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Distributed optimization, economic dispatch (ED), consensus ADMM, Newton-Raphson method.

1. INTRODUCTION

The economic dispatch (ED) is one of the most important problems in power system operation, where generator outputs are decided by minimizing system generation cost subject to power balance constraints. Sun et al., Yang et al., and Chen et al. introduced interconnected systems for economic cooperation. Due to the expansion of the electricity market, the models and data within the region of the system became a commercial secret, which makes the data exchange between regions difficult if not impossible. From these reasons, system operators need to solve sub-problems with limited information(1-3).

These changes converted the traditional centralized system operation decentralized and distributed. To solve the ED problem, Boyd et al. suggested simple and powerful alternating direction method of multipliers (ADMM) suited for distributed convex optimization(4). Yang et al. applied it to the optimization of large-scale power system; a fully distributed and robust algorithm for alternating current optimal power flow (AC OPF) is proposed. The algorithm is based upon ADMM which is customized as a region-based optimization procedure(5).

Erseghe, Peng et al., and Ma et al. have applied ADMM to solve ED and OPF problems(6-8). In the distributed AC OPF, entire power system is separated into multiple areas and each area solves its own ED or OPF problem with the minimal information of the other areas. Especially consensus ADMM and proximal ADMM were suggested for the ED and AC OPF with second-order conic programming (SOCP) relaxation. Also, Yang et al. proposed a distributed consensus and a supply–demand balance approach based on quadratic cost functions to solve the ED under switching topologies and guaranteed the global feasibility of the algorithm(9). Chen et al. solved ED problem with general convex cost functions using an ADMM-based distributed algorithm(10). Moreover, the traditional centralized ADMM is extended to a distributed implementation problem by using the center-free algorithm and projection method.

This paper presents an approach for an ED problem based on consensus ADMM algorithm that does not require any form of central coordination. The solution of a local or regional optimization is exchanged only between neighboring areas. The convergence speed depends upon the number of sub-problems and decision variables, hence in order to improve the convergence speed and number of iterations, the optimal solution of a consensus area and its average are calculated using the decision variables at every iteration. As a result, the solution of the overall system can be found more efficiently based on sub-problems.

The proposed techniques have been applied to a 10-bus system to demonstrate their effectiveness and compared to the Newton-Raphson method.

2. PROBLEM FORMULATION

This section defines the formulation for the economic dispatch (ED) which is used by Saadat (1999), and modifies it so that each area containing the overlapping buses can be optimized through consensus ADMM which is used by Ma et al. (2016)(8).

2.1 Objective Function

The objective function for the conventional generating plants consists of quadratic cost functions as follows:

(1)
$\min . C_{G}=\Sigma_{i=1}^{n}\left(a_{i}+b_{i}P_{G,\:i}+c_{i}P_{G,\:i}^{2}\right)$

where

$C_{G}$ : sum of conventional generation cost [$/h],

$P_{G,\:i}$: generation output at bus-i [MW],

$a_{i}$, $b_{i}$, $c_{i}$ : coefficients of generation cost bus-i,

$n$ : number of generators.

2.2 Constraints

2.2.1 Power Balance Equation

(2)
$∑_{i=1}^{n}P_{G,\:i}= P_{load}+ P_{loss}$

where the load balance equation consists of the load $P_{load}$ and power losses $P_{loss}$ of generators such as fossil-fuel generation, wind-turbine generation, and BESS.

2.2.2 Power Generation Capacity Constraints

(3)
$P_{G,\:i}^{\min}≤ P_{G,\:i}≤ P_{G,\:i}^{\max}$

where $P_{G,\:i}$ is the power generation at bus-$i$, and $\min$ and $\max$ is the lower and upper limits of the power generation, respectively.

2.3 Expressions for Area containing Overlapping Buses

2.3.1 Objective Function

(4)
$\min . C_{G,\:\cos t}\left(P_{G,\:A,\:i}\right),\: A=1,\:\cdots ,\:m$

where $P_{G,\:A,\:i}$ is the power generation at bus-$i$ of an area that contains overlapping buses in consensus area.

2.3.2 Equality Constraints

(5)
$\Sigma_{i=1}^{ng}P_{G,\:A,\:i}=P_{load,\:A}+P_{loss,\:A},\: A=1,\:\cdots ,\:m$

where ng is the number of generators in each area. $P_{load,\:A}$and $P_{loss,\:A}$ is the system load and power losses of an area, respectively.

2.3.3 Inequality Constraints

(6)
$P_{G,\:A,\:i}^{\min}\le P_{G,\:A,\:i}\le P_{G,\:A,\:i}^{\max},\: A=1,\:\cdots ,\:m$

where $P_{G,\:A,\:i}$ is the power generation at bus-$i$ of an area, and $\min$ and $\max$ is the lower and upper limits of the power generation, respectively.

3. ADMM ALGORITHM

3.1 General ADMM

Boyd et al. suggested that ADMM is an algorithm that solves problems in the form of an objective function and constraints(1).

