Grid-supporting inverters with significantly reduced storage requirements
Patent Information
- Authority / Receiving Office
- CA · CA
- Patent Type
- Patents
- Current Assignee / Owner
- Filing Date
- 2018-09-26
- Publication Date
- 2026-08-04
Abstract
Description
CA 3018726 Date reçue / Received date 2025-04-11 grid faults and grid frequency swings. Current-controlled (CC) inverters can efficiently control current and ride through the grid faults, but current-controlled inverters have issues with operating in standalone mode and / or weak grid conditions due to the fact that their control is directed towards the current (1 ). (0005] Voltage-controlled (VC) inverters based on the concept of droop characteristics, based on the operation principles of synchronous generators (SG ), have also been recently developed for use as inverters (2), (3) to overcome some of the shortcomings of the currentcontrolled inverters. The voltage-controlled inverters generally offer a direct voltage support and work in both grid-connected and standalone modes with minor adjustments (such as addition of a virtual impedance and a self-synchronizing scheme). However, voltage-controlled inverters need additional logics to ensure the inverter current transients, as well as current magnitude during voltage sag conditions remain within acceptable limits. (0006] The high penetration level of renewable energy sources into the existing power system also demands the energy sources static and dynamic behaviors to be supportive and stabilizing to the grid. 1t is commonly stated, for example, that the photovoltaic inverters do not have a physical inertia and do not respond to the grid transients in a stabilizing way similar to a synchronous generator. The dynamic performance of a synchronous generator naturally releases (or absorbs) inertia power during grid transients leading to improved stability. During a transient increase in the grid's active power, for instance, the frequency of the grid active power goes up. This leads to the increase in the rotor speed of the synchronous generator because the frequency of the grid power and the rotor speed arc coupled. This results in the synchronous generator 2 CA 3018726 2018-09-26 absorbing some of the excessive power and storing it as kinetic energy into the rotor mass of the synchronous generator. Conversely, when there is a shortage in the grid's active power and the active power frequency falls, the rotor speed decreases and releases some of its kinetic energy into the grid. In addition to this interesting inertial behavior of the synchronous generator, the governor and exciter control systems of the synchronous generator respond to longer-term transients of the grid in a similar stabilizing way because the governor and exciter simply adopt and modify a similar inertia nature into their logic.
[0007] Due to these aforementioned properties of the synchronous generator, the concept of virtual synchronous machine (under various names such as VSM, VISMA, Synchronverter, etc.) has been introduced (4)-(10) to reproduce or mimic the same dynamics of the synchronous generator in an inverter. The virtual synchronous machine (VSM) method model the inertial effect that uses a virtual inertia concept and builds the concept inside the controller. The inverter that uses a VSM will then reproduce that inertia response. This approach implies that the inverter has access to a source of power that can actually supply that inertia. That source is either a battery storage system (BSS), a large de-link clement, the kinetic energy of the wind turbine, etc.
[0008] In an inverter where a battery storage system or a large de-link element is not present, the existing VSM methods cannot successfully work to their potential. The method of (11) adds a control of the de link voltage for bidirectional dc / ac converters and high-voltage, direct current (HVDC) electric power transmission system applications, while the method (11) still assumes a virtual inertia in the contr'oller. This is a contradictory approach since it is not clear whether the de voltage controller detennincs the inertia response or the VSM controller, and these two 3 CA 3018726 2018-09-26 controllers will compete and oscillate against each other and lead to instability.
[0009] Implementing a virtual synchronous machine for an inverter (without a battery storage system or a large capacitor) requires a de bus voltage controller coupled with an inertia controller that reflects the physical constraints (such as its size and voltage) associated with this de clement. This is clear by considering the system block diagram as shown in prior art FIG. 1. FIG. 1 shows single-line diagram of a three-phase double-stage system ( 12). The Stage I normally harvests the maximum power called maximum power point tracking (MPPT). Stage 11 is an invetier to harvest the energy from the intermediate de bus capacitor, Cctc, and deliver the energy to supply a local load and / or to be injected into the grid.
