Static Characteristics of Synchronous Generators Equipped with Excitation and Active Power Control Systems Based on Multiple Parameters
Abstract Generated abstract
The paper develops static characteristics for synchronous generators equipped with automatic excitation control and turbine active power control acting on several operating parameters. Starting from steady-state active and reactive power relations, the authors formulate incremental equations that include control by voltage, frequency, absolute rotor angle, relative angle, active power, and reactive power, then derive slope coefficients as determinant ratios for use in stability calculations. The analysis identifies which control laws create dependences of generator active and reactive power on terminal voltage magnitude, voltage phase, and frequency. It concludes that most existing generators, lacking angle-based regulation, effectively retain only three nonzero static slope coefficients: active power with respect to frequency, and reactive power with respect to voltage and frequency.
Full Text
Static Characteristics of Synchronous Generators Equipped with Excitation and Active-Power Control Systems Based on Several Parameters
R. P. Borisov, Yu. V. Khrushchev
(Presented by the Department of Electrical Systems and Networks)
Calculations of the aperiodic stability of power systems by the sign of the free term of the characteristic equation, using generalized static characteristics of generators and load nodes in the form of regulating effects, have found expression and development in works [1, 2].
Further development and application of this method, especially for long-distance AC transmissions with intermediate connections, proves to be very promising.
The static characteristics of generators can be refined by taking into account automatic excitation-control devices (AEC) and automatic turbine-control devices (ATC). Regulation of the turbine torque may be carried out by frequency, by deviation of the absolute rotor angle, and by the relative angle between the emf vectors and the terminal voltage. Excitation regulation may be carried out by any operating parameters or by a group of parameters simultaneously. Taking these factors into account becomes necessary in assessing the stability of modern power systems.
As has been shown, the dependences of the active and reactive powers of a generator as functions of the terminal voltage, emf, the angle between them, and frequency can be determined as follows:
\[ P=\frac{e'_d u}{x'_d}\sin\delta_r+\frac{u^2}{2}\frac{x'_d-x_q}{\omega x'_d\cdot x_q}\sin 2\delta_r, \tag{1} \]
\[ Q=\frac{e'_d u}{x'_d}\cos\delta_r-\frac{u^2}{\omega x'_d}\left(1+\frac{x'_d-x_q}{x_q}\sin^2\delta_r\right). \tag{2} \]
With proportional ATC according to the indicated parameters, the turbine torque may be determined by the following equation:
\[ M_t=M_{t0}+n_\omega(\omega-\omega_0)+n_\delta(\delta-\delta_0)+n_{\delta r}(\delta_r-\delta_{r0}), \tag{3} \]
where \(n_\omega\), \(n_\delta\), \(n_{\delta r}\) are the gain coefficients with respect to frequency, absolute angle, and relative angle.
For an AEC system acting on deviations of a group of parameters, the equation
\[ e'_d=e'_{d0}+k_u(u-u_0)+k_\omega(\omega-\omega_0)+k_\delta(\delta-\delta_0)+ \]
\[ +k_{\delta r}(\delta_r-\delta_{r0})+k_p(P-P_0)+k_Q(Q-Q_0), \tag{4} \]
is valid.
where \(k\) denotes the total amplification coefficients with respect to deviations of the parameters indicated in the subscripts.
Let us rewrite these equations, regarding the power delivered by the generator as a negative load,
\[ P=-\omega M_{\mathrm{g}}=-\omega M_{\mathrm{em}} \tag{5} \]
and taking into account the equality
\[ \dot{\delta}_{\mathrm{g}}=\dot{\delta}-\dot{\delta}_{u}. \tag{6} \]
Then
\[ P=-\omega\left[M_{\mathrm{T0}}+n_{\omega}(\omega-\omega_0)+n_{\dot{\delta}}(\dot{\delta}-\dot{\delta}_0)+n_{\dot{\delta}_{\mathrm{g}}}(\dot{\delta}-\dot{\delta}_{u}-\dot{\delta}_{\mathrm{g0}})\right], \]
\[ P=-\frac{e_d u}{x_d}\sin(\delta-\delta_u) -\frac{u^2}{2}\frac{x_d-x_q}{\omega x_d x_q}\sin 2(\delta-\delta_u), \]
\[ Q=-\frac{e_d u}{x_d}\cos(\delta-\delta_u) +\frac{u^2}{\omega x_d}\left[1+\frac{x_d-x_q}{x_q}\sin^2(\delta-\delta_u)\right], \]
\[ \begin{aligned} e_d={}&e_{d0}+k_u(u-u_0)+k_{\omega}(\omega-\omega_0)+k_{\dot{\delta}}(\dot{\delta}-\dot{\delta}_0)+{}\\ &+k_{\dot{\delta}_{\mathrm{g}}}(\dot{\delta}_{\mathrm{g}}-\dot{\delta}_{\mathrm{g0}}) +k_p(P-P_0)+k_Q(Q-Q_0). \end{aligned} \]
The obtained system of equations (7) makes it possible to construct four characteristics in three variables. Analysis of the influence of the initial conditions on the static characteristics of the generator in the general case requires a large amount of computational work and is evidently expedient only in specifically posed problems.
