Again, this assumption is consistent with the previous works. If the line impedanceis known, the voltages of the surrounding nodes can be calculated. This includes smart gridtechnologies and increasing the sources of clean power. The test point, 8, will find every combination of places in those 31 spaces. To further explain this, a walkthrough will be done for the IEEE node network. Additionally, I am grateful for the funding I received through Mitacs Accelerate, a collaborationwith Enbala Power Networks, for without which would have made it difficult for me pursue thisresearch.

Comparatively, for the proposedalgorithm, it is fairly equally likely that either one, two, or three nodes will be unobserved. The algorithmwas tested on standard IEEE distribution feeders: Comparatively for the proposed algorithm, there is a highprobability of at least four nodes being unobserved. Disrespecting of consideration to normal condition and explanation about existent different cases for minimum unobservability which is proposed a method for that here to find minimum value for single PMU outage suggested objective function as the best and single line outage in a system as a contingency are the disadvantages of this one. These important topics are imperative for understanding certain aspects of theliterature review as well as the proposed solution later in the thesis.

# Optimal PMU Placement with Uncertainty Using Pareto Method

Note that this method is a little more complex than the summarized steps listed and the fullmethod can be read in [29]. After testing thesecombinations, the number 8 would either become the new upper limit or the new lower limit,and another iteration would be done.

This algorithm is based on a greedy algorithm which has manybenefits such as fast computation time and high reliability. Clearly, the placement setin Figure 2. The optimization process includes three main operations: Stage 1 can be summarizedbelow in one step: When placed at a node, it is capable of measuring the voltage phasor at the node and all incident current phasors to the node.

The aim was to keep plxcement as low as possible in order to be feasible forreal world distribution networks with many nodes.

## Mathematical Problems in Engineering

Now, if the switch is open or closed, completeobservability is achieved. This slowing of frequency decline allows operators to find a solution beforeother components, ones that cannot operate at a lower frequency, trip offline. Percent Unobserved NodesTo further illustrate the benefits of the proposed algorithm over the minimal case, an analysis wasdone to look at the percentage of unobserved nodes. Lastly, the customized greedy algorithm outperformedthe proposed algorithms in previous literature.

The result is a vector of values as given by 3. Thessis placed at a node, it is capable of measuring the voltage phasor at thenode and all incident current phasors to the node. When the switch is closed, complete observabilityis achieved. The addition of renewable energy sources in the distribution system isnow causing bidirectional power flow to occur.

## Incorporation of PMUs in power system state estimation

These important topics are imperative for understanding certain aspects of theliterature review as well as the proposed solution later placeemnt the thesis. Since this algorithm takes into account network reconfiguration, the switch was assumedto be open.

Therefore, a pre-greedy algorithm step wastaken. When the52switches are closed, nodes to are observed. How-ever, due to the fact that distribution networks can have hundreds of thousands of buses, it isimportant to decrease the computation time as much as possible in order to find a solution in areasonable amount of time for real-world networks.

By considering uncertainty, a multiobjective optimization exercise is hence formulated. It was assumed thatthe worst-case scenario was in effect: The second issue is that power demand from utility users has been increasing due to the in-creasing number of electronic devices being used and operated.

Therefore, network reconfigurationrefers to the changing configurations that occur when switches change their status from open toclosed, or vice versa. Additionally, these two papers and [27] were the only papers that considered network recon-figuration. Some ofthe previous works did not include the computation time, therefore it is not listed on the table.

It states what assumptionswere used, how network reconfiguration was considered, and explains the proposed algo-rithm step by step. At every generation GDE maintains a population P G of vectors of candidate solutions to the oltimal, which evolve throughout placemen optimization process to find global solutions [ 25 ]: Therefore, NUBP can be further broken down into 2. Continue step 2 until complete observability has been achieved.

The most recent notable conference took place in and was held in Paris, France. The transmission and distribution systems compose the delivery part ofthe power system.

However, the voltage angle differencesbetween locations on distribution systems are up to two orders of magnitude smaller than those ontransmission networks [13]. The distribution system typically contains switches which canchange its topology if the status of the switches change.

The test point will be the middle of the bound which is 8.