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This paper aims to find strategic locations for additional Phasor Measurement Units (PMUs) installation while considering resiliency of existing PMU measurement system. A virtual attack agent is modeled based on an optimization framework. The virtual attack agent targets to minimize observability of power system by coordinated attack on a subset of critical PMUs. A planner agent is then introduced which analyzes the attack pattern of virtual attack agent. The goal of the planner agent is to mitigate the vulnerability posed by the virtual attack agent by placing additional PMUs at strategic locations. The ensuing problem is formulated as an optimization problem. The proposed framework is applied on 14, 30, 57 and 118 bus test systems, including a large 2383 node western polish test system to demonstrate the feasibility of proposed approach for large systems.

PMUs play a significant role in wide area monitoring and control. PMUs are capable of measuring node voltages and line currents as phasors. The measured quantities are time stamped based on global positioning satellite (GPS) signal. The time stamp allows analysis of measurement data that is geographically dispersed. Physical properties of power network enable computing the voltage and current phasors across the entire network by installing PMUs at only a subset of nodes.

The PMUs placement in strategic locations has been the vital research topic for PMU application and various methodologies have been introduced by power engineers all across the world [

1) Heuristic approaches. They have been widely adopted in this area. Simulated annealing is used in [

2) Mathematical approaches. Mathematical approach has been gaining popularity from recent years. They are easy to apply in the situation where a definite solution is required. They are based on formulae derived from mathematical calculations. Integer linear programming is a common approach as presented in [

Due to the critical nature of power systems, complete observability of all nodes at all times is required. However, the networked PMUs might be rendered out of service by natural disasters such as hurricanes or PMUs can be intentionally taken down by malicious attacks. Enough attention should be given to PMU vulnerability while placing PMUs in the system. The concept of economically deploying PMUs considering resiliency of existing system post attack is missing in the above literatures. Hence, this paper highlights a considerable interest in improving PMU redundancy at minimum cost. In order to ascertain a subset of nodes which are most likely to be attacked, a virtual attack agent is modeled. The aim of the virtual attack agent is to reduce system observability to a minimum while carrying out a coordinated attack on a subset of PMU installation nodes. This virtual attack is used by the operator agent to identify a set of critical nodes whose redundancy needs to be increased. The planner agent then finds strategic locations to place additional PMUs in order to increase redundancy of critical nodes while minimizing incurred cost.

This paper is organized as follows: Section 2 introduces two agents including the attacker and the planner to design a framework for classifying critical PMUs and planning scheme. Section 3 establishes a mathematical model to corporate on their objectives: the attacker aims to disable critical PMUs while the planner tries to design remedial measures. In Section 4, the model is applied to different standard test systems. Finally, Section 5 summaries the paper.

An uncertainty constraint PMU placement problem can be expressed in three different agent based stages:

・ Attacker: A virtual attack agent is introduced whose goal is to take down a set of installed PMUs to reduce system observability. Uncertain events like intentional attacks are an important aspect that needs to be considered while making PMU placement decision. Due to geographical span of interconnected power systems planning a coordinated attack on all of the installed PMUs is improbable. Hence, the virtual attack agent will carry out coordinated attacks on a subset of installed PMUs that are deemed critical. Here, the set of critical PMUs are the ones which when taken out of service minimizes system observability. Cardinality of the critical set is assumed to vary depending on the resources available to virtual attack agent.

・ Operator: At this stage, the operator has to take corrective measures to mitigate the possible damage caused by the attacker. The operator agent identifies a set of critical nodes based on virtual attack agents attack plan. The operator agent then relays the corrective measure, which in this case is to increase the redundancy of critical nodes, to the planner agent.

・ Planner: The task of planner is to deploy additional PMUs to increase redundancy of critical nodes at minimum cost.

Schematic representation of the three cyclic stages is shown in

Each undesired PMU outage caused by the virtual attack agent is an optimization scenario for the operator. These undesired outages can be single, double or multiple based on virtual attack agent’s resources. Let P be the number of PMUs deployed into the system and Ψ be the scenario which corresponds to the number PMUs to be attacked by the attacker. The total scenario can be represented as combinatorial number _{P}C_{Ψ} as:

Since there are hundreds of thousands of possible attack scenarios, it is impossible to enumerate all scenarios for large systems due to computational burden. Instead, by adopting the approach in (2) a worst case scenario can be obtained.

where ƞ Î [0, 100]―representing the percentage of installed PMUs that are attacked. As a worst-case scenario, an assumption has been made that the attacker can attack up to 50% of the total deployed PMUs. Depending upon ƞ value, a set of attacked PMUs

Development of agent models as an optimization problem is discussed in this section. The initial deployment locations for PMUs, which act as the starting point for the proposed agent based framework are obtained using optimal PMU placement algorithm from [

The objective of virtual attack agent is to attack a subset of installed PMUs in the system such that the system bus observability is minimized. The attack agent is modeled using binary integer programming.

