Grid-supporting inverters with significantly reduced storage requirements

Abstract

A method for controlling an inverter, and in particular a double stage inverter, for implementing a model of a synchronous generator is provided including implementing a rotor inertia using an intermediate dc-link capacitor without duplicating the emulated inertia in the controller, simulating the rotor speed based on a measured voltage of the dc-link capacitor, while allowing the voltage to change in a defined range, and mapping the changing voltage of the dc-link capacitor into the inverter as an internal frequency. A system for connecting a power generator to a power grid is also provided including a control device for an inverter, the control device implementing a model of a synchronous generator. The control device including a computer processor in electrical communication with a storage device with instructions stored thereon, that when executed on the computer processor, perform the method for controlling an inverter for implementing a model of a synchronous generator.

Claims

1. A method for implementing a controller for control of an inverter for implementation of a model of a synchronous generator with a virtual rotor with a rotor speed represented by an internal frequency (ω.sub.m) to convert a dc input to an ac output, said method comprising: implementing the virtual rotor to emulate an inertia (J) of an actual rotor using an intermediate dc-link capacitor (C.sub.dc) of the inverter, the implementing comprising: controlling a voltage of the intermediate dc-link capacitor (C.sub.dc); simulating a rotor speed of the virtual rotor based on the voltage of the intermediate dc-link capacitor (C.sub.dc), while allowing the voltage applied to change in a defined range; and mapping a changing voltage of the intermediate dc-link capacitor (C.sub.dc) into the inverter as the internal frequency (ω.sub.m), where the controller for the inverter does not duplicate the emulated inertia (J).

2. The method of claim 1 further comprising providing self-synchronization prior to connecting the inverter to a grid.

3. The method of claim 1 further comprising providing a virtual impedance at an output of the inverter as a gain value that represents a virtual resistance that enables achieving a stability margin.

4. The method of claim 1 further comprising applying a power dampening term P.sub.dmp as a voltage amplitude V.sub.dmp of a grid voltage v.sub.g, where $V_{\mathrm{dmp}}=-\frac{2}{3}D\frac{d}{\mathrm{dt}}\left(\tilde{\cos }\theta .Math.v_g\right)=-{\mathrm{DV}}_g\frac{d}{\mathrm{dt}}\left[\sin \left(\theta -{\theta }_g\right)\right]\approx {\mathrm{DV}}_g\left({\omega }_g-{\omega }_m\right)\cos \left(\theta -{\theta }_g\right)$ that introduces a damping effect into an internal voltage magnitude E which in turn translates into P.sub.dmp.

5. The method of claim 4 wherein a calculated reference Q* is used to derive the internal voltage amplitude E given by
Q*=Q.sub.n−k.sub.v(V.sub.g−V.sub.n),Ė=k.sub.q(Q*−Q) where the reactive power Q is given by $Q=-E\tilde{\cos }\theta .Math.i=-E\cos \theta i_a-E\cos \left(\theta -\frac{2\pi }{3}\right)i_b-E\cos \left(\theta +\frac{2\pi }{3}\right)i_c.$

6. The method of claim 1 wherein the method of self-synchronization comprises: calculating a cross product of a grid voltage v.sub.g and an inverter voltage v.sub.o as a signal sync $\mathrm{Sync}=\left({\mu }_1+\frac{{\mu }_2}{s}\right)\left(v_{o\alpha }{v}_{g\beta }-v_{o\beta }{v}_{g\alpha }\right)=\left({\mu }_1+\frac{{\mu }_2}{s}\right)V_o{V}_g\sin \left({\varphi }_g-{\varphi }_o\right)$ where α and β indices denote variables transformed to aft domain by a Clarke transformation and Vo is an output voltage amplitude and ϕ.sub.g and ϕ.sub.o are grid and output voltage phase angles.

7. The method of claim 6 further comprising adjusting the signal Sync to make ϕ.sub.g and ϕ.sub.o to be equal prior to closing a grid connection switch.

8. A system for connecting a power generation unit to a power grid comprising: a control device for an inverter, the control device implementing a model of a synchronous generator, the control device comprising: a computer processor in electrical communication with a storage device with instructions stored thereon, that when executed on the computer processor, perform a method for implementing a controller for control of an inverter for implementation of a model of a synchronous generator with a virtual rotor with a rotor speed represented by an internal frequency (ω.sub.m) to convert a dc input to an ac output, said method comprising: implementing a virtual rotor to emulate an inertia (J) of an actual rotor using an intermediate dc-link capacitor (C.sub.dc) of the inverter, the implementing comprising: controlling a voltage of the intermediate dc-link capacitor (C.sub.dc); simulating the rotor speed of the virtual rotor based on the measured voltage of the intermediate dc-link capacitor (C.sub.dc), while allowing the voltage applied to change in a defined range; and mapping the changing capacitor voltage of the intermediate dc-link capacitor (C.sub.dc) into the inverter as the internal frequency (ω.sub.m), where the controller for the inverter does not duplicate the emulated inertia (J).

9. The system of claim 8 further comprising providing a virtual impedance at an output of the inverter as a gain value that represents a virtual resistance that enables achieving a stability margin.

10. The system of claim 8 further comprising applying a power dampening term P.sub.dmp as a voltage amplitude V.sub.dmp of a grid voltage v.sub.g, where $V_{\mathrm{dmp}}=-\frac{2}{3}D\frac{d}{\mathrm{dt}}\left(\tilde{\cos }\theta .Math.v_g\right)=-{\mathrm{DV}}_g\frac{d}{\mathrm{dt}}\left[\sin \left(\theta -{\theta }_g\right)\right]\approx {\mathrm{DV}}_g\left({\omega }_g-{\omega }_m\right)\cos \left(\theta -{\theta }_g\right)$ that introduces a damping effect in to an internal voltage magnitude E which in turn translates into P.sub.dmp.

