Power Quality in Power Systems and Electrical Machines, 2nd Edition

Problem 7.8: Harmonic Power Flow with one Nonlinear Bus and Two Sets of Harmonic Nonlinear Device Currents

To the three-bus system of Fig. P7.8 apply the Newton–Raphson harmonic load flow analysis technique. Assume that bus 1 is the swing or slack bus, bus 2 is a linear PQ bus, and bus 3 is a nonlinear bus, where the fundamental real and reactive powers are specified. In addition, at bus 3 the real and imaginary harmonic nonlinear device currents are given as

$g_{r,3}^{\left(5\right)}=\left|{\tilde{V}}_3^{\left(1\right)}\right|\cos \left({\delta }_3^{\left(1\right)}\right)+\left|{\tilde{V}}_3^{\left(5\right)}\right|2^{}\cos \left(2{\delta }_3^{\left(5\right)}\right)+\left|{\tilde{V}}_3^{\left(7\right)}\right|cos\left({\delta }_3^{\left(7\right)}\right)\text{,}$

  (P8-1a)

$g_{i,3}^{\left(5\right)}=\left|{\tilde{V}}_3^{\left(1\right)}\right|\sin \left({\delta }_3^{\left(1\right)}\right)+\left|{\tilde{V}}_3^{\left(5\right)}\right|2^{}\sin \left(2{\delta }_3^{\left(5\right)}\right)+\left|{\tilde{V}}_3^{\left(7\right)}\right|sin\left({\delta }_3^{\left(7\right)}\right)\text{,}$

  (P8-1b)

$g_{i,3}^{\left(7\right)}=\left|{\tilde{V}}_3^{\left(1\right)}\right|\cos \left({\delta }_3^{\left(1\right)}\right)+\left|{\tilde{V}}_3^{\left(5\right)}\right|\cos \left({\delta }_3^{\left(5\right)}\right)+\left|{\tilde{V}}_3^{\left(7\right)}\right|2^{}cos\left(2{\delta }_3^{\left(7\right)}\right)\text{,}$

  (P8-2a)

$g_{i,3}^{\left(7\right)}=$