(7)
$$ \begin{aligned} & \text { Min. } f(x)+g(z) \\ & \text { s.t. } B x+D z=c \end{aligned} $$

where $x\in R_{n}$, $z\in R_{m}$, $B\in R_{p\times n}$, $D\in R_{p\times m}$ and $c\in R_{p}$. The objective functions, and $f$, are $g$ convex. As in the method of multipliers, the augmented Lagrange function is defined as follows:

(8)
$\begin{aligned} L_{\rho}(x, z, y)= & f(x)+g(z)+y^{T}(B x+D z-c) \\ & +(\rho / 2)\|B x+D z-c\|_{2}^{2} \end{aligned}$

ADMM consists of the iterations

(9)
$$ \begin{aligned} x^{k+1} & :=\underset{x}{\arg \min } L_{\rho}\left(x, z^{k}, y^{k}\right), \\ \end{aligned} $$

(10)
$$ \begin{aligned} z^{k+1} & :=\underset{z}{\arg \min } L_{\rho}\left(x^{k+1}, z, y^{k}\right), \end{aligned} $$

(11)
$y^{k+1}:= y^{k}+ρ\left(Bx^{k+1}+ Dz^{k+1}-c\right),\:$

where $\rho > 0$. The algorithm is very similar to the dual ascent and the method of multipliers: it consists of an $x$-minimization step (9), a $z$-minimization step (10), and a dual variable update (11). As in the method of multipliers, the dual variable update uses a step size equal to the augmented Lagrange parameter .

ADMM can be written in a slightly different form, which is often more convenient, by combining the linear and quadratic terms in the augmented Lagrange function and scaling the dual variable. Defining the residual $r=Bx+Dz-c$, second and third terms of (8) can be transformed as follows:

(12)
$\begin{aligned} y^{T} r & +(\rho / 2)\|r\|_{2}^{2} \\ & =(\rho / 2)\|r+(\rho / 2) y\|_{2}^{2}-((1 / 2 \rho))\|y\|_{2}^{2} \\ & =(\rho / 2)\|r+u\|_{2}^{2}-(\rho / 2)\|u\|_{2}^{2} \end{aligned}$

where is the scaled dual variable. Using the scaled dual variable, ADMM can be expressed as follows:

(13)
$$ x^{k+1}:=\underset{z}{\arg \min }\left(f(z)+(\rho / 2)|| B x+D z^{z}-c+u^{k}||_{2}^{2}\right. $$

(14)
$$ z^{k+1}:=\underset{z}{\arg \min }\left(g(z)+(\rho / 2)|| B x^{k+1}+D z-c+u^{k}||_{2}^{2}\right. $$

(15)
$u^{k+1}:= u^{k}+ Bx^{k+1}+ Dz^{k+1}-c.$

Finally, the ADMM iteration should converge to the following results.

(16)
$$ \lim _{k \rightarrow \infty}\left[\begin{array}{c} B x^{k}+D z^{k}-c \\ f\left(x^{k}\right)+g\left(z^{k}\right) \\ u^{k} \end{array}\right]=\left[\begin{array}{c} 0 \\ p^{*} \\ u^{*} \end{array}\right] $$

where $p^{*}$ is the optimal value of the objective function, and $u^{*}$ is the optimal dual variable.

3.2 Consensus ADMM

Consensus ADMM is a special form of ADMM which is proposed by Boyd et al.(1), Ma et al.(8), and Kar et al.(11) as follows:

(17)
$$ \begin{aligned} & \text { Min. } \quad \sum_{A=1}^{m} f_{A}\left(x_{A}\right) \\ & \text { s.t } \quad x_{A}-z=0, A=1, \cdots, m \end{aligned} $$

where $x_{A}$ denotes variables in Area $A$. This is called global consensus algorithm, since all the local variables should be equal to the global v.riable at convergence. The augmented Lagrange function can be derived for (17) as follows:

(18)
$L_{\rho}\left(x_{1},\:\cdots ,\:x_{m},\:z,\:y\right)$ $$ =\sum_{A=1}^{m}\left(f_{A}\left(x_{A}\right)+y_{A}^{T}\left(x_{A}-z\right)+(\rho / 2)\left\|x_{A}-z\right\|_{2}^{2}\right) $$

Thus, the common global variable is solved as follows:

(19)
$$ x_{i}^{k+1}:=\underset{x_{i}}{\arg \min }\left(f_{i}\left(x_{i}\right)+y_{i}^{T}\left(x_{i}-z\right)+\rho / 2\right)|| x_{i}-z||_{2}^{2} $$

(20)
$z^{k+1}:=\dfrac{1}{m}∑_{i=1}^{m}\left(x_{i}^{k+1}+(1/ρ)y_{i}^{k}\right)$,

(21)
$y_{i}^{k+1}:= y_{i}^{k}+ρ\left(x_{i}^{k+1}- z^{k+1}\right)$.

In this paper, the consensus ADMM is used to solve the ED problem. Each area only needs to handle its own objective function and constraints for a given global $z^{k}$. All the areas update their decision variables until it converges.