11. The system of claim 10 further comprising a calculated reference Q* that is used to derive the internal voltage amplitude E given by
Q*=Q.sub.n−k.sub.v(V.sub.g−V.sub.n),Ė=k.sub.q(Q*−Q) where the reactive power Q is given by $Q=-E\tilde{\cos }\theta .Math.i=-E\cos \theta i_a-E\cos \left(\theta -\frac{2\pi }{3}\right)i_b-E\cos \left(\theta +\frac{2\pi }{3}\right)i_c.$

12. The system of claim 8 further comprising providing self-synchronization prior to connecting the inverter to a grid.

13. The system of claim 12 wherein the self-synchronization further comprises: calculating a cross product of a grid voltage v and an inverter voltage v.sub.o as a sync signal $\mathrm{Sync}=\left({\mu }_1+\frac{{\mu }_2}{s}\right)\left(v_{o\alpha }{v}_{g\beta }-v_{o\beta }{v}_{g\alpha }\right)=\left({\mu }_1+\frac{{\mu }_2}{s}\right)V_o{V}_g\sin \left({\varphi }_g-{\varphi }_o\right)$ where α and β indices denote variables transformed to aft domain by a Clarke transformation and Vo is an output voltage amplitude and ϕ.sub.g and ϕ.sub.o are grid and output voltage phase angles.

14. The system of claim 13 wherein the self-synchronization further comprises adjusting the signal Sync to make ϕ.sub.g and ϕ.sub.o to be equal prior to closing a grid connection switch between an output of a power generation unit and a power grid.

15. The system of claim 14 wherein the power generation unit is a distributed and renewable energy source.

16. The system of claim 15 wherein the power generation distributed and renewable energy source further comprises photovoltaic (PV) or wind systems.

Description

BRIEF DESCRIPTION OF THE DRAWINGS

(1) The subject matter that is regarded as the invention is particularly pointed out and distinctly claimed in the claims at the conclusion of the specification. The foregoing and other objects, features, and advantages of the invention are apparent from the following detailed description taken in conjunction with the accompanying drawings in which:

(2) FIG. 1 is a block diagram of a prior art double-stage inverter;

(3) FIG. 2A is an expanded view of a prior art current controller (CC) block in a rotating frame of FIG. 2C;

(4) FIG. 2B is an expanded view of a prior art current controller (CC) block in stationary frame of FIG. 2C;

(5) FIG. 2C illustrates a block diagram of a prior art inverter using a current control method;

(6) FIG. 2D illustrates a block diagram of a prior art inverter using a voltage control method;

(7) FIG. 3A is a block diagram of a prior art synchronous generator;

(8) FIG. 3B is a block diagram of an equivalent model of the synchronous generator of FIG. 3A;

(9) FIG. 3C is a block diagram of a prior art virtual synchronous machine;

(10) FIG. 3D is a block diagram of a prior art virtual synchronous machine applied to a two-stage converter;

(11) FIGS. 4A and 4B illustrate the prior art frequency response for inertial power and governor power for a slow frequency swing and a fast frequency swing, respectively;

(12) FIG. 5A is a block diagram of a converter and controller in accordance with embodiments of the invention;

(13) FIG. 5B is a block diagram of an overall equivalent model of the converter and controller of FIG. 5A in accordance with embodiments of the invention;

(14) FIG. 6 is a block diagram of a converter and controller with axillary control blocks in accordance with embodiments of the invention;

(15) FIGS. 7A-C are a series of graphs showing the loci of eigenvalues of linearized system J versus three controller gains in accordance with embodiments of the invention;

(16) FIG. 8 is a block diagram the enhanced virtual synchronous machine (eVSM) applied to a two-stage photovoltaic (PV) system her accordance with an embodiment of the invention

(17) FIG. 9 is a series of graphs illustrating simulation results comparing inertial behavior of the eVSM controller with SG (6<t<11.5) in accordance with embodiments of the invention;

(18) FIG. 10 is a series of graphs illustrating simulation results of the eVSM for input power, grid frequency, and grid amplitude jump (t<6);

(19) FIG. 11 is a series of graphs illustrating simulation results of the eVSM during standalone (SA) and transition to grid-connected (GC) operation (12<t<18);

(20) FIGS. 12A and 12B illustrate the results of the performance of an embodiment the proposed controller during SA, GC, and transition between these two modes of operation; and

(21) FIG. 13 is a photograph of an experimental set up;

(22) FIG. 14 illustrates the results showing the standalone load change, with the top line showing inverter LN voltages, the middle line showing inverter currents, and the bottom line showing Vdc (50 ms/div), and where the left side of the graph shows the load increased from 150 W to 500 W and the right side of the graph shows the load decreased;

(23) FIGS. 15A-15C illustrate the results of the grid frequency jump, with the top of 15A and 15B showing inverter LN voltages, the middle of 15A and 15B showing inverter currents, and the bottom of 15A and 15B showing Vdc (100 ms/div). Pmpp=450 W and f jumps (15A) left: 60 Hz to 60.1 Hz, right: 60.1 Hz to 60 Hz, (15B) left: 60 Hz to 59.9 Hz, right: 59.9 Hz to 60 Hz, and 15C showing pout during these jumps;

(24) FIGS. 16A-16C illustrate experimental results for the grid amplitude jump, with the top of 16A and 16B showing inverter LN voltages, the middle of 16A and 16B showing inverter currents, and the bottom of 16A and 16B showing Vdc (100 ms/div), (16A) nominal grid 2.5% increase and back to nominal grid (16B) nominal grid 2.5% decrease and back to nominal grid right, and 16C showing output active and reactive power; and

(25) FIG. 17 illustrates experimental results for GC to SA, with the top line showing inverter LN voltages, the middle line showing inverter currents, and the bottom line showing Vdc (50 ms/div).

DESCRIPTION OF THE INVENTION

(26) The present invention has utility as an improved method and system for grid-supporting inverters for integration of distributed and renewable energy sources to a power grid. Unlike existing virtual synchronous machine (VSM) approaches, embodiments of the inventive inverter control approach, herein referred to as enhanced virtual synchronous machine (eVSM), do not duplicate the inertial loop or emulate the virtual inertia inside the controller, but deploys the physically existing inertia of the dc-link element. The eVSM employs an innovative method for enlarging the inertia utilization range which obviates the need for having a large dc-link element or a dedicated battery storage system, while still providing the same inertia response of an equivalent synchronous machine. Embodiments of the eVSM inverter controller use the physical dc-link capacitor dynamics for emulating the inertial response rather than relying on a dedicated battery storage or large dc element. Thus, embodiments of the eVSM implement the virtual inertia outside the controller using only the dc link capacitor, whereas existing VSM methods implement the inertia inside the controller. Existing VSM duplicate the inertia in the controller and the bus which will deviate from the synchronous machine model.

