# Mathematical Models for the Single-Channel and Multi-Channel PMU Allocation Problem and Their Solution Algorithms

^{1}

^{2}

^{3}

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## Abstract

**:**

## 1. Introduction

## 2. Consideration of This Study

- (1)
- To achieve the proper PMU number by optimizing a linear or a quadratic cost function under a topological constraint function considering the limitation of the PMU channel limit capacity.
- (2)
- (3)
- We focus on finding the proper PMU number with a branch assignment set solution with a fixed number of channels.

## 3. Contribution of This Study

- We create an algorithmic connection between the OPP based-model with a restricted channel capacity and the optimal solutions produced by the suggested mathematical and heuristic algorithms.
- The branch-and-bound algorithm is used in a comparison study with nonlinear and heuristic algorithms.
- We present a state-of-the-art solution produced by the $0/1$ integer program solving with a zero absolute gap calculated with zero percentage gap tolerances; this means that this solution is a global one returned by the solver.
- We extend the ILP model into a heuristic search of the feasible set to show that evolutionary algorithms can efficiently perform the optimization following the BBA’s solution.
- Finally, a novel nonlinear program is created and solved by either continuous algorithms or a BBA embedded in the SCIP optimizer performed in MATLAB.

## 4. Materials and Methods

## 5. Optimization Models Studied for the OPP Model

- ✓
**Observability of the graph**

**Definition 1.**

**Definition 2.**

#### 5.1. Binary-Integer Programming Model Considering an Unlimited Number of Channel Capacities

**Definition 3.**

**Definition 4.**

**Definition 5.**

Algorithm 1: Steps of BBA methodology |

1. Calculate the lower bound: A procedure is used to achieve the lower bound on the objective function to solve the given sub-problem and deliver an estimated point. |

2. Calculate the upper Bound: An upper bound is evaluated to give the solution. |

3. Branching method: The sub-problem is portioned to deliver more than two children by a procedure strategy |

4. Methodology search: A search order is programmed to construct the binary tree. |

5. If the stopping criterion is satisfied, then end up the iterative scheme. |

6. The BBA ends delivering a solution as an output. |

Algorithm 2: Steps of GA implementation |

1 Set $k:=0$; starting from the population $P\left(0\right)$; |

2 Estimate $P\left(k\right)$; |

3 If the stopping criterion is completed, then end up the iterations; |

4 Choose $M\left(k\right)$ from $P\left(k\right)$; |

5 Evolve $M\left(k\right)$ to create $P(k+1)$; |

6 Set $k:=k+1$, go to step 2. |

Algorithm 3: Steps of PSO implementation |

Objective function $f\left(\overrightarrow{y}\right),\overrightarrow{y}=\left[{y}_{1},{y}_{2},\dots ,{y}_{n}\right]$. |

Initialize locations ${y}_{i}$ and velocity ${v}_{i}$ of $p$ particles. |

Determine $Gbest$ from $\mathit{min}\left\{f\left({y}_{1}\right),\dots ,f\left({y}_{p}\right)\right\}$ at ($m=0$). |

while (criterion) |

$t=t+1$ (Iteration counter) |

For loop over all $p$ particles and all $n$ dimensions |

Produce new velocity ${V}^{t+1}={W}^{t}\times {V}_{i}^{t}+{C}_{1}\times {r}_{1}\times ({P}_{i}^{t}-{X}_{i}^{t})+{C}_{2}\times {r}_{2}\times ({G}^{t}-{X}_{i}^{t})$ |

Compare new locations ${X}_{i}^{t+1}={X}_{i}^{t}+{{V}_{i}}^{t+1}$ |

Assess objective functions at new locations ${X}_{i}^{t+1}$ |

Deliver the current best for each particle $Pbest$ |

end ε |

Deliver the current global best $Gbest$ |

end while |

output the final results $Pbest$ and $Gbest$ |

#### 5.2. Constrained Binary-Integer Programming Model with a Fixed Number of Channels

Algorithm 4: Steps of MILP Solver implementation |

MILP Algorithm: Optimal Placement of PMU with multi-channel capacity |

1. $w:$ The set of implementation costs, ${w}^{T}={\left|\mathrm{1,1},\dots ,1\right|}^{m}$; |

2. $m:$ Set of possible combinations including a number of nodes; |

3. Define a set of variables Y $={\left\{{y}_{1},{y}_{2}\dots ,{y}_{\mathrm{m}}\right\}}^{T},{y}_{i}\in {\left\{\mathrm{0,1}\right\}}^{m}$; |

4. Define binary connectivity matrix; |

5. $J\left(y\right)={w}^{T}y$; objective function considering costs; |

6. /* solving the problem of mixed integers; |

7. programming*/ |

8. Invoke MILP solver in MATLAB environment; |

9. solve placement minimizing using mip; |

10. $y\to optimalplacementofcombinationsforselectingPMUslocationsandbranchassigments$ |

11. Solution $y$ (display combinations by which a PMU location is the Center and the other locations representing the PMUs directly observed); |

12. Solve linear relaxation problems of the initial MILP model |

13. Tighten LP via cutting planes and domain propagation |

14. Search for feasible solutions via primal heuristics for the MILP model |

15. Branch on variables |

16. Output: a binary-feasible solution to the MILP model |

✓ Exit: MILP status (optimal or infeasible) and solution |

✓ Absolute gap: $\mathrm{U}\u2013\mathrm{L}\le AbsoluteGapTolerance$ |

✓ Relative gap: 100 $\times $ (U − L)/(abs(U) + 1) $\le \mathrm{R}\mathrm{e}\mathrm{l}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{v}\mathrm{e}\mathrm{G}\mathrm{a}\mathrm{p}\mathrm{T}\mathrm{o}\mathrm{l}\mathrm{e}\mathrm{r}\mathrm{a}\mathrm{n}\mathrm{c}\mathrm{e}$ |

- Formulation so that the absolute gap is small;
- Heuristic computations are needed to find a good lower bound;
- Branching;
- Node selection.

#### 5.3. Constrained Binary-Integer Programming Model with a Single-Line PMU

**Definition 6.**

#### 5.4. Constrained Mixed-Integer Nonlinear Programming Formulation

#### 5.5. Constrained Nonlinear Programming Model

**Definition 7.**

**Definition 8.**

**Definition 9.**

Algorithm 5: Steps of nonlinear algorithms |

Step 0: initial estimate ${x}_{o}\epsilon \Omega $, we set the iterative number equal to zero, $i=0$ |

Step 1: Computation of the search direction ${h}_{i}=h\left({x}_{i}\right)$ |

Step 2: Computation of step-length ${\lambda}_{i}$ such as: |

$z\left({x}_{i}+{\lambda}_{i}\times {h}_{i}\right)={min}_{\lambda}\left\{\left\{z\left({x}_{i}+\lambda \times {h}_{\iota}\right)\right\}:\lambda \ge 0\right\}$, ${x}_{i}+{\lambda}_{i}\times {h}_{i}\epsilon \Omega $ |

Step 3: ${x}_{i+1}={x}_{i}+{\lambda}_{i}{h}_{i}$ |

Step 4: if $\Vert {x}_{i+1}-{x}_{i}\Vert \le \epsilon $, then stop. Otherwise, set $i=i+1$ and go to step 1. |

- Instead of searching local solution points for the optimization models, we examine all the sufficient and guaranteed optimality conditions for finding global solutions.
- Hence the testing of optimality yields the exact solutions by utilizing the BBA, SQP, IPMs, GPSA, and BPSO for simulation purposes.

