Modeling and mechanism analysis of inertia and damping issues for wind turbines PMSG grid-connected system

Abstract

Aiming at the problem of global climate change and energy crisis, wind power has become the focus of energy sustainable development in all countries. The wind turbines (WTs) power system is connected to the grid via the power electronic converter, causing the system inertia level to drop. In this paper, the direct-drive WT system is considered as the research object, and the whole-system frequency response model is established. The inertia and damping characteristics of the WT converter systems with virtual inertia control are analyzed. With the support of fan rotor kinetic energy and the energy saved in a capacitor, the simple control can also make the system exhibit different degrees of inertia and damping features. The results show that the equivalent inertia and the WT inertia time constant, capacitance parameters and virtual control parameters k_d are related; the equivalent damping parameter is related to the steady-state operating point parameters and the virtual control parameter k_p; the equivalent synchronization parameter is related to the steady-state operating point parameters and the virtual inertia control parameter k_i. Finally, the correctness of the inertial and damping characteristics of the WT grid-connected system is verified by simulation, which provides a theoretical reference for studying the inertial damping of power electronic dominant systems.

Appendix

Figure 4 shows a schematic diagram of the charging and discharging of a DC bus capacitor. The i_dc, i_c, i_s and u_dc in the figure are the input current, capacitor current, output current and capacitor voltage, respectively.

According to the charging and discharging rules of the capacitor, we can get:

$$ i_{\text{c}} = C_{{\text{dc}}} \frac{{\text{d}u_{{\text{dc}}} }}{{\text{d}t}} = i_{{\text{dc}}} - i_{\text{s}} $$

(22)

Selecting the reference voltage as the DC bus voltage U_dc and the power selection system-rated capacity S_B, the reference current I_B can be obtained as:

$$ I_{\text{B}} = \frac{{S_{\text{B}} }}{{U_{{\text{dc}}} }} = \frac{{P_{\text{B}} }}{{U_{{\text{dc}}} }} $$

(23)

Rewrite the current i_c in equation (22) as (where E_k is the energy stored by the capacitor):

$$ i_{\text{c}} = C_{{\text{dc}}} \frac{{\text{d}u_{{\text{dc}}} }}{{\text{d}t}} = \frac{{\frac{1}{2}C_{{\text{dc}}} U_{{\text{dc}}}^{2} }}{{\frac{1}{2}C_{{\text{dc}}} U_{{\text{dc}}}^{2} }}C\frac{{\text{d}u_{{\text{dc}}} }}{{\text{d}t}} = \frac{{2E_{\text{k}} }}{{U_{{\text{dc}}}^{2} }}\frac{{\text{d}u_{{\text{dc}}} }}{{\text{d}t}} $$

(24)

Combining equation (22) and dividing both sides of the equation by the current reference, we get:

$$ i_{{_{\text{c}} }}^{ * } = \frac{{i_{\text{c}} }}{{I_{\text{B}} }} = \frac{{2E_{\text{k}} }}{{I_{\text{B}} U_{{_{{\text{dc}}} }} }}\frac{{\text{d(}\frac{{u_{{\text{dc}}} }}{{U_{{_{{\text{dc}}} }} }})}}{{\text{d}t}} = \frac{{2E_{\text{k}} }}{{P_{\text{B}} }}\frac{{\text{d}u_{{\text{dc}}}^{ * } }}{{\text{d}t}} = \frac{{i_{{\text{dc}}} }}{{I_{\text{B}} }} - \frac{{i_{\text{s}} }}{{I_{\text{B}} }} = i_{{_{{\text{dc}}} }}^{ * } - i_{{_{\text{s}} }}^{ * } $$

(25)

Referring to the definition of the inertia time constant of the power system, the inertia time constant of the capacitor can be defined as:

$$ T^{\prime}_{\text{J}} (s) = \frac{{CU_{{_{{\text{dc}}} }}^{2} }}{{S_{\text{B}} }} = = \frac{{CU_{{_{{\text{dc}}} }}^{2} }}{{P_{\text{B}} }} $$

(26)

This style (22) can be rewritten as:

$$ T^{\prime}_{\text{J}} \frac{{\text{d}u_{{\text{dc}}}^{ * } }}{{\text{d}t}} = i_{{_{{\text{dc}}} }}^{ * } - i_{{_{\text{s}} }}^{ * } $$

(27)