(27) Embodiments of the eVSM provide stabilizing support to the grid that is similar to a synchronous machine despite the small size of the dc-link element in the eVSM. Moreover, transient responses of embodiments of the eVSM may be improved beyond those of conventional synchronous machines with the enhanced flexibility provided by adjusting damping and governor functions.

(28) Embodiments of the invention provide control of an inverter. A simple yet effective method of controlling the inverter is provided that is based on the synchronous generator (SG) model, thereby providing dynamic performance similar to an SG. In this method, the rotor inertia is implemented using the intermediate dc-link capacitor, and rotor speed is mimicked with the capacitor voltage. By allowing the dc-link voltage to change within an enlarged yet admissible range and mapping the dc-link voltage into the controller as the internal frequency, the inventive method does not require a battery storage system (BSS) nor a large capacitor to emulate the rotor inertia. A self-synchronizing property prior to grid connection is achieved and improved stability margins, beyond those of synchronous machines, is made possible with embodiments of the eVSM.

(29) As shown the existing VSM approach with ideal input power source and with proper governor and damper models can reproduce a virtual inertia behavior with the moment of inertia J that is implemented by the integrator

(30) $\frac{1}{\mathrm{Js}}$
inside the VSM controller. However, when the existing VSM is applied to a converter with a bus capacitor and its related control systems, the incurred additional dynamics will deviate the emulated system from the targeted SG dynamics and its correct inertia response. It is further noted that in an SG, the moment of inertia corresponds to the physical rotor mass. To accurately emulate the SG in an inverter with a bus capacitor, the source of energy for the inertia should be a single component similar to the rotor mass in SG. Considering and comparing the two equations
p.sub.in−p.sub.out=J ω.sub.m{dot over (ω)}.sub.m,p.sub.in−p.sub.out=C.sub.dcV.sub.dc{dot over (V)}.sub.dc  Eq. 3
it can be immediately observed that the bus capacitor in a two-stage inverter relates to the rotor mass in the SG. Therefore, unlike the existing VSMs, in embodiments of the eVSM there is no need for an additional integrator to emulate the rotating inertia as the capacitor in the hardware itself will emulate this inertia properly. This is the starting point to derive embodiments of a controller for an inverter in general and in particular in a double-stage inverter.

(31) A correspondence is established between the couplet (J, ω.sub.m) and (C.sub.dc, V.sub.dc) using a linear transformation as shown in FIG. 5A which is expressed by

(32) $\begin{array}{cc}{\omega }_m={\omega }_n+\frac{1}{k}\left(V_{dc}-V_{dc,n}\right).& \mathrm{Eq}.4\end{array}$

(33) It is shown below that this choice results in a moment of inertia equal to

(34) $J=k\left(\frac{V_{dc,n}}{{\omega }_n}\right)C_{dc}.$
As shown in FIG. 5B, the dc link capacitor dynamics becomes identical to that of the rotor inertia in SG, while the rest of the system is identical to the SG, where ω.sub.m is used to generate an inverter voltage with the amplitude coming from the QC block. This inverter voltage along with the grid voltage determines the inverter current passing through the filter's impedance. Therefore, the proposed method directly integrates the physical inertia coming from the dc bus capacitor. This is in contrast to the existing VSM methods where inertia is implemented inside the controller and duplicates that of the dc link capacitor.

(35) It is further noted that as V.sub.dc is linked to the output frequency in embodiments of the inventive controller, the bus voltage will vary with the grid frequency. Therefore, the modulation index is scaled using the actual V.sub.dc value (not a nominal value) decouple the dc link voltage variations from the smooth and sinusoidal output voltage (23).

(36) It is to be understood that in instances where a range of values are provided that the range is intended to encompass not only the end point values of the range but also intermediate values of the range as explicitly being included within the range and varying by the last significant figure of the range. By way of example, a recited range of from 1 to 4 is intended to include 1-2, 1-3, 2-4, 3-4, and 1-4.

(37) It is also appreciated that the virtual controls of the eVSM are software based, where the software is stored in a memory device in electrical communication with a computer processing unit. The instructions contained in the software are carried out by the processor of a computing device. The computing device may be a standalone computing device, or a computer on a chip illustratively including a microprocessor. The software may be stored in updatable random access memory or as firmware in read only memory.

(38) Referring now to the figures, FIG. 6 illustrates a block diagram of an inventive embodiment of an eVSM controller with detailed auxiliary blocks including the QC block with voltage droop controller and the damping strategy. As may also be seen in FIG. 6, the concept of virtual impedance (Z(s)) may also be introduced to the embodiments of the inventive technique to provide a desired impedance at the output of the inverter. In the simplest form, Z(s) is a gain that represents a virtual resistance that enables achieving desirable stability margins without using physical lossy resistors at the output. It is also worth noting that embodiments of the inventive method do not need any phase lock loop (PLL) or any other measurement from the output angle or frequency

(39) In embodiments of the inventive eVSM controller, the SG damping term in the swing equation (2), P.sub.dmp=D(ω.sub.m−ω.sub.g) may also be added. Unlike the VSM methods, however, the point of applying P.sub.dmp is not available in embodiments of the inventive controller because the inertia-generating integrator is not inside the controller, but rather in the actual system. Therefore, P.sub.dmp has to be applied differently. As shown in FIG. 6, the damping scheme P.sub.dmp is translated into voltage amplitude, namely V.sub.dmp, and is implemented at the output of the QC block. A shown in FIG. 6

(40) $\begin{array}{cc}{V}_{\mathrm{dmp}}=-\frac{2}{3}D\frac{d}{\mathrm{dt}}\left(c\tilde{o}s\theta .Math.v_g\right)=-{\mathrm{DV}}_g\frac{d}{\mathrm{dt}}\left[\sin \left(\theta -{\theta }_g\right)\right]\approx {\mathrm{DV}}_g\left({\omega }_g-{\omega }_m\right)\cos \left(\theta -{\theta }_g\right).& \mathrm{Eq}.5\end{array}$
This introduces a damping effect in the voltage magnitude E which in turn translates into P.sub.dmp. It is worth noting that in embodiments of the inventive method the term (ω.sub.m−ω.sub.g) is calculated instantaneously with no dynamics without directly measuring or calculating ω.sub.g.

