Pragmatic multi-stage simulated annealing for optimal placement of synchrophasor measurement units in smart power grids

Abstract

Conventional power grids across the globe are reforming to smart power grids with cutting edge technologies in real time monitoring and control methods. Advanced real time monitoring is facilitated by incorporating synchrophasor measurement units such as phasor measurement units (PMUs) to the power grid monitoring system. Several physical and economic constraints limit the deployment of PMUs in smart power grids. This paper proposes a pragmatic multi-stage simulated annealing (PMSSA) methodology for finding the optimal locations in the smart power grid for installing PMUs in conjunction with existing conventional measurement units (CMUs) to achieve a complete observability of the grid. The proposed PMSSA is much faster than the conventional simulated annealing (SA) approach as it utilizes controlled uphill and downhill movements during various stages of optimization. Moreover, the method of integrating practical phasor measurement unit (PMU) placement conditions like PMU channel limits and redundant placement can be easily handled. The efficacy of the proposed methodology has been validated through simulation studies in IEEE standard bus systems and practical regional Indian power grids.

Graphical abstract

Keywords

phasor measurement units (PMUs) / pragmatic PMU placement / simulated annealing (SA) / western region Indian power grid (WRIPG)

1 Introduction

The exhilarating advancements in computational and communication technologies revolutionize the conventional power grids to smart power grids with advanced monitoring and control. Synchrophasor measurement units, such as phasor measurement units (PMUs), are the principal segment of smart power grid monitoring system [1]. PMUs deliver GPS synchronized real time positive sequence voltage and current phasor measurements at the buses where they are installed [2]. These synchronous measurements along with the time stamp are transmitted to the system protection center (SPC) through high speed telecommunication system. This facilitates the SPC to estimate the state of the grid with better accuracy and with enhanced speed of operation compared to the conventional approaches using remote telemetry units (RTUs) [2−6].

Although installing PMUs at all the buses in the smart power grid will deliver the direct measurement of the system, it may be either impracticable or undesirable due to physical and economic constraints [7−11]. Physical constraints include practical infeasibility of replacing all CMUs in the grid with PMUs and redundancy of placing PMUs at all buses as the PMU installed at a bus can observe neighboring connected buses using Ohm’s law. Economic constrains arise due to the heavy investment required for PMUs.

Therefore, the optimal placement of PMUs in smart power grids is becoming a great research interest in recent years. Many optimization algorithms are being published for finding the optimal locations for the PMUs in a power grid. Nuqui et al. [9] presented a tree search method based on the spanning trees of a power system graph for optimal PMU placement (OPP). But the heavy computational requirements and high time consumption limit the methodology to small power grids. Borka and Miroslav [10] presented a genetic algorithm (GA)-based approach for OPP. The classical GA was modified with the bus ranking methodology for attaining optimal solution. But he did not consider the existing CMUs in the power grid. Gou [11] proposed an integer linear programming (ILP) for solving the optimal PMU placement problem. But ILP leads to single solution even if multiple solutions are available. Besides, when considering the existing CMUs, buses have to be merged with each other and become problematic if the algorithm proposes to place the PMU at the merged bus. Mohammadi-Ivatloo [12] proposed a topology-based GA for optimal placement of PMUs. But he did not consider the existing CMUs in the power system network. Baldwin et al. [13] proposed a dual search method to find the optimal PMU locations in a power system network without considering the existing conventional measurements. The modified bisecting method was used to fix the number of PMUs and simulated annealing (SA) had been used for randomly selecting PMU sets. But the longer time requirement for converging to optimal solution limits the methodology for small power grids. Redundant placement of the PMU which ensures the observability under contingencies, such as failure of measurement unit or loss of transmission line using ILP, was presented in Ref. [14]. The algorithm first finds the optimal PMU locations using ILP and then redundancy is increased by solving a slave ILP. But the proposed method of optimization is not preferable for large power system networks. The PMU placement technique was proposed in Ref. [15] based on the triangular factorization of Jacobian matrix for analyzing the redundancy and observability of a power grid. But no method was proposed to improve the redundancy of measurement. The optimal placement methodology based on ILP for channel limited PMUs was presented in Ref. [16]. From literature review, it can be concluded that, either quality of the solution or computational simplicity has to be compromised when applying the majority of the existing optimal PMU placement methods to practical power grids. In addition, many papers assume that PMUs can measure any number of transmission lines connected to the bus. But practical PMUs have only limited number of channels. Hence, the majority of the PMU placement methodologies are not feasible for practical application.

