State Estimation

State estimation [23][24][26], part of system monitoring, is the process using mea-surements to best estimate the smart grid status. The estimates are state variables including bus voltage magnitudes and angles. In this thesis, we consider steady-state DC power system with n + 1 buses, where one of buses is set to reference bus and the rest n buses are with n unknown state variables. We assume the bus voltage magnitudes are given, and the state variable of each bus i is simply a bus voltage angle denoted as x_i, i = 1, ..., n. There are m measurements consisting of the measurements of real power flows along branches and the measurements of real injections at buses. Each measure-ment is denoted as z_i, i = 1, ..., m. Note that, in DC power flow model, we don’t consider reactive power flows and injections.

Figure 2.3: Two-port π−model of a branch [26]

According to the two-port π-model shown in Fig. 2.3 [26], the real power flow from bus i to bus j is given by

Pi,j = V_i²(gsi+ gij)− ViVj(gijcos(xi− xj) + bijsin(xi− xj)), (2.1) and the real power injection at bus i is

P_i = V_i ∑

j∈Nⁱ

V_j(G_ijcos(x_i− xj) + B_ijsin(x_i− xj)), (2.2) where V_i, x_i is bus voltage magnitude and angle at bus i, G_ij + jB_ij is the ijth element of the complex bus admittance matrix, g_ij + jb_ij is the admittance of the series branch

between bus i and bus j, g_si+ jb_si is the admittance of the shunt branch linked at bus i, and Ni is the set of all buses linked to bus i.

We apply several assumptions for DC model:

• We ignore the shunt admittance.

• In steady-state DC power system, the phase angle difference xi− xj is small, ∀i, j.

• The resistance of each branch is typically smaller than its reactance.

• There are no losses.

The simplified real power flow from bus i to bus j becomes

P_i,j^s = V_iV_jb_ij(x_i− xj), (2.3) and the simplified real power injection at bus i becomes

P_i^s = V_i ∑

j∈Ni

V_jb_ij(x_i− xj) = ∑

j∈Ni

P_i,j^s . (2.4)

In the control center, we assume both bus voltage magnitudes and branch reactance are given or can be measured. According to (2.3) and (2.4), the linear equations between measurements P_i,j^s , P_i^s and bus voltage angles x_i are derived.

The DC state estimation can be formulated as matrix form as follow:

z = Hx + e, (2.5)

where x = (x₁, x₂, ..., x_n+1)^T, z = (z₁, z₂, ..., z_m)^T is a measurement vector consisting of P_i^s and P_i,j^s , e = (e1, e2, ..., em)^T is a random measurement error vector with zero mean Gaussian Distribution, and H is a m× (n + 1) Jacobian matrix derived from (2.3) and (2.4). In the matrix H, the columns correspond to the measurements and the rows correspond to the state variables (buses).

It is typically that the number of measurements is greater than the number of state variables, i.e., m > n; thus, the state estimation problem is over determined equations.

We want to minimize the sum of the square errors with different weight w_ias the function of x

minx J (x) =

∑m i=1

wie²_i = e^TR⁻¹e = (z− Hx)^TR⁻¹(z− Hx), (2.6) where R⁻¹ = diag(w1, w2, ..., wm).

We perform noise whitening and choose the weight w_i = σ_i⁻², i = 1, ..., m so that R is covariance matrix of e and the one step solution of weighted least-square (WLS) problem above [25] is

ˆ

x_r = (H^T_rR⁻¹H_r)⁻¹H^T_rR⁻¹z, (2.7) where H_ris derived by removing last column of H by assuming that last bus is reference bus. The state estimates x_r = [ˆx^T_r 0]^T are then used to smart grid configuration. We assume the smart grid network is observable (Rank(H) = Rank(H_r) = n), i.e., ˆx can be uniquely determined by (2.7). Note that the identical estimator can be proved using maximum likelihood criterion and minimum variance criterion under the the assumption that measurement errors are Gaussian distributed with zero mean [25].

We give two examples of a 5-bus power system to illustrate how to derive Jacobian matrix H and determine if the network is observable. Consider the 5-bus power network in Fig. 2.4. where we assume that the network operates in steady-state and calculate the Jacobian matrix of DC model. For simplification, without loss of generality, we assume all bus voltages V_i and all admittances b_i,j be 1. For the first example in Fig. 2.5, the network contains two meters, one injection meter and one branch meter.

The measurements reported by those meters are as follow:

z1 = P₂^s = ∑

j∈N²={1,4}

(x2− xj) = 2x2− x1− x4 (2.8)

z₂ = P_1,2^s = x₁− x2 (2.9)

Figure 2.4: An example of 5-bus power network. Each bus owns a voltage V_i and a

Using the last bus as the reference bus is equivalent to removing last column of Hex1

to get Hr,ex1. Since H^T_r,ex1Hr,ex1 is not invertible, the state estimator can not to obtain an unique state estimate ˆx. Therefore, we call the network configuration of Fig. 2.5 unobservable.

For the second example shown in Fig. 2.6, by following the same procedure above, we have

Figure 2.5: An unobservable example with two meters in the network.

Again, since H^T_r,ex2H_r,ex2 is invertible, there exists an unique state estimate. In other words, the network of Fig. 2.6 is observable.

Another way to define the Jacobian matrix is by using a graph model [14]. Regarding the buses in the network as as nodes and transmission lines as edges, we convert a power grid into a graph with n nodes and m_aedges. Note that m_ais the number of transmission lines and in general ma ̸= m. Define the incidence matrix A ∈ R^n×ma representing the graph (network) as

A(i, j) =





1 if jth edge starts at ith node

−1 if jth edge ends at ith node 0 otherwise

,∀j = 1, . . . , ma. (2.12)

Use the diagonal matrix D ∈ R^ma×ma to describe the reactance of transmission lines whose diagonal entries are the reciprocal of the reactance of the edges. As a result, the Jacobian matrix H can now be expressed as

H,



 P₁DA^T

−P2DA^T P₃ADA^T





m×n+1

, m = 2m_a+ n, (2.13)

Figure 2.6: An observable example with six meters in the network.

where P₁, P₂ and P₃ consist of subsets of rows of identity matrices of proper dimensions, indicating which measurements are actually taken. The sub-matrix P₁DA^T represents the measurements putting on the same direction of the directed edge. Similarly, the sub-matrix −P2DA^T represents the measurements putting on opposite direction of the directed edge. The sub-matrix P₃ADA^T represents the injection measurements putting on the buses. In the rest of this paper, we use this representation of Jacobian matrix, and select measured meters by setting P1, P2 and P3.