(41) Continuing with FIG. 6, the reactive power control (QC) block controls the reactive power flow according to the voltage magnitude the same way an SG exciter does. The adopted voltage droop controller is given by
Q*=Q.sub.n−k.sub.v(V.sub.g−V.sub.n),Ė=k.sub.q(Q*−Q)  Eq. 6

(42) The calculated reference Q* from equation 6 is then used to obtain the desired internal voltage amplitude E through an integrating unit the same way popular VSM methods such as (5), (6) do. The reactive power Q may be calculated using internal voltage variable according to

(43) $Q=-Ec\tilde{o}s\theta .Math.i=-E\cos \theta i_a-E\cos \left(\theta -\frac{2\pi }{3}\right)i_b-E\cos \left(\theta +\frac{2\pi }{3}\right)i_c$
and the voltage amplitude V.sub.g is calculated according to

(44) 0$V_g=\sqrt{-\frac{4}{3}\left(v_a{v}_b+v_b{v}_c+v_c{v}_a\right)}$
or any other methods to estimate reactive power and voltage magnitude.

(45) A self-synchronization procedure may be employed in embodiments of the invention. The self-synchronization procedure ensures smooth transition to GC mode. The method of (9), (10) is modified and used in embodiments of the inventive method as a cross product of the grid voltage and inverter voltage

(46) $\begin{array}{cc}\mathrm{Sync}=\left({\mu }_1+\frac{{\mu }_2}{s}\right)\left(v_{o\alpha }{v}_{g\beta }-v_{o\beta }{v}_{g\alpha }\right)=\left({\mu }_1+\frac{{\mu }_2}{s}\right)V_o{V}_g\sin \left({\varphi }_g-{\varphi }_o\right)& \mathrm{Eq}.7\end{array}$
is added as shown in FIG. 6 where α and β indices denote the variables transformed to αβ domain by the Clarke transformation, Vo is the output voltage amplitude, and ϕ.sub.g and ϕ.sub.o are the grid and output voltage phase angles. The signal sync in equation 7 is an index of synchronization, which is used to synchronize the inverter output voltage v.sub.o to grid voltage v.sub.g prior to connecting to the grid. Prior to the grid connection, due to the correspondence between the active power and ω.sub.m, the signal Sync may be adjusted so that ϕ.sub.g and ϕ.sub.o become equal. At this time, a grid connection switch can be closed. Subsequently, v.sub.o=v.sub.g and the signal Sync will reset to zero.

(47) In inventive embodiments of the eVSM controller, the internal frequency and the dc-link voltage are coupled through equation 4. There are two key points to this relationship: (i) the dc-link voltage is shaped to follow the grid frequency because the internal frequency has to closely follow the grid frequency, and (ii) the dc-link voltage will satisfy {dot over (V)}.sub.dc=k {dot over (ω)}.sub.m which can be shown to amplify the inertia utilization. To clarify the second point, assume that the center dc-link voltage is V.sub.dc,n=k.sub.oω.sub.n. Then, the inertia power of the proposed PV system is equal to
C.sub.dcV.sub.dc{dot over (V)}.sub.dc≈C.sub.dcV.sub.dc,n{dot over (V)}.sub.dc=C.sub.dck.sub.okω.sub.n {dot over (ω)}.sub.m≈(C.sub.dck.sub.ok)ω.sub.m {dot over (ω)}.sub.m=Jω.sub.m {dot over (ω)}.sub.m,  Eq. 8
where
J=(k.sub.ok)C=KC.  Eq. 9

(48) Equation (9) signifies an inertia utilization amplification with gain K=k.sub.ok. The first gain, k.sub.o, depends on the value of the center dc-link voltage V.sub.dc,n. For a 500 V dc-link voltage in a 60 Hz system, for instance, k.sub.o is equal to 500/377=1.33. The second gain, k, depends on the allowable range of swings of the dc-link voltage. In practice, this gain can be selected relatively large without violating the practical constraints of the system. For instance, for a 120 V (rms) line to neutral 3-phase grid and a center dc-link voltage of V.sub.dc,n=500 V, the physical limitations of the VSC require that the grid peak voltage 120√{square root over (2)}≈170 V remain below

(49) $\frac{V_{dc,n}}{2}=250V.$
Even if a margin of about 20 V is left for this limit, the dc-link voltage can still have a dynamic range of ±60 V around its center value. If the grid frequency range of variations is ±0.5 Hz, that is ±3.14 rad/s, this means k is approximately equal to 20. With this set of numbers, the moment of inertia amplification gain K will be about 26.

(50) Inertia energy source management may be conducted as follows in embodiments of the inventive eVSM. The source of energy to supply the inertia in an SG is dominantly the rotor mass and normally this inertia energy cannot be taken from the input mechanical source as its time constant does not allow it. Embodiments of the inventive controller have made this distinction and follows the same rule and takes the inertia energy from the bus capacitor. However, in cases where the input source time constant allows it, such as the case of photovoltaic (PV) or battery sources, the inertia energy may be managed to be partially provided from the input source. Assume that the desired total emulated moment of inertia is J.sub.total and the amount of the moment of inertia provided by bus capacitor in the proposed controller is J=KC. Therefore, J.sub.total=J+J.sub.in, where J.sub.in is the moment of inertia supply by the input stage. In this case, the desired swing equation for the system is p.sub.in−p.sub.out=J.sub.totalω.sub.m{dot over (ω)}.sub.m=Jω.sub.m{dot over (ω)}.sub.m+J.sub.inω.sub.m{dot over (ω)}.sub.m. This can be achieved by adding −J.sub.inω.sub.m{dot over (ω)}.sub.m to the reference of the input power.