This paper proposes a pragmatic multi-stage simulated annealing (PMSSA) approach for practical placement of PMUs in conjunction with existing CMUs in a power grid. The proposed method considers PMUs to have limited number of current measurement channels and hence, it can be directly applied to industrial available PMUs. The PMSSA applies multiple stages in the optimization procedure with controlled uphill and downhill movements which facilitate faster convergence to OPP configuration in contrast with the classical SA. Numerous case studies have been conducted in standard IEEE bus systems for verifying the effectiveness of the proposed algorithm. Practical case studies have been performed in southern and western regional Indian power grids and optimal geographical bus locations for PMU placement in these regions are presented.

2 PMU assisted monitoring system

The concept of complete observability in smart power grids using monitoring system incorporated with PMUs at optimal locations is presented in this section. A smart power grid is completely observable if all the bus voltage and current phasors can be either directly measured or estimated with high accuracy so that the power flow values in the grid can be estimated [2]. The PMUs can be installed in a smart power grid in two ways for attaining complete observability. The first approach is to install the PMUs at all the buses, neglecting all the existing CMUs, so that smart power grid will be completely observable. But this approach may not be practically feasible due to physical and economic constraints explained in the preceding section. The second approach is to optimally incorporate the PMUs along with the existing CMUs, so that the measurement from both the CMUs and PMUs can be used for achieving complete observability of the smart power grid [8−11]. Since the update rate of the PMUs and the CMUs are different, the synchronous operation of both units requires proper time stamping [17]. Extensive system specific studies are to be conducted for finding the best time for synchronization of measurements from both units [17,18].

The concept of complete observability and optimal placement of PMUs is illustrated with the help of a 7-bus system as shown in Fig. 1 [19−21].

By applying OPP methods [8−12], it can be seen that installing one PMU at bus-3 and another at bus-5 makes the 7-bus system completely observable. Hence, with optimal placement, only two PMUs are sufficient for achieving complete observability of the 7-bus system. Complete observability with these two PMUs is achieved as follows: the PMU at bus-3 directly observes bus-3 and indirectly observes buses 1, 2 and 4 with the help of Ohm’s law. Similarly, the PMU at bus-5 directly observes bus-5 and indirectly observes buses 4, 6 and 7. Hence, buses 3 and 5 are called directly-observed buses and the rest are called indirectly-observed buses or associate buses.

Optimal placement of the PMUs along with the existing CMUs is elucidated with the same 7-bus system with the CMUs, as shown in Fig. 2. The rectangular box in the system represents the existing power flow measurement (PFM) unit and the arrow with a circle indicates the injection measurement (IM) unit.

In this case, only one PMU, the one at bus-5, is sufficient for the complete observability of the seven bus system. The PMU at bus-5 can directly observe bus-5 and indirectly observe buses 4, 6 and 7 with the help of Ohm’s law. Buses 1, 2 and 3 can be made observable with the help of CMU measurements as follows: from the estimated voltage phasor at bus-4 (V₄), the line current through the branch 3-4 (I₃₄) can be estimated using the PFM, as given in Eq. (1).

(1)I_{\mathrm{34}}={\displaystyle \frac{P_{\mathrm{34}}-j{Q}_{\mathrm{34}}}{V_{\mathrm{4}}}},

where P₃₄ and Q₃₄ are the active and reactive power flows in branch 3−4 respectively measured by the PFM unit. Now, the voltage phasor at bus-3 can be estimated using Eq. (2).