## 6. The IEEE-14-Bus System for Illustrating the Mathematical Models and Numerical Results Obtained by the Optimization Procedure

Algorithm 6: Steps of optimizing YALMIP’s BBA |

+ Solver chosen : BMIBNB |

+ Processing objective function |

+ Processing constraints |

+ Branch and bound started |

* Starting YALMIP global branch & bound. |

* Upper solver : INTLINPROG |

* Lower solver : INTLINPROG |

* LP solver : INTLINPROG |

* -Extracting bounds from model |

* -Performing root-node bound propagation |

* -Calling upper solver + Calling INTLINPROG |

(Found a solution!) |

* -Branch-variables : 0 |

* -More root-node bound-propagation |

* -Performing LP-based bound-propagation |

* -And some more root-node bound-propagation |

* Starting the b&b process |

Node Upper Gap (%) Lower Open Time |

+ Calling INTLINPROG |

1 : 4.00000E+00 0.00 4.00000E+00 0 0s Poor lower bound |

* Finished. Cost: 4 (lower bound: 4, relative gap 0%) |

* Termination with all nodes pruned |

* Timing: 38% spent in upper solver (1 problems solved) |

* 19% spent in lower solver (1 problems solved) |

* 1% spent in LP-based domain reduction (0 problems solved) |

* 1% spent in upper heuristics (0 candidates tried) |

ans = |

Columns 1 through 15 |

0 1 0 0 0 0 0 0 0 0 0 1 0 1 0 |

Columns 16 through 18 |

0 1 0 |

ans = |

2 12 14 17 |

Algorithm 7: Steps of optimizing YALMIP’s BBA |

+ Solver chosen : BMIBNB |

+ Processing objective function |

+ Processing constraints |

+ Branch and bound started |

* Starting YALMIP global branch & bound. |

* Upper solver : INTLINPROG |

* Lower solver : INTLINPROG |

* LP solver : INTLINPROG |

* -Extracting bounds from model |

* -Performing root-node bound propagation |

* -Calling upper solver + Calling INTLINPROG |

(Found a solution!) |

* -Branch-variables : 0 |

* -More root-node bound-propagation |

* -Performing LP-based bound-propagation |

* -And some more root-node bound-propagation |

* Starting the b&b process |

Node Upper Gap (%) Lower Open Time |

+ Calling INTLINPROG |

1 : 2.64286E+00 0.00 2.64286E+00 0 0s Poor lower bound |

* Finished. Cost: 2.6429 (lower bound: 2.6429, relative gap 0%) |

* Termination with all nodes pruned |

* Timing: 27% spent in upper solver (1 problems solved) |

* 12% spent in lower solver (1 problems solved) |

* 1% spent in LP-based domain reduction (0 problems solved) |

* 1% spent in upper heuristics (0 candidates tried) |

Columns 1 through 15 |

0 1 0 0 0 0 0 0 0 1 1 0 1 0 0 |

Columns 16 through 18 |

0 0 0 |

2 10 11 13 |

^{−6}or 1 × 10

^{−10}. The optimal solution was found within a minimum value [72,74,75,76,79,80,81]. The upper and the lower bounds are computed in this instance. YALMIP’s BBA invokes local and MILP optimizer routines to build the BBA tree [37,38,74,75,76,79]. Fmincon delivers the upper bound whereas the lower bound is solved by the SCIP optimizer through linear programming relaxations [37,38,75]. Furthermore, the lower bound can be calculated using intlinprog, MOSEK, and Gurobi [72,74,75,76,79,80,81].

## 7. Results

^{−6}) declared in [72,76,79,80].

^{−4}) declared in [72,76,79,80,81,84,85,86,87,88].

## 8. Discussion

## 9. Final Remarks and Future Study

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Appendix A

## Appendix B

+ Processing objective function |

+ Processing constraints |

+ Branch and bound started |

* Starting YALMIP global branch & bound. |

* Upper solver : MOSEK |

* Lower solver : MOSEK |

* LP solver : MOSEK |

* -Extracting bounds from model |

* -Performing root-node bound propagation |

* -Calling upper solver + Calling Mosek |

(no solution found) |

* -Branch-variables : 0 |

* -More root-node bound-propagation |

* -Performing LP-based bound-propagation |

* -And some more root-node bound-propagation |

* Starting the b&b process |

Node Upper Gap (%) Lower Open Time |

+ Calling MOSEK |

1 : 1.00000E+01 0.00 1.00000E+01 0 0s Solution found by heuristics |

* Finished. Cost: 10 (lower bound: 10, relative gap 0%) |

* Termination with all nodes pruned |

* Timing: 33% spent in upper solver (2 problems solved) |

* 16% spent in lower solver (1 problems solved) |

* 1% spent in LP-based domain reduction (0 problems solved) |

* 3% spent in upper heuristics (1 candidates tried) |

ans = |

1 5 42 43 58 61 70 75 77 79 |

Optimal PMU numbers: 10 |

best function value: 10 |

L | n_{PMU} | PMU Locations | PMU Channels |
---|---|---|---|

4 | 10 | 1, 2, 6, 9, 10, 12, 18, 23, 25, 27 | {1-2, 1-3} {2-5, 2-7} {6-8, 6-28} {9-6, 9-10, 9-11} {10-17, 10-20, 10-21, 10-22} {12-4, 12-13, 12-14, 12-16} {18-15, 18-19} {23-25, 23-24} {25-24, 25-26, 25-27} {27-25, 27-28, 27-29, 27-30} |

+ Solver chosen : BMIBNB |

+ Processing objective function |

+ Processing constraints |

+ Branch and bound started |

* Starting YALMIP global branch & bound. |

* Upper solver : MOSEK |

* Lower solver : MOSEK |

* LP solver : MOSEK |

* -Extracting bounds from model |

* -Performing root-node bound propagation |

* -Calling upper solver + Calling Mosek |

(no solution found) |

* -Branch-variables : 0 |

* -More root-node bound-propagation |

* -Performing LP-based bound-propagation |

* -And some more root-node bound-propagation |

* Starting the b&b process |

Node Upper Gap (%) Lower Open Time |

+ Calling MOSEK |

1 : 8.46667E+00 0.00 8.46667E+00 0 0s Solution found by heuristics |

* Finished. Cost: 8.4667 (lower bound: 8.4667, relative gap 0%) |

* Termination with all nodes pruned |

* Timing: 45% spent in upper solver (2 problems solved) |

* 21% spent in lower solver (1 problems solved) |

* 1% spent in LP-based domain reduction (0 problems solved) |

* 2% spent in upper heuristics (1 candidates tried) |

ans = |

2 4 6 43 58 61 67 70 77 79 |

Optimal PMU numbers: 10 |

best function value: 8.466667e+00 |

L | n_{PMU} | PMU Locations | PMU Channels |
---|---|---|---|

4 | 10 | 2, 4, 6, 9, 10, 12, 15, 18, 25, 27 | {2-1, 2-4, 2-5, 2-6} {4-2, 4-3, 4-6, 4-12} {6-2, 6-4, 6-7, 6-8} {9-6, 9-10, 9-11} {10-17, 10-20, 10-21, 10-22} {12-4, 12-13, 12-14, 12-16} {15-12, 15-14, 15-18, 15-23} {18-15, 18-19} {25-24, 25-26, 25-27} {27-25, 27-28, 27-29, 27-30} |

Mixed-Integer Nonlinear Program (MINLP) Optimization |

min f(x) |

s.t. lb <= x <= ub |

cl <= c(x) <= cu |

xi = Integer/Binary |

Problem Properties: |

# Decision Variables: 82 |

# Constraints: 276 |

# Bounds: 164 |

# Binary Variables: 82 |

# Nonlinear Equality: 30 |

Solver Parameters: |

Solver: SCIP |

Objective Gradient: @(x)mklJac(fun,x) |

Constraint Jacobian: @(x)mklJac(nlcon,x) |

Jacobian Structure: Supplied |

time | node | left |LP iter|LP it/n| mem |mdpt |frac |vars |cons |cols |rows |cuts |confs|strbr| dual bound | primal bound | gap |