(51) Design of synchronization block gains, μ.sub.1 and μ.sub.2, may be carried out used phase lock loop (PLL) theory, because it is readily observed that the VSM is very similar to a PLL during pre-synchronization. Therefore, the set of values for μ.sub.1 and μ.sub.2 may be derived as

(52) $\begin{array}{cc}\left({\mu }_1+\frac{{\mu }_2}{s}\right)=\frac{1}{V_n^2}\left(100+\frac{1}{0.05s}\right)& \mathrm{Eq}.10\end{array}$

EXAMPLES

Example 1

(53) Embodiments of the proposed method for an eVSM may be applied to any two-stage converter system. As an example, a two-stage photovoltaic (PV) system is shown in FIG. 8. One challenge in renewable applications in general is the uncertainty and variability of the input available power. In PV systems, it is desirable to operate as close as possible to the unknown maximum power point (MPP) during normal system operations. To address this issue, the embodiment method of FIG. 8 uses a control loop over the variable

(54) $\frac{\mathrm{dp}}{\mathrm{dv}}$
to regulate the PV system to

(55) ${\left(\frac{\mathrm{dp}}{\mathrm{dv}}\right)}_n+k_{\mathrm{pv}}\left({\omega }_n-{\omega }_m\right).$
This mechanism will operate the PV at a point slightly on the right side of the MPP determined by

(56) ${\left(\frac{\mathrm{dp}}{\mathrm{dv}}\right)}_n$
as long as the grid frequency is close to the nominal value. Similar to the governor of an SG, this loop will adjust the input power according to a droop characteristic (k.sub.pv) with respect to the internal grid frequency variable (ω.sub.m). A first PI.sub.1 with a slow time constant is used to emulate the slow behavior of an SG governor.

(57) As observed in FIG. 8, the optional input inertia term −J.sub.inω.sub.m{dot over (ω)}.sub.m can be added to the input power reference p.sub.in* and a fast PI.sub.2 controller is used to allow tracking of a power reference up to the desired frequency band of the inertia.

Example 2

(58) A stability analysis of the embodiment of the inventive control system shown in FIG. 6 is performed to obtain system parameters.

(59) Differential equations of the system shown in FIG. 6 are summarized as follows:

(60) $\begin{array}{cc}{\stackrel{.}{W}}_{dc}=p_{\mathrm{in}}-p_{\mathrm{out}}=p_{\mathrm{in}}\left(E+V_{\mathrm{dmp}}\right)i_d\stackrel{.}{E}=k_q\left(Q^{*}-Q\right)=k_q\left(Q^{*}+{\mathrm{Ei}}_q\right)\frac{d}{\mathrm{dt}}{i}_d=-\frac{R}{L}{i}_d+\frac{1}{L}\left(E+V_{\mathrm{dmp}}\right)-\frac{1}{L}{V}_{\mathrm{gd}}+{\omega }_m{i}_q\frac{d}{\mathrm{dt}}{i}_q=-\frac{R}{L}{i}_q-\frac{1}{L}{V}_{\mathrm{gd}}-{\omega }_m{i}_d\stackrel{.}{\delta }={\omega }_g-{\omega }_m,\delta ={\theta }_g-\theta & \mathrm{Eq}.11\end{array}$
where W.sub.dc=½C.sub.dcV.sub.dc.sup.2, the variables are transformed to the dq-frame using the internal angle θ, and it is assumed that the virtual impedance is equal to R. The damping term is calculated according to

(61) $V_{\mathrm{dmp}}=-\frac{2}{3}D\frac{d}{\mathrm{dt}}{v}_g.Math.\stackrel{•}{\cos }\theta =-\frac{2}{3}D\frac{d}{\mathrm{dt}}\left(\frac{3}{2}{V}_g\sin \left({\theta }_g-\theta \right)\right)=-{\mathrm{DV}}_g\left({\omega }_g-{\omega }_m\right)\cos \delta$
and the grid voltage dq components are V.sub.gd=V.sub.g cosδ,V.sub.gq=V.sub.g sin δ. It is also noted that {dot over (W)}.sub.dc=CV.sub.dc{dot over (V)}.sub.dc=Ck[V.sub.dc,n+k(ω.sub.m−ω.sub.n)]{dot over (ω)}.sub.m, and substituting in equation 10 gives,

(62) $\begin{array}{cc}{\stackrel{.}{\omega }}_m=\frac{1}{\mathrm{CK}}\left[p_{\mathrm{in}}-{\mathrm{Ei}}_d+{\mathrm{DV}}_g{i}_d\left({\omega }_g-{\omega }_m\right)\cos \delta \right]\text{/}\left[V_{\mathrm{dc},n}+k\left({\omega }_m-{\omega }_n\right)\right]\stackrel{.}{E}=k_q\left(Q^{*}-Q\right)=k_q\left[Q_n+k_v\left(V_n-V_g\right)+{\mathrm{Ei}}_q\right]\stackrel{.}{\delta }={\omega }_g-{\omega }_m{\stackrel{.}{i}}_d=-\frac{R}{L}{i}_d+\frac{1}{L}\left[E-{\mathrm{DV}}_g\left({\omega }_g-{\omega }_m\right)\cos \delta \right]-\frac{1}{L}{V}_g\cos \delta +{\omega }_m{i}_q\frac{d}{\mathrm{dt}}{i}_q=-\frac{R}{L}{i}_q-\frac{1}{L}{V}_g\sin \delta -{\omega }_m{i}_d.& \mathrm{Eq}.12\end{array}$

(63) Equation set 12 represents a fifth order nonlinear system with state variables (ω.sub.m, E, δ, i.sub.a, i.sub.g). For every given set of values of (p.sub.in, Q.sub.n), the equilibrium point of the equations may be derived and used in linear stability analysis using the Jacobian linearization method. Assuming that the grid frequency/voltage is at its rated, ω.sub.g=ω.sub.n, V.sub.g=V.sub.n, and the equilibrium point is given by (ω.sub.g, E.sub.o, δ.sub.o, i.sub.ao, i.sub.qo), the Jacobian matrix will be given by