The mathematical formulations for attacker’s objective is as follows:

S.t.

The objective function (3) ξ_{k} is the decision variable that tends to give the observability of each bus in terms of binary variable. If the bus is observable by PMUs remaining in the system after the coordinated attack by virtual attack agent then ξ_{k} will take the value of 1 and if the bus is not observable by any of the PMUs then ξ_{k} will take the value ‘0’. In general, observability of a bus can be 0 in which case the bus is not observable or observability can be a positive number which means the bus is observable.

Since the available PMUs were placed based on system network topology, it becomes necessary to define a network connectivity matrix A.

Elements in matrix A are defined as follows:

In constraint (4), x_{i} is an auxiliary binary variable of PMU placement. If the PMU is present at the i^{th} bus then x_{i} is regarded as 1 otherwise 0. Before the attack, the observability of the i^{th} bus denoted by left-hand side of (4) should be equal to the product of connectivity matrix of bus i and PMU placement variable x_{i}. Since the attacker already know the exact location of the PMUs, the attacker agent tries to enumerate all the possibilities to destroy or damage the PMU which are critical. This procedure is presented in (5). The word ‘critical’ defines those set of PMUs whose installation in the system increases the system observability. Post attack the variable x_{i} is zero for the disabled or attacked PMU. In this case, the constraint (4) will act as inequality constraint because the observability of the bus at right hand side will be greater than left hand side. The connectivity matrix is always fixed as long as all the transmission lines in the system are in service. The variable x_{p} is the PMU placement variable post attack. Depending upon the auxiliary variable x_{p}, the attacker performs all combinatorial number and checks the observability of each bus one by one. Those combination sets where the observability of bus shows the maximum number, the attacker tries to attack on those particular sets of PMUs. Constraint (4) helps the attacker to judge the most attractive set of PMUs to act on.

Computational complexity of this optimization model increases substantially when dealing with large number of system buses. From (4), the total number of inequality constraints is equal to the number of system buses N and the equality constraint (5) is split into two sections, one for the set of the buses where PMUs were installed and other for the set of buses where PMUs were not installed. Therefore the total number of constraints is N + 1 + 1. Similarly the total numbers of variables are twice the number of system buses M. This is because the first half M/2 denotes the auxiliary variable of PMU placement post attack and the other half M/2 denotes the bus observability.

The responsibility of the operator is to identify vulnerable nodes based on the behavior of virtual attack agent. Vulnerable nodes in this context are a set of critical buses whose observability is compromised by the virtual attack agent. Critical buses are the buses include critical PMU installation buses and buses that are observable by critical PMUs.

The number of PMUs attacked by virtual attack agent is a percentage of the total number of installed PMUs. Since, larger systems have larger number of installed PMUs, the number of critical buses also tends to increase with system size. Since various sets of PMUs were obtained depending upon the availability of attacker’s resources. Now, with the concern of PMU’s and their installation cost, from those several sets of classified critical PMUs, the planner has to choose only the most repeated PMUs among all sets of critical PMUs. To obtain this, following formulation is used.

where

The critical buses are those buses that are observable from the set of critical PMUs.

where _{w} and θ is the index of buses which are adjacent to critical PMU located buses.

The objective of the planner agent is to install additional PMUs in strategic locations to mitigate the vulnerability posed by virtual attack agent. The optimal PMU placement considering the critical PMUs is as follows:

Subject to

The objective function (13) implies that minimum number of PMU is placed in the system and _{i} as described earlier in attacker’s model. In this model, b_{i} is observability constraint for non critical buses and is considered equivalent to one. Whereas for critical buses, the observability constraint

The performance of proposed model is tested on 14, 30, 57 and 118 IEEE test bus systems including large power system 2383 bus Western Polish system [

The number of critical PMUs depends upon the size of the system and the system topology. The set of PMUs that poses a higher influence in increasing the system bus observability are shown in the _{min} is the number of attacked PMUs. Similarly,

To further analyze strictly critical PMUs, only one set of PMUs per system is evaluated. The PMUs that happens to be critical for more than twice among the differentiated level of resources availability are only considered as most critical PMUs.