(2)V_{\mathrm{3}}=V_{\mathrm{4}}-I_{\mathrm{34}}{Z}_{\mathrm{34}},

where Z₃₄ represents the impedance of branch 3−4. In similar way, the voltage phasor at bus-1 also can be estimated. For calculating the voltage phasor at bus-2, the line current in branch 2−3 (I₂₃) is required. I₂₃ can be estimated by applying KCL at bus-3 which can be expressed as Eq. (3):

(3)I_{\mathrm{23}}=I_{\mathrm{34}}-I_{\mathrm{13}}-I_{\mathrm{3}}

where I₃ represents the current measurement from the IM unit at bus-3. With this estimated value of branch current I₂₃, the voltage phasor at bus-2 can be estimated similar to Eq. (2).

Hence the joint placement of PMUs along with the CMUs renders complete observability with fewer numbers of PMUs compared to the monitoring system with the PMUs alone.

3 Proposed PMSSA algorithm

The proposed PMSSA algorithm for optimal placement of PMUs in conjunction with the existing CMUs in a smart power grid for the complete observability is described in this section. The basic objective function for OPP can be expressed mathematically in Eqs. (4) and (5)，

(4)\min {\displaystyle {\sum }_{k=\mathrm{1}}^N{x}_k},

subject to the constraint that

(5)\mathit{M}\mathit{X}\ge \mathit{T},

where N is the total number of buses in the smart power grid, M is the incidence matrix, T is the observability matrix, a column matrix of size (N× 1). All values of T are equal to ‘one’ if PMU placement without redundancy is considered. The depth of redundancy of the ith bus can be changed by changing the ith value of T.

As outlined in the preceding section, the PMSSA consists of multiple stages during the optimization process. The optimization starts with a random PMU configuration P_N of size (1 × N).The ith element of the configuration is defined in such a way that P_N(1, i) is equal to ‘one’ if the PMU is present at the ith bus of the smart power grid and zero otherwise. The number of PMUs in the configuration P_N is represented by N_P. The proposed algorithm optimizes this PMU configuration such that the smart power grid will be completely observable with the minimum number of channel limited PMUs and the existing CMUs. The first stage of the PMSSA starts with a randomly generated PMU configuration P_N. In contrast to the classical SA approach for OPP, the proposed PMSSA does not necessitate the starting PMU configuration to deliver complete observability.

Before starting the first stage of optimization, the cooling schedule for the PMSSA is developed based on the number of buses and average nodal degree (D_g) of the buses in the smart power grid. The cooling schedule consists of starting temperature (T₀) and cooling rate (T_n+1) considered for the optimization. The cooling rate decides the rate at which the starting temperature has to be reduced during the optimization. The mathematical expressions for starting temperature and cooling rate are given in Eqs. (6) and (7) respectively.

(6)\alpha ,\beta \ge \mathrm{1},

(7)\begin{array}{lll}{T}_{n+\mathrm{1}}={\displaystyle \frac{T_n}{(\mathrm{1}+e^{-\delta N})}}\hfill & \mathrm{0}\hfill & \delta \le 1\hfill \end{array},

where α, β and δ are constants. Since PMSSA optimization consists of multiple stages, a cutoff temperature is set for each stage. Once the temperature reaches cutoff temperature, the optimization switches over to next stage.

In first stage of optimization, the PMU configuration P_N is perturbed to generate neighboring configuration P_N + ΔP_N [8]. The number of PMUs in the newly formed configuration is represented as N_P^{\prime }. The newly formed PMU configuration is accepted or rejected based on the observability constant (B₀) and redundant observability constant (R₀) as given in Eqs. (8) and (9) respectively. B₀ will be greater than zero only if the power grid network is completely observable with the PMU configuration. R₀ is a measure of redundancy of PMU configuration. R₀ can take a value between zero and one.