(64) 0$\hspace{1em}\left[\begin{array}{ccccc}-\frac{{\mathrm{DV}}_g{i}_{\mathrm{do}}}{{\mathrm{CkV}}_{\mathrm{dc},n}}\cos {\delta }_o& -\frac{i_{\mathrm{do}}}{{\mathrm{CkV}}_{\mathrm{dc},n}}& 0& -\frac{E_o}{{\mathrm{CkV}}_{\mathrm{dc},n}}& 0\\ 0& k_q{i}_{\mathrm{qo}}& 0& 0& k_q{E}_o\\ -1& 0& 0& 0& 0\\ i_{\mathrm{qo}}+\frac{{\mathrm{DV}}_g}{L}\cos \delta & \frac{1}{L}& \frac{V_g}{L}\sin {\delta }_o& -\frac{R}{L}& {\omega }_g\\ -i_{\mathrm{do}}& 0& -\frac{V_g}{L}\cos {\delta }_o& -{\omega }_g& -\frac{R}{L}\end{array}\right]$
To obtain the equilibrium point, it is noted that the first equation implies E.sub.oi.sub.do=p.sub.in, the second one implies E.sub.oi.sub.qo=−Q.sub.n, the fourth and fifth equations imply E.sub.o−Ri.sub.do+Lω.sub.gi.sub.qo=V.sub.g cos δ.sub.o and Lω.sub.gi.sub.do+Ri.sub.qo=−V.sub.g sin δ.sub.o. Thus,
(E.sub.o−Ri.sub.do+Lω.sub.gi.sub.qo).sup.2+(Lω.sub.gi.sub.do+Ri.sub.qo).sup.2=V.sub.g.sup.2.Math.(E.sub.o.sup.2−Rp.sub.in−Lω.sub.gQ.sub.n).sup.2+(Lω.sub.gp.sub.in−RQ.sub.n).sup.2=V.sub.g.sup.2E.sub.o.sup.2
which leads to the second-order polynomial versus E.sub.o.sup.2. Thus, E.sub.o is calculated according

(65) $E_o=\sqrt{\left(A+\frac{1}{2}{V}_g^2\right)+\sqrt{{\left(A+\frac{1}{2}{V}_g^2\right)}^2-A^2-B}},A={\mathrm{Rp}}_{\mathrm{in}}+L{\omega }_g{Q}_n,B={\left(L{\omega }_g{p}_{\mathrm{in}}-{\mathrm{RQ}}_n\right)}^2.$
When E.sub.o is calculated, other variables are also easily calculated from the above relationships.

(66) Three controller parameters are R, D and k.sub.q. Let us define

(67) $R={\beta }_{1}X\mathrm{and}{k}_q={\beta }_2\frac{{\omega }_g}{3V_g}X.$
These definitions normalize the three controller parameters and facilitates a more general design stage. With this definition, it was shown that 0.25<β.sub.1<1.25 and 0.025<β.sub.2<0.125 result in desirable performances. The damping gain D is already normalized. Here a value between 2 to 5%, 0.02<D<0.05, is a desired selection. FIG. 7A shows the loci of eigenvalues of J when β.sub.1=0.75, β.sub.2=0.075 and D is varied from 0 to 0.05. The strong stabilizing effect of D is clearly observed from the graph. FIG. 7B shows the case where β.sub.1=0.75, D=0.025 and β.sub.2 is varied from 0.025 to 0.125. This parameter mainly shifts the automatic voltage regulator (AVR) real pole to the left side (as expected). However, it will also slightly shift the two other low frequency complex poles to the right. Finally, FIG. 7C shows the case where β.sub.2=0.075, D=0.025 and β.sub.1 varied from 0.25 to 1.25. This is the impact of virtual resistance which pushes all the poles towards the left. However, excessive increase of this parameter will cause a return of two low-frequency complex poles towards the right side. This study concludes a suggested set of values β.sub.1=1, β.sub.2=0.1, D=0.03 that results in the location of poles at −340±j418, −20±j19, −45.

Example 3

(68) A stability analysis of the control system when the output filter is extended to an LCL topology was performed. Similar to the L filter case, differential equations of proposed system of FIG. 6 with LCL filter are obtained as:

(69) $\begin{array}{cc}{\stackrel{.}{\omega }}_m=\frac{1}{C_{\mathrm{dc}}k}\frac{p_{\mathrm{in}}-{\mathrm{Ei}}_d-V_{\mathrm{dmp}}{i}_d}{V_{\mathrm{dc},n}+k\left({\omega }_m-{\omega }_n\right)},\stackrel{.}{E}=k_q\left(Q^{*}-Q\right)=k_q\left[Q_n+k_v\left(V_n-V_g\right)+{\mathrm{Ei}}_q\right],\stackrel{.}{\delta }={\omega }_g-{\omega }_m,{\stackrel{.}{i}}_d=-\frac{R}{L}{i}_d+\frac{1}{L}\left(E+V_{\mathrm{dmp}}\right)-\frac{1}{L}{v}_{\mathrm{cd}}+{\omega }_m{i}_q,{\stackrel{.}{i}}_q=-\frac{R}{L}{i}_q-\frac{1}{L}{v}_{\mathrm{cq}}-{\omega }_m{i}_d,{\stackrel{.}{i}}_{\mathrm{gd}}=\frac{1}{L_g}\left(v_{\mathrm{cd}}-V_g\cos \delta \right)+{\omega }_m{i}_{\mathrm{gq}},{\stackrel{.}{i}}_{\mathrm{gq}}=\frac{1}{L_g}\left(v_{\mathrm{cq}}-V_g\sin \delta \right)-{\omega }_m{i}_{\mathrm{gd}},{\stackrel{.}{v}}_{\mathrm{cd}}=\frac{1}{C}\left(i_d-i_{\mathrm{gd}}\right)+{\omega }_m{v}_{\mathrm{cq}},{\stackrel{.}{v}}_{\mathrm{cq}}=\frac{1}{C}\left(i_q-i_{\mathrm{gq}}\right)-{\omega }_m{v}_{\mathrm{cd}},{\stackrel{.}{V}}_{\mathrm{dmp}}=-\frac{1}{\tau }{V}_{\mathrm{dmp}}-\frac{1}{\tau }{\mathrm{DV}}_g\left({\omega }_g-{\omega }_m\right)\cos \delta .& \mathrm{Eq}.13\end{array}$