T 0.1s| 1 | 0 | 0 | - |2520k| 0 | - | 690 | 942 | 690 | 637 | 0 | 0 | 0 | -- |6.500000e+001 | Inf |

b 0.1s| 1 | 0 | 0 | - |2457k| 0 | - | 690 | 942 | 690 | 637 | 0 | 0 | 0 | -- |1.800000e+001 | Inf |

0.1s| 1 | 0 | 307 | - |2453k| 0 | 7 | 690 | 942 | 690 | 637 | 0 | 0 | 0 |1.000000e+001 |1.800000e+001 | 80.00 |

0.1s| 1 | 0 | 324 | - |2542k| 0 | 13 | 690 | 942 | 690 | 641 | 4 | 0 | 0 |1.000000e+001 |1.800000e+001 | 80.00 |

0.1s| 1 | 0 | 327 | - |2612k| 0 | 0 | 690 | 942 | 690 | 643 | 6 | 0 | 0 |1.000000e+001 |1.800000e+001 | 80.00 |

* 0.1s| 1 | 0 | 327 | - |2620k| 0 | - | 690 | 942 | 690 | 643 | 6 | 0 | 0 |1.000000e+001 |1.000000e+001 | 0.00 |

SCIP Status : problem is solved [optimal solution found] |

Solving Time (sec) : 0.12 |

Solving Nodes : 1 |

Primal Bound : +1.00000000000000e+001 (3 solutions) |

Dual Bound : +1.00000000000000e+001 |

Gap : 0.00 |

1 2 18 43 58 64 67 71 77 82 |

Optimal PMU numbers: 10 |

best function value: 10 |

BBNodes: 1 |

BBGap: 0 |

Time: 0.5361 |

Algorithm: ‘SCIP: Spatial Branch and Bound using IPOPT and SoPlex’ |

Status: ‘Globally Integer Optimal’ |

L | n_{PMU} | PMU Locations | PMU Channels |
---|---|---|---|

4 | 10 | 1, 2, 6, 9, 10, 12, 15, 19, 25, 27 | {1-2, 1-3} {2-1, 2-4, 2-5, 2-6} {6-2, 6-7, 6-8, 6-28} {9-6, 9-10, 9-11} {10-17, 10-20, 10-21, 10-22} {12-13, 12-14, 12-15, 12-16} {15-12, 15-14, 15-18, 15-23} {19-18, 19-20} {25-24, 25-26, 25-27} {27-29, 27-30} |

Mixed-Integer Nonlinear Program (MINLP) Optimization |

min f(x) |

s.t. lb <= x <= ub |

cl <= c(x) <= cu |

xi = Integer/Binary |

Problem Properties: |

# Decision Variables: 82 |

# Constraints: 276 |

# Bounds: 164 |

# Binary Variables: 82 |

# Nonlinear Equality: 30 |

Solver Parameters: |

Solver: SCIP |

Objective Gradient: @(x)mklJac(fun,x) |

Constraint Jacobian: @(x)mklJac(nlcon,x) |

Jacobian Structure: Supplied |

time | node | left |LP iter|LP it/n| mem |mdpt |frac |vars |cons |cols |rows |cuts |confs|strbr| dual bound | primal bound | gap |

T 0.1s| 1 | 0 | 0 | - |2550k| 0 | - | 690 | 942 | 690 | 637 | 0 | 0 | 0 | -- |5.540000e+001 | Inf |

b 0.2s| 1 | 0 | 0 | - |2487k| 0 | - | 690 | 942 | 690 | 637 | 0 | 0 | 0 | -- |1.560000e+001 | Inf |

* 0.2s| 1 | 0 | 319 | - |2492k| 0 | - | 690 | 942 | 690 | 637 | 0 | 0 | 0 |8.466667e+000 |8.466667e+000 | 0.00 |

0.2s| 1 | 0 | 319 | - |2492k| 0 | - | 690 | 942 | 690 | 637 | 0 | 0 | 0 |8.466667e+000 |8.466667e+000 | 0.00 |

SCIP Status : problem is solved [optimal solution found] |

Solving Time (sec) : 0.17 |

Solving Nodes : 1 |

Primal Bound : +8.46666666666667e+000 (3 solutions) |

Dual Bound : +8.46666666666667e+000 |

Gap : 0.00 |

2 4 18 43 56 64 67 72 77 79 |

Optimal PMU numbers: 10 |

BBNodes: 1 |

BBGap: 0 |

Time: 0.3718 |

Algorithm: ‘SCIP: Spatial Branch and Bound using IPOPT and SoPlex’ |

Status: ‘Globally Integer Optimal’ |

L | n_{PMU} | PMU Locations | PMU Channels |
---|---|---|---|

4 | 10 | 2, 4, 6, 9, 10, 12, 15, 20, 25, 27 | {2-1, 2-4, 2-5, 2-6} {4-2, 4-3, 4-6, 4-12} {6-2, 6-7, 6-8, 6-28} {9-6, 9-10, 9-11} {10-9, 10-17, 10-21, 10-22} {12-13, 12-14, 12-15, 12-16} {15-12, 15-14, 15-18, 15-23} {20-10, 20-19} {25-24, 25-26, 25-27} {27-25, 27-28, 27-29, 27-30} |

Binary Swarm Optimization | |||
---|---|---|---|

L | n_{PMU} | PMU Locations | PMU Channels |

4 | 10 | 1, 2, 6, 10, 11, 12, 19, 24, 26, 29 | {1-2, 1-3 {2-1, 2-4, 2-5, 2-6 {6-7, 6-8, 6-9, 6-28 {10-6, 10-17, 10-21, 10-22 {11-9 {12-13, 12-14, 12-15, 12-16 {19-18, 19-20 {24-22, 24-23, 24-25 {26-25 {29-27, 29-30} |

4 | 10 | 2, 4, 6, 9, 10, 12, 15, 18, 25, 27 | {2-1, 2-4, 2-5, 2-6 {4-2, 4-3, 4-6, 4-12 {6-2, 6-7, 6-8, 6-28 {9-6, 9-10, 9-11 {10-17, 10-20, 10-21, 10-22 {12-4, 12-13, 12-15, 12-16 {15-12, 15-14, 15-18, 15-23 {18-15, 18-19 {25-24, 25-26, 25-27 {27-29, 27-30} |

Genetic Algorithms | |||

4 | 10 | 3, 6, 7, 10, 11, 12, 15, 19, 25, 29 | {3-1, 3-4 {6-2, 6-8, 6-9, 6-28 {7-5, 7-6 {10-17, 10-20, 10-21, 10-22 {11-9 {12-4, 12-13, 12-15, 12-16 {15-12, 15-14, 15-18, 15-23 {19-20, 19-18 {25-24, 25-26, 25-27 {29-27, 29-30} |

4 | 10 | 2, 4, 6, 9, 10, 12, 15, 19, 25, 27 | {2-1, 2-4, 2-5, 2-6 {4-2, 4-3, 4-6, 4-12 {6-4, 6-7, 6-8, 6-28 {9-6, 9-10, 9-11 {10-17, 10-20, 10-21, 10-22 {12-13, 12-14, 12-15, 12-16 {15-12, 15-14, 15-18, 15-23 {19-18, 19-20 {25-24, 25-26, 25-27 {27-25, 27-28, 27-29, 27-30} |