(8)B_{\mathrm{0}}={\displaystyle \frac{{\prod }_{i=\mathrm{1}}^N[{\displaystyle {\sum }_{j=\mathrm{1}}^N{\mu }_{i}M(i,j)\times P_N(\mathrm{1},j)}/N]}{N}},

(9)R_{\mathrm{0}}={\displaystyle \frac{{\displaystyle {\sum }_{i=\mathrm{1}}^N[{\displaystyle {\sum }_{j=\mathrm{1}}^N{\mu }_{i}M(i,j)\times P_N(\mathrm{1},j)}/N]}}{N}},

where μ_i is the channel limit factor (CLF), defined as

{\mu }_i=\{\begin{array}{ll}\mathrm{1}\hfill & if\hspace{0.17em}PMU\hspace{0.17em}is\hspace{0.17em}installed\hspace{0.17em}at\hspace{0.17em}ith\hspace{0.17em}bus,\hfill \\ \mathrm{1}\hfill & if\hspace{0.17em}ith\hspace{0.17em}bus\hspace{0.17em}is\hspace{0.17em}an\hspace{0.17em}associate\hspace{0.17em}bus,\hfill \\ \mathrm{0}\hfill & otherwise.\hfill \end{array}

CLF assists in placing PMUs with channel limits, as it is a direct indication of indirectly observed buses. The pseudo code of selection criteria for Stage1 optimization is given as

\mathrm{\Delta }{N}_P=N_P^{\prime }-N_P

If\hspace{0.17em}{B}_{\mathrm{0}}\ne \mathrm{0}

Accept the new PMU configuration

If\hspace{0.17em}{B}_{\mathrm{0}}=\mathrm{0}

Accept the new PMU configuration if

R_{\mathrm{0}}\ge \mathrm{0.5}

\mathrm{e}\mathrm{l}\mathrm{s}\mathrm{e}\hspace{0.17em}\mathrm{R}\mathrm{e}\mathrm{j}\mathrm{e}\mathrm{c}\mathrm{t}

If\hspace{0.17em}\mathrm{\Delta }{N}_P\ge \mathrm{0}

Accept the new PMU configuration if

R_{\mathrm{0}}\ge \mathrm{0.8}

\mathrm{e}\mathrm{l}\mathrm{s}\mathrm{e}\hspace{0.17em}\mathrm{R}\mathrm{e}\mathrm{j}\mathrm{e}\mathrm{c}\mathrm{t}

During this stage, the perturbed PMU configuration can have either more than or less than or equal number of PMUs compared to previous configuration. If the perturbed PMU configuration has lesser number of PMUs and deliver complete observability (B₀ greater than zero), the solution is accepted. If it does not deliver complete observability, then it is accepted only if the new configuration (perturbed PMU configuration) has a R₀ value greater than 0.5. The higher the value of R₀ is, the more the redundancy in the measurements is and hence the higher the number of PMUs is. If R₀ is less than 0.5, the new configuration is rejected and the previous configuration is kept. If the new PMU configuration has more or equal number of PMUs than the previous configuration, that configuration is accepted only if its R₀ is greater than 0.8, otherwise, the new configuration is rejected and the previous configuration is kept. This enables a controlled uphill movement during the optimization.

The first optimization is repeated till the temperature reaches Stage-1 cutoff temperature. After completing the first optimization, the new PMU configuration is modified to cope with the existing CMUs in the grid. This is achieved by applying joint placement criteria (JPC) as follows:

Criteria 1: It is redundant to place PMUs at the buses having CMUs.

Criteria 2: For a parent bus (bus installed with PMU) having y connected buses and in these buses, if y–1 buses are observable, then the yth bus voltage can be estimated.

Criteria 3: In Criteria 2, only one bus in y–1 buses is required to be directly observed by a PMU and the other y–2 buses can be indirectly observed buses.

Criteria 4: For an associate bus (bus observed by a PMU in the neighboring bus), all of its neighboring buses, except the parent bus, can be indirectly observed buses.

JPC ensures that placement of the PMUs along with the existing CMUs provides complete observability of the smart power grid. These four criteria can be expressed mathematically in Eqs. (10)–(12)

(10)[{\mathit{P}}_N].{[{\mathit{S}}_n^i]}^T=\mathbf{0}

(11)[{\mathit{S}}_n^p].{[{\mathit{P}}_N]}^T=\mathbf{0}

(12)[{\mathit{S}}_n^a].{[{\mathit{P}}_N]}^T=\mathbf{0}

where [{\mathit{S}}_n^i], [{\mathit{S}}_n^p] and [{\mathit{S}}_n^a] are the injection measurement matrix, power flow measurement matrix and associate bus matrix, respectively. These matrices are defined as

