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The traditional weighted least of squares (WLS) state estimation problem has the form

(2.1)

subject to

(2.2)

(2.3)

where x is the n × 1 state variable vector, J(x) the objective function (weighted quadratic error), h (x) a m × 1 functional vector expressing the measurements as a function of the state variables, w a m × 1 weighting factor vector, z the m × 1 measurement vector, f ( x ) the p × 1 equality constraint vector mainly enforcing conditions at transit buses (no generation and no demand), and g (x ) the q × 1 inequality constraint vector enforcing physical limits of the system.

Among the components of vector h (x ), note that active and reactive and power injections at bus i are computed as

(2.4)

(2.5)

where P_i and Q_i are the active and reactive power injection at bus i, respectively; v_i and θ_i are the voltage magnitude and angle at bus i, respectively; G_ij and B_ij are the real and imaginary part of the bus admittance matrix, respectively; and Ξ is the set of all buses. G_ii and B_ii can be computed as indicated in [7].

The active and reactive power flow ij are computed as

(2.6)

(2.7)

where P_ij and Q_ij are the active and reactive power flow from bus i to bus j, respectively, and the shunt susceptance of line ij.

As it is customary, measurements are considered to be independent Gaussian‐distributed random variables.