Abstract
Micro-phasor measurement units (μPMUs) with a micro-second resolution and milli-degree accuracy capability are expected to play an important role in improving the state estimation accuracy in the distribution network with increasing penetration of distributed generations. Therefore, this paper investigates the problem of how to place a limited number of μPMUs to improve the state estimation accuracy. Combined with pseudo-measurements and supervisory control and data acquisition (SCADA) measurements, an optimal μPMU placement model is proposed based on a two-step state estimation method. The E-optimal experimental criterion is utilized to measure the state estimation accuracy. The nonlinear optimization problem is transformed into a mixed-integer semidefinite programming (MISDP) problem, whose optimal solution can be obtained by using the improved Benders decomposition method. Simulations on several systems are carried out to evaluate the effective performance of the proposed model.
PHASOR measurement units (PMUs) can provide real-time magnitude and phase angle information of voltage and current with high accuracy, from which many applications such as network observability, state estimation (SE), and safety protection and coordinated control can benefit [
Most of existing OPP methods ensure complete network observability with a minimum number of installed PMUs or a minimum installation cost. The network observability is mainly divided into topological observability [
Furthermore, some other possible scenarios [
However, it is worth noting that studying the full rank of gain matrix or Jacobian matrix of SE can obtain network observability and ensure the executability of SE [
Therefore, on the basis of the existing studies, this paper proposes an optimal µPMU placement method to obtain the optimal solution, thus maximizing SE accuracy in distribution networks. A covariance matrix of SE error in rectangular coordinates is formulated based on mixed measurements and the two-step SE method [
The main contributions of this paper are listed as follows.
1) An optimal µPMU placement model is proposed to minimize the worst error variance of SE based on E-optimal experimental criterion for a given number of PMUs, thus improving the SE accuracy. Pseudo-measurements, SCADA measurements, and zero injection buses (ZIBs) are collectively utilized for a two-step SE to reduce the number of required µPMUs.
2) The proposed nonlinear optimization model is transformed to an MISDP problem by introducing an auxiliary variable and the improved Benders decomposition method is used to obtain the optimization solution.
3) Global optimal solutions can be obtained by utilizing the improved Benders decomposition method compared with convex relaxation and greedy approach, and the solving time is much less than CUTSDP solver, conventional Benders decomposition method, and branch and bound method, especially when the number of PMUs is large.
The rest of the paper is organized as follows. Section II introduces the proposed methodology for SE and an evaluation index of estimation accuracy. Section III establishes the optimal PMU placement model and proposes the solution methodology. Section IV presents the results of case study. Finally, Section V draws the conclusions.
Among SE methods utilized in the OPP problem, the most common one is WLS method [
For a three-phase network with N nodes (including Nsrc active nodes and Nload passive nodes), the brief calculation steps of the two-step SE method in rectangular coordinates are presented as follows. For more details, please refer to [
In the first step, prior estimation is carried out by using pseudo-measurements, and the voltage formula at iteration k can be expressed as:
(1) |
where and are the real and imaginary parts of , respectively; is the matrix of pseudo-measurements that are known beforehand; is the voltage matrix under zero loads in rectangular coordinates; and is the coefficient matrix at iteration k.
The iteration stops only when the gap between the calculated and the real is less than the given value. After the first step of SE, the estimation error covariance of can be obtained from (2).
(2) |
where is the standard deviation of pseudo-measurements; and is the initial coefficient matrix.
As analyzed in [
In the second step, SCADA and µPMU measurements are utilized to carry out a linear post-estimate combined with and obtained in the first step. The post-estimate voltage can be expressed as:
(3) |
where and are the vectors of Npm µPMU measurements and Nsm SCADA measurements, respectively; is the matrix mapping state voltages to µPMU measurements; is the matrix mapping state voltages to SCADA measurements; and is the gain matrix obtained by minimizing the error covariance , which can be expressed as:
(4) |
(5) |
(6) |
where and are the covariance matrices of the µPMU and SCADA measurement noises, respectively; and is the state voltage vector.
The relationship between the voltage and µPMU measurement i at phase p of bus k is shown in (7), and the relationship between the voltage and SCADA measurement i at phase p of bus k is shown in (8).
(7) |
(8) |
where is the voltage at phase p of bus k; denotes the
Finally, the first-order approximation of the estimation error covariance can be obtained from (9).
(9) |
The error covariance matrix is obtained through a two-step SE method in the previous subsection and indicates the estimation accuracy that the measurement system can achieve. The scalar-valued function is usually chosen as an indicator to evaluate the accuracy of SE in the estimation field [
(10) |
where is the maximum eigenvalue of and its physical meaning is the worst error variance of an estimate. In this paper, the worst error variance, i.e., E-optimal design, is selected as the evaluation index of SE accuracy and the optimal µPMU placement model is established based on this strategy.
According to [
(11) |
