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An implicit pole generator without regulators

Description of the model

This example examines the operation of a turbogenerator-transformer unit via a two-circuit overhead line (overhead line) to an infinite-power bus without regulators of excitation and rotation speed. At a time of 5 s. in the model, a three-phase short circuit (short circuit) occurs at the beginning of one of the overhead line circuits. An analytical calculation of the terminal shutdown time using the area method is carried out and compared with the results of mathematical modeling [1-3].

The process of launching and configuring the model from the script development environment using command control, processing simulation results, visualization of simulation results, and scenarios for independent work with the model are proposed. The signals are logged and their time schedules are shown. The model's appearance:

synchronous_generator_round_rotor--1737723205823.png

The main blocks used in this example and their parameters:

  1. Synchronous Machine Round Rotor - an implicit-pole synchronous generator (,).
  2. Voltage Source (Three-Phase) is a three-phase voltage source, used to simulate the BWM, the steady-state mode is set by setting the current line voltage and phase shift (,).
  3. Fault (Three-Phase) - short circuit.
  4. From library of passive elements Three-phase power transmission line units Three-Phase PI Section Line (, AC 500/64, 2 circuits) and a two-winding three-phase transformer Two-Winding Transformer (Three-Phase) (TD-630000/500).

Calculation of the cut-off angle using the area method

Importing the necessary modules for working with graphs:

In [ ]: 
using Plots;
gr();

SBM Parameters:

In [ ]: 
# номинальное напряжение, В
Us = 500e3
# мощность КЗ, ВА
Skz = 10e9
# полное сопротивление системы, Ом
Zkz = Us ^ 2 / Skz
# соотношение активного и индуктивного сопротивления
XR = 15
# индуктивное сопротивление системы, Ом
Xkz = Zkz * sin(atan(XR))
# активное сопротивление системы, Ом
Rkz = Zkz * cos(atan(XR))
# комплексное сопротивление системы, Ом
Zs = Rkz + 1im * Xkz;

Power line parameters:

In [ ]: 
# удельное сопротивление прямой последовательности, Ом/км
Z0 = 0.01967 + 0.2899im
# количество цепей
n = 2
# длина ВЛ, км
L = 200
# удельная емкостная проводимость, См/км
b0 = 3.908e-6
bc = b0 * L * n
# полное сопротивление ВЛ, Ом
Zl = Z0 * L / n
# емкостное сопротивление половины ВЛ, Ом
Zc = -2im / bc;

Transformer Parameters:

In [ ]: 
# активное сопротивление ТР, Ом
Rt = 0.4514
# индуктивное сопротивление ТР, Ом
Xt = 30.62
Zt = Rt + Xt * 1im;

Generator Parameters:

In [ ]: 
# номинальное напряжение, В
Ug = 24e3
# номинальная мощность, ВА
Sg = 500e6
# активное сопротивление, о.е.
Ra = 0.003
# синхронное индуктивное сопротивление по оси d, о.е.
Xd = 1.81
# переходное индуктивное сопротивление по оси d, о.е.
Xdd = 0.3
# комплексное переходное сопротивление генератора, Ом
Zg = (Ra + Xdd * 1im) * Us ^ 2 / Sg;

Load Parameters:

In [ ]: 
# активная мощность нагрузки, Вт
Pn = 20e6
# активное сопротивление нагрузки, Ом
Zn = Us ^ 2 / Pn;

Calculation of own and mutual resistances and in normal mode, using the single current method:

In [ ]: 
I2 = 1
Uc = I2 * Zs
Ic0 = Uc / Zc
Ibc = I2 + Ic0
Ub = Uc + Zl * Ibc
Ib0 = Ub / Zc
Iab = Ib0 + Ibc
Ua = Ub + Iab * Zt
Ia0 = Ua / Zn
I1 = Ia0 + Iab
U1 = Ua + I1 * (Xdd * 1im * Us ^ 2 / Sg)
Z12d = (Ua + I1 * (Xd * 1im * Us ^ 2 / Sg)) / I2
Z12 = U1 / I2
Z11 = U1 / I1
alpha11 = pi / 2 - angle(Z11)
alpha12 = pi / 2 - angle(Z12);

Calculation of own and mutual resistances and in the post-emergency mode using the single current method:

In [ ]: 
I2 = 1
Uc = I2 * Zs
Ic0 = Uc / (2 * Zc)
Ibc = I2 + Ic0
Ub = Uc + 2 * Zl * Ibc
Ib0 = Ub / (2 * Zc)
Iab = Ib0 + Ibc
Ua = Ub + Iab * Zt
Ia0 = Ua / Zn
I1 = Ia0 + Iab
U1 = Ua + I1 * (Xdd * 1im * Us ^ 2 / Sg)
Z12_pav = U1 / I2
Z11_pav = U1 / I1
alpha11_pav = pi / 2 - angle(Z11_pav)
alpha12_pav = pi / 2 - angle(Z12_pav);