To establish the influence of the amplification coefficients of the AVR and the active-power controller on the static characteristics, write the system of equations (7) in finite increments:
\[ \left. \begin{aligned} \Delta P &=-(M_{\mathrm{T0}}+n_{\omega}\omega_0)\Delta\omega -\omega_0 n_{\dot{\delta}}\Delta\dot{\delta} -\omega_0 n_{\dot{\delta}_{\mathrm{g}}}\Delta\dot{\delta} +\omega_0 n_{\dot{\delta}_{\mathrm{g}}}\Delta\dot{\delta}_{u} \\ &=m_0\Delta\omega-\omega_0(n_{\dot{\delta}}+n_{\dot{\delta}_{\mathrm{g}}})\Delta\dot{\delta} +\omega_0 n_{\dot{\delta}_{\mathrm{g}}}\Delta\dot{\delta}_{u},\\ \Delta P &=\frac{\partial P}{\partial e_d}\Delta e_d +\frac{\partial P}{\partial \delta}\Delta\delta +\frac{\partial P}{\partial u}\Delta u +\frac{\partial P}{\partial \omega}\Delta\omega,\\ \Delta Q &=\frac{\partial Q}{\partial e_d}\Delta e_d +\frac{\partial Q}{\partial \delta}\Delta\delta +\frac{\partial Q}{\partial u}\Delta u +\frac{\partial Q}{\partial \omega}\Delta\omega,\\ \Delta e_d &=(k_{\dot{\delta}}+k_{\dot{\delta}_{\mathrm{g}}})\Delta\dot{\delta} +k_p\Delta P+k_Q\Delta Q+k_u\Delta u -k_{\dot{\delta}_{\mathrm{g}}}\Delta\dot{\delta}_{u} +k_{\omega}\Delta\omega, \end{aligned} \right\} \tag{8} \]
where
\[ m_0=-(M_{\mathrm{T0}}+\omega_0 n_{\omega}), \]
\[ \frac{\partial P}{\partial\ }, \quad \frac{\partial Q}{\partial\ } \]
are the partial derivatives of the generator active and reactive powers.
The zero subscript denotes the parameters of the initial operating condition. Additions due to mismatch between the parameters of the initial operating condition and the settings of the measuring elements are assigned to \(M_{\mathrm{T0}}\).
Transform the system of equations (8), eliminating from it the increments \(\Delta\delta\) and \(\Delta e_d\):
\[ \left. \begin{aligned} \Delta P&=\rho_u\Delta u+\rho_{\delta u}\Delta\dot{\delta}_{u}+\rho_{\omega}\Delta\omega,\\ \Delta Q&=q_u\Delta u+q_{\delta u}\Delta\dot{\delta}_{u}+q_{\omega}\Delta\omega. \end{aligned} \right\} \tag{9} \]
In an arbitrarily chosen system of units, the quantities \(\rho\) and \(q\) are the slope coefficients of the static characteristics [3] of the generator active and reactive powers with respect to the voltage at the terminals,
phase and frequency. To determine the slope coefficients of the controlled generator, we write the system of equations in matrix form:
\[ \begin{bmatrix} 0 & -\omega_0(n_\delta+n_{\delta \mathrm{g}}) & -1 & 0\\ \dfrac{\partial P}{\partial e_d} & \dfrac{\partial P}{\partial \delta} & -1 & 0\\ \dfrac{\partial Q}{\partial e_d} & \dfrac{\partial Q}{\partial \delta} & 0 & -1\\ -1 & \kappa_\delta+\kappa_{\delta \mathrm{g}} & \kappa_p & \kappa_Q \end{bmatrix} \begin{bmatrix} \Delta e_d\\ \Delta \delta\\ \Delta P\\ \Delta Q \end{bmatrix} = -\begin{bmatrix} 0 & \omega_0 n_{\delta \mathrm{g}} & m_o\\ \dfrac{\partial P}{\partial u} & \dfrac{\partial P}{\partial \delta_u} & \dfrac{\partial P}{\partial \omega}\\ \dfrac{\partial Q}{\partial u} & \dfrac{\partial Q}{\partial \delta_u} & \dfrac{\partial Q}{\partial \omega}\\ \kappa_u & -\kappa_{\delta \mathrm{g}} & \kappa_\omega \end{bmatrix} \begin{bmatrix} \Delta u\\ \Delta \delta_u\\ \Delta \omega \end{bmatrix} \tag{10} \]
The slope coefficients are obtained in the form of ratios of two determinants:
\[ p_{\delta u}=\frac{1}{\Delta} \begin{vmatrix} 0 & -\omega_0(n_\delta+n_{\delta \mathrm{g}}) & -\omega_0 n_{\delta \mathrm{g}} & 0\\ \dfrac{\partial P}{\partial e_d} & \dfrac{\partial P}{\partial \delta} & -\dfrac{\partial P}{\partial \delta_u} & 0\\ \dfrac{\partial Q}{\partial e_d} & \dfrac{\partial Q}{\partial \delta} & -\dfrac{\partial Q}{\partial \delta_u} & -1\\ -1 & \kappa_\delta-\kappa_{\delta \mathrm{g}} & \kappa_{\delta \mathrm{g}} & \kappa_Q \end{vmatrix} \tag{11} \]
\[ q_{\delta u}=\frac{1}{\Delta} \begin{vmatrix} 0 & -\omega_0(n_\delta+n_{\delta \mathrm{g}}) & -1 & -\omega_0 n_{\delta \mathrm{g}}\\ \dfrac{\partial P}{\partial e_d} & \dfrac{\partial P}{\partial \delta} & -1 & -\dfrac{\partial P}{\partial \delta_u}\\ \dfrac{\partial Q}{\partial e_d} & \dfrac{\partial Q}{\partial \delta} & 0 & \dfrac{\partial Q}{\partial \delta_u}\\ -1 & \kappa_\delta+\kappa_{\delta \mathrm{g}} & \kappa_p & \kappa_{\delta \mathrm{g}} \end{vmatrix} \tag{12} \]
The remaining coefficients are determined analogously. Analysis of relations (11) and (12), as well as of the other slope coefficients, makes it possible to draw the following general conclusions for controlled synchronous generators.