In order to facilitate the analysis, the formula (27) is rewritten into the power expression as:

$$ T^{\prime}_{\text{J}} \frac{{\text{d}u_{{\text{dc}}}^{ * } }}{{\text{d}t}} = i_{{_{{\text{dc}}} }}^{ * } - i_{{_{\text{s}} }}^{ * } = \frac{{P_{{_{{\text{dc}}} }}^{ * } }}{{u_{{\text{dc}}}^{ * } }} - \frac{{P_{{_{\text{s}} }}^{ * } }}{{u_{{\text{dc}}}^{ * } }} $$

(28)

Under the control of the voltage outer loop, the u_dc change is small. It can be considered to be approximately 1.

$$ T^{\prime}_{\text{J}} \frac{{\text{d}u_{{\text{dc}}}^{ * } }}{{\text{d}t}} = P_{{_{{\text{dc}}} }}^{ * } - P_{{_{\text{s}} }}^{ * } $$

(29)

The vector diagram of grid-connected current control using grid voltage orientation is shown in Fig. 11. According to the figure, the grid-connected current expression can be obtained (ignoring the filter capacitor current component):

$$ i_{\text{s}} = i_{\text{d}} = \frac{{U_{\text{s}} }}{X}\sin \delta $$

(30)

Fig. 11

Inverter grid-connected voltage vector

When considering the effect of voltage control, we can get from equation (30):

$$ i_{\text{d}} = \frac{{U_{\text{s}} }}{X}\sin \delta = (u_{{\text{dc}}} - U_{{\text{dcref}}} )\left(K_{{\text{pu}}} + \frac{{K_{{\text{iu}}} }}{s}\right) $$

(31)

Linearizing equation (10) near the steady-state operating point can obtain:

$$ sK\Delta \delta = (sK_{{\text{pu}}} + K_{{\text{iu}}} )\Delta u_{{\text{dc}}} $$

(32)

In the formula, K is a system structure parameter, and K = U_s / cosδ₀.

When the input power change is not considered, that is, δi_dc = 0, linearize equation (29), consider $s\Delta \delta = \Delta \omega$ and consider equation (32) to eliminate δu_dc, and a dynamic model of the DC capacitor of the grid-connected inverter under the time scale of the DC voltage for:

$$ T^{\prime}_{\text{J}} (s)\frac{{\text{d}\Delta \omega }}{{\text{d}t}} = - T^{\prime}_{\text{D}} (s)\Delta \omega - T^{\prime}_{\text{S}} (s)\Delta \delta $$

(33)

and:

$$ \left\{ {\begin{array}{*{20}l} {T_{{\text{Jc}}} (s) = {{CU_{{_{{\text{dc}}} }}^{2} } \mathord{\left/ {\vphantom {{CU_{{_{{\text{dc}}} }}^{2} } {S_{\text{B}} }}} \right. \kern-0pt} {S_{\text{B}} }}} \hfill \\ {T_{{\text{Dc}}} (s) = 1.5K_{{\text{pu}}} U_{\text{g}} } \hfill \\ {T_{{\text{Sc}}} (s) = 1.5K_{{\text{iu}}} U_{\text{g}} } \hfill \\ \end{array} } \right. $$

(34)

Referring to the definition of the kinetic energy analysis of the power system, the electric field energy stored by the capacitor can also provide certain inertia characteristics for the system. The electric field energy change is:

$$ \Delta E_{\text{k}} (t) = \frac{1}{2}C_{\text{dc}} (U_{\text{dc}}^{2} - U_{\text{dc}}^{2} (t)) $$

(35)

Then, the amount of change in the output electromagnetic power of the capacitor at time t is:

$$ P_{\text{ec}} = \frac{{{\text{d}}\Delta E_{\text{k}} (t)}}{{{\text{d}}t}} = - CU_{\text{dc}} (t)\frac{{{\text{d}}U_{\text{dc}} (t)}}{{{\text{d}}t}} $$

(36)

Combine the equivalent inertia time constant of the (34) capacitor link and consider equations (35) and (22) to obtain the inertia support power of the capacitor when the grid frequency changes:

$$ P_{\text{ec}} = \frac{{{\text{d}}\Delta E_{\text{k}} (t)}}{{{\text{d}}t}} \approx - \frac{{S_{\text{B}} T^{\prime}_{\text{J}} }}{{\omega_{0} }}\frac{{{\text{d}}\Delta \omega (t)}}{{{\text{d}}t}} $$

(37)