where denotes the complex conjugate transpose. Further, since the measurements at different nodes are independent of each other, can be rewritten as:
(12) |
where and are the binary variables, means that a µPMU is installed at node i, and means that other measurement device such as feeder terminal unit (FTU) is installed at node i, 0 otherwise; im is the serial number of the
In this paper, we only focus on the optimal placement of µPMUs with unlimited number of channels, so the variable is known by default and only the variable is optimized. Therefore, the latter formula in (12) related to can be replaced with a constant and can be expressed as (13) which is only related to the variable .
(13) |
The objective of the optimal µPMU placement model in this paper is to improve the SE accuracy, and the constraint is the number of installed µPMUs. Therefore, combined with the E-optimal standard design in Section II-B as well as the SE error covariance matrix (13), the model can be written as:
(14) |
where is the maximum number of µPMUs.
As shown in (13), in (14) involves the inversion of variable matrix, which makes (14) be a nonlinear optimization problem and hard to obtain the largest eigenvalue of directly. As we know, the maximum eigenvalue of a matrix equals to the inverse of the minimum eigenvalue of its inverse matrix. In this way, the original problem of minimizing the maximum eigenvalue of can be converted to a problem of how to maximize the minimum eigenvalue of :
(15) |
where is the minimum eigenvalue of .
By introducing the auxiliary variable t [
(16) |
where is an identity matrix.
The proof of transformation is presented as follows.
Proof Constraint in (16) means that all eigenvalues of are greater than or equal to 0, which means that when , always holds. In order to achieve this, the value of t needs to satisfy that the solution of the inequality with as a variable is greater than or equal to 0. In this case, when , t gets the maximum value which is equal to the minimum eigenvalue of , i.e., . Therefore, (16) is equivalent to (15).
Considering the mathematical characteristics of the MISDP model (16), Benders decomposition is utilized in this paper to decompose the model into a simple mixed-integer linear programming (MILP) master problem and an SDP subproblem with continuous variables [
The subproblem is an SDP model where the discrete variables are given, and the expression is as follows:
(17) |
where is the µPMU placement result of node i at iteration k; and is the dual variable corresponding to the constraint at iteration k and it can be obtained when the subproblem is solved.
The master problem is an MILP model, and the expression is as follows:
(18) |
where is the objective function value of the subproblem with at iteration l; and is an auxiliary variable added in the solving process. During the process of solving the MISDP model based on the Benders decomposition for optimal µPMU placement, Benders cuts formed by and play the role of connecting the master problem and the subproblem.
The master problem is formed by relaxing the constraints of the original problem, while the subproblem is a restricted version of the original problem. Therefore, the Benders decomposition method regards the objective function values of master problem and subproblem as lower and upper bounds of original problem, respectively. During iterations, the objective function value of the master problem increases with the Benders cuts provided by the subproblem. Finally, the method converges until the gap between objective function values of master problem and subproblem is less than a given value. However, it is worth noting that the new master problem must be solved from scratch when the Benders cut generated by the subproblem is added as a new constraint in each iteration. As the number of iterations increases, the more Benders cuts are added to (18), the longer it takes to solve the master problem.
Therefore, the improved Benders decomposition method based on lazy constraints is utilized to solve this problem in this paper [
The procedure of the improved Benders decomposition method using lazy constraints can be summarized as follows.
Step 1: . Start to solve the MILP master problem (18) without Benders cut. When a feasible solution is found, go to Step 2.
Step 2: solve the SDP subproblem (17) using feasible solution to obtain the objective function value of subproblem and the dual variable .
Step 3: form the Benders cut according to the feasible solution , the dual variable , and the objective function value of subproblem. Add the Benders cut to the master problem as lazy constraint.
Step 4: . Continue to solve the master problem. If a feasible solution is found, return to Step 2. If the convergence precision (1
With lazy constraints, there is no need to solve the master problem from scratch after each cut is added, which greatly accelerates the solving process. Therefore, using the improved Benders decomposition method to solve the MISDP problem ensures the solution efficiency.
In addition, it is known that the convergence of the improved Benders decomposition method for the MISDP problem is guaranteed as long as the envelope of function is convex, where is the function that relaxes all 0-1 variables into continuous variables and provides the optimal objective function value of problem (19) for given values of [
(19) |
Since the SDP function is convex according to the definition of convex programming, the convergence of the proposed MISDP problem can be guaranteed, which ensures the optimal solution of discrete variables. Together with the fact that the SDP model (17) can ensure the global optimality of the solution of continuous variables, the proposed method can finally obtain -global optimal solution, i.e., the error between the obtained optimal solution and the true global optimal solution does not exceed .
To implement the proposed method, a complete flowchart is presented in