The angular characteristics are calculated using a transient EMF ,
which remains unchanged at the initial time of the short circuit and after it is turned off:

In [ ]: 
# вырабатываемая генератором мощность в нормальном режиме
P0 = 400e6;
Q0 = 0;
# синхронная ЭДС по оси q
Eq = sqrt((Us + Q0 * abs(Z12d) / Us) ^ 2 + (P0 * abs(Z12d) / Us) ^ 2)
# угол ротора в исходном режиме
d0 = atan(P0 * abs(Z12d) / (Us ^ 2 + Q0 * abs(Z12d)))
# переходная ЭДС
Ep = sqrt((Us + Q0 * abs(Z12) / Us) ^ 2 + (P0 * abs(Z12) / Us) ^ 2)
# угол переходной ЭДС
d0p = atan(P0 * abs(Z12) / (Us ^ 2 + Q0 * abs(Z12)))
# переходная ЭДС по оси q в нормальном режиме
Eqq = Ep * cos(d0p - d0)
# переходная ЭДС по оси q в аварийном и послеаварийном режимах
Eqq_pav = Eqq
Eqq_av = Eqq
# максимальная активная мощность в нормальном режиме
Pmax = (Eqq ^ 2 / abs(Z11) * sin(alpha11) + Eqq * Us / abs(Z12)) / 1e6
# угол переходной ЭДС по оси q
d0p = asin(P0 / Pmax / 1e6)
# максимальная мощность в аварийном режиме
Pmax_av = 0
# максимальная мощность в послеаварийном режиме
Pmax_pav = (Eqq_pav ^ 2 / abs(Z11_pav) * sin(alpha11_pav) + Eqq_pav * Us / abs(Z12_pav)) / 1e6
# угол ротора при работе на послеаварийной характеристике
d0_pav = asin(P0 / Pmax_pav / 1e6)
# критический угол
dkr = pi - d0_pav
# предельный угол отключения
d_pr = acos((P0 * 1e-6 * (pi - d0_pav - d0p) + Pmax_pav * cos(pi - d0_pav) - Pmax_av * cos(d0p)) / (Pmax_pav - Pmax_av));

Visualization of the area method:

In [ ]: 
# функции угловых характеристик
delta = 0:0.1:180
Pmax = x -> (Eqq ^ 2 / abs(Z11) * sin(alpha11) .+ (Eqq * Us / abs(Z12)) * sin.(x .- alpha12)) / 1e6
Pmax_av = x -> fill(0, size(x))
Pmax_pav = x -> (Eqq_pav ^ 2 / abs(Z11_pav) * sin(alpha11_pav) .+ Eqq_pav * Us / abs(Z12_pav) * sin.(x .- alpha12_pav)) / 1e6 
Pt = x -> fill(P0 * 1e-6, size(x))
# визуализация метода площадей
plot(delta, [Pmax(delta .* pi ./ 180) Pmax_av(delta .* pi ./ 180) Pmax_pav(delta .* pi ./ 180) Pt(delta)],
label = [L"Норм.реж." L"Авар.реж." L"П/АВ.реж." L"Мощн.турбины"], ylabel = L"P, МВт", xlabel = L"\delta\prime, град", legend=false,
xaxis = (0:30:180), size = (900,440), left_margin=5Plots.mm, bottom_margin=5Plots.mm)
plot!((d0p:0.01:d_pr) .* 180 / pi, Pmax_av((d0p:0.01:d_pr)), fillrange = Pt((d0p:0.01:d_pr)), fillalpha = 0.2, c = 2, label = "Fторм")
plot!((d_pr:0.01:dkr) .* 180 / pi, Pmax_pav((d_pr:0.01:dkr)), fillrange = Pt((d_pr:0.01:dkr)), fillalpha = 0.2, c = 1, label = "Fторм")
vline!([d0p,d_pr,dkr] .* 180 / pi, c = 5)
annotate!([(90, Pmax(pi / 2) - 50, (L"Норм.реж.", 10, :black)),
        (50, Pmax_av(pi / 2) .+ 30, (L"Авар.реж.", 10, :black)),
        (90, Pmax_pav(pi / 2) - 50, (L"ПА/В.реж.", 10, :black)),
        (d0p * 180 / pi + 20, P0 * 1e-6 - 200, (L"F_{уск}", 15, :black, :left)),
        (d_pr * 180 / pi + 15, P0 * 1e-6 + 200, (L"F_{торм}", 15, :black, :left)),
        (0, P0 * 1e-6 + 50, (L"P_{турб}", 10, :black, :left)),
        (d0p * 180 / pi, 30, (L"\delta\prime_0  = %$(round(d0p * 180 / pi, digits=2))", 12, :black, :right)),
        (d_pr * 180 / pi, 30, (L"\delta\prime_{пред} = %$(round(d_pr * 180 / pi, digits=2))", 12, :black, :left)),
        (dkr * 180 / pi, 30, (L"\delta\prime_{крит} = %$(round(dkr * 180 / pi, digits=2))", 12, :black, :right))])
Out[0]:

The maximum shutdown time of the short circuit:

In [ ]: 
Tj = 7.05
tpr = sqrt((d_pr - d0p) * 2 * Tj / (0.8 * 2 * pi * 50))
print("t_пред = $(round(tpr, digits = 3))")

t_пред = 0.235

Simulation

Loading the model:

In [ ]: 
model_name = "synchronous_generator_round_rotor";
model_name in [m.name for m in engee.get_all_models()] ? engee.open(model_name) : engee.load( "$(@__DIR__)/$(model_name).engee");

Setting the model using command control for the time of line disconnection and short circuit equal to 0.24 seconds, at which dynamic stability will be maintained:

In [ ]: 
step = "Отключение КЗ";
# отключение цепи ВЛ
engee.set_param!(model_name * "/" * step, "Time" => 5.24);

Launching the uploaded model:

In [ ]: 
results = engee.run(model_name);

To import the simulation results, logging of the necessary signals was enabled in advance and their names were set. Converting the simulation results from the results variable:

In [ ]: 
# вектор времени симуляции
sim_time = results["w"].time;
# вектор скорости вращения ротора
w1 = results["w"].value;
# вектор тока генератора
P1 = results["P"].value;

Simulation with an increase in the short circuit duration to 0.2402 s:

In [ ]: 
# отключение цепи ВЛ
engee.set_param!(model_name*"/"*step, "Time" => 5.2402);
results = engee.run(model_name);
# вектор скорости вращения ротора
w2 = results["w"].value;
# вектор тока генератора
P2 = results["P"].value;

Graphs of the generator's active power:

In [ ]: 
plot(sim_time, P1, title = L"Мощность\; генератора", ylabel = L"P, о.е.", xlabel="Время, c",  label = L"t_{КЗ} = 0.24 с", legendfontsize = 10)
plot!(sim_time, P2,  label = L"t_{КЗ} = 0.2402 с", ylims = (-1, 2),size = (900,440), left_margin=5Plots.mm, bottom_margin=5Plots.mm)
Out[0]:

Rotor speed charts:

In [ ]: 
plot(sim_time, w1, title = L"Скорость\; ротора", ylabel = L"\omega, о.е.", xlabel=L"Время, c", label = L"t_{КЗ} = 0.24 с", legendfontsize = 10)
plot!(sim_time, w2, label = L"t_{КЗ} = 0.2402 с", ylims = (0.95, 1.1),size = (900,440), left_margin=5Plots.mm, bottom_margin=5Plots.mm)
Out[0]:

In the second experiment, the dynamic stability of the generator was disrupted, so the maximum short-circuit shutdown time in the model was empirically found. The result is consistent with the theoretical calculation with a small margin of error.

In [ ]: 
println("Предельное время отключения аналитически = "*string(round(tpr, digits=3)) * "с")
println("Предельное время отключения практически = "*string(0.24) * "с")

Предельное время отключения аналитически = 0.235с
Предельное время отключения практически = 0.24с

Addition

Try to change the following model parameters yourself and explore how this affects dynamic stability:

  1. increase overhead line length by 200 km;
  2. Type of fault in the Fault (Three-Phase) block;
  3. Reduce the active power of the generator in the initial mode by 200 MW.

Conclusions

In this example, tools were used for command management of the Engee model and uploading simulation results, and work with the Plots module is shown. The operation of an implicit-pole generator through a transformer and a two-circuit overhead line on a BWM without regulators of excitation and rotation speed was considered. The analytical calculation of dynamic stability was carried out and the results were compared with the modeling data.

Links

  1. Venikov V. A. Transient electromechanical processes in electrical systems: Textbook for electric power industry. spec. universities. — 4th ed., revised and dop. — M.: Higher School, 1985
  2. Transient processes of electrical systems in examples and illustrations: A textbook for universities (V. V. Yezhkov, N. I. Zelenokhat, I. V. Litkens, and others; edited by V. A. Stroev). Moscow: Znak, 1996. 224s., ill.
  3. Dolgov A.P. Stability of electrical systems : a textbook / Dolgov A.P.. Novosibirsk : Novosibirsk State Technical University, 2010. 177 p. ISBN 978-5-7782-1320-3.

Blocks used in example