[0010] As described briefly above, in general there arc two different approaches to design control systems for inverters namely cunent control and voltage control methods. In current control methods, a cuncnt control loop based on a feedback of the output current is used where, the reference for the loop is formed based on a phase locked loop (PLL) that operates on the output voltage. In voltage control methods, the reference for the voltage is generated based on either the droop principles or virtual synchronous machine (VSM) approaches. In these methods, additional internal loops may or may not exist. (0011 I FIG. 2C illustrates a prior art block diagram of a commonly used inverter current control method (13 ). Converter of Stage I regulates the input power (Pin) and forwards the input power to the intennediate bus capacitor. A proportional integral (PI) controller regulates the de bus voltage and generates the active current reference value ( / ~). Reactive current reference U;) in this configuration is determined by the reactive power control (QC) block which operates based 4 CA 3018726 2018-09-26 on the grid voltage amplitude ( Vg) and the reactive power nominal value (Qn). These reference currents are then controlled by the current controller (CC) block. The current controller could be implemented with two well-known methods, as shown in expanded view prior a11 FIG. 2A (in rotating frame) and in expanded view prior art FIG. 2B (in stationary frame). In rotating or DQ frame, shown in prior art FIG. 2A, grid current i is transformed into the DQ frame and proportional integral controllers are used to generate the inverter reference voltage e*. Feed forward and decoupling terms arc also shown in this figure which help for better dynamic performance of the controller. In stationary frame, shown in prior art FTG. 28, on the other hand, , the reference signal is generated using sin (0) = [sin(0) sin(0 - zrr) sin(0 + zrr)] and 3 3 cos (0) = -[cos(0) cos(0 - zrr) cos(0 + zrr)], which are the in-phase and quadrature-phase 3 3 signals of the grid voltage from the PLL. The grid current is then regulated using proportional resonant (PR) controllers to generate the output reference voltage c •. In three phase systems, the control signal e* is then divided by half of the de link nominal voltage (O.SVdc,n) forming the modulation index that is used to generate the gating signals for the inverter switches. Although the block diagrams of prior art PIGS. 2A-C are drawn for a three-phase system application, similar controller structures can be used to control a single-phase double stage inverter (14 ), (15). The inverter controllers of prior a1t PIGS. 2A-C are limited to operate the inverter in grid-connected (GC) mode because the inverter controllers need an already established voltage.
[0012] FIG. 2D illustrates a prior art block diagram of a commonly used inve1ter voltage control method. Voltage control approaches can be categorized into two wide groups namely direct droop-based methods and virtual synchronous machine (VSM) methods. In the direct droop 5 CA 3018726 2018-09-26 method the voltage reference is generated based on the amplitude and frequency which are obtained from w = Wn + kp(Pn - P) and V = Vn + kq(Qn - Q), where Wn and Vn are the nominal grid voltage frequency and amplitude, Pn is the nominal active power, and kp and kq are the active and reactive power droop coefficients. Active power (P) and reactive power (Q) are averaged values which are calculated from the output voltage and current signals and low-pass filters. Close correspondence between this method and VSM approach is shown in (16). [0013) The use of synchronous generators (SG) is the well-established in the existing power systems, where the natural and controlled properties of synchronous generators have been extensively studied. The technically attractive properties of the SGs may be summarized as follows. i) SG is primarily a voltage source and can establish a voltage. ii) mechanical speed and electrical frequency are directly coupled and are synchronized. This leads to the possibility of interconnecting SGs to form a grid with a stable frequency. iii) Kinetic energy of the rotor acts as a stabilizing inertia during transients and disturbances. iv) Droop-based governor and excitation (or automatic voltage regulator (A YR)) control systems make the SG respond efficiently to output power variations while maintaining the frequency and voltage within practically acceptable limits. Due to these properties, the concept of VSM has been developed (in slightly varying formats) in order to furnish an inverter with appropriate control systems such that the inverter can display the virtual synchronous machine (VSM) or synchronous generator (SG) prope1ties during operation. Ideally, a VSM can also improve stability margins of the SG thanks to software and thus flexible implementation of dampers. [0014) A circuit diagram and an equivalent model of the SG is presented in prior art FIGS. 6 CA 3018726 2018-09-26 3A and 3B, respectively. According to a rotor speed ( or frequency) drooping characteristics, the governor adjusts the flow of input mechanical power into the machine. The SG converts this power to an output electrical power. According to a voltage magnitude droop characteristics, the exciter adjusts the