-term or real-time cooperative reactive power optimal control process of transmission and distribution, thus promoting further DER consumption.