Power Quality in Power Systems and Electrical Machines

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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

$g_{r, 3}^{(5)} = | {\tilde{V}}_{3}^{(1)} | \cos (δ_{3}^{(1)}) + | {\tilde{V}}_{3}^{(5)} |^{2} \cos (2 δ_{3}^{(5)}) + | {\tilde{V}}_{3}^{(7)} | cos (δ_{3}^{(7)}),$ $g_{r, 3}^{(5)} = | {\tilde{V}}_{3}^{(1)} | \cos (δ_{3}^{(1)}) + | {\tilde{V}}_{3}^{(5)} |^{2} \cos (2 δ_{3}^{(5)}) + | {\tilde{V}}_{3}^{(7)} | cos (δ_{3}^{(7)}),$

(P8-1a)

$g_{i, 3}^{(5)} = | {\tilde{V}}_{3}^{(1)} | \sin (δ_{3}^{(1)}) + | {\tilde{V}}_{3}^{(5)} |^{2} \sin (2 δ_{3}^{(5)}) + | {\tilde{V}}_{3}^{(7)} | sin (δ_{3}^{(7)}),$ $g_{i, 3}^{(5)} = | {\tilde{V}}_{3}^{(1)} | \sin (δ_{3}^{(1)}) + | {\tilde{V}}_{3}^{(5)} |^{2} \sin (2 δ_{3}^{(5)}) + | {\tilde{V}}_{3}^{(7)} | sin (δ_{3}^{(7)}),$

(P8-1b)

$g_{i, 3}^{(7)} = | {\tilde{V}}_{3}^{(1)} | \cos (δ_{3}^{(1)}) + | {\tilde{V}}_{3}^{(5)} | \cos (δ_{3}^{(5)}) + | {\tilde{V}}_{3}^{(7)} |^{2} cos (2 δ_{3}^{(7)}),$ $g_{i, 3}^{(7)} = | {\tilde{V}}_{3}^{(1)} | \cos (δ_{3}^{(1)}) + | {\tilde{V}}_{3}^{(5)} | \cos (δ_{3}^{(5)}) + | {\tilde{V}}_{3}^{(7)} |^{2} cos (2 δ_{3}^{(7)}),$

(P8-2a)

$g_{i, 3}^{(7)} = | {\tilde{V}}_{3}^{(1)} | \sin (δ_{3}^{(1)}) + | {\tilde{V}}_{3}^{(5)} | \sin (δ_{3}^{(5)}) + | {\tilde{V}}_{3}^{(7)} |^{2} sin (2 δ_{3}^{(7)}) .$ $g_{i, 3}^{(7)} = | {\tilde{V}}_{3}^{(1)} | \sin (δ_{3}^{(1)}) + | {\tilde{V}}_{3}^{(5)} | \sin (δ_{3}^{(5)}) + | {\tilde{V}}_{3}^{(7)} |^{2} sin (2 δ_{3}^{(7)}) .$

(P8-2b)

f07-35-9780128007822 — Figure P7.8 Three-bus power system consisting of two linear (swing bus = bus 1, PQ bus = bus 2) buses and one nonlinear bus (bus 3). The negative sequence impedance (≈ subtransient reactance at 60 Hz) of the swing bus is Z₂ = jX⁻ = j0.10 pu and there are two sets of harmonics.

The 5th and 7th nonlinear currents are referred to the harmonic voltages ${\tilde{V}}_{3}^{(5)}$ ${\tilde{V}}_{3}^{(5)}$ and ${\tilde{V}}_{3}^{(7)}$ ${\tilde{V}}_{3}^{(7)}$ , respectively.

a) Find the fundamental and harmonic (5th and 7th) bus admittance matrices in polar form.