Sequential Quadratic Programming | |||

L | n_{PMU} | PMU locations | PMU channels |

3 | 10 | 1, 7, 8, 10, 11, 12, 15, 20, 25, 29 | {1-2, 1-3 {7-5, 7-6 {8-6, 8-28 {10-17, 10-21, 10-22 {11-9 {12-4, 12-13, 12-16 {15-14, 15-18, 15-23 {20-10, 20-19 {25-24, 25-26, 25-27 {29-27, 29-30} |

4 | 10 | 3, 5, 10, 11, 12, 19, 23, 26, 27, 28 | {3-1, 3-4 {5-2, 5-7 {10-17, 10-20, 10-21, 10-22 {11-9 {12-13, 12-14, 12-15, 12-16 {19-18, 19-20 {23-15, 23-24 {26-25 {27-25, 27-28, 27-29, 27-30 {28-6, 28-8, 28-27} |

Interior-Point Methods | |||

3 | 10 | 1, 5, 8, 10, 11, 12, 19, 23, 26, 29 | {1-3, 1-4 {5-2, 5-7 {8-6, 8-28 {10-17, 10-21, 10-22 {11-9 {12-13, 12-14, 12-16 {19-18, 19-20 {23-15, 23-24 {26-25 {29-27, 29-30} |

4 | 10 | 2, 4, 6, 10, 11, 12, 19, 23, 25, 27 | {2-1, 2-4, 2-5, 2-6 {4-2, 4-3, 4-6, 4-12 {6-2, 6-7, 6-8, 6-28 {10-6, 10-17, 10-21, 10-22 {11-9 {12-13, 12-14, 12-15, 12-16 {19-18, 19-20 {23-15, 23-24 {25-24, 25-26, 25-27 {27-25, 27-28, 27-29, 27-30} |

Generalized Pattern Search Algorithm | |||

3 | 10 | 2, 4, 6, 10, 11, 12, 19, 23, 25, 27 | {2-1, 2-4, 2-5 {4-2, 4-3, 4-6 {6-7, 6-8, 6-28 {10-17, 10-21, 10-22 {11-9 {12-13, 12-14, 12-16 {19-18, 19-20 {23-15, 23-24 {25-24, 25-26, 25-27 {27-25, 27-29, 27-30} |

4 | 10 | 1, 2, 6, 10, 11, 12, 18, 24, 25, 29 | {1-2, 1-3 {2-1, 2-4, 2-5, 2-6 {6-7, 6-8, 6-10, 6-28 {10-17, 10-20, 10-21, 10-22 {11-9 {12-13, 12-14, 12-15, 12-16 {18-15, 18-19 {24-22, 24-23, 24-25 {25-24, 25-26, 25-27 {29-27, 29-30} |

Genetic Algorithms | |||

3 | 10 | 3, 5, 8, 10, 11, 12, 18, 24, 26, 27 | {3-1, 3-4 {5-2, 5-7 {8-6, 8-28 {10-17, 10-20, 10-21 {11-9 {12-13, 12-14, 12-16 {18-15, 18-19 {24-22, 24-23, 24-25 {26-25 {27-25, 27-29, 27-30} |

Genetic Algorithms | |||

4 | 10 | 2, 3, 6, 9, 10, 12, 15, 19, 25, 27 | {2-1, 2-4, 2-5, 2-6 {3-1, 3-4 {6-4, 6-7, 6-8, 6-9 {9-6, 9-10, 9-11 {10-9, 10-17, 10-21, 10-22 {12-4, 12-13, 12-15, 12-16 {15-12, 15-14, 15-18, 15-23 {19-18, 19-20 {25-24, 25-26, 25-27 {27-25, 27-28, 27-29, 27-30} |

Binary Particle Swarm Optimization | |||

3 | 10 | 2, 4, 6, 9, 10, 12, 18, 24, 26, 29 | {2-1, 2-4, 2-5 {4-2, 4-3, 4-12 {6-7, 6-8, 6-28 {9-6, 9-10, 9-11 {10-17, 10-20, 10-21 {12-13, 12-14, 12-16 {18-15, 18-19 {24-22, 24-23, 24-25 {26-25 {29-27, 29-30} |

4 | 10 | 3, 6, 7, 9, 10, 12, 18, 24, 26, 29 | {3-1, 3-4 {6-2, 6-8, 6-9, 6-28 {7-5, 7-6 {9-6, 9-10, 9-11 {10-6, 10-17, 10-20, 10-21 {12-13, 12-14, 12-15, 12-16 {18-15, 18-19 {24-22, 24-23, 24-25 {26-25 {29-27, 29-30} |

CBC Branch and Cut Algorithm | |||

3 | 10 | 2, 4, 6, 9, 10, 12, 18, 24, 25, 27 | {2-1, 2-4, 2-5 {4-2, 4-3, 4-6 {6-2, 6-7, 6-8 {9-6, 9-10, 9-11 {10-17, 10-20, 10-21 {12-13, 12-14, 12-16 {18-15, 18-19 {24-22, 24-23, 24-25 {25-24, 25-26, 25-27 {27-28, 27-29, 27-30} |

4 | 10 | 2, 4, 6, 9, 10, 12, 15, 19, 25, 27 | {2-1, 2-4, 2-5, 2-6 {4-2, 4-3, 4-6, 4-12 {6-2, 6-7, 6-8, 6-10 {9-6, 9-10, 9-11 {10-6, 10-17, 10-21, 10-22 {12-13, 12-14, 12-15, 12-16 {15-12, 15-14, 15-18, 15-23 {19-18, 19-20 {25-24, 25-26, 25-27 {27-25, 27-28, 27-29, 27-30} |

SCIP Spatial Branch-and-Bound Algorithm | |||

3 | 10 | 3, 5, 6, 9, 10, 12, 15, 19, 25, 27 | {3-1, 3-4 5-2, 5-7 6-4, 6-7, 6-8 9-6, 9-10, 9-11 10-17, 10-21, 10-22 12-13, 12-15, 12-16 15-12, 15-14, 15-23 19-18, 19-20 25-24, 25-26, 25-57 27-28, 27-29, 27-30} |

4 | 10 | 3, 5, 6, 9, 10, 12, 15, 18, 25, 27 | {3-1, 3-4 5-2, 5-7 6-2, 6-4, 6-7, 6-8 9-6, 9-10, 9-11 10-17, 10-20, 10-21, 10-22 12-13, 12-14, 12-15, 12-16 15-12, 15-14, 15-18, 15-23 18-15, 18-19 25-24, 25-26, 25-27 27-25, 27-28, 27-29, 27-30} |

Gurobi: Branch-and-Bound Algorithm | |||

3 | 10 | 2, 3, 6, 9, 10, 12, 18, 24, 26, 27 | {2-1, 2-5, 2-6 3-1, 3-4 6-7, 6-8, 6-28 9-6, 9-10, 9-11 10-17, 10-20, 10-21 12-13, 12-14, 12-16 18-15, 18-19 24-22, 24-23, 24-25 26-25 27-25, 27-29, 27-30} |

4 | 10 | 1, 2, 6, 9, 10, 12, 18, 24, 25, 27 | {1-2, 1-3 2-1, 2-4, 2-5, 2-6 6-7, 6-8, 6-9, 6-10 9-6, 9-10, 9-11 10-17, 10-20, 10-21, 10-22 12-4, 12-13, 12-14, 12-16 18-15, 18-19 24-22, 24-23, 24-25 25-24, 25-26, 25-27 27-25, 27-28, 27-29, 27-30} |

