# Linear System State

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Observability analysis using linear programming

The set of available measurements are used by the state estimator to estimate the system state. As the measurement and the location of the system is given, the network observability analysis can be determined if the system is observable or not. If the system is not observable then additional meters may have to be placed on the particular location to make the system observable. Prior to the state estimation the observability analysis should be done to ensure that there are enough set of measurements.

*Different types of errors and change in topology sometimes may lead to the case where the entire system cannot be estimated. Then the system may contain some isolated observable island. The observability analysis detects all the observable island before the execution of the state estimator.*

The methods for the network observability analysis can be divided into two : numerical and topological.

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1.2 State Estimation, Observability Analysis and Meter Placement

Electric power system state estimation [3-8] was introduced by Fred Schweppe of MIT in 1969. The operating state of a power system is determined by the state

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estimation (SE) function using a redundant set of real-time measurement data. The SE function is the basis for all advanced applications of an Energy Management System (EMS). The results of state estimation are used to compute various estimates for the line flows, losses and net bus injections. The deregulation of the electric power industry has transformed state estimation from an important application into a critical one [5]. Many critical commercial issues in the power market, such as congestion management, need to be formulated and addressed based on a precise model of the power system, which is derived from the state estimation process. Failure to obtain these quantities in real-time or miscalculating them, should be avoided in order to ensure proper accounting of power transactions as well as for system security. This implies that the state estimator should be made very robust against the topology changes (including branch outages and bus splitting) and temporary loss of measurements or remote terminal units (RTU). Observability analysis is another important procedure closely related to state estimation. State estimation will not be possible if there are not enough measurements. A system is determined to be observable if all the state variables (bus voltage magnitudes and relative phase angles) can be estimated using the available measurements. Various methods proposed for network observability analysis have been well documented in the literature [9-14]. If the power system is unobservable there will be a need to install new meters to make the system observable. Meter placement requires making decisions as to where and what types of meters should be placed. Many approaches are based on Integer Programming (IP) and heuristic solution techniques [15-19].

When installing a new state estimator or upgrading an existing one, measurement configuration will have to be considered in order to ensure that the system will be observable. The paper [20] investigates the meter placement problem with an objective of ensuring network observability against single branch outages. It presented a topological method to install new meters around the system for single branch outages. The papers [21][22] present a systematic procedure by which measurement systems can be optimally upgraded. The proposed method yields a measurement configuration that can withstand any single branch outage or loss of single measurement, without losing

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network observability. It is a numerical method based on the measurement Jacobian and sparse triangular factorization, making its implementation easy in existing state estimators. However, that method is valid only for loss of single measurements and single branch outages, and also assumes that the original measurement system is observable. In a given practical power system there will be some specified contingencies which will include simultaneous loss of several measurements and/or outages of several branches. A new method [23] will be proposed in Chapter II, which greatly improves the unified approach presented in [21][22] to account for cases involving such contingencies. This is accomplished by extending the IP problem to consider more than one candidate for a given contingency.

1.3 Loss Allocation

The electric power industry is experiencing important changes brought about by the deregulation. Electric power generators and users engage in power transactions which take place over the transmission system and create losses. Transmission losses represent up to 5.10% of the total generation, and are worth millions of dollars per year. Consequently, the problem of “who should pay for losses” arises and the satisfactory sharing of the transmission system utilization costs among all market participants has become a key issue. Unfortunately, losses are expressed as a nonlinear function of line flows, and it becomes almost impossible to calculate exactly the losses that are incurred by each generator, load or transaction in the system.

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A number of loss allocation schemes have been proposed in the literature to allocate the system losses to generators / loads in a pool market or to individual transactions in a bi-lateral contracts market. Based on different assumptions and approximations there are mainly four families of schemes: Pro rata methods [24], incremental transmission loss (ITL) methods [25-30], proportional sharing procedures [31-41], and loss formula methods [42-51]. Different loss allocation methods have been compared in [52-55].

http://idea.library.drexel.edu/bitstream/1860/519/9/Dafis_Chris.pdf

1.2.2 Observability in Power Systems

Historically, the observability problem in power systems considers the following

question: Given the output data on a time interval [t0,t1], what is the procedure to

compute the state of the system? In other words, power system observability requires

that enough measurements exist and they are distributed throughout the network in such a

way that the solution to the state estimation problem is possible. As a result, the derived

observability criterion is the solvability condition of the state estimation problem.