b) Make an initial guess for the bus vector ${\bar{U}}^{0} = {[δ_{2}^{(1)}, |{\tilde{V}}_{2}^{(1)}|, δ_{3}^{(1)}, |{\tilde{V}}_{3}^{(1)}|, δ_{1}^{(5)}, |{\tilde{V}}_{1}^{(5)}|, δ_{2}^{(5)}, |{\tilde{V}}_{2}^{(5)}|, δ_{3}^{(5)}, |{\tilde{V}}_{3}^{(5)}|, δ_{1}^{(7)}, |{\tilde{V}}_{1}^{(7)}|, δ_{2}^{(7)}, |{\tilde{V}}_{2}^{(7)}|, δ_{3}^{(7)}, |{\tilde{V}}_{3}^{(7)}|]}^{t}$ ${\bar{U}}^{0} = {[δ_{2}^{(1)}, |{\tilde{V}}_{2}^{(1)}|, δ_{3}^{(1)}, |{\tilde{V}}_{3}^{(1)}|, δ_{1}^{(5)}, |{\tilde{V}}_{1}^{(5)}|, δ_{2}^{(5)}, |{\tilde{V}}_{2}^{(5)}|, δ_{3}^{(5)}, |{\tilde{V}}_{3}^{(5)}|, δ_{1}^{(7)}, |{\tilde{V}}_{1}^{(7)}|, δ_{2}^{(7)}, |{\tilde{V}}_{2}^{(7)}|, δ_{3}^{(7)}, |{\tilde{V}}_{3}^{(7)}|]}^{t}$ si717_e and compute the nonlinear load device currents $G_{r, 3}^{(1)}, G_{i, 3}^{(1)}, G_{r, 3}^{(5)}, G_{i, 3}^{(5)}, G_{r, 3}^{(7)}$ $G_{r, 3}^{(1)}, G_{i, 3}^{(1)}, G_{r, 3}^{(5)}, G_{i, 3}^{(5)}, G_{r, 3}^{(7)}$ and $G_{i, 3}^{(7)}$ $G_{i, 3}^{(7)}$ , which are referred to the swing bus.

c) Evaluate the initial mismatch vector $Δ {\bar{M}}^{0} = {[Δ \bar{W}, Δ {\bar{I}}^{(5)}, Δ {\bar{I}}^{(7)}, Δ {\bar{I}}^{(1)}]}^{t} .$ $Δ {\bar{M}}^{0} = {[Δ \bar{W}, Δ {\bar{I}}^{(5)}, Δ {\bar{I}}^{(7)}, Δ {\bar{I}}^{(1)}]}^{t} .$

d) Find the Jacobian ${\bar{J}}^{0}$ ${\bar{J}}^{0}$ for this power system configuration and compute the bus correction vector ΔŪ⁰. The Jacobian ${\bar{J}}^{0}$ ${\bar{J}}^{0}$ is of the form

${\bar{J}}^{0} = [\begin{array}{c} {\bar{J}}^{(1)} & {\bar{J}}^{(5)} & {\bar{J}}^{(7)} \\ {\bar{YG}}^{(5, 1)} & {\bar{YG}}^{(5, 5)} & {\bar{YG}}^{(5, 7)} \\ {\bar{YG}}^{(7, 1)} & {\bar{YG}}^{(7, 5)} & {\bar{YG}}^{(7, 7)} \\ {\bar{YG}}^{(1, 1)} & {\bar{YG}}^{(1, 5)} & {\bar{YG}}^{(1, 7)} \end{array}] .$ ${\bar{J}}^{0} = [\begin{array}{c} {\bar{J}}^{(1)} & {\bar{J}}^{(5)} & {\bar{J}}^{(7)} \\ {\bar{YG}}^{(5, 1)} & {\bar{YG}}^{(5, 5)} & {\bar{YG}}^{(5, 7)} \\ {\bar{YG}}^{(7, 1)} & {\bar{YG}}^{(7, 5)} & {\bar{YG}}^{(7, 7)} \\ {\bar{YG}}^{(1, 1)} & {\bar{YG}}^{(1, 5)} & {\bar{YG}}^{(1, 7)} \end{array}] .$

si723_e (P8-3)