MOSEK: Branch-and-Bound Algorithm | |||

3 | 10 | 1, 2, 6, 9, 10, 12, 15, 20, 25, 27 | {1-2, 1-3 2-1, 2-5, 2-6 6-4, 6-7, 6-8 9-6, 9-10, 9-11 10-17, 10-21, 10-22 12-13, 12-14, 12-16 15-12, 15-18, 15-23 20-10, 20-19 25-24, 25-26, 25-27 27-28, 27-29, 27-30} |

4 | 10 | 1, 5, 6, 9, 10, 12, 18, 23, 25, 27 | {1-2, 1-3 5-2, 5-7 6-8, 6-28 9-6, 9-10, 9-11 10-17, 10-20, 10-21, 10-22 12-4, 12-13, 12-14, 12-16 18-15, 18-19 23-15, 23-24 25-24, 25-26, 25-27 27-25, 27-28, 27-29, 27-30} |

GLPK: Revised Simplex to Build the BBA Tree | |||

3 | 10 | 2, 4, 6, 10, 11, 12, 19, 23, 25, 27 | {2-1, 2-4, 2-5 4-2, 4-3, 4-6 6-7, 6-8, 6-28 10-17, 10-21, 10-22 11-9 12-13, 12-14, 12-16 19-18, 19-20 23-15, 23-24 25-24, 25-26, 25-27 27-25, 27-29, 27-30} |

4 | 10 | 3, 5, 6, 9, 10, 12, 18, 24, 26, 29 | {3-1, 3-4 5-2, 5-7 6-2, 6-7, 6-8, 6-28 9-6, 9-10, 9-11 10-9, 10-17, 10-20, 10-21 12-4, 12-13, 12-14, 12-16 18-15, 18-19 24-22, 24-23, 24-25 26-25 29-27, 29-30} |

GLPK: Revised Simplex to Build the BBA Tree | |||

Optimizing the b-objective function leading to a set with increased redundancy | |||

3 | 10 | 2, 4, 6, 9, 10, 12, 15, 20, 25, 27 | {2-1, 2-4, 2-5 4-2, 4-3, 4-6 6-7, 6-8, 6-28 9-6, 9-10, 9-11 10-17, 10-21, 10-22 12-4, 12-13, 12-16 15-14, 15-18, 15-23 20-10, 20-19 25-24, 25-26, 25-27 27-25, 27-29, 27-30 |

4 | 10 | 2, 4, 6, 9, 10, 12, 15, 18, 25, 27 | {2-1, 2-4, 2-5, 2-6 4-2, 4-3, 4-6, 4-12 6-2, 6-7, 6-8, 6-28 9-6, 9-10, 9-11 10-17, 10-20, 10-21, 10-22 12-4, 12-13, 12-14, 12-16 15-12, 15-14, 15-18, 15-23 18-15, 18-19 25-24, 25-26, 25-27 27-25, 27-28, 27-29, 27-30} |

GLPK: Revised Simplex to Build the BBA Tree | |||

Optimizing the b-objective function leading to a set with increased redundancy | |||

3 | 10 | 2, 4, 6, 9, 10, 12, 15, 20, 25, 27 | {2-1, 2-4, 2-5 4-2, 4-3, 4-6 6-7, 6-8, 6-28 9-6, 9-10, 9-11 10-17, 10-21, 10-22 12-4, 12-13, 12-16 15-14, 15-18, 15-23 20-10, 20-19 25-24, 25-26, 25-27 27-25, 27-29, 27-30 |

4 | 10 | 2, 4, 6, 9, 10, 12, 15, 18, 25, 27 | {2-1, 2-4, 2-5, 2-6 4-2, 4-3, 4-6, 4-12 6-2, 6-7, 6-8, 6-28 9-6, 9-10, 9-11 10-17, 10-20, 10-21, 10-22 12-4, 12-13, 12-14, 12-16 15-12, 15-14, 15-18, 15-23 18-15, 18-19 25-24, 25-26, 25-27 27-25, 27-28, 27-29, 27-30} |

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**Figure 6.**Plots derived by the intlinprog optimizer in the MATLAB optimization library. Subsets: {3, 12, 19, 80, 87, 90, 93, 95, 100, 104, 118, 124, 127, 133, 135, 138 143} [60].

**Figure 7.**Plots derived by the intlinprog optimizer in the MATLAB optimization library. Subsets: {3, 13, 38, 81, 87, 90, 94, 97, 98, 100, 104, 112, 118, 126, 130, 137, 143} [60].

**Figure 8.**Plots derived by the IPOPT optimizer in the MATLAB optimization library. PMU subsets: {1, 4, 6, 18, 49, 56, 60, 64, 67, 68, 72, 77, 81, 87, 91, 93, 97}.

**Figure 9.**Plots derived by the GA optimizer in the MATLAB optimization library. Subsets: {1, 5, 12, 48, 55, 58, 62, 65, 66, 68, 72, 77, 79, 81, 86, 91, 94}.

**Figure 10.**Plots derived by BPSO. PMU subsets: {1, 4, 6, 23, 49, 56, 60, 64, 67, 68, 72, 77, 81, 87, 91, 93, 97}.

L | n_{PMU} | PMU Locations | PMU Channels |
---|---|---|---|

4 | 4 | 2, 8, 10, 13 | {2-1, 2-3, 2-4, 2-5} {8-7} {10-9, 10-11} {13-6, 13-12, 13-14} |

L | n_{PMU} | PMU Locations | PMU Channels |
---|---|---|---|

4 | 4 | 2, 6, 7, 9 | {2-1, 2-3, 2-4, 2-5} {6-5, 6-11, 6-12, 6-13} {7-4, 7-8, 7-9} {9-4, 9-7, 9-10, 9-14} |

IEEE Power Systems | Branches | PMU Number | Solution Algorithm | |
---|---|---|---|---|

Primal Simplex | Dual Simplex | |||

14-bus | 20 | 7 | 1, 6, 10, 14, 17, 18, 19 | 2, 3, 9, 12, 14, 18, 20 |

30-bus | 41 | 15 | 2, 3, 8, 10, 13, 16, 20, 21, 23, 25, 29, 32, 34, 36, 39 | 1, 4, 8, 10, 13, 16, 20, 21, 23, 25, 29, 32, 34 36, 39 |

57-bus | 80 | 29 | 2, 4, 6, 8, 14, 17, 26, 29, 31, 33, 35, 38, 40, 43, 45, 47, 50, 52, 55, 59, 61, 63, 65, 68, 70, 71, 72, 74, 76 | 2, 4, 6, 7, 9, 13, 17, 26, 29, 31, 33, 36, 38, 40, 43, 45, 47, 49, 55, 57, 58, 60, 62, 64, 68, 70, 71, 73, 76 |

118-bus | 186 | 61 | 1, 4, 6, 9, 10, 18, 19, 22, 24, 27, 29, 32, 34, 37, 40, 46, 48, 55, 58, 60, 61, 64, 73, 76, 79, 81, 83, 91, 93, 95, 96, 103, 105, 111, 113, 114, 120, 122, 126, 129, 132, 134, 137, 140, 144, 147, 152, 157, 159, 162, 165, 170, 172, 173, 176, 177, 179, 182, 183, 184, 186 | 1, 4, 6, 9, 10, 17, 18, 22, 24, 27, 29, 32, 34 37, 40, 46, 48, 55, 58, 60, 61 64, 73, 78, 81, 83, 87, 90, 94, 96, 99, 101, 105, 111, 113, 114, 118, 122, 126, 129, 132, 134, 137, 140, 144, 147, 153, 157, 158, 162, 168, 172, 173, 174 176, 177, 179, 182, 183, 184, 185 |