Fig. 1 Flowchart of proposed method.
The proposed optimal µPMU placement model is evaluated in two balanced systems, namely the IEEE 33-bus and 123-bus systems [

Fig. 2 Topology of IEEE 33-bus system.

Fig. 3 Topology of IEEE 123-bus system.
It is worth noting that balanced systems are considered for simplicity in this paper whereas the proposed method is also applicable to unbalanced distribution systems. The pseudo-measurements utilized in the test cases include active and reactive power injection of each node. Since these measurements are estimations rather than actual values, the corresponding noise is modeled with a relatively large standard deviation with [
In this subsection, approximate substitution method which is utilized to deal with ZIBs is compared with linear transformation method [
No. of ZIBs | Approximate substitution method | Linear transformation method [ | ||||
---|---|---|---|---|---|---|
µPMU placement result | Objective value | Solving time (s) | µPMU placement result | Objective value | Solving time (s) | |
None | 2, 6, 11, 25 | 62880 | 122 | 2, 6, 11, 25 | 62880 | 122 |
6 | 2, 7, 12, 26 | 75640 | 121 | 2, 7, 12, 26 | 75640 | 120 |
6, 11 | 2, 7, 11, 26 | 78000 | 128 | 2, 7, 12, 26 | 75780 | 112 |
6, 11, 14 | 1, 4, 9, 25 | 79710 | 116 | 1, 4, 9, 25 | 79710 | 108 |
6, 11, 14, 25 | 1, 5, 10, 25 | 90610 | 130 | 1, 4, 9, 26 | 82570 | 101 |
6, 11, 14, 25, 31 | 1, 5, 8, 13 | 114920 | 126 | 1, 3, 7, 9 | 97580 | 95 |
The sensitivity of the µPMU placement results to the SCADA measurement placement is tested in IEEE 33-bus system. The number of nodes with SCADA measurements Ns ranges from 0 to 4. On the basis of the SCADA measurements in the previous round, one of the nodes configured with µPMUs from the previous round is selected to be equipped with SCADA measurements in the current round. ZIBs are not considered here.

Fig. 4 µPMU placement results when Nset=4 under different SCADA measurement placements using proposed method. (a) . (b) . (c) . (d) . (e) .
It can be observed that the proposed method can take good account of the SCADA measurements in all cases, thus making the distribution of measurements in the system more uniform. Additionally,

Fig. 5 Optimization results under different SCADA measurement placements and different numbers of µPMUs using proposed method.

