Network Matrix Reduction

This chapter presents a method of observability analysis of the power network using symbolic matrix reduction method. Slutsker and Scudder introduced this method which doesn't need numerical calculation and it is simple and fast compared to the other methods which needs calculation.

As mentioned in earlier chapter, observability analysis has to be done before state estimation to find the observable island, i.e. the part of the network where state estimation can be performed.

The measurement Jacobian matrix is reduced using symbolic matrix reduction.

The state estimation determines network state x_k for the observable island k with N_kbuses, from the given set of noisy measurement z_k

Z_k = hk(x_k) + v_k (1)

where v_k is the measurement noise vector and hk(x_k) is the island measurement matrix.

The equation (1) can be solved using the least square fit method

x_k= (h_ktR_k-1H_k)-1 H_ktR_k-1z_k

Where

H_k = is island measurement Jacobian matrix and R_k is the island measurement covariance matrix.

The condition for network to be observable is

Rank(H_k^t) = n_k(3)

Where n_k= 2N_k-1 is the number of independent state variables.

The rank of H_k doesn't depend on the measurement set. If there is no measurement noise the network observability will not change. When v_k = 0, the equation (1) becomes

Z_k = h(x_k) (4)

For the complete network the equation becomes

Z = H(x) (5)

Equations (4) and (5) are non linear and linearizing these equations, we get

Z_k = h(x_k) (6)

Z = H(x)(7)

Equation (6) can be solved or identified as the observable island is equivalent to equation (2).

Equations (6) can be solved by identifying and eliminating redundant equations. The Jacobian matrix H(x_k) is represented in the symbolic form. The elimination is carried out by eliminating the row of island variables and the column of island measurement equations.

Every observable island has got one or more solvable subsystems. Eliminating the observable island from (7) is carried out by detecting and eliminating its solvable subsystem.

Once all the subsystems of a given island are eliminated from the equation (7), Hk contains none of the island variable except the reference variable. Thus the observable island is identified.

Buses related to the eliminated variables of solvable system are referred as an observable group. During reduction process, observable group are merged into bigger observable group. The observable island is identified by adding the new observable groups in each step of the reduction process to the previously found island.

Before reduction of Hk matrix, the valencies of all the buses are set to one. In each reduction step the valency of the eliminated group buses is added to the valency of the reference bus, and the valency of the eliminated bus is set to zero. The observable group of reference bus includes all the eliminated buses. If the eliminated bus has valency greater than one then it was the reference bus of observable group eliminated previously and that group must be added to the current reference bus.

Once the reduction process is complete, the valency which is greater than one is the reference bus of the final observable island. The value of the reference variable is the size of an island. Observable group of reference bus contains all island buses.

There should be at least one voltage measurement in one of the buses to the network island to be observable. It is assumed that the measurement is located in the reference bus.

Method description:

It is the step by step matrix reduction process by eliminating observable group. Each reduction process is describes by eliminating the redundant equation automatically as well as identifying and augmenting the observable group to the previously found islands.

This method consists of 3 steps.

In step 1, the symbolic Ht bus -measurement incident matrix is constructed. In step 2, branch measurement flow is eliminated by eliminating a column of Ht corresponding to the equation and a row corresponding to the eliminated variable. All the redundant equations are removed and reduce to the reference variable. When the step 2 is completed the Ht matrix is reduced and it represents the set of linearly independent injection measurement equations. In step 3 reduction of Ht continues with respect of injection measurement equations. Two variable injection equations are removed as the same way the branch flow equations are removed in step 2.

When no more two variable equations can be found, search begins for solvable sub system with more than one equation. In reduced Ht matrix observable group can be found using simple search technique.

The nearest bus to the reference bus is chosen as a starting bus which will point through non zero entries of its Ht row to equations of a solvable subsystem in which variable participates. Those equations will point through the new busses of the solvable subsystem through non zeros in the column of the Ht matrix. New buses point through new equation and new equation point through new equation. This process continues till the solvable subsystem is found.

The buses found through analysis of Ht column are place on a Master Bus List (MBL). The buses from MBL are analyzed sequentially and placed in the Processed Bus List. The buses which are in PBL are placed in Processed Equation List (PEL).

When the size of PBL is equal to the size of PEL, then the solvable subsystem is found. The equality makes sure that the subsystem is observable and is the main criteria to search for the solvable subsystem. Buses which share equation with the reference bus play important role finding the solvable subsystem. Those buses are referred as the neighbor buses. In step 3, each bus is referred as potential reference bus. The search begins by using neighbor bus as the starting bus. This entire process assures that all the observable subsystem is identified and eliminate from Ht.

After completion of step 3, the reduced Ht includes only bused whose variable doesn't contain relationship (6). The remaining buses cannot be added to the already identifies island and thus the observability analysis is complete.

The steps involved in symbolic reduction matrix are:

Build the symbolic Jacobian matrix (Ht) for the network to be analyzed.

Reduce Ht using branch flow measurement equation. Identify and eliminate redundant equations.

Reduce Ht using two variable injection measurement equations.

Choose the bus with highest valency count or the highest measurement count to be reference bus, if all the buses have been processed go to step 11.

Identify first neighbor buses of the reference bus.

Choose a starting bus.

Search for a solvable subsystem.

if the solvable system is not found, go to the next step. If the solvable system is found, reduce Ht by eliminating the subsystem. Increase the valency of the reference bus. Combine previously detected observable groups of subsystem buses with the observable group of the reference bus.

If all first neighbor buses have been tried as starting buses, go to next step. Otherwise go to step 6.

If all remaining buses have been tried as reference bus, go to next step. Otherwise mark the reference bus as processed and go to step 4.

Mark each of the identified islands as observable if it has at least one voltage measurement. Observability analysis is complete.

The observability algorithm doesn't use any floating point arithmetic, the time required to determine the network observability is less than any other techniques. The Jacobian matrix is stored in symbolic form only. Only the nonzero positions are recorded. It doesn't take much memory to find the observability as it uses reduce matrix to find the solvable system.

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