(70) The capacitor voltage of LCL filter is v.sub.c and its capacitance is C. The grid side current is i.sub.g and its inductance is L.sub.g. Equation set (13) represents a tenth order nonlinear system with state variables (ω.sub.m, E, δ, i.sub.d, i.sub.q, i.sub.gd, i.sub.gq, v.sub.ed, v.sub.eq, V.sub.dmp). Similar to the L filter analysis, the Jacobian linearization method is used. Assuming that the grid frequency/voltage is at its rated, the equilibrium point is given by (ω.sub.g, E.sub.o, δ.sub.o, i.sub.do, i.sub.qo, i.sub.gdo, i.sub.gqo, v.sub.cdo, v.sub.cqo, 0). To obtain the equilibrium point, the equations imply:

(71) $E_o{i}_{\mathrm{do}}=p_{\mathrm{in}},E_o{i}_{\mathrm{qo}}=-Q_n,\begin{array}{c}{v}_{\mathrm{cdo}}=E_o-{\mathrm{Ri}}_{\mathrm{do}}+L{\omega }_g{i}_{\mathrm{qo}}=E_o-\left({\mathrm{Rp}}_{\mathrm{in}}+L{\omega }_g{Q}_n\right)E_o^{-1}\\ =E_o-{\mathrm{AE}}_o^{-1},\end{array}$$v_{\mathrm{cqo}}=-{\mathrm{Ri}}_{\mathrm{qo}}-L{\omega }_g{i}_{\mathrm{do}}=\left({\mathrm{RQ}}_n-L{\omega }_g{p}_{\mathrm{in}}\right)E_o^{-1}={\mathrm{bE}}_o^{-1},b={\mathrm{RQ}}_n-L{\omega }_g{p}_{\mathrm{in}},v_{\mathrm{cdo}}+L_g{\omega }_g{i}_{\mathrm{gqo}}=V_g\cos {\delta }_o,v_{\mathrm{cqo}}-L_g{\omega }_g{i}_{\mathrm{gdo}}=V_g\sin {\delta }_o,i_{\mathrm{gdo}}=i_{\mathrm{do}}+C{\omega }_g{v}_{\mathrm{cqo}}=\left(p_{\mathrm{in}}+C{\omega }_{g}b\right)E_o^{-1},i_{\mathrm{gqo}}=i_{\mathrm{qo}}-C{\omega }_g{v}_{\mathrm{cdo}}=-C{\omega }_g{E}_o-\left(Q_n-C{\omega }_{g}A\right)E_o^{-1}.Math.{\left(v_{\mathrm{cdo}}+L_g{\omega }_g{i}_{\mathrm{gqo}}\right)}^2+{\left(v_{\mathrm{cqo}}-L_g{\omega }_g{i}_{\mathrm{gdo}}\right)}^2=V_g^2.Math.{\left\{\left(1-L_{g}C{\omega }_g^2\right)E_o-\left[A\left(1-L_{g}C{\omega }_g^2\right)+L_g{\omega }_g{Q}_n\right]E_o^{-1}\right\}}^2+{\left\{\left[\left(1-L_{g}C{\omega }_g^2\right)b-L_g{\omega }_g{p}_{\mathrm{in}}\right]E_o^{-1}\right\}}^2=V_g^2.Math.{\left\{\left(1-L_{g}C{\omega }_g^2\right)E_o^2-\left[A\left(1-L_{g}C{\omega }_g^2\right)+L_g{\omega }_g{Q}_n\right]\right\}}^2+{\left\{\left[\left(1-L_{g}C{\omega }_g^2\right)b-L_g{\omega }_g{p}_{\mathrm{in}}\right]\right\}}^2=V_g^2{E}_0^2.Math.{\left(\gamma E_o^2-A\gamma -L_g{\omega }_g{Q}_n\right)}^2+{\left(\gamma b-L_g{\omega }_g{p}_{\mathrm{in}}\right)}^2=V_g^2{E}_o^2,\gamma =1-L_{g}C{\omega }_g^2.Math.{\left(E_o^2-A-L_g{\omega }_g{Q}_n{\gamma }^{-1}\right)}^2+{\left(b-L_g{\omega }_g{p}_{\mathrm{in}}{\gamma }^{-1}\right)}^2=V_g^2{\gamma }^{-2}{E}_o^2$

(72) Comparing with the similar equation derived for L filter in Example 2 above and its solution, here the solution is

(73) $E_o=\sqrt{\left(\stackrel{\_}{A}+\frac{1}{2}{\stackrel{\_}{V}}_g^2\right)+\sqrt{{\left(\stackrel{\_}{A}+\frac{1}{2}{\stackrel{\_}{V}}_g^2\right)}^2-{\stackrel{\_}{A}}^2-\stackrel{\_}{B}}}$
where
V.sub.g=V.sub.gγ.sup.−1,A=A+L.sub.gω.sub.gQ.sub.nγ.sup.−1,B=(b−L.sub.gω.sub.gp.sub.inγ.sup.−1).sup.2.

(74) By calculating E.sub.O, other variables are obtained from the above relationship. The Jacobian matrix (Γ) is given by:

(75) $\left[\begin{array}{cccccccccc}0& -\frac{i_{\mathrm{do}}}{C_{\mathrm{dc}}k{V}_{\mathrm{dc},n}}& 0& -\frac{E_o}{C_{\mathrm{dc}}k{V}_{\mathrm{dc},n}}& 0& 0& 0& 0& 0& -\frac{i_{\mathrm{do}}}{C_{\mathrm{dc}}k{V}_{\mathrm{dc},n}}\\ 0& k_q{i}_{\mathrm{qo}}& 0& 0& k_q{E}_o& 0& 0& 0& 0& 0\\ -1& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ i_{\mathrm{qo}}& \frac{1}{L}& 0& -\frac{R}{L}& {\omega }_g& 0& 0& -\frac{1}{L}& 0& \frac{1}{L}\\ -i_{\mathrm{do}}& 0& 0& -{\omega }_g& -\frac{R}{L}& 0& 0& 0& -\frac{1}{L}& 0\\ i_{\mathrm{gqo}}& 0& \frac{V_g}{L_g}\sin {\delta }_o& 0& 0& 0& {\omega }_g& \frac{1}{L_g}& 0& 0\\ -i_{\mathrm{gdo}}& 0& -\frac{V_g}{L_g}\cos {\delta }_o& 0& 0& -{\omega }_g& 0& 0& \frac{1}{L_g}& 0\\ v_{\mathrm{cqo}}& 0& 0& \frac{1}{C}& 0& -\frac{1}{C}& 0& 0& {\omega }_g& 0\\ -v_{\mathrm{cdo}}& 0& 0& 0& \frac{1}{C}& 0& -\frac{1}{C}& -{\omega }_g& 0& 0\\ \frac{D}{\tau }{V}_g\cos {\delta }_o& 0& 0& 0& 0& 0& 0& 0& 0& -\frac{1}{\tau }\end{array}\right].$

