Harmonic State Estimation

State estimation [32] is one of the main tools in energy management systems. The states are the bus voltage magnitudes and phase angles. The state estimation problem is to estimate the system states from the over-determined system equations. Generally the problem is formulated as a weighted least squares problem where the estimates of states

minimize the weighted sum of the squares of the measurement error. The general structure of the state estimation is based on the single phase and single frequency model, assuming that the system is balanced and symmetrical, voltage and current waveforms are pure sinusoidal with a constant frequency. The measurements used are the active power, reactive power, and voltage measurements in the power system. For the non- linear power flow model, the solutions are obtained iteratively.

The traditional state estimation technique estimates the complex bus voltages which are generally used as state variables at the fundamental frequency and needs to be extended to estimate the harmonic distortion levels in the electric power systems. The harmonic state estimation (HSE) technique is developed to determine harmonic generation and propagation throughout the power system [9-14]. HSE is a generalized state estimation algorithm in the sense that the estimation is carried out over several harmonic frequencies and generally in three phases where the system is assumed to be unbalanced and asymmetrical. One of the distinctions of HSE from traditional state estimation is the measurement requirements. The measurements for HSE include the harmonic voltage and current measurements. Current and voltage waveform and phase angle measurements are required for the harmonic state estimation. Power measurements can also be used but in general these measurements are not favored in HSE algorithms.

The mathematical model which relates the measurements to the state variables can be formulated as

( )

= +

where, z is the p-dimensional measurement vector, x is the q-dimensional state vector, h(·) is a set of nonlinear equations relating measurements to states and e is the p- dimensional error vector. If the measurements can be related to the states with linear equations, then the linear measurement equations can be used:

=

z Hx + e (2.23)

where z is the p-dimensional measurement vector, x is the q-dimensional state vector, H is the p×q dimensional measurement matrix and e is the p-dimensional error vector at harmonic order h. Using state estimation, state variables are estimated from the available measurements. If bus voltages are selected as state variables to be estimated which are similar to the traditional state estimation, then the harmonic voltage and current measurements are related to the state variables with the following equations

h = h h+ h I Y V e (2.24) h = h h+ h V PV e (2.25) Lh = Lh h+ h I Y V e (2.26)

where P is the permutation matrix with entries 1 and 0, IL is the line current measurement

and YL is the line to bus admittance matrix with proper dimension. Equations (2.24),

(2.25) and (2.26) show the relation of bus current injection measurement, bus voltage measurement and line current measurement to the state variables at harmonic order h, respectively.

The harmonic state estimation problem is to find the system states from the over- determined system of equations (2.23) where q < p. The problem formulation with weighted least squares (WLS) where the objective function to be minimized for the WLS estimator is given as,

( )

In (2.27) xi are the components of the state vector x, T is the weighting matrix, and H is the Hermitian transpose. The elements of the weighting matrix represent the measurement accuracy and reliability of corresponding measurement. The common choice for T is the inverse of the error covariance matrix of the measurements. The harmonic order h is dropped in (2.27), however the estimation is carried out for each harmonic order of interest. The solution of (2.27) for the state vector x is

(

)

ˆ T − T

x = H TH H Tz (2.28)

In (2.28) (HTTH)-1 is called the gain matrix. The measured values, states and the measurement matrix are complex valued. The optimization problem in (2.27) can be modified by separating the real and imaginary parts of the complex variables [10, 14]. Instead of WLS, other performance criteria can also be used, such as Weighted Least Absolute Value, Least Median of Squares or Non-quadratic Estimators.

In a three-phase harmonic state estimation up to harmonic order h, the number of states is 3h times larger than the number of states in traditional state estimation. A complete system-wide state estimation in harmonic domain is not practical because of the large

number of measurements required and the cost of the instrumentation. In reality there is a limited number of measurements, and the harmonic state of the network is estimated partially [13]. The state estimator will provide a unique solution if the measurement matrix has a full rank. Underdetermined equations don’t allow all variables to be estimated. In this case the gain matrix does not have full rank.

The system is said to be observable if a unique solution of states can be obtained for a given set of measurements [32]. Classical observability analysis can be used to determine the observability of the system for HSE problem. It is important to determine the observable/unobservable states of the network and the redundant measurements. Existing observability analysis methods can be collected into three groups: topological, numerical, and symbolic approaches.

Topological observability [33] is based on the graph theory principles. “The network is topologically observable if there exists a spanning tree of full rank” [33]. A tree is a connected loop-free set of branches of the network. A spanning tree is maximal tree such that all nodes are connected by branches of the tree. Topological methods are combinatorial, and require significant computation effort. However, state estimation results are not necessary to determine the observability of the system.

The system is said to be algebraically observable if the gain matrix has a full rank.

Numerical observability [34, 35] is based on the solvability of state variables . Even if the system is algebraically observable, a numerically ill conditioned set of equations and rounding errors of floating point calculations may prevent the determination of the state variables with acceptable accuracy.

Symbolic observability [36] is also based on the measurement matrix but not on the numerical values of entries of the matrix. Each element of the measurement matrix which is different than zero is replaced by 1 eliminating numerical problems related by floating point operations.

Hybrid methods [37] combining the topological and numerical methods are also developed for observability analysis. An overview on observability approaches can be found in [38].