Fig. 6 Optimization results using different methods in IEEE 33- and 123-bus systems. (a) IEEE 33-bus system. (b) IEEE 123-bus system.
Before tests, IEEE 33- and 123-bus systems are both configured with SCADA measurements and ZIBs randomly. Particularly, there are 11 SCADA measurements and 2 ZIBs in IEEE 33-bus system, and 43 SCADA measurements and 5 ZIBs in IEEE 123-bus system.
Item | IEEE 33-bus | IEEE 123-bus |
---|---|---|
No. of ZIBs | 14, 30 | 14, 26, 36, 76, 110 |
No. of nodes with SCADA measurements | 16, 19, 32 | 3, 12, 28, 42, 57, 69, 84, 89, 101, 122 |
Number of voltage measurements | 3 | 10 |
Number of injected current measurements | 3 | 10 |
Number of branch current measurements | 5 | 23 |
Firstly, the IEEE 33-bus system is tested in this case. Nset ranges from 1 to 8. Several methods are compared with the proposed method to solve the E-optimal µPMU placement problem.
The greedy approach in [
Then, the node with the minimum value is added to PMU placement set and removed from candidate nodes. The approach terminates until the number of PMUs reaches the upper limit.
Convex relaxation method in [
Enumeration method can list and compare all possible PMU placement schemes, so it can ensure the optimality of final results. However, this method can only be utilized in small-scale systems considering long computation time.
The results of each method are shown in
Then, the IEEE 123-bus system is considered in this case. Nset is set as 1 to 30 and two methods, i.e., greedy approach and convex relaxation method, are compared with the proposed method.
Additionally,
Number of µPMUs | Optimal µPMU placement result |
---|---|
1 | 5 |
2 | 2, 7 |
3 | 2, 5, 9 |
4 | 2, 6, 11, 25 |
5 | 1, 4, 7, 13, 28 |
6 | 1, 2, 5, 7, 13, 29 |
7 | 1, 2, 5, 7, 13, 25, 29 |
8 | 1, 2, 5, 7, 11, 14, 27, 29 |
Since the ultimate goal of optimal µPMU placement in this paper is to improve the SE accuracy, the effect of placement results on the SE accuracy is also investigated when . Errors of voltage magnitude and phase angle in the IEEE 33-bus system are shown in

Fig. 7 Errors of voltage magnitude and phase angle in IEEE 33-bus system. (a) Voltage magnitude. (b) Phase angle.
A practical distribution network with 446 nodes is tested to discuss the applicability of the proposed method to practical power systems. A total number of 890 pseudo-measurements and 127 SCADA measurements are considered. Nset is set to be , respectively. Greedy approach and convex relaxation method are compared with the proposed method. It can be observed from

Fig. 8 Optimization results using different methods in a practical distribution network.
Finally, the E-optimal µPMU placement problem modeled as an MISDP problem is also solved by utilizing the following three methods: ① the conventional Benders decomposition (Method 1) introduced in Section III; ② the CUTSDP solver [
Method 2 is provided by a MATLAB toolbox named YALMIP [
Nset | Proposed method | Method 1 | Method 2 | Method 3 | ||||
---|---|---|---|---|---|---|---|---|
Solving time (s) | Objective value | Solving time (s) | Objective value | Solving time (s) | Objective value | Solving time (s) | Objective value | |
1 | 2 | 3800 | 8 | 3800 | 1 | 3800 | 47 | 3800 |
2 | 18 | 9000 | 86 | 9000 | 11 | 9000 | 255 | 9000 |
3 | 33 | 54300 | 183 | 54300 | 51 | 54300 | 637 | 54300 |
4 | 130 | 65800 | 1104 | 65800 | 1654 | 65800 | 2883 | 65800 |
5 | 399 | 123100 | 53180 | 123100 | 17860 | 123100 | 7290 | 123100 |
6 | 724 | 192500 | 285700 | 192500 | 136300 | 192500 | 18500 | 192500 |
7 | 809 | 211800 | 48510 | 211800 | ||||
8 | 2775 | 323200 | 70200 | 323200 |

Fig. 9 Convergence diagram of four methods in IEEE 33-bus system.
Moreover, the solving time of the proposed method in the case of 30 µPMUs is 3.1 hours and 11.5 hours for the IEEE 123-bus system and the practical distribution network, respectively. While Methods 1, 2, and 3 cannot complete the solution of the same OPP problem within 24 hours. In summary, the proposed method shows higher solution efficiency than Methods 1, 2, and 3. It proves that the proposed method can deal with discrete variables and continuous variables efficiently and can obtain the global optimal solution at the same time.
This paper presents an optimal µPMU placement method in distribution networks. A new model for optimal µPMU placement is established based on a two-step SE method, which can not only improve the SE accuracy, but also take into account the impact of SCADA measurements and ZIBs. The nonlinear optimization problem is transformed into an MISDP problem and an improved Benders decomposition method is utilized to solve it to obtain global optimal solution. Simulations on several test cases verify the feasibility, optimality, and high efficiency of the proposed method by contrast with other methods.
Future work could include studying the optimal µPMU placement problem by combining SE with other applications such as fault location to meet multi-application requirements of distribution networks.
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