(76) FIG. 10 shows the loci of system eigenvalues with LCL filter versus changes in the grid inductance. In this figure, L.sub.g changes from 1 mH to 0.1 mH and from 1 mH to 5 mH. It is observed that although L.sub.g changes over a wide range, eigenvalues of the system are not drastically changed and it remains stable. Notice that the decrease in the damping of resonance pole does not cause a problem as it is accompanied with an increase in the resonance frequency [26].

Example 4

(77) A simulation was performed to determine the performance of an embodiment of the eVSM controller. System parameter are presented in Table 1.

(78) TABLE-US-00001 TABLE 1 System parameters Dc link nominal voltage V.sub.dc, n 430 V Dc link capacitor C.sub.dc 880 μF Grid LN voltage (peak) V.sub.g 120√{square root over (2)} V Inverter power rating S 1 kVA Grid frequency f 60 Hz Switching frequency f.sub.s 10 kHz Filter Parameters L.sub.i 5 mH C 2 μF

(79) The simulation of the embodiment of the eVSM controller was conducted under the following conditions: 0<t<1: MPPT operation

(80) $\left(\frac{\mathrm{dp}}{\mathrm{dv}}=0,P=P_{\mathrm{mpp}}=1000W\right)$ t=1:

(81) ${\left(\frac{\mathrm{dp}}{\mathrm{dv}}\right)}^{*}\mathrm{jump}\mathrm{of}-10.$ t=2: grid frequency jump of −0.1 Hz t=3: grid frequency jump of 0.1 Hz t=4: grid magnitude jump of −2.5% t=5: grid magnitude jump of 2.5% 6<t<9: slow grid frequency swing (0.5 Hz frequency, 0.2 Hz peak) 9.5<t<11.5: fast grid frequency swing (5 Hz frequency, 0.2 Hz peak) 10<t<11: input inertia command enabled t=12: islanding (initial resistive load of 700 W) t=13: 100 W load shedding t=14: 255 W/255 Var load addition t=16: pre-synchronization enabled t=18: reconnection to grid

(82) Based on the supplied parameter it was observed as shown in FIGS. 9-11, the controller perfectly responded to the grid and the source transients in a stabilizing manner and settled in the new operating point. Correspondence between the ω.sub.m and V.sub.dc can also be seen in this test. It is also observed that the controller regulates its output power when there is a jump in the input power, while V.sub.dc and ω.sub.m return to their steady-state values prior to this jump. The inertia utilization amplification, the inertia source management stage, distinction between the inertia and governor transients, grid-connected and standalone operations, and seamless transition between these modes were observed.

Example 5

(83) An inventive embodiment of the proposed controller was experimentally validated using a test the grid that was emulated with a Chroma three-phase ac source with a 60V L-N voltage. A buck converter was implanted as the stage I converter, which connected a Chroma PV simulator to the inverter's dc link capacitor. Controllers for both stage I and the proposed eVSM were implemented on a float point Texas Instrument TMS320F28335 DSP. A dc link center voltage was designed to be at 250V, and the rest of the system parameters are as presented in TABLE I. A view of the experimental set up is shown in FIG. 13.

(84) FIGS. 12A and 12B illustrate the results of the performance of the proposed controller during SA, GC, and transition between these two modes of operation. Channels Ch.1 and Ch.2 are inverter currents i.sub.a and i.sub.b (2 A/div), Ch.3 and Ch.4 are inverter phase voltages v.sub.oa and v.sub.ob (100V/div). In FIG. 12A, the inverter is initially supplying local loads in the SA mode, and then the inverter is connected to the grid (GC mode) to inject power from the PV source to the grid and finally in FIG. 12B the inverter is disconnected from the grid (GC mode) and goes back to the SA mode. FIGS. 12A and 12B both show the stable operation of the inventive inverter during both SA and GC and also the smooth transition of the inverter from SA to GC and vice versa, respectively.

(85) Load jump scenarios are presented in FIG. 14, where the load in SA mode jumps from 150 W to 500 W and vice versa. The dc link voltage variations in response to the frequency variations are also shown in FIG. 14.

(86) The results shown in FIGS. 15a-c show the invertor working in GC mode, where the grid frequency jumps in the following sequence: from 60 Hz to 60.1 Hz to 59.9 Hz and back to 60 Hz. It can be observed that the controller reacts to frequency jumps and the active power as well as the dc link voltage change accordingly.

(87) Grid voltage amplitude is also changed by +/−2.5% and results are presented in FIGS. 16a-c. Change in the output reactive power can be observed in these results.

(88) The controller also provides a seamless transient between SA and GC modes of operation. Results in FIG. 17 show the inverter voltage and current during transition from GC to SA mode, where the voltage remains stable and the inverter continues supplying local loads.

(89) While at least one exemplary embodiment has been presented in the foregoing detailed description, it should be appreciated that a vast number of variations exist. It should also be appreciated that the exemplary embodiment or exemplary embodiments are only examples, and are not intended to limit the scope, applicability, or configuration of the described embodiments in any way. Rather, the foregoing detailed description will provide those skilled in the art with a convenient road map for implementing the exemplary embodiment or exemplary embodiments. It should be understood that various changes can be made in the function and arrangement of elements without departing from the scope as set forth in the appended claims and the legal equivalents thereof.

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