The PMU placement planning scheme is presented in this section. The goal of the planning scheme is to place additional PMUs in order to mitigate the loss of observability in the event of an attack. From the previous section, the set of critical buses with respect to loss of observability was obtained. The planner agent uses this

information to obtain PMU placement scheme for installing additional PMUs with least cost to increase redundancy of critical nodes. For the most critical buses as shown in

IEEE System | PMU location | Resources available to the attacker | |||||||||
---|---|---|---|---|---|---|---|---|---|---|---|

10% | 20% | 30% | 40% | 50% | |||||||

N_{min} | Ψ_{1} | N_{min} | Ψ_{2} | N_{min} | Ψ_{3} | N_{min} | Ψ_{4} | N_{min} | Ψ_{5} | ||

14 | 2, 7, 10, 13 | _ | _ | 1 | 2 | _ | _ | _ | _ | 2 | 2, 13 |

30 | 1, 2, 6, 10, 11, 12, 15, 19, 25, 29 | 1 | 10 | 2 | 6, 10 | 3 | 6, 10, 25 | 4 | 6, 10, 12, 15 | 5 | 6, 10, 12, 15, 19 |

57 | 2, 6, 12, 19, 22, 25, 27, 32, 36, 39, 41, 45, 46, 49, 51, 52, 55 | 2 | 6, 41 | 3 | 6, 32, 41 | 5 | 6, 22, 32, 41, 46 | 7 | 6, 12, 22, 32, 41, 49, 55 | 9 | 6, 12, 22, 32, 36, 39, 41, 49, 55 |

118 | 1, 5, 9, 12, 15, 17, 21, 25, 28, 34, 37, 40, 45, 49, 52, 56, 62, 64, 68, 70, 71, 76, 77, 80, 85, 87, 91, 94, 101, 105, 110, 114 | 3 | 56, 105, 110 | 6 | 49, 56, 80, 85, 105, 110 | 10 | 5, 12, 17, 49, 56, 80, 85, 105, 110 | 13 | 5, 12, 15, 17, 34, 37, 40, 49, 56, 80, 85, 105, 110 | 16 | 5, 12, 15, 17, 34, 37, 40, 45, 49, 56, 62, 80, 85, 94, 105, 110 |

IEEE Test System | Total installed PMUs | Selected critical PMU buses |
---|---|---|

300 bus system | 87 | 315 109 112 190 268 269 270 272 |

2383 polish | 746 | 6 18 29 133 246 309 310 321 322 353 354 361 365 366 374 425 456 494 511 525 526 527 546 556 613 644 645 679 694 717 750 754 755 796 797 870 923 944 978 979 1050 1096 1120 1138 1190 1201 1212 1213 1216 1217 1245 1483 1504 1524 1647 1664 1669 1680 1761 1822 1882 1883 1885 1919 1920 2112 2113 2166 2195 2196 2235 2258 2261 2274 2323 |

IEEE Test System | Critical Buses |
---|---|

14 | 1 2 3 4 5 |

30 | 2 4 6 7 8 9 10 17 20 21 22 28 |

57 | 4 5 6 7 8 11 21 22 23 31 32 33 34 38 41 42 43 56 |

118 | 2 3 4 5 6 7 8 11 12 14 15 16 17 18 30 31 42 45 47 48 49 50 51 54 55 56 57 58 59 66 69 77 79 80 81 83 84 85 86 88 89 96 97 98 99 103 104 105 106 107 108 109 110 111 112 113 117 |

IEEE Test System | No. of Optimal PMUs | % of Additional PMUs compared with original placement | Additional PMU Placement Location considering critical buses | |
---|---|---|---|---|

Normal Operating Condition | More weight age to critical PMUs | |||

14 | 4 | 6 | 50% | 1 4 |

30 | 10 | 14 | 40% | 5 8 12 16 22 |

57 | 17 | 26 | 53% | 4 7 11 21 23 30 33 34 42 |

118 | 32 | 51 | 59% | 4 6 8 18 32 46 54 57 58 78 83 88 96 100 106 108 111 112 117 |

This paper proposes a planning approach for optimal PMU placement making the system more resilient to PMU failure. The likelihood of undesired events is analyzed by creating a virtual attack agent which intends to damage some of the critical PMUs in the system. Operator agent is used to obtain a subset of buses that are critical based on the attack pattern of virtual attack agent. Simulation results illustrate the ability of the planner agent to place additional PMUs at strategic locations to increase the redundancy of critical buses. The developed framework was tested on several test systems including a 2383 bus western polish system and optimal results were obtained in all cases. Future work will consider the account of zero-injection measurement and branch flow measurement for more economical solution.

Jyoti Paudel,Xufeng Xu,Karthikeyan Balasubramaniam,Elham B. Makram, (2015) A Strategy for PMU Placement Considering the Resiliency of Measurement System. Journal of Power and Energy Engineering,03,29-36. doi: 10.4236/jpee.2015.311003