Traditionally the state estimation problem is based on a power system model that neglects

the dynamics of the system and assumes the system is at an equilibrium point. Therefore,

the derived observability criteria are also based on the assumption that the system is at

equilibrium and neglects any system dynamics. The general structure for the

observability determination in power systems is given in Figure 7. Algorithms that

compute observability accept inputs from the measurement system and the network

processors to determine known system variables and the topology of the system

respectively. Once the algorithm qualifies whether the system is observable, the next

stage is to quantify the result. The two main approaches in qualifying observability are

numerical and topological approaches.

The topological approach is based on graph theoretic principles and focuses on

determining the maximum spanning tree (a tree that contains every node in the network),

of full rank, where each branch of the tree is assigned to a different measurement [1,2].

With this configuration, all branch flows can be determined from the measurements,

making the network observable (assuming the entire network is incorporated in the

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spanning tree). For branches where a meter does not exist, the branch is assigned to an

injection measurement at an incident node. A further introduction to these methods is

provided in Chapter 2.

Observability Formulation

Quantifying Result

Network Processor Measurement

System

Figure 7: General Structure of Observability Determination

Contrary to the topological analysis that centers upon the network topology and the

inherent measurement number and placement, the numerical approaches center upon the

Jacobian matrix defined as [1,2]:

*i i*

*k k*

*i i*

*k k*

*P P*

*V*

*J*

*Q Q*

*V*

ϑ

ϑ

∂ ∂

= ∂ ∂ ∂ ∂

∂ ∂

(1.1)

in the case of real (*P**i*) and reactive (*Q**i*) power measurements, where the states of the

system are the bus voltages *V**i *and angles è

*i*. The Jacobian matrix is introduced in

solving the state estimation problem, and in general, when the Jacobian matrix is of full15

rank the network is said to be observable. A Jacobian of full rank renders the state

estimation problem solvable, providing the link between a solvable state estimation

problem and the observability determination of the system. This Jacobian also includes

information necessary for the solution of the power flow problem in power systems. The

power flow Jacobian, in the case of real and reactive measurements, is typically a subset

of the state estimation Jacobian. A further introduction to numerical approaches

addressing observability based on the state estimation Jacobian is provided in Chapter 2.

Both topological and numerical methods have been applied for large systems with

successful results. However, both methods are deficient in the following areas:

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Examples of our work• The information is based only on the algebraic system variables and the dynamic

states of the system are ignored. A way to calculate these dynamic states even at

system equilibrium points is not provided. A change in the power system model

is obviously needed to accommodate this capability.

• The inability to track the observability determination of the system during

transitions from one equilibrium to the next.

1.2.3 Observability in System Theory

The study of observability was motivated by the desire to know the entire state of the

system without measuring all the system states, but rather having measurements that are

functions of the system states, or subsets. The idea of a system description as system

states, inputs and outputs (state-space description), as well as the system properties of

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controllability and observability were first formulated by Kalman in the early 1960s

[10,11].

The early investigations on the observability property of systems focused on linear timeinvariant

systems (LTI), providing a relationship between the system coefficient matrix

(*A *in Table 1) and the measurement matrix (*C *in Table 1), where *x *is the system states

vector, *u *are the system inputs and *p *is the system measurements vector. Later studies

evolved to linear time-varying systems (LTV) [12,13] and nonlinear systems [14,15]. A

brief description of the observability properties of these systems is provided in Table 1

(further details can be found in [13]).

The mathematical models used to describe the dynamics vary greatly, and Table 1

provides a mere subset of these models to outline the progression of observability

research. Therefore various other observability tools exist based on the selected

mathematical model to represent the physical system. In general, the observability

property, and other system properties such as stability and controllability, is global for

linear systems, but local in nature for nonlinear systems. Therefore, evaluating a single

operating point in terms of observability can be sufficient for linear systems, and the

entire family of system operating points needs to be examined for nonlinear systems.

