Figure 1.
Mapping of sinusoidal $abc$ signals to constant $dq0$ signals.
Figure 1.
Mapping of sinusoidal $abc$ signals to constant $dq0$ signals.
Figure 2.
Example of symmetric and nonsymmetric configurations. (a) Symmetrically configured network; (b) Network with a nonsymmetric configuration.
Figure 2.
Example of symmetric and nonsymmetric configurations. (a) Symmetrically configured network; (b) Network with a nonsymmetric configuration.
Figure 3.
Threephase unit connected to the network.
Figure 3.
Threephase unit connected to the network.
Figure 4.
Symmetrically configured threephase inductor.
Figure 4.
Symmetrically configured threephase inductor.
Figure 5.
Example—symmetrically configured $RL$ transmission line.
Figure 5.
Example—symmetrically configured $RL$ transmission line.
Figure 6.
Example—linear network with mutual ground resistances.
Figure 6.
Example—linear network with mutual ground resistances.
Figure 7.
While a singlephase system provides alternating power, a balanced threephase system provides constant power over a line cycle.
Figure 7.
While a singlephase system provides alternating power, a balanced threephase system provides constant power over a line cycle.
Figure 8.
Ideal power source connected to an infinite bus (singlephase diagram).
Figure 8.
Ideal power source connected to an infinite bus (singlephase diagram).
Figure 9.
Schematic diagram of a threephase synchronous machine with 2 poles.
Figure 9.
Schematic diagram of a threephase synchronous machine with 2 poles.
Figure 10.
Simulating a sudden 3phase short circuit at the armature terminals.
Figure 10.
Simulating a sudden 3phase short circuit at the armature terminals.
Figure 11.
Example: synchronous machine connected to a medium length transmission line and load. (a) Oneline diagram; (b) Singlephase diagram.
Figure 11.
Example: synchronous machine connected to a medium length transmission line and load. (a) Oneline diagram; (b) Singlephase diagram.
Figure 12.
Signal flow diagram for a synchronous machine connected to a medium length transmission line and load.
Figure 12.
Signal flow diagram for a synchronous machine connected to a medium length transmission line and load.
Figure 13.
The simplified machine model is equivalent to a voltage source behind a series inductance and resistance.
Figure 13.
The simplified machine model is equivalent to a voltage source behind a series inductance and resistance.
Figure 14.
Energy conversion in the machine, based on the simplified model.
Figure 14.
Energy conversion in the machine, based on the simplified model.
Figure 15.
Transformation from one reference frame to another.
Figure 15.
Transformation from one reference frame to another.
Figure 16.
Signalflow diagram: synchronous machine connected to an infinite bus.
Figure 16.
Signalflow diagram: synchronous machine connected to an infinite bus.
Figure 17.
Signalflow diagram: two machines connected to each other, and feeding a resistive load.
Figure 17.
Signalflow diagram: two machines connected to each other, and feeding a resistive load.
Figure 18.
A threephase inverter.
Figure 18.
A threephase inverter.
Figure 19.
A basic control scheme for grid forming inverters.
Figure 19.
A basic control scheme for grid forming inverters.
Figure 20.
A basic threephase inverter stage.
Figure 20.
A basic threephase inverter stage.
Figure 21.
Grid forming inverter, with additional current and voltage control loops.
Figure 21.
Grid forming inverter, with additional current and voltage control loops.
Figure 22.
The inner current loop: a typical control scheme that eliminates the cross terms ${\omega}^{*}{i}_{q}$ and ${\omega}^{*}{i}_{d}$.
Figure 22.
The inner current loop: a typical control scheme that eliminates the cross terms ${\omega}^{*}{i}_{q}$ and ${\omega}^{*}{i}_{d}$.
Figure 23.
By eliminating the cross terms, the current loop is modeled by two singleinput singleoutput systems.
Figure 23.
By eliminating the cross terms, the current loop is modeled by two singleinput singleoutput systems.
Figure 24.
A basic control scheme for grid feeding inverters.
Figure 24.
A basic control scheme for grid feeding inverters.
Figure 25.
A basic implementation of a Phase Locked Loop (PLL).
Figure 25.
A basic implementation of a Phase Locked Loop (PLL).
Figure 26.
Grid feeding inverter, connected to a photovoltaic (PV) source.
Figure 26.
Grid feeding inverter, connected to a photovoltaic (PV) source.
Figure 27.
Conceptual operation of the frequency droop mechanism.
Figure 27.
Conceptual operation of the frequency droop mechanism.
Figure 28.
Droop characteristics in steady state: active power as a function of frequency.
Figure 28.
Droop characteristics in steady state: active power as a function of frequency.
Figure 29.
The generator is modeled in steady state as a voltage source behind a series reactance.
Figure 29.
The generator is modeled in steady state as a voltage source behind a series reactance.
Figure 30.
Reactive power as a function of voltage. Variations in $\leftV\right$ may result in high and unpredictable reactive power flow.
Figure 30.
Reactive power as a function of voltage. Variations in $\leftV\right$ may result in high and unpredictable reactive power flow.
Figure 31.
Droop characteristics in steady state: reactive power as a function of voltage.
Figure 31.
Droop characteristics in steady state: reactive power as a function of voltage.
Figure 32.
Example: Reactive power sharing between two generators.
Figure 32.
Example: Reactive power sharing between two generators.
Figure 33.
Grid supporting inverter operating as a voltage source (conceptual control scheme).
Figure 33.
Grid supporting inverter operating as a voltage source (conceptual control scheme).
Figure 34.
Equivalent circuit for a permanent magnet synchronous motor with a round rotor (assuming ${L}_{d}={L}_{q}={L}_{s}$).
Figure 34.
Equivalent circuit for a permanent magnet synchronous motor with a round rotor (assuming ${L}_{d}={L}_{q}={L}_{s}$).
Figure 35.
A basic control scheme for a permanent magnet synchronous motor.
Figure 35.
A basic control scheme for a permanent magnet synchronous motor.
Table 1.
Comparison of approaches for dynamic modeling.
Table 1.
Comparison of approaches for dynamic modeling.
Model  Operating Point  SmallSignal  High Frequencies  NonSymmetric Networks 