L | N_{PMU} | Primal-Simplex | Dual-Simplex | ||
---|---|---|---|---|---|

PMU Locations | PMU Channels | PMU Locations | PMU Channels | ||

1 | 7 | 2, 4, 5, 6, 8, 10, 13 | {2-1} {4-9} {5-1} {6-12} {8-7} {10-11} {13-14} | 2, 4, 6, 8, 9, 10, 13 | {2-1} {4-3} {6-5} {8-7} {9-14} {10-11} {13-12} |

2 | 5 | 2, 5, 7, 10, 13 | {2-3, 2-5} {5-1, 5-6} {7-4, 7-8} {10-9, 10-11} {13-12, 13-14} | 2, 3, 7, 11, 13 | {2-1, 2-5} {3-2, 3-4} {7-8, 7-9} {11-6, 11-10} {13-12, 13-14} |

3 | 4 | 2, 6, 8, 9 | {2-1, 2-3, 2-5} {6-11, 6-12, 6-13} {8-7} {9-4, 9-10, 9-14} | 2, 7, 10, 13 | {2-1, 2-3, 2-5} {7-4, 7-8, 7-9} {10-9, 10-11} {13-6, 13-12, 13-14} |

4 | 4 | 2, 7, 11, 13 | {2-1, 2-3, 2-4, 2-5} {7-4, 7-8, 7-9} {11-6, 11-10} {13-6, 13-12, 13-14} | 2, 8, 10, 13 | {2-1, 2-3, 2-4, 2-5} {8-7} {10-9, 10-11} {13-6, 13-12, 13-14} |

5 | 4 | 2, 7, 11, 13 | {2-1, 2-3, 2-4, 2-5} {7-4, 7-8, 7-9} {11-6, 11-10} {13-6, 13-12, 13-14} | 2, 8, 10, 13 | {2-1, 2-3, 2-4, 2-5} {8-7} {10-9, 10-11} 13-6, 13-12, 13-14} |

L | N_{PMU} | Primal-Simplex | Dual-Simplex | ||
---|---|---|---|---|---|

PMU Locations | PMU Channels | PMU Locations | PMU Channels | ||

1 | 7 | 2, 5, 6, 7, 9, 10, 13 | {2-3} {5-1} {6-12} {7-8} {9-4} {10-11} {13-14} | 2, 4, 6, 7, 9, 10, 13 | {2-1} {4-3} {6-5} {7-8} {9-4} {10-11} {13-12} |

2 | 5 | 2, 6, 7, 10, 13 | {2-1, 2-3} {6-5, 6-13} {7-4, 7-8} {10-9, 10-11} {13-6, 13-12} | 2, 4, 7, 11, 13 | {2-1, 2-5} {4-2, 4-3} {7-8, 7-9} {11-6, 11-10} {13-12, 13-14} |

3 | 4 | 2, 6, 7, 9 | {2-1, 2-3, 2-5} {6-11, 6-12, 6-13} {7-4, 7-8, 7-9} {9-4, 9-10, 9-14} | 2, 6, 7, 9 | {2-1, 2-3, 2-5} {6-11, 6-12, 6-13} {7-4, 7-8, 7-9} {9-7, 9-10, 9-14} |

4 | 4 | 2, 6, 7, 9 | {2-1, 2-3, 2-4, 2-5} {6-5, 6-11, 6-12, 6-13} {7-4, 7-8, 7-9} {9-4, 9-7, 9-10, 9-14} | 2, 6, 7, 9 | {2-1, 2-3, 2-4, 2-5} {6-5, 6-11, 6-12, 6-13} {7-4, 7-8, 7-9} {9-4, 9-7, 9-10, 9-14} |

5 | 4 | 2, 6, 7, 9 | {2-1, 2-3, 2-4, 2-5} {6-5, 6-11, 6-12, 6-13} {7-4, 7-8, 7-9} {9-4, 9-7, 9-10, 9-14} | 2, 6, 7, 9 | {2-1, 2-3, 2-4, 2-5} {6-5, 6-11, 6-12, 6-13} {7-4, 7-8, 7-9} {9-4, 9-7, 9-10, 9-14} |

* Starting YALMIP global branch & bound. |

* Upper solver : fmincon |

* Lower solver : SCIP |

* LP solver : SCIP |

* -Extracting bounds from model |

* -Performing root-node bound propagation |

* -Calling upper solver (no solution found) |

* -Branch-variables : 18 |

* -More root-node bound-propagation |

* -Performing LP-based bound-propagation |

* -And some more root-node bound-propagation |

* Starting the b&b process |

Node Upper Gap (%) Lower Open Time |

1 : 4.00000E+00 0.00 4.00000E+00 2 5s Solution found by heuristics |

2 : 4.00000E+00 0.00 4.00000E+00 0 6s Poor lower bound | Pruned stack based on new upper bound |

* Finished. Cost: 4 (lower bound: 4, relative gap 0%) |

* Termination with all nodes pruned |

* Timing: 14% spent in upper solver (2 problems solved) |

* 11% spent in lower solver (2 problems solved) |

* 8% spent in LP-based domain reduction (36 problems solved) |

* 1% spent in upper heuristics (1 candidates tried) |

Columns 1 through 15 |

0 1 0 0 0 0 0 0 0 1 0 1 1 0 0 |

Columns 16 through 18 |

0 0 0 |

2 10 12 13 |

Linear scalar (real, 18 variables) |

Current value: 4 |

Coefficients range: 1 to 1 |

Optimal PMU numbers: 4 |

L | n_{PMU} | PMU Locations | PMU Channels |
---|---|---|---|

4 | 4 | 2, 6, 8, 9 | {2-1, 2-3, 2-4, 2-5} {6-5, 6-11, 6-12, 6-13} {8-7} {9-4, 9-7, 9-10, 9-14} |

+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ |

* Starting YALMIP global branch & bound. |

* Upper solver : fmincon |

* Lower solver : SCIP |

* LP solver : SCIP |

* -Extracting bounds from model |

* -performing root-node bound propagation |

* -Calling upper solver (no solution found) |

* -Branch-variables : 18 |

* -More root-node bound-propagation |

* -Performing LP-based bound-propagation |

* -And some more root-node bound-propagation |

* Starting the b&b process |

Node Upper Gap (%) Lower Open Time |

1 : 2.64286E+00 0.00 2.64286E+00 2 11s Solution found by heuristics |

2 : 2.64286E+00 0.00 2.64286E+00 0 12s Poor lower bound | Pruned stack based on new upper bound |

* Finished. Cost: 2.6429 (lower bound: 2.6429, relative gap 0%) |

* Termination with all nodes pruned |

* Timing: 19% spent in upper solver (2 problems solved) |

* 3% spent in lower solver (2 problems solved) |

* 8% spent in LP-based domain reduction (36 problems solved) |

* 1% spent in upper heuristics (1 candidates tried) |

Columns 1 through 15 |

0 1 0 0 0 0 0 0 0 1 1 0 1 0 0 |

Columns 16 through 18 |

0 0 0 |

2 10 11 13 |

Linear scalar (real, 18 variables) |

Current value: 2.6429 |

Coefficients range: 0.64286 to 0.85714 |

Optimal PMU numbers: 4 |

best function value: 2.642857e+00 |

L | n_{PMU} | PMU Locations | PMU Channels |
---|---|---|---|

4 | 4 | 2, 6, 7, 9 | {2-1, 2-3, 2-4, 2-5} {6-5, 6-11, 6-12, 6-13} {7-4, 7-8, 7-9} {9-4, 9-7, 9-10, 9-14} |

L | N_{PMU} | SCIP Optimizes the Single Objective Function | SCIP Optimizes the b-Objective Function | ||
---|---|---|---|---|---|

