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Example 2.2 Classical Solution Example

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The system in Figure 2.1 is considered in this example. In order to solve the nonlinear system by the Newton method, matrices H and F and vectors Δ z and f (x ) should be computed at each iteration. The diagonal terms of the measurement weight matrix W correspond to the inverse of the variance of the measurement errors. Expressions for these matrices are



The minimum and maximum active/reactive power injections in puMW/puMVar are , , , and . Measurements are generated by solving the power flow with a height accuracy and then adding independent Gaussian‐distributed errors to the exact values of the measurements. These measurements and the exact values are provided in Tables 2.2–2.4. In these tables, superscript “true” indicates true value, superscript “m” measurement, and a “∧” estimated value.

The initial solution considered for state variables is the flat voltage level, and a convergence tolerance of 10−5 on the largest Δxi is considered. The convergence, illustrated in Table 2.5, is obtained after four iterations.

TABLE 2.2 Voltages: measured, true and estimated values.

Bus no. (p.u.) (p.u.) (p.u.) (rad) (rad)
1 1.0934 1.0965 1.0960 0.1819 0.1820
2 1.0252 1.0220 1.0222 0.1122 0.1128
3 — 0.9531 0.9540 0.0320 0.0334
4 — 0.9174 0.9175 0.0000 0.0000

TABLE 2.3 Power injections: measured, true and estimated values.

Bus no.
(MW p.u.) (MVAr p.u.)
1 2.6093 2.6000 2.5940 2.9597 2.9743 2.9578
3 −0.4634 −0.5000 −0.4819 −0.2606 −0.2800 −0.2674

TABLE 2.4 Power flows: measured, true and estimated values.

Line no.
(MW p.u.) (MVAr p.u.)
1–4 1.8120 1.8198 1.8200 — — —
3–2 −0.7615 −0.7802 −0.7740 — — —
3–4 0.2650 0.2802 0.2921 0.3448 0.3447 0.3524

TABLE 2.5 State‐variable updates and convergency summary.

State variable Iterations
0 1 2 3 4
θ 1 0.0000 0.1824 0.1821 0.1820 0.1820
θ 2 0.0000 0.1056 0.1130 0.1128 0.1128
θ 3 0.0000 0.0287 0.0336 0.0334 0.0334
v 1 1.0000 1.1003 1.0959 1.0960 1.0960
v 2 1.0000 1.0183 1.0221 1.0222 1.0222
v 3 1.0000 0.9363 0.9538 0.9540 0.9540
v 4 1.0000 0.8895 0.9174 0.9175 0.9175
49 850.5529 562.7272 4.3639 4.2674 4.2674
Advances in Electric Power and Energy

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