In particular, this matrix (Eq. P8-3 or Eq. P8-4) has the following 16 unknowns: $δ_{2}^{(1)}, |{\tilde{V}}_{2}^{(1)}|, δ_{3}^{(1)}, |{\tilde{V}}_{3}^{(1)}|, δ_{1}^{(5)}, |{\tilde{V}}_{1}^{(5)}|, δ_{2}^{(5)}, |{\tilde{V}}_{2}^{(5)}|, δ_{3}^{(5)}, |{\tilde{V}}_{3}^{(5)}|, δ_{1}^{(7)}, |{\tilde{V}}_{1}^{(7)}|, δ_{2}^{(7)}, |{\tilde{V}}_{2}^{(7)}|, δ_{3}^{(7)}, |{\tilde{V}}_{3}^{(7)}|,$ $δ_{2}^{(1)}, |{\tilde{V}}_{2}^{(1)}|, δ_{3}^{(1)}, |{\tilde{V}}_{3}^{(1)}|, δ_{1}^{(5)}, |{\tilde{V}}_{1}^{(5)}|, δ_{2}^{(5)}, |{\tilde{V}}_{2}^{(5)}|, δ_{3}^{(5)}, |{\tilde{V}}_{3}^{(5)}|, δ_{1}^{(7)}, |{\tilde{V}}_{1}^{(7)}|, δ_{2}^{(7)}, |{\tilde{V}}_{2}^{(7)}|, δ_{3}^{(7)}, |{\tilde{V}}_{3}^{(7)}|,$ (see identification of columns of Eq. P8-4) and the following 16 equations: ΔP₂⁽¹⁾, ΔQ₂⁽¹⁾, ΔI_r,1⁽⁵⁾, ΔI_i,1⁽⁵⁾, ΔI_r,2⁽⁵⁾, ΔI_i,2⁽⁵⁾, ΔI_r,3⁽⁵⁾, ΔI_i,3⁽⁵⁾, ΔI_r,1⁽⁷⁾, ΔI_i,1⁽⁷⁾, ΔI_r,2⁽⁷⁾, ΔI_i,2⁽⁷⁾, ΔI_r,3⁽⁷⁾,ΔI_i,3⁽⁷⁾, ΔI_r,3⁽¹⁾, ΔI_i,3⁽¹⁾, (see identification of rows of Eq. P8-4). Note there are no $\bar{H}$ $\bar{H}$ matrices (e.g., ${\bar{H}}^{(1)}, {\bar{H}}^{(5)}$ ${\bar{H}}^{(1)}, {\bar{H}}^{(5)}$ and ${\bar{H}}^{(7)}$ ${\bar{H}}^{(7)}$ ) related to α₃ and β₃. This reduces the equation system by two equations. In this case we can define the matrix ${\bar{J}}^{0}$ ${\bar{J}}^{0}$ (Eq. P8-4).

e) Update the bus vector ${\bar{U}}^{1} = ({\bar{U}}^{0} - Δ {\bar{U}}^{0})$ ${\bar{U}}^{1} = ({\bar{U}}^{0} - Δ {\bar{U}}^{0})$ and recompute the updated mismatch vector $Δ {\bar{M}}^{1} .$ $Δ {\bar{M}}^{1} .$

Hint: The top row – not part of the Jacobian – of Eq. P8-4 lists the variables:

$[δ_{2}^{(1)} |{\tilde{V}}_{2}^{(1)}| δ_{3}^{(1)} |{\tilde{V}}_{3}^{(1)}| δ_{1}^{(5)} |{\tilde{V}}_{1}^{(5)}| δ_{2}^{(5)} |{\tilde{V}}_{2}^{(5)}| δ_{3}^{(5)} |{\tilde{V}}_{3}^{(5)}| δ_{1}^{(7)} |{\tilde{V}}_{1}^{(7)}| δ_{2}^{(7)} |{\tilde{V}}_{2}^{(7)}| δ_{3}^{(7)} |{\tilde{V}}_{3}^{(7)}|],$ $[δ_{2}^{(1)} |{\tilde{V}}_{2}^{(1)}| δ_{3}^{(1)} |{\tilde{V}}_{3}^{(1)}| δ_{1}^{(5)} |{\tilde{V}}_{1}^{(5)}| δ_{2}^{(5)} |{\tilde{V}}_{2}^{(5)}| δ_{3}^{(5)} |{\tilde{V}}_{3}^{(5)}| δ_{1}^{(7)} |{\tilde{V}}_{1}^{(7)}| δ_{2}^{(7)} |{\tilde{V}}_{2}^{(7)}| δ_{3}^{(7)} |{\tilde{V}}_{3}^{(7)}|],$

and the right-hand side column – not part of the Jacobian – lists the mismatch quantities which must be satisfied:

${[Δ P_{2}^{(1)} Δ Q_{2}^{(1)} Δ I_{r, 1}^{(5)} Δ I_{i, 1}^{(5)} Δ I_{r, 2}^{(5)} Δ I_{i, 2}^{(5)} Δ I_{r, 3}^{(5)} Δ I_{i, 3}^{(5)} Δ I_{r, 1}^{(7)} Δ I_{i, 1}^{(7)} Δ I_{r, 2}^{(7)} Δ I_{i, 2}^{(7)} Δ I_{r, 3}^{(7)} Δ I_{i, 3}^{(7)} Δ I_{r, 3}^{(1)} Δ I_{i, 3}^{(1)}]}^{t} .$ ${[Δ P_{2}^{(1)} Δ Q_{2}^{(1)} Δ I_{r, 1}^{(5)} Δ I_{i, 1}^{(5)} Δ I_{r, 2}^{(5)} Δ I_{i, 2}^{(5)} Δ I_{r, 3}^{(5)} Δ I_{i, 3}^{(5)} Δ I_{r, 1}^{(7)} Δ I_{i, 1}^{(7)} Δ I_{r, 2}^{(7)} Δ I_{i, 2}^{(7)} Δ I_{r, 3}^{(7)} Δ I_{i, 3}^{(7)} Δ I_{r, 3}^{(1)} Δ I_{i, 3}^{(1)}]}^{t} .$

These two identifications help in defining the partial derivatives of the Jacobian. For example,

$J_{3, 5}^{0} = \frac{\partial (Δ I_{r, 1}^{(5)})}{\partial (δ_{1}^{(5)})}, J_{3, 10}^{0} = \frac{\partial (Δ I_{r, 1}^{(5)})}{\partial (V_{3}^{(5)})}, J_{8, 5}^{0} = \frac{\partial (Δ I_{i, 3}^{(5)})}{\partial (δ_{1}^{(5)})},$ $J_{3, 5}^{0} = \frac{\partial (Δ I_{r, 1}^{(5)})}{\partial (δ_{1}^{(5)})}, J_{3, 10}^{0} = \frac{\partial (Δ I_{r, 1}^{(5)})}{\partial (V_{3}^{(5)})}, J_{8, 5}^{0} = \frac{\partial (Δ I_{i, 3}^{(5)})}{\partial (δ_{1}^{(5)})},$

si733_e

$J_{8, 10}^{0} = \frac{\partial (Δ I_{i, 3}^{(5)})}{\partial (V_{3}^{(5)})}, J_{9, 11}^{0} = \frac{\partial (Δ I_{r, 1}^{(7)})}{\partial (δ_{1}^{(7)})}, and$ $J_{8, 10}^{0} = \frac{\partial (Δ I_{i, 3}^{(5)})}{\partial (V_{3}^{(5)})}, J_{9, 11}^{0} = \frac{\partial (Δ I_{r, 1}^{(7)})}{\partial (δ_{1}^{(7)})}, and$

si734_e

$J_{12, 16}^{0} = \frac{\partial (Δ I_{i, 2}^{(7)})}{\partial (V_{3}^{(7)})} .$ $J_{12, 16}^{0} = \frac{\partial (Δ I_{i, 2}^{(7)})}{\partial (V_{3}^{(7)})} .$