PMU Locations | PMU Channels | PMU Locations | PMU Channels | ||

1 | 7 | 2, 4, 6, 8, 9, 10, 13 | {2-1} {4-3} {6-5} {8-7} {9-14} {10-11} {13-12} | 2, 4, 5, 6, 8, 9, 10 | {2-1} {4-3} {5-6} {6-12} {8-7} {9-14} {10-11} |

2 | 5 | 2, 4, 7, 11, 13 | {2-1, 2-3} {4-2, 4-5} {7-8, 7-9} {11-6, 11-10} {13-12, 13-14} | 2, 5, 7, 11, 13 | {2-3, 2-4} {5-1, 5-4} {7-8, 7-9} {11-6, 11-10} {13-12, 13-14} |

3 | 4 | 2, 6, 8, 9 | {2-1, 2-3, 2-5} {6-11, 6-12, 6-13} {8-7} {9-4, 9-10, 9-14} | 2, 6, 7, 9 | {2-1, 2-3, 2-5} {6-11, 6-12, 6-13} {7-4, 7-8, 7-9} {9-7, 9-10, 9-14} |

4 | 4 | 2, 7, 10, 13 | {2-1, 2-3, 2-4, 2-5} {7-4, 7-8, 7-9} {10-9, 10-11} {13-6, 13-12, 13-14} | 2, 6, 7, 9 | {2-1, 2-3, 2-4, 2-5} {6-5, 6-11, 6-12, 6-13} {7-4, 7-8, 7-9} {9-4, 9-7, 9-10, 9-14} |

5 | 4 | 2, 6, 8, 9 | {2-1, 2-3, 2-4, 2-5} {6-5, 6-11, 6-12, 6-13} {8-7} {9-4, 9-7, 9-10, 9-14} | 2, 6, 7, 9 | {2-1, 2-3, 2-4, 2-5} {6-5, 6-11, 6-12, 6-13} {7-4, 7-8, 7-9} {9-4, 9-7, 9-10, 9-14} |

Max | |||||
---|---|---|---|---|---|

Iter | f-Count | f(x) | Constraint | MeshSize | Method |

0 | 1 | 11 | 1 | 0.25 | |

1 | 184 | 5 | 0 | 0.009772 | Update multipliers |

2 | 473 | 4.71875 | 0.09375 | 0.001 | Increase penalty |

3 | 1197 | 3.99805 | 0.001953 | 9.333e-07 | Update multipliers |

4 | 2094 | 3.99934 | 0.0003319 | 8.71e-10 | Update multipliers |

5 | 3378 | 4 | 4.545e-07 | 8.128e-13 | Update multipliers |

6 | 4887 | 4 | 0 | 7.586e-16 | Update multipliers |

Norm of First-Order | ||||||
---|---|---|---|---|---|---|

Iter | F-Count | f(x) | Feasibility | Steplength | Step | Optimality |

0 | 19 | 1.000000e+01 | 1.000e+00 | 2.000e+00 | ||

1 | 38 | 1.500000e+00 | 1.000e+00 | 1.000e+00 | 3.082e+00 | 1.000e+00 |

2 | 58 | 2.952500e+00 | 3.000e-01 | 7.000e-01 | 1.050e+00 | 1.021e+00 |

3 | 77 | 4.250000e+00 | 0.000e+00 | 1.000e+00 | 4.500e-01 | 1.066e+00 |

4 | 96 | 4.080921e+00 | 0.000e+00 | 1.000e+00 | 2.160e-01 | 9.631e-01 |

5 | 115 | 4.000027e+00 | 0.000e+00 | 1.000e+00 | 2.845e-01 | 5.909e-01 |

6 | 134 | 4.000001e+00 | 0.000e+00 | 1.000e+00 | 5.261e-03 | 2.367e-03 |

7 | 153 | 4.000000e+00 | 0.000e+00 | 1.000e+00 | 1.112e-03 | 1.108e-03 |

8 | 172 | 4.000000e+00 | 0.000e+00 | 1.000e+00 | 1.691e-05 | 9.899e-06 |

9 | 191 | 4.000000e+00 | 0.000e+00 | 1.000e+00 | 1.024e-33 | 0.000e+0 |

Iter Objective inf_pr inf_du lg(mu) ||d|| lg(rg) alpha_du alpha_pr ls |
---|

0 9.0000000e+00 0.00e+00 2.00e+00 0.0 0.00e+00 - 0.00e+00 0.00e+00 0 |

1 2.0000000e+00 1.00e+00 1.00e+00 -11.0 2.00e+00 - 1.00e+00 5.00e-01f 2 |

2 2.9090909e+00 2.84e-01 5.08e-01 -11.0 5.45e-01 - 1.00e+00 1.00e+00h 1 |

3 4.1838231e+00 7.36e-02 6.46e-01 -11.0 3.05e-01 - 1.00e+00 1.00e+00h 1 |

4 5.0510211e+00 1.75e-02 7.35e-01 -11.0 1.85e-01 - 1.00e+00 1.00e+00h 1 |

5 5.5122440e+00 3.35e-03 1.20e+00 -11.0 1.08e-01 - 1.00e+00 1.00e+00h 1 |

6 5.5457963e+00 2.60e-03 2.09e+00 -11.0 5.85e-02 - 1.00e+00 1.00e+00h 1 |

7 5.2306455e+00 1.61e-03 1.51e+00 -11.0 6.58e-02 - 1.00e+00 1.00e+00f 1 |

8 4.3870988e+00 8.62e-03 1.08e+00 -11.0 3.93e-01 - 1.00e+00 1.00e+00f 1 |

9 3.9647785e+00 1.07e-02 6.53e-01 -11.0 3.64e-01 - 1.00e+00 1.00e+00f 1 |

10 4.0022318e+00 2.36e-04 5.75e-02 -11.0 2.86e-02 - 1.00e+00 1.00e+00h 1 |

11 4.0013014e+00 8.39e-07 5.23e-02 -11.0 2.46e-03 - 1.00e+00 1.00e+00h 1 |

12 4.0417359e+00 2.28e-07 4.36e+00 -11.0 1.19e-01 - 1.00e+00 1.00e+00h 1 |

13 4.0012661e+00 3.64e-08 2.84e-01 -11.0 1.16e-01 - 1.00e+00 1.00e+00f 1 |

14 4.0012370e+00 4.73e-12 9.64e-02 -11.0 1.64e-03 - 1.00e+00 1.00e+00h 1 |

15 4.0010381e+00 1.49e-12 1.12e+00 -11.0 5.34e-03 - 1.00e+00 1.00e+00f 1 |

16 4.0005805e+00 1.50e-11 1.21e+00 -11.0 1.25e-02 - 1.00e+00 1.00e+00f 1 |

17 4.0002220e+00 1.19e-11 7.46e-01 -11.0 1.44e-02 - 1.00e+00 1.00e+00f 1 |

18 4.0000000e+00 4.46e-12 3.65e+00 -11.0 1.17e-02 - 1.00e+00 1.00e+00f 1 |

19 4.0000000e+00 2.22e-16 6.79e-04 -11.0 3.78e-06 - 1.00e+00 1.00e+00H 1 |

20 4.0000000e+00 3.00e-15 4.60e-09 -11.0 6.91e-11 - 1.00e+00 1.00e+00h 1 |

21 4.0000000e+00 0.00e+00 7.88e-11 -11.0 4.64e-13 - 1.00e+00 1.00e+00h 1 |

Obtained results using generalized pattern search algorithm | |||

L | n_{PMU} | PMU locations | PMU channels |

4 | 4 | 2, 6, 8, 9 | {2-1, 2-3, 2-4, 2-5} {6-5, 6-11, 6-12, 6-13} {8-7} {9-4, 9-7, 9-10, 9-14} |