(13)S_n^i(\mathrm{1},k)=\{\begin{array}{ll}\mathrm{1}\hfill & if\hspace{0.17em}injection\hspace{0.17em}measurement\hspace{0.17em}is\hspace{0.17em}present\hspace{0.17em}in\hspace{0.17em}kth\hspace{0.17em}bus,\hfill \\ \mathrm{0}\hfill & otherwise;\hfill \end{array}

(14)S_n^p(\mathrm{1},j)=\{\begin{array}{ll}\mathrm{1}\hfill & if\hspace{0.17em}flow\hspace{0.17em}measurement\hspace{0.17em}is\hspace{0.17em}present\hspace{0.17em}between\hspace{0.17em}buses\hspace{0.17em}i\hspace{0.17em}and\hspace{0.17em}j,\hfill \\ \mathrm{0}\hfill & otherwise;\hfill \end{array}

(15)S_n^a(\mathrm{1},b)=\{\begin{array}{ll}\mathrm{1}\hfill & if\hspace{0.17em}b\hspace{0.17em}is\hspace{0.17em}an\hspace{0.17em}associate\hspace{0.17em}bus,\hfill \\ \mathrm{0}\hfill & otherwise.\hfill \end{array}

After modification using JPC, if the PMU configuration does not deliver complete observability, then Stage-2 operation is conducted. In Stage-2, additional PMUs are placed at unobservable buses based on their nodal degrees. One PMU is placed at a time on an unobservable bus having the highest nodal degree and checked for complete observability. This is repeated till the PMU configuration delivers complete observability of the smart power grid. This forms Stage-2 operation and is exempted if the PMU configuration after JPC satisfies complete observability. The pseudo code for Stage-2 operation is

j=\mathrm{0}

Do

j\to j+\mathrm{1}

P_N(\mathrm{1},S_{ub}(\mathrm{1},j))=\mathrm{1}

Apply joint placement criteriaCalculate

\mathrm{C}\mathrm{a}\mathrm{l}\mathrm{c}\mathrm{u}\mathrm{l}\mathrm{a}\mathrm{t}\mathrm{e}{B}_{\mathrm{0}}

While (B_{\mathrm{0}}>\mathrm{0})

After acquiring a PMU configuration with complete observability, Stage-3 optimization is started. Similar to Stage-1 optimization, PMU configuration is perturbed to find neighboring PMU configurations and the newly formed PMU configuration is accepted if it satisfies complete observability regardless of the number of PMUs in the configuration. This makes the algorithm traverse uphill and downhill. The pseudo code for Stage-3 optimization is

Calculate\hspace{0.17em}\mathrm{\Delta }{N}_P

Calculate\hspace{0.17em}{B}_{\mathrm{0}}

If\hspace{0.17em}{B}_{\mathrm{0}}\ne \mathrm{0}

Accept the new PMU configuration

If\hspace{0.17em}{B}_{\mathrm{0}}=\mathrm{0}

Reject

If\hspace{0.17em}\mathrm{\Delta }{N}_P\ge \mathrm{0}

If\hspace{0.17em}{B}_{\mathrm{0}}\ne \mathrm{0}

Accept the new PMU configuration

If\hspace{0.17em}{B}_{\mathrm{0}}=\mathrm{0}

Reject

After Stage-3 optimization is completed once the temperature reduces to Stage-3 cutoff value, Stage-4 optimization initiates. In Stage-4, the new PMU configuration is accepted only if it has lesser number of PMUs than previous configurations and satisfies complete observability. Stage-4 is continued till the temperature cools down to Stage-4 cutoff value. The pseudo code for Stage-4 criteria is

Calculate\hspace{0.17em}\mathrm{\Delta }{N}_P

Calculate\hspace{0.17em}{B}_{\mathrm{0}}

If\hspace{0.17em}{B}_{\mathrm{0}}\ne \mathrm{0}

Accept the new PMU configuration

If\hspace{0.17em}{B}_{\mathrm{0}}=\mathrm{0}

Reject

If\hspace{0.17em}\mathrm{\Delta }{N}_P\ge \mathrm{0}

Reject

The flowchart of complete optimization procedure is given in Fig. 3

The redundancy of the PMU configuration can be easily varied by varying the value of redundancy constant R_O and incorporating the redundancy condition in Stages 3 and 4.