3.3 Consensus ADMM Implementation

For the extension of ADMM to instances of (7), Yang et al. suggested the augmented Lagrange function of this problem needs to be transformed as follows(9):

(22)
$$ \begin{aligned} & L_{A \rho}\left(x_{A}, z^{k}, y_{C A}^{k}\right)=C_{G, \operatorname{cost}}\left(P_{G, A, i}\right)+\left(y_{C A}^{k}\right)^{T}\left(x_{C A}-z^{k}\right) \\ & +(\rho / 2)\left\|x_{C A}-z^{k}\right\|_{2}^{2}, \quad A=1, \ldots, m \end{aligned} $$

Equality and inequality constraints use (5) and (6), respectively. Grant et al. developed an algorithm using MATLAB-based CVX, which is a modeling system for constructing and solving disciplined convex programs (DCPs)(12). The CVX is used to solve each area’s the ED problem. Algorithm for the ED is as follows:

4. SIMULATION RESULTS AND DISCUSSION

This section verifies the effectiveness of the proposed consensus ADMM formulation for the ED problem in a10-bus system. The solution is compared with the Newton-Raphson (N-R) method which is a centralized technique and used by Saadat(13). The system is shown in Fig. 1 and partitioned into 3 areas. The dotted line denotes the boundary of each area. Area 1 includes buses 11, 12, 13, 14 and 15. Area 2 includes buses 21, 22, 23, 24 and 25. The consensus area includes buses 12,15, 22 and 25. Each bus has their own load. Six generators are connected on bus 11, 12 and 13 at Area 1, and 21, 22 and 23 at Area 2, respectively. Table 1 shows the generation capacity and coefficients of energy costs. The simulations were executed in a sequential computational environment, using MATLAB R2020a(14).

Fig. 1. One-line diagram of a 10-bus system.

../../Resources/kiee/KIEE.2021.70.11.1640/fig1.png

The consensus ADMM is compared to the centralized N-R method for the convergence performance and the results in Figs. 2 and 3 show that the two algorithms converge to their optimal values fast.

Fig. 2. Consensus ADMM ().

../../Resources/kiee/KIEE.2021.70.11.1640/fig2.png

Fig. 3. Power generation and total cost of the centralized N-R method.

../../Resources/kiee/KIEE.2021.70.11.1640/fig3.png

Table 2 shows the converged outputs of the consensus ADMM and the centralized N-R method. From the results, it can be seen that the outputs from the consensus ADMM are superior to those from the centralized N-R method. Furthermore, there is an additional merit in consensus ADMM compared with the centralized N-R method. The centralized N-R method needs a central operator that collects and transfers the power output commands and updated value of each area, which violates the privacy of each system operator. However, the proposed consensus ADMM utilizes the local information only, hence the privacy of each system operator can be protected.

5. CONCLUSION

In this paper, an approach for the ED between interconnected utilities based on consensus ADMM is proposed. The proposed method was applied to a 10-bus system to demonstrate their effectiveness and compared to the centralized Newton-Raphson method. The proposed consensus ADMM approach showed a superior performance compared to the N-R method. Furthermore, it utilizes only the local information so that the privacy of each system operator can be guaranteed and the regional sub-problems can be solved more efficiently. For future research, the security issues such as the thermal limits of transmission line and voltage magnitude when the outage is occurred in the system can be considered in large power system connected to multiple zones.

Table 1. The generation capacity and coefficients of energy costs.

Bus

a

b

c

Pmin[MW]

Pmax[MW]

11

200

7.0

0.008

10

150

12

180

6.3

0.009

10

100

13

140

6.8

0.007

10

120

21

200

7.0

0.008

10

150

22

170

6.3

0.006

10

100

23

140

6.8

0.007

10

120

Table 2. Comparison of Consensus ADMM and N-R method.

Area

No. of Gen.

Consensus ADMM

Centralized N-R Method

Generation [MW]

Cost [$/h]

Generation [MW]

Cost [$/h]

1

11

23.65

370.02

27.31

397.11

13

38.72

413.76

55.25

537.09

Sum of Area 1

62.37

783.78

82.56

934.20

2

21

28.46

405.68

21.29

352.63

23

44.05

453.11

51.65

509.88

Sum of Area 2

72.51

858.79

72.94

862.51

Consensus

12

59.96

590.13

69.11

658.38

22

89.95

785.20

98.35

847.67

Sum of Consensus Area

149.91

1375.33

167.46

1506.05

Total

284.79

3017.90

322.96

3302.78

Acknowledgements

References

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저자소개

김규호(Kyu-Ho Kim)
../../Resources/kiee/KIEE.2021.70.11.1640/au1.png

Kyu-Ho Kim received the B.S., M.S. and Ph.D. degrees from Hanyang University, South Korea, in 1988, 1990 and 1996, respectively.

He is a Professor in the Department of Electrical Engineering at Hankyong National University, South Korea.

He was a Visiting Scholar at Baylor University for 2011-2012 and Unversity of Colorado Denver for 2020-2021.

His research interests include power system control and operation, optimal power flow and the development of control techniques for wind power plants.