timevarying phasors  √  √  X  X 
$abc$  X  X  √  √ 
$dq0$  √  √  √  X 
Table 2.
Typical operation modes of gridconnected inverters.
Table 2.
Typical operation modes of gridconnected inverters.
Grid Forming  Also called Voltage Source Inverters. The inverter operates as a voltage source. The voltage amplitude $\leftE\right$ and the frequency $\omega $ are directly controlled. The active power P and reactive power Q are determined by the interaction of the inverter with the grid. The inverter usually cannot operate in parallel to other grid forming inverters, since the frequency $\omega $ is constant. Typical applications are standby UPS systems and isolated small networks. In power flow studies: represented as a reference bus (slack bus), with a constant voltage amplitude $\leftE\right$ and angle $\delta =0$.

Grid Feeding  Also called Grid Following Inverters, or inverters with PQ control. The inverter operates as a power source. The active power P and reactive power Q are directly controlled. The frequency $\omega $ and voltage amplitude $\leftE\right$ are determined by the interaction of the inverter with the grid. Suitable for parallel operation. Cannot operate in isolation. The system must include other generators (e.g., other inverters, synchronous machines) that control the voltage amplitude and frequency. Typical applications are renewable energy systems, and distributed generation systems. In power flow studies: represented as a PQ bus, with constant active power P and reactive power Q.

Grid Supporting  Delivers power to the grid, while promoting stability and reliability. Regulates the frequency and voltage, and balances the active and reactive power generation. Implements a linear relationship between P and $\omega $, and between Q and $\leftE\right$. Suitable for parallel operation. Suitable for isolated operation. Combines well with energy storage systems, and online UPS systems. In power flow studies: in general, cannot be represented as a standard bus. However, if the frequency $\omega $ is known, and the voltage droop mechanism can be ignored, then the inverter is represented as a PV bus with constant active power and voltage amplitude $\leftE\right$, similarly to a synchronous machine.
