Question 4.AE.4: For the analysis of subtransient phenomena, a simulation of ......

Following the notation of [7] one obtains the voltage equations:

e_a=p \Psi_a-r \cdot i_a        (E4.4-1a)

e_b=p \Psi_b-r \cdot \dot{i}_b          (E4.4-1b)

e_c=p \Psi_c-r \cdot i_c              (E4.4-1c)

e_{f d}=p \Psi_{f d}+r_{f d} \cdot i_{f d}          (E4.4-1d)

0=p \Psi_{k d}-r_{k d}-i_{k d}+v_{k d}          (E4.4-1e)

0=p \Psi_{k  q}-r_{k  q}-i_{k  q}+v_{k q }          (E4.4-1f)

for k=1, 2, 3, …, 8 damper windings in d- or q-axes.
The flux linkage equations are

\Psi_a=-L_{a a} i_a-L_{a b} i_b-L_{a c} i_c+L_{a f d} i_{f d}+L_{a k d} i_{k d}+L_{a k q} i_{k q}              (E4.4-2a)

\Psi_b=-L_{b a} i_a-L_{b b} i_b-L_{b c} i_c+L_{b f d} i_{f d}+L_{b k d} i_{k d}+L_{c k q} i_{k q}          (E4.4-2b)

\Psi_c=-L_{c a} i_a-L_{c b} i_b-L_{c c} i_c+L_{c f d} i_{f d}+L_{c k d} i_{k d}+L_{c k q} i_{k q}          (E4.4-2c)

\Psi_{f d}=-L_{f d a} i_a-L_{f d b} i_b-L_{f d c} i_c+L_{f d f d} i_{f d}+L_{f d k d} i_{k d}                (E4.4-2d)

\Psi_{k d}=-L_{k q a} i_a-L_{k q b} i_b-L_{k q c} i_c+L_{k d f d} i_{f d}+L_{k d 1 d} i_{1 d}+L_{k d 2 d} i_{2 d}+\ldots+L_{k d l l d} i_{l d}            (E4.4-2e)

\Psi_{k q}=-L_{k q a} i_a-L_{k q b} i_b-L_{k q c} i_c+L_{k k 1 q} i_{1 q}+L_{k q 2 q} i_{2 q}+\ldots+L_{k q l q} i_q         (E4.4-2f)

for k=1, 2, …, 8 and l=1, 2, …, 8.
Following the notation of Fig. E4.4.2, the self-inductance of the kth damper winding is

L_{k d}^R=2 \cdot \ell_R \cdot A_{R k d} / I_{k d}      (E4.4-3)

and the leakage inductance with respect to stator winding is

or referred to the stator reference frame, where m_1\ and\ m_2 are the number of stator and rotor phases, respectively

L_{k d}=m_1\left(2 \cdot \ell_R \cdot A_{R k d} / I_{k d}\right) M_{k d a}^2 / m_2            (E4.4-5)

L_{k d l}=m_1\left\{2 \cdot \ell_R \cdot\left(A_{R k d}-A_{S k d}\right) / I_{k d}\right\} M_{k d d}^2 / m_2 .                (E4.4-6)

The mutual inductances between the kth and the lth d-axis windings are

L_{k d  l d}=m_1 \cdot L_{k d  l d}^{k d} \cdot M_{k d l d}^2 \cdot M_{l d a}^2 / m_2            (E4.4-7a)

L_{l d k d}=m_1 \cdot L_{l d k d}^{l d} \cdot M_{l d k d}^2 \cdot M_{k d  a}^2 / m_2                (E4-4.7b)

Correspondingly the ohmic resistance is

r_{k d}=m_1 \cdot R_{k d} \cdot M_{k d a}^2 / m_2            (E4.4-8)

The same procedure applies to the q-axis parameters, as indicated in Fig. E4.4.3. In these equations \mathscr{L} _R is the rotor length. M_{kda},\ M_{kdld},\ M_{lda},\ and\ M_{ldkd} are transformation ratios. The calculation of the stator inductances (Fig. E4.4.4) is performed in the same manner as for the rotor inductances.

If Eqs. E4.4-2 to E4.4-8 are introduced in Eq. E4.4-1 one obtains

[L] \cdot\left[\frac{d i}{d t}\right]=[V]            (E4.4-9)

where [L] is an inductance matrix with time-varying coefficients, \left[\frac{d i}{d t}\right] is the derivative of the solution vector [i] and [V] represents the forcing sources. Using Gaussian elimination one gets

\left[\frac{d i}{d t}\right]=[L]^{-1} \cdot[V]            (E4.4-10)

Subsequently a Runge-Kutta integration procedure of the fourth order provides the first starting steps for the Adams-Moulton method, which is a predictor-corrector integration procedure, and one obtains the solution vector [i]. A line-to-line short-circuit including the generator-transformer leakage impedance is performed. The line-to-neutral voltages in per unit (referred to the rated voltage of the generator) are shown in Fig. E4.4.5. The stator currents are depicted in Fig. E4.4.6, and field and damper bar currents are shown in Figs. E4.4.7 and E4.4.8a,b. The resulting electrical and mechanical torques are depicted in Figs. E4.4.9 and E4.4.10.

The integral \int_1^{t_2}\left(i_{k b a r}\right)^2 d t is a measure for the energy absorbed by the damper bar k due to the current ikbar. Therefore, the energy dissipated in the damper bar k during time (t_2 – t_1) is

E_{k \mathrm{bar}}=\left\{\int_1^{t_2}\left(i_{k \mathrm{bar}}\right)^2 d t\right\} R_{k \mathrm{bar}} .          (E4.4-11)

It is well known that the bar (e.g., #16) of the amortisseur (see Fig. E4.4.11) located before the rotor pole as viewed in the direction of rotation incurs greater loss than the bar (e.g., #1) of the amortisseur located behind the rotor pole as viewed in the direction of rotation. Figure E4.4.4.12 shows the values E_{kbar} for a sudden line-to-line short-circuit at the terminals including transformer impedance for various time intervals (t_2 – t_1) . Figure E4.4.13 depicts the subtransient field of a generator during the first cycle of a three phase short-circuit.
Table E4.4.1 summarizes the dissipated energies in the 1st and 16th amortisseur bars during various fault conditions of a generator including the generator-transformer leakage impedance:

1) line-to-line short-circuit,
2) balanced three-phase short-circuit,
3) out-of-phase synchronization with a 120° leading rotor angle,
4) out-of-phase synchronization with a 120° lagging rotor angle, and
5) unbalanced stator currents resulting in an inverse stator current of \left|\widetilde{I}_2\right| \mid=8.7 \%.