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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 2.1: two voltage measurements at bus 1 and 2; two active/reactive power injection measurements at bus 1 and 3; three active power flow measurements at lines 1–4, 3–2, and 3–4; and one reactive power flow measurement at line 3–4. The standard deviation of any measurement other than voltage is 0.02, while the standard deviation of voltage measurements is 0.01. The considered measurement configuration provides a redundancy ratio of (10 + 2)/7 = 1.71. Note that the measurement set includes 10 actual measurements plus two exact measurements corresponding to active and reactive power injections at the transit bus.

The state estimation problem for this example has the form

minimize_x

subject to 2 equality constraints for the transit bus and 7 inequality constraints enforcing physical limits:

Figure 2.1 One‐line diagram and measurement configuration of a 4‐bus power system.

TABLE 2.1 Line parameters.

Bus i	Bus j	Bij (p.u.)
1	2	10
1	4	10
2	3	10
3	4	10