current into the filed winding of the machine, which in turn adjusts the delivered reactive power of the machine. The internally induced voltage in FIG. 3B is shown by e with magnitude E and angle 0, where 0 = n; 0m and np is the number of pole pairs and 0m is the mechanical angle of the rotor. The damper windings are located on the rotor to stabilize the SG during transients and disturbances. In a simplified form, governing electromechanical (also called the swing) equation of an SG is expressed as Eq. l where J denotes the moment of inertia of the rotor mass, Tm and Te the mechanical and electrical torque on the machine shaft, Wg the grid angular frequency, and Wm the rotor electrical speed (a 2-pole SG is considered for simplicity). The term D'( Wm - wg) in this equation is a simplified model of the damper winding. Considering Pin and Pout as the input and output powers and noting (rm - re)Wm = Pin - Pout, (Eq. I) can be expressed as Jw ~~'A+ D(w -OJ ) = p. -p Eq. 2 m dt m g m ont where Pdmp = D ( Wm - wg) models the damper windings in terms of power instead of torque (17), which is the common way of modeling the damper windings without engaging the circuit level equations of those windings. As shown in prior art FIGS. 3A and 3B, (J)m is the rotor speed and the frequency of the SG's internal voltage e. The rotor speed, (J)m, becomes equal to (J)g in the 7 CA 3018726 2018-09-26 steady state in a stable grid. The interaction between the internal voltage and the grid voltage via the SG's total equivalent inductance generates the output current.
[0015] There are two different type of dynamics in the swing equation (2). The input power Pin is adjusted by the governor control system whose bandwidth is in the range of 0.3-3.0 Hz and the inertia behavior is represented by the term / Wm Wm whose dominant frequency range covers 3-30 Hz (18). The governor response is proportional to Wn - Wm and the inertia response is prop01tional to - Wm. As shown in prior att FIGS. 4A and 4B, the inertia power and the governor power will behave differently in response to a grid frequency swing at different rates. Since the governor is slow, the response of the governor to low-frequency swings is more dominant than the natural inertia response of the rotor as shown in prior att FIG. 4A. Conversely, when a quick swing happens, the rotor inertia response is more dominant than the governor response as shown in prior art FIG. 413. It is worthwhile noting that the distinction between these two behaviors must be taken into consideration when a VSM intends to fully mimic a synchronous generator in order to be able to be fully compatible with the existing grid and be smoothly integrated.
[0016] FIG. 3C illustrates a typical prior art model of the vi1tual synchronous machine (VSM) control approach to supplying power to a grid. It is clearly observed that the VSM copies the synchronous generator (SG) principles to a large extent. The reactive power control (QC) block in the VSM is equivalent to the Exciter of an SG that controls the internal voltage amplitude. There are critical factors and aspects related to the damping and inertia properties of existing VSMs which illustratively include the following:
[0017] 1) The damper winding is not easily emulated. The point denoted by cox needs to be 8 CA 3018726 2018-09-26 exactly equal to the grid frequency. In (8), the measured grid frequency is used and the interaction between the frequency estimation dynamics and the VSM could be an issue especially in weak grid situations and in fast frequency changing conditions. In (5), the nominal value of the grid frequency is used and thus unlike the actual SG, in this approach the output of the damper winding model generates a steady state power when a frequency offset exists. In (19), a transfer function f(s) is applied to Wm to provide this damping. In (20) a first order low pass transfer function is used for / (s). The advantage of the methods based on Wm is that the methods do not need a grid frequency measurement. However, such methods totally alter the swing equation and the methods must be carefully designed (if possible at all) so as not to alter the desired response properties of the methods. In (9), (10) another approach is used which is equivalent to adding the derivative of power as the damping term. This is similar to what is also presented in (21) and is shown to offer improved damping properties. [0018) 2) The "vi1iual'' moment of ine11ia is given by]. This implies that the system is physically able to supply this level of inertia. In other words, there is an implicit assumption that the system is equipped with adequate storage capacity being in the form of battery storage, capacitor, rotating inertia such as wind or a combination of multiple sources. [0019) 3) For the VSM to accurately mimic a SG, for a fast disturbance in the grid frequency in the frequency range of the inertia, the input power P;n which is controlled by the governor should be almost constant. Otherwise, the dynamics of the input power will interfere with the ine11ia and the system dynamics will deviate from that of the SG. The existing VS Ms appear to overlook this fact and they do not exactly mimic the inertia dynamics of an actual SG as the 9 CA 3018726 2018-09-26 existing VSM do not model the dynamics of the governor.