For power systems, various mathematical models exist, however, since the objective is to

track the system as it moves along equilibrium points, only nonlinear system models are

considered.

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Table 1: Sample Observability Methods in System Theory

1.2.4 Differential Algebraic Models of Systems

The dynamics of a power system can be modeled with a combination of nonlinear

differential equations, and nonlinear algebraic equations. The nonlinear differential

equations correspond to the nonlinear dynamics of the system, and the nonlinear

algebraic equations correspond to the algebraic constraints of the system. In a sense, the

algebraic equations provide the region upon which the trajectories defined by the

solutions of the differential equations lie (solution manifold).

Type Model Observability Test Comments

Linear

Time

Invariant

*x Ax Bu*

*p Cx*

= +

=

&

[ -1]

( )

*n T*

*O*

*O*

*J C CA CA*

*rank J n*

=

=

L

*x*∈¡*n **p*∈¡*q*

, & - Constant Coefficient

Matrices

*A B C*

Linear

Time

Varying

( ) ( )

( )

*x A t x B t u*

*p Ct x*

= +

=

&

1

1 ( , ) ( , ) ( )

( ) ( , )

is nonsingular

*o*

*t*

*T T*

*O o o*

*t*

*o*

*O*

*J t t t C*

*C td*

*J*

ô ô

ô ô ô

= Ö

Ö

∫

*x*∈¡*n **p*∈¡*q*

:System Transition

Matrix

Ö

Nonlinear

ODE

Based

( ) ( )

( )

*x f x v x u*

*p h x*

= +

=

&

1

1 1

1

[ ( ) ( )

( ) ( )]

( )

*o o*

*O f f q*

*n n T*

*f f q*

*O*

*J dL h dL h*

*dL h dL h*

*rank J n*

− −

=

=

L L

L

(.), (.), (.) :

Nonlinear Functions in

*x **n**p **q*

*f v h*

*x*

∈¡ ∈¡

( ) ( )

( )

*f*

*L h x h f x*

*x*

*Lie Derivative*

= ∂

∂

The electricity markets of England, Spain and Brazil are currently using Pro rata schemes to allocate the losses to

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generators and/or consumers. This type of methods is simple to understand and implement. However, the network topology is never taken into account. Obviously it is not fair for two identical loads, which locate near generators and far away from generators respectively, to be allocated with the same amount of losses.

ITL methodologies use the sensitivities of losses to bus injections to allocate the losses to generators and loads. The paper [28] provides analyses and test results from a practical implementation of an incremental allocation procedure in the Norwegian electric system. The paper [29] solves a system of differential equations by using numerical integration where a distributed slack bus concept is used. The ITL methods depend on the selection of the slack bus and also the slack bus is allocated with no losses.

Proportional sharing methods, sometimes called flow-tracing schemes, assume that the power injections are proportionally shared among the outflows of each bus and trace the electricity down from the generation sources or up from the load sinks. The assumption here “the power flow reaching a bus from any power line splits among the lines evacuating power from the bus proportionally to their corresponding power flows” is neither provable nor disprovable. Also, it is not possible to allocate losses to generators and loads at the same time.

Recently, some loss-formula based methods have been presented. A quadratic loss formula is proposed in [42] to allocate transmission losses among trades. A "physical-power-flow-based" approach expresses the quadratic loss approximation with individual transactions in a multiple-transaction framework in [43]. Another loss allocation method is based on the bus impedance Z-bus matrix [44] and allocates transmission losses among loads and generators assuming a pool dispatch. A

http://www.pserc.org/cgi-pserc/getbig/publicatio/reports/2006report/abur_state_estimators_s22_reports.pdf

1.2 State Estimation Literature Review

Schweppe was one of the first to formulate static state estimation for a power network

based on the power flow model [1]. The idea is to estimate the electrical states of the power network,

mainly voltage magnitudes and phase angles. These states might not be directly observable

based on physical relationships between the measurements and the desired unknown states.

