Stability Problems in Power Engineering

Abstract

The equations describing the electromechanical transient behaviour of a synchronous machine with flux linkage variations, machine damping and transient saliency included, in the case of balanced generator operation, are the following (B.Adkins, 1962):

$$\begin{gathered} {u_d} = - {R_s}{i_d} - \dot \psi d - \omega \left( {1 + s} \right){\psi _q} \hfill \\ {u_q} = - {R_s}{i_q} - {{\dot \psi }_q} + \omega \left( {1 + s} \right){\psi _d} \hfill \\ {u_f} = {R_f}{i_f} + {{\dot \psi }_f} \hfill \\ 0 = {R_D}{i_D} + {{\dot \psi }_D} \hfill \\ 0 = {R_Q}{i_Q} + {{\dot \psi }_Q} \hfill \\ \dot \delta = \omega s; T\dot s + Ds = {P_{mec}} - \omega \left( {{\psi _d}{i_q} - {\psi _q}{i_d}} \right) \hfill \\ \end{gathered} $$

(3.1.1)

together with the linear algebraic relations between currents and flux linkages (The linearity of these relations follows from the assumption that the machine is unsaturated):

$$\begin{array}{*{20}{c}} {{{\psi }_{d}} = {{L}_{d}}{{i}_{d}} + {{M}_{{ad}}}{{i}_{f}} + {{M}_{{ad}}}{{i}_{D}}} \hfill \\ {{{\psi }_{f}} = {{M}_{{ad}}}{{i}_{d}} + {{L}_{f}}{{i}_{f}} + {{M}_{{ad}}}{{i}_{D}}} \hfill \\ {{{\psi }_{D}} = {{M}_{{ad}}}{{i}_{d}} + {{M}_{{ad}}}{{i}_{f}} + {{L}_{D}}{{i}_{D}}} \hfill \\ {{{\psi }_{q}} = {{L}_{q}}{{i}_{q}} + {{M}_{{aq}}}{{i}_{Q}}} \hfill \\ {{{\psi }_{Q}} = {{M}_{{aq}}}{{i}_{q}} + {{L}_{Q}}{{i}_{Q}}} \hfill \\ \end{array}$$

(3.1.2)

If the inductances of the above equations are replaced by the corresponding synchronous, field mutual, and damper reactances

$$\begin{gathered} {x_d} = \omega {L_d}, {x_f} = \omega {L_f}, {x_D} = \omega {L_D}, {x_q} = \omega {L_q}, \hfill \\ {x_Q} = \omega {L_Q}, {x_{ad}} = \omega {M_{ad}}, {x_{aq}} = \omega {M_{aq}} \hfill \\ \end{gathered} $$

and if the transient, sub-transient and leakage reactances ${{x'}_{d}}{{x''}_{d}}{{x''}_{q}}{{x}_{\sigma }}$ are defined from the relations

$$\begin{array}{*{20}{c}} {{{x}_{\sigma }} = {{x}_{d}} - {{x}_{{ad}}} = {{x}_{q}} - {{x}_{{aq}}},} \hfill \\ {{{x}_{f}} = {{{\left( {{{x}_{d}} - {{x}_{\sigma }}} \right)}}^{2}}/\left( {{{x}_{d}} - {{{x'}}_{d}}} \right)} \hfill \\ {{{x}_{D}} = {{x}_{d}} - {{{x'}}_{d}} + {{{\left( {{{{x'}}_{d}} - {{x}_{\sigma }}} \right)}}^{2}}/\left( {{{{x'}}_{d}} - x''d} \right),} \hfill \\ {{{x}_{Q}} = {{{\left( {{{x}_{q}} - {{x}_{\sigma }}} \right)}}^{2}}/\left( {{{x}_{q}} - {{{x''}}_{q}}} \right)} \hfill \\ \end{array}$$

then the flux linkages are expressed as follows

$$\begin{array}{*{20}{c}} {\omega {{\psi }_{d}} = {{x}_{d}}{{i}_{d}} + \left( {{{x}_{d}} - {{x}_{\sigma }}} \right){{i}_{f}} + \left( {{{x}_{d}} - {{x}_{\sigma }}} \right){{i}_{D}}} \hfill \\ {\omega {{\psi }_{f}} = \left( {{{x}_{d}} - {{x}_{\sigma }}} \right){{i}_{d}} + \frac{{{{{\left( {{{x}_{d}} - {{x}_{\sigma }}} \right)}}^{2}}}}{{{{x}_{d}} + {{{x'}}_{d}}}}{{i}_{f}} + \left( {{{x}_{d}} - {{x}_{\sigma }}} \right){{i}_{D}}} \hfill \\ \end{array}$$

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