4 Results and discussion

The proposed PMSSA algorithm has been validated by case studies conducted in various IEEE standard bus systems (IEEE-14 and IEEE-30) and practical Indian regional power grids. The results of these case studies are discussed in this section. The PMSSA algorithm and test systems have been modeled in Matlab/Simulink platform. The single line diagrams of IEEE-14 and IEEE-30 systems with CMUs are demonstrated in Figs. 4 and 5 respectively. The IMs in the network are shown by dark circles and PFMs by rectangular boxes. The detailed system data are given in Table 1. The optimal locations in the system for installing PMU without channel limitations are tabulated in Table 2. The same locations have been illustrated in Figs. 4 and 5 as circled buses. The optimal location for installing channel limited PMUs are also given in Table 2. Analogous to many practical industrial grade PMUs, the channel limit of three is considered in this paper.

Practical case studies have been conducted on Western Region Indian Power Grid (WRIPG) and Southern Region Indian Power Grid (SRIPG). The WRIPG comprises of four states (Gujarat, Madhya Pradesh, Maharashtra, and Chhattisgarh) and covers 15% of the total geographical area of India [20,21]. The WRIPG consists of 77 buses of UHV, EHV and HV which are interconnected through 145 branches [22].

The SRIPG is the second largest of the five regional power grids in India. It constitutes four states (Andhra Pradesh, Karnataka, Kerala, and Tamil Nadu) and covers 25% of the total geographical area of India. The SRIPG consists of 208 buses of UHV, EHV and HV which are interconnected through 312 branches [23].

The single line diagrams of the two regional power grids are depicted in Figs. 6 and 7. The geographical names of the buses in both regions are tabulated in Table 3. Buses No. 1 to No. 208 belong to the SRIPG and buses No. 417 to No. 493 belong to the WRIPG. The CMU locations assumed in this paper are given in Table1. Similar to the representations in IEEE bus systems, the CMUs and PMU locations are marked in Figs. 6 and 7.

The convergence of the proposed methodology toward optimal number of PMUs for all four case studies (IEEE-14, IEEE-30, WRIPG and SRIPG) are displayed in Figs. 8 to 11. The number of PMUs after each stage of optimization for the PMUs without channel limits are listed in Table 4. The results of the case study conducted with the PMUs with channel limitation of three are given in Table 5. Computational evaluation of the proposed methodology has been performed in a computer with the system configuration as follows. Intel i3 processor with a clock speed of 2.4GHz, 4GB RAM and 3MB of L2 Cache. The processor usage during optimization and simulation time taken for completing the optimization is taken as performance criteria. These criteria along with the starting temperature of the PMSSA for various test systems are tabulated for evaluating the performance. The evaluation results for optimization of the PMUs without channel limits and the PMUs with three channel limits are presented in Tables 6 and 7, respectively.

A comparative study of the proposed PMSSA has been conducted with the existing OPP methodologies such as ILP, classical GA and classical SA. The number of PMUs proposed for various test systems by these methodologies for complete observability of the smart power grid along with the simulation time taken for optimization is chosen for the comparative study. The results are tabulated in Table 8. It is seen from Table 8 that the proposed methodology outstands the existing methodologies for OPP.

5 Conclusions

Optimal placement of PMUs for the complete observability of smart power grids is essential owing to many physical and economic constraints. A PMSSA has been proposed in this paper for the optimal joint placement of PMUs in conjunction with the existing CMUs in smart power grids. Distinctively controlled uphill and downhill movements during multiple stages in the proposed methodology result in faster convergence to optimal PMU configuration than conventional approaches. The proposed optimization methodology considers the channel limit of industrial PMUs and hence can be directly applied for practical application. Case studies have been conducted in standard IEEE bus systems for verifying the efficacy of the algorithm for finding the optimal PMU configuration. Practical case studies are performed in regional Indian power grids and geographical locations are proposed for the OPP.

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