Another advancement in the field of state estimation was the introduction of a weight matrix

to increase the accuracy of the results. Weighting is done to enhance the “input” of accurate

measurements, and de-emphasize the less accurate measurements. It can be shown that the

maximum likelihood estimate utilizes weights that are based on the covariance of the measurement

devices [2]. The more accurate a measurement, the greater is the selected weight in the

state estimator. Weighting is the practice of accounting for the confidence in a measurement.

Over time, the confidence in a measurement may change. A solution to this problem is to auto

tune of the weights of measurements. The suggested method of auto tuning the weights is to look

at recent error variances of the measurements and use these to recalculate the weights of measurements

from a short history [3]. References [2, 4-10] further relate to ideal weighting of measurements

for power system state estimation.

Measurement errors are typically assumed to be statistically distributed with a zero mean

[11]. Due to increase use in “sensorless” technologies such as A/D converters the zero mean as2

sumption is not always true [11]. A suggested method of overcoming this problem is to combine

measurement calibration [12-15] with state estimation. Calibration of the measurements can be

done in parallel with state estimation by noting the error of measurement over several scans of

the measurement. The calibration error will be a constant compared to measurement error which

is typically normally distributed [15].

The process of overcoming measurement noise is inherent in taking physical measurements,

but there are situations in which the data is grossly erroneous. The data that are erroneous

must be identified and eliminated. One method for the detection of bad data is the examination of

the measurements and if the measurements deviate from expected values by some preset threshold

the measurement can be assumed to be bad [16]. Another problem that causes state estimators

inaccuracies is the power system model itself. Generally the simple linear model *Hx=z *is

used where the *H *is the measurement model (processing matrix), *x *is the state vector, and *z *is the

measurements. If the process matrix is incorrect, the model does not represent what is physically

happening in the system. The detection of both erroneous data or improper formation of the

process matrix may be done by examining the residual of the equation *Hx=z *[2]. A further modeling

‘error' is a result of linearization. Since the process matrix is truly a function of operating

state, *H=h(x)*. The linearization of the problem results in constant *H*.

References [4, 6, 10, 17] are textbooks relating to state estimation in power engineering;

references [5, 8, 9, 18] are representative of solutions methods; and [16, 19] are case studies.

1.3 The Pseudoinverse and Least-Squares Estimation

The commonly used model for a linear static system is

*Hx=z *(1.1)

with *H *as the process matrix (*m *by *s *matrix), *x *is the state vector (dimension *s*), and *z *is the

measurement vector (dimension *m*) is overdetermined when *m *is larger than *s*. References [4, 6,

10, 17, 20] describe Equation (1.1). Equation (1.1) can be “solved” in the least-square sense by

minimizing *||r||**2*,

*r=Hx-z *(1.2)

where || ● ||2 refers to the 2-norm [17]. Properties of norms appear in [17] and Appendix A. It

can be shown that *|| r ||**2 *is minimized when

*x *= *x*ˆ = *H*+*z *. (1.3)

The notation *x*ˆ is the “estimate” of vector *x, H**+ *pseudoinverse of *H*. References [4, 6, 10, 17,

20] describe the properties of the pseudoinverse. Equation (1.3) is known as an unbiased least

squares estimator.

Other methods of determining the state variables are under study. One such method is

weighted least absolute value. Unlike weighted least squares there is no explicit formula for the

solution to linear weighted least absolute value. The weighted least absolute value is found by

linear programming [21]. Another method suggested is to find the maximum agreement with

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measurements. The state estimate agrees with the majority of the measurements taken in the system

[7].

The least squares method of state estimation requires the system to be observable. Observability

can be defined as: given a set of measurements and their locations (i.e., given *z *and

*H*), then a unique estimate of the system state vector *x, *i.e. *x*ˆ , can be found. A basis of observability

analysis is graph theory. To determine which states are unobservable, set the measurement

vector, *z*, to zero,

*Hx*ˆ = 0 .