Obtained results by using Sequential Quadratic Programming | |||

4 | 4 | 2, 7, 10, 13 | {2-1, 2-3, 2-4, 2-5 7-4, 7-8, 7-9 10-9, 10-11 13-6, 13-12, 13-14} |

Obtained results by using Interior-Point Methods | |||

4 | 4 | 2, 8, 10, 13 | {2-1, 2-3, 2-4, 2-5} {8-7} {10-9, 10-11} {13-6, 13-12, 13-14} |

Obtained results by using Genetic Algorithms | |||

4 | 4 | 2, 6, 8, 9 | {2-1, 2-3, 2-4, 2-5 6-5, 6-11, 6-12, 6-13 8-7 9-4, 9-7, 9-10, 9-14} |

Obtained results by using Binary Swarm Optimization | |||

4 | 4 | 2, 7, 11, 13 | {2-1, 2-3, 2-4, 2-5 7-4, 7-8, 7-9 11-6, 11-10 13-6, 13-12, 13-14} |

IEEE-Power Systems | Channel of PMUs | PMUs Number | Objective with One Product Redundancy Term | b-Objective Redundancy Term |
---|---|---|---|---|

IEEE-14 bus | 1 | 7 | 14 | 14 |

2 | 5 | 15 | 15 | |

3 | 4 | 14 | 16 | |

4 | 4 | 16 | 19 | |

5 | 4 | 16 | 19 | |

IEEE-30 bus | 1 | 15 | 30 | 30 |

2 | 11 | 32 | 33 | |

3 | 10 | 33 | 39 | |

4 | 10 | 38 | 46 | |

5 | 10 | 40 | 49 | |

6 | 10 | 40 | 51 | |

7 | 10 | 37 | 52 | |

IEEE-57 bus | 1 | 29 | 58 | 58 |

2 | 19 | 57 | 57 | |

3 | 17 | 61 | 62 | |

4 | 17 | 64 | 68 | |

5 | 17 | 69 | 71 | |

6 | 17 | 64 | 72 | |

7 | 17 | 64 | 72 | |

IEEE-118 bus | 1 | 61 | 122 | 122 |

2 | 41 | 123 | 123 | |

3 | 33 | 121 | 123 | |

4 | 32 | 134 | 140 | |

5 | 32 | 148 | 153 | |

6 | 32 | 155 | 159 | |

7 | 32 | 158 | 162 | |

8 | 32 | 161 | 163 | |

9 | 32 | 162 | 164 |

L | n_{PMU} | PMU Locations | PMU Channels |
---|---|---|---|

3 | 17 | 1, 6, 9, 15, 19, 22, 25, 27, 32, 36, 41, 44, 47, 50, 52, 54, 57 | {1-2, 1-16, 1-17 6-4, 6-5, 6-7 9-8, 9-10, 9-12 15-3, 15-13, 15-14 19-18, 19-20 22-21, 22-23, 22-38 25-24, 25-30 27-26, 27-28 32-31, 32-33, 32-34 36-35, 36-37, 36-40 41-11, 41-42, 41-43 44-39, 44-45 47-46, 47-48 50-49, 50-51 52-29, 52-53 54-9, 54-55 57-39, 57-56} |

L | n_{PMU} | PMU Locations | PMU Channels |
---|---|---|---|

3 | 17 | 1, 6, 10, 15, 19, 22, 26, 29, 30, 32, 36 38, 41, 46, 49, 54, 57 | {1-2, 1-16, 1-17 6-4, 6-5, 6-8 10-9, 10-12, 10-51 15-3, 15-13, 15-45 19-18, 19-20 22-21, 22-23, 22-38 26-24, 26-27 29-7, 29-28, 29-52 30-25, 30-31 32-31, 32-33, 32-34 36-35, 36-37, 36-40 38-37, 38-44, 38-45 41-11, 41-42, 41-43 46-14, 46-47 49-13, 49-38, 49-50 54-53, 54-55 57-39, 57-56} |

Interior-Point Methods | |||
---|---|---|---|

L | n_{PMU} | PMU Locations | PMU Channels |

4 | 17 | 1, 4, 6, 9, 15, 20, 24, 28, 31, 32, 36, 38, 41, 47, 51, 53, 57 | {1-2, 1-15, 1-16, 1-17 4-3, 4-5, 4-6, 4-18 6-4, 6-5, 6-7, 6-8 9-8, 9-12, 9-13, 9-55 15-1, 15-3, 15-14, 15-45 20-19, 20-21 24-23, 24-25, 24-26 28-27, 28-29 31-30, 31-32 32-31, 32-33, 32-34 36-35, 36-37, 36-40 38-22, 38-44, 38-48, 38-49 41-11, 41-42, 41-43, 41-56 47-46, 47-48 51-10, 51-50 53-52, 53-57 57-39, 57-56} |

Genetic Algoritjms | |||
---|---|---|---|

L | n_{PMU} | PMU Locations | PMU Channels |

4 | 17 | 1, 5, 9, 15, 19, 22, 26, 29, 30, 32, 36, 38, 39, 41, 46, 51, 54 | {1-2, 1-15, 1-17 5-4, 5-6 9-8, 9-10, 9-12, 9-13 15-1, 15-3, 15-13, 15-45 19-18, 19-20 22-21, 22-23, 22-38 26-24, 26-27 29-27, 29-28, 29-52 30-25, 30-31 32-31, 32-33, 32-34 36-35, 36-37, 36-40 38-22, 38-44, 38-48. 38-49 39-37, 39-57 41-11, 41-42, 41-43, 41-56 46-14, 46-47 51-10, 51-50 54-53, 54-55} |

Binary Swarm Optimization | |||
---|---|---|---|

L | n_{PMU} | PMU Locations | PMU Channels |

4 | 17 | 1, 4, 6, 9, 15, 20, 24, 28, 31, 32, 36, 38, 41, 47, 51, 53, 57 | {1-2, 1-15, 1-16, 1-17 4-3, 4-5, 4-6, 4-18 6-4, 6-5, 6-7, 6-8 9-11, 9-12, 9-13, 9-55 15-1, 15-3, 15-14, 15-45 20-19, 20-21 24-23, 24-25, 24-26 28-27, 28-29 31-30, 31-32 32-31, 32-33, 32-34 36-35, 36-37, 36-40 38-22, 38-44, 38-48, 38-49 41-11, 41-42, 41-43, 41-56 47-46, 47-48 51-10, 51-50 53-52, 53-57 57-39, 57-56} |

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## Share and Cite

**MDPI and ACS Style**

Theodorakatos, N.P.; Babu, R.; Theodoridis, C.A.; Moschoudis, A.P.
Mathematical Models for the Single-Channel and Multi-Channel PMU Allocation Problem and Their Solution Algorithms. *Algorithms* **2024**, *17*, 191.
https://doi.org/10.3390/a17050191

**AMA Style**

Theodorakatos NP, Babu R, Theodoridis CA, Moschoudis AP.
Mathematical Models for the Single-Channel and Multi-Channel PMU Allocation Problem and Their Solution Algorithms. *Algorithms*. 2024; 17(5):191.
https://doi.org/10.3390/a17050191

**Chicago/Turabian Style**

Theodorakatos, Nikolaos P., Rohit Babu, Christos A. Theodoridis, and Angelos P. Moschoudis.
2024. "Mathematical Models for the Single-Channel and Multi-Channel PMU Allocation Problem and Their Solution Algorithms" *Algorithms* 17, no. 5: 191.
https://doi.org/10.3390/a17050191