The active power of the generator may vary as a function of the phase of the voltage at the terminals only when the turbine torque is controlled with respect to the absolute angle, or when the active-power automatic regulator acts simultaneously with respect to the relative angle and the automatic excitation regulator acts with respect to the absolute angle.
Reactive power depends on the voltage phase if, for the generator’s automatic excitation regulator or active-power automatic regulator, the control parameter is the absolute angle.
A dependence of the generator active power on the voltage magnitude occurs only when the active-power automatic regulator is implemented with respect to the absolute or relative angles,
Active power depends on frequency under any laws of the active-power regulator (APR) and automatic voltage regulator (AVR), unless the condition
\[ M_{\mathrm{t}0}=n_{\omega}=n_{\delta}=n_{\delta_{\mathrm{g}}}=0 \]
is satisfied.
The reactive power of the generator changes in accordance with changes in the voltage magnitude at the terminals and in the frequency, likewise under all APR and AVR laws.
The problems of regulating the turbine torque by absolute and relative angles are currently at the stage of intensive development and study [4]. Excitation regulation by these parameters is also still not widespread. Therefore, the overwhelming majority of existing generators have only three nonzero slope coefficients: active power with respect to frequency, and reactive power with respect to voltage and frequency. Unregulated generators have the same dependences [1].
The need to use regulation according to different parameters must be studied for various operating conditions of synchronous generators in interconnected power systems.
Appendix
We shall derive the expressions for the active and reactive powers of a synchronous generator from the equations of transient processes [5] in the stator winding,
\[ \begin{aligned} u\sin\delta_{\mathrm{g}}+p e_d+r(1+pT_d)i_d+x_q i_q p\gamma&=0,\\ u\cos\delta_{\mathrm{g}}-e_d p\gamma-x_d i_d p\gamma+r(1+pT_q)i_q&=0, \end{aligned} \tag{13} \]
as well as from the equations of electromagnetic torque and reactive power,
\[ M_{\mathrm{em}}=i_d x_q i_q-i_q(x_d i_d+e_d), \tag{14} \]
\[ Q=i_d u\cos\delta_{\mathrm{g}}-i_q u\sin\delta_{\mathrm{g}}. \tag{15} \]
Putting \(p\gamma=\omega\), \(r=0\) in equations (13) and passing to the steady-state condition, we find
\[ \begin{aligned} u\sin\delta_{\mathrm{g}}+\omega x_q i_q&=0,\\ u\cos\delta_{\mathrm{g}}-\omega e_d-\omega x_d i_d&=0. \end{aligned} \tag{16} \]
Eliminating the currents \(i_d\) and \(i_q\) from equations (14), (15) with the aid of (16), we obtain
\[ M_{\mathrm{em}}=\frac{e_d u}{\omega x_d}\sin\delta_{\mathrm{g}} +\frac{u^2}{2}\frac{x_d-x_q}{\omega^2 x_d x_q}\sin 2\delta_{\mathrm{g}}, \tag{17} \]
\[ Q=\frac{e_d u}{x_d}\cos\delta_{\mathrm{g}} -\frac{u^2}{\omega x_d} \left(1+\frac{x_d-x_q}{x_q}\sin^2\delta_{\mathrm{g}}\right). \tag{18} \]
Multiplying (17) by \(\omega\), we find the equation of the generator active power:
\[ P=\omega M_{\mathrm{em}}=\frac{e_d u}{x_d}\sin\delta_{\mathrm{g}} +\frac{u^2}{2}\frac{x_d-x_q}{\omega x_d x_q}\sin 2\delta_{\mathrm{g}}. \tag{19} \]
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