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CHAPTER 2 STATE ESTIMATION IN POWER SYSTEMS BASED ON A MATHEMATICAL PROGRAMMING APPROACH
Eduardo Caro and Araceli Hernández
Universidad Politécnica de Madrid, Madrid, Spain
2.1 INTRODUCTION
This chapter revisits the state estimation problem in power system and develops a direct approach of a mathematical programming solution for the most relevant estimators. Traditionally, power system state estimation is tackled by solving the system of nonlinear equations corresponding to the first‐order necessary optimality conditions of the estimation problem.
Besides the traditional approach, the estimation problem can be solved directly, which has a number or advantages such as (i) including inequality constraints representing physical hard limits, (ii) taking advantage of currently available state‐of‐the‐art nonlinear programming solvers, (iii) treating sparsity in an efficient and implicit manner, etc.
For the interested reader, pioneering state estimation works include [1–6]. A detailed description of state estimation in power system is provided in [7], which includes an appropriate literature review.
2.2 FORMULATION
The traditional weighted least of squares (WLS) state estimation problem has the form
(2.1)
subject to
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 Pi and Qi are the active and reactive power injection at bus i, respectively; vi and θi are the voltage magnitude and angle at bus i, respectively; Gij and Bij are the real and imaginary part of the bus admittance matrix, respectively; and Ξ is the set of all buses. Gii and Bii can be computed as indicated in [7].
The active and reactive power flow ij are computed as
(2.6)
(2.7)
where Pij and Qij are the active and reactive power flow from bus i to bus j, respectively, and
As it is customary, measurements are considered to be independent Gaussian‐distributed random variables.
Example 2.1 Traditional Formulation
The 4‐bus power system depicted in Figure 2.1 is considered throughout this chapter for illustrative purposes. The network data is provided in Table 2.1. This example includes a generating bus, two demand buses, and a transit bus. Considering bus 4 as the reference bus, the state variable vector x has the form
Ten measurements are considered as shown in Figure