si735_e

$\begin{array}{l} \begin{array}{c} \underset{↓}{δ_{2}^{(1)}} & \underset{↓}{|{\tilde{V}}_{2}^{(1)}|} & \underset{↓}{δ_{3}^{(1)}} & \underset{↓}{|{\tilde{V}}_{3}^{(1)}|} & \underset{↓}{δ_{1}^{(5)}} & \underset{↓}{|{\tilde{V}}_{1}^{(5)}|} & \underset{↓}{δ_{2}^{(5)}} & \underset{↓}{|{\tilde{V}}_{2}^{(5)}|} & \underset{↓}{δ_{3}^{(5)}} & \underset{↓}{|{\tilde{V}}_{3}^{(5)}|} & \underset{↓}{δ_{1}^{(7)}} & \underset{↓}{|{\tilde{V}}_{1}^{(7)}|} & \underset{↓}{δ_{2}^{(7)}} & \underset{↓}{|{\tilde{V}}_{2}^{(7)}|} & \underset{↓}{δ_{3}^{(7)}} & \underset{↓}{|{\tilde{V}}_{3}^{(7)}|} \end{array} \\ {\bar{J}}^{0} = |\begin{array}{c} 19.8 & 1.98 & - 9.9 & - 0.99 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ - 1.98 & 19.8 & 0.99 & - 9.9 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & J_{3, 5}^{0} & J_{3, 6}^{0} & J_{3, 7}^{0} & J_{3, 8}^{0} & J_{3, 9}^{0} & J_{3, 10}^{0} & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & J_{4, 5}^{0} & J_{4, 6}^{0} & J_{4, 7}^{0} & J_{4, 8}^{0} & J_{4, 9}^{0} & J_{4, 10}^{0} & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & J_{5, 5}^{0} & J_{5, 6}^{0} & J_{5, 7}^{0} & J_{5, 8}^{0} & J_{5, 9}^{0} & J_{5, 10}^{0} & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & J_{6, 5}^{0} & J_{6, 6}^{0} & J_{6, 7}^{0} & J_{6, 8}^{0} & J_{6, 9}^{0} & J_{6, 10}^{0} & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & J_{7, 5}^{0} & J_{7, 6}^{0} & J_{7, 7}^{0} & J_{7, 8}^{0} & J_{7, 9}^{0} & J_{7, 10}^{0} & 0 & 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 1 & J_{8, 5}^{0} & J_{8, 6}^{0} & J_{8, 7}^{0} & J_{8, 8}^{0} & J_{8, 9}^{0} & J_{8, 10}^{0} & 0 & 0 & 0 & 0 & 0.1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & J_{9, 11}^{0} & J_{9, 12}^{0} & J_{9, 13}^{0} & J_{9, 14}^{0} & J_{9, 15}^{0} & J_{9, 16}^{0} \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & J_{10, 11}^{0} & J_{10, 12}^{0} & J_{10, 13}^{0} & J_{10, 14}^{0} & J_{10, 15}^{0} & J_{10, 16}^{0} \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & J_{11, 11}^{0} & J_{11, 12}^{0} & J_{11, 13}^{0} & J_{11, 14}^{0} & J_{11, 15}^{0} & J_{11, 16}^{0} \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & J_{12, 11}^{0} & J_{12, 12}^{0} & J_{12, 13}^{0} & J_{12, 14}^{0} & J_{12, 15}^{0} & J_{12, 16}^{0} \\ 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & J_{13, 11}^{0} & J_{13, 12}^{0} & J_{13, 13}^{0} & J_{13, 14}^{0} & J_{13, 15}^{0} & J_{13, 16}^{0} \\ 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & J_{14, 11}^{0} & J_{14, 12}^{0} & J_{14, 13}^{0} & J_{14, 14}^{0} & J_{14, 15}^{0} & J_{14, 16}^{0} \\ - 9.90 & - 0.99 & 21.3 & 1.48 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ - 0.99 & 9.9 & - 2.51 & - 18.3 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \end{array}| \begin{array}{c} \leftarrow Δ P_{2}^{(1)} \\ \leftarrow Δ Q_{2}^{(1)} \\ \leftarrow Δ I_{r, 1}^{(5)} \\ \leftarrow Δ I_{i, 1}^{(5)} \\ \leftarrow Δ I_{r, 2}^{(5)} \\ \leftarrow Δ I_{i, 2}^{(5)} \\ \leftarrow Δ I_{r, 3}^{(5)} \\ \leftarrow Δ I_{i, 3}^{(5)} \\ \leftarrow Δ I_{r, 1}^{(7)} \\ \leftarrow Δ I_{i, 1}^{(7)} \\ \leftarrow Δ I_{r, 2}^{(7)} \\ \leftarrow Δ I_{i, 2}^{(7)} \\ \leftarrow Δ I_{r, 3}^{(7)} \\ \leftarrow Δ I_{i, 3}^{(7)} \\ \leftarrow Δ I_{r, 3}^{(1)} \\ \leftarrow Δ I_{i, 3}^{(1)} \end{array} . \end{array}$ $\begin{array}{l} \begin{array}{c} \underset{↓}{δ_{2}^{(1)}} & \underset{↓}{|{\tilde{V}}_{2}^{(1)}|} & \underset{↓}{δ_{3}^{(1)}} & \underset{↓}{|{\tilde{V}}_{3}^{(1)}|} & \underset{↓}{δ_{1}^{(5)}} & \underset{↓}{|{\tilde{V}}_{1}^{(5)}|} & \underset{↓}{δ_{2}^{(5)}} & \underset{↓}{|{\tilde{V}}_{2}^{(5)}|} & \underset{↓}{δ_{3}^{(5)}} & \underset{↓}{|{\tilde{V}}_{3}^{(5)}|} & \underset{↓}{δ_{1}^{(7)}} & \underset{↓}{|{\tilde{V}}_{1}^{(7)}|} & \underset{↓}{δ_{2}^{(7)}} & \underset{↓}{|{\tilde{V}}_{2}^{(7)}|} & \underset{↓}{δ_{3}^{(7)}} & \underset{↓}{|{\tilde{V}}_{3}^{(7)}|} \end{array} \\ {\bar{J}}^{0} = |\begin{array}{c} 19.8 & 1.98 & - 9.9 & - 0.99 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ - 1.98 & 19.8 & 0.99 & - 9.9 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & J_{3, 5}^{0} & J_{3, 6}^{0} & J_{3, 7}^{0} & J_{3, 8}^{0} & J_{3, 9}^{0} & J_{3, 10}^{0} & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & J_{4, 5}^{0} & J_{4, 6}^{0} & J_{4, 7}^{0} & J_{4, 8}^{0} & J_{4, 9}^{0} & J_{4, 10}^{0} & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & J_{5, 5}^{0} & J_{5, 6}^{0} & J_{5, 7}^{0} & J_{5, 8}^{0} & J_{5, 9}^{0} & J_{5, 10}^{0} & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & J_{6, 5}^{0} & J_{6, 6}^{0} & J_{6, 7}^{0} & J_{6, 8}^{0} & J_{6, 9}^{0} & J_{6, 10}^{0} & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & J_{7, 5}^{0} & J_{7, 6}^{0} & J_{7, 7}^{0} & J_{7, 8}^{0} & J_{7, 9}^{0} & J_{7, 10}^{0} & 0 & 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 1 & J_{8, 5}^{0} & J_{8, 6}^{0} & J_{8, 7}^{0} & J_{8, 8}^{0} & J_{8, 9}^{0} & J_{8, 10}^{0} & 0 & 0 & 0 & 0 & 0.1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & J_{9, 11}^{0} & J_{9, 12}^{0} & J_{9, 13}^{0} & J_{9, 14}^{0} & J_{9, 15}^{0} & J_{9, 16}^{0} \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & J_{10, 11}^{0} & J_{10, 12}^{0} & J_{10, 13}^{0} & J_{10, 14}^{0} & J_{10, 15}^{0} & J_{10, 16}^{0} \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & J_{11, 11}^{0} & J_{11, 12}^{0} & J_{11, 13}^{0} & J_{11, 14}^{0} & J_{11, 15}^{0} & J_{11, 16}^{0} \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & J_{12, 11}^{0} & J_{12, 12}^{0} & J_{12, 13}^{0} & J_{12, 14}^{0} & J_{12, 15}^{0} & J_{12, 16}^{0} \\ 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & J_{13, 11}^{0} & J_{13, 12}^{0} & J_{13, 13}^{0} & J_{13, 14}^{0} & J_{13, 15}^{0} & J_{13, 16}^{0} \\ 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & J_{14, 11}^{0} & J_{14, 12}^{0} & J_{14, 13}^{0} & J_{14, 14}^{0} & J_{14, 15}^{0} & J_{14, 16}^{0} \\ - 9.90 & - 0.99 & 21.3 & 1.48 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ - 0.99 & 9.9 & - 2.51 & - 18.3 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \end{array}| \begin{array}{c} \leftarrow Δ P_{2}^{(1)} \\ \leftarrow Δ Q_{2}^{(1)} \\ \leftarrow Δ I_{r, 1}^{(5)} \\ \leftarrow Δ I_{i, 1}^{(5)} \\ \leftarrow Δ I_{r, 2}^{(5)} \\ \leftarrow Δ I_{i, 2}^{(5)} \\ \leftarrow Δ I_{r, 3}^{(5)} \\ \leftarrow Δ I_{i, 3}^{(5)} \\ \leftarrow Δ I_{r, 1}^{(7)} \\ \leftarrow Δ I_{i, 1}^{(7)} \\ \leftarrow Δ I_{r, 2}^{(7)} \\ \leftarrow Δ I_{i, 2}^{(7)} \\ \leftarrow Δ I_{r, 3}^{(7)} \\ \leftarrow Δ I_{i, 3}^{(7)} \\ \leftarrow Δ I_{r, 3}^{(1)} \\ \leftarrow Δ I_{i, 3}^{(1)} \end{array} . \end{array}$

si736_e (P8-4)

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Table of Contents for Power Quality in Power Systems and Electrical Machines

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

Table of Contents for
Power Quality in Power Systems and Electrical Machines