[0020] 4) The power source can be implemented using a battery where there is no need for an extra de bus control block diagram and there will be no extra dynamics due to the bus capacitor. In this case, the overall VSM behavior is identical to an SG if the governor and damper dynamics are also accurately modeled. However, when a VSM approach is used for a photovoltaic (PV) or other power sources where a two-stage power converter is needed, the de bus controller and bus capacitors have to be designed such that they do not add significant dynamics to the already achieved dynamics of the SG. In some approaches, such as (11 ), (22) a de bus controller is added into the VSM method whose equivalent control block diagram is shown in prior art FIG. 3D. It is clearly observed from prior art FIG. 3D that bus capacitor dynamics are added to the original VSM. This extra dynamic appears as an outer loop with respect to the internal VSM loop. Thus, this outer loop (that is shown in the dotted box A in prior art FIG. 3D) should be much slower so as not to alter the VSM dynamics, and this demands an excessively large capacitor compared to the emulated ine11ia. This setup will completely alter the dynamics of the system and deviates from the original VSM responses if the de-bus capacitor is not "very large".
[0021] Thus, there is a need for improved grid-supporting inverters for integration of distributed and renewable energy sources to a power grid. There is a further need for virtual synchronous machine (VSM) models with an improved frequency response to more closely mimic a synchronous generator. 10 CA 3018726 2018-09-26 SUMMARY OF THE INVENTION
[0022] A method for control of an inverter for implementing a model of a synchronous generator is provided. The method includes implementing a rotor inertia using an intermediate de-link capacitor without duplicating the emulated inettia in the controller, simulating the rotor speed based on a measured capacitor voltage of the intermediate de-link capacitor, while allowing the capacitor voltage to change in a defined range, and mapping the changing capacitor voltage of the intermediate de-link capacitor into the double-stage inverter as an internal frequency. The inverter can be a double-stage inverter. A system for connecting a power generation unit to a power grid is also provided. The system includes 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 steps of implementing a rotor inertia using an intennediate de-link capacitor, simulating the rotor speed based on a measured capacitor voltage of the intermediate de-link capacitor, while allowing the capacitor voltage to change in a defined range, and mapping the changing capacitor voltage of the intermediate delink capacitor into an inve1ier as an internal frequency. BRIEF DESCRIPTION OF THE DRAWINGS
[0023] 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 11 CA 3018726 2018-09-26 description taken in conjunction with the accompanying drawings in which:
[0024]
[0025] FIG. 1 is a block diagram of a prior art double-stage inverter; FIG. 2A is an expanded view of a prior art current controller (CC) block in a rotating frame of FIG. 2C;
[0026] FIG. 2B is an expanded view of a prior art cunent controller (CC) block in stationary frame of FIG. 2C;
[0027] method;
[0028] method; FIG. 2C illustrates a block diagram of a prior art inverter using a current control FIG. 20 illustrates a block diagram of a prior art inverter using a voltage control
[0029] FIG. 3A is a block diagram of a prior art synchronous generator;
[0030] FIG. 3B is a block diagram of an equivalent model of the synchronous generator of FIG. 3A;
[0031] FIG. 3C is a block diagram of a prior art virtual synchronous machine;
[0032] FIG. 3D is a block diagram of a prior art virtual synchronous machine applied to a two-stage converter;
[0033] 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;
[0034] FIG. 5A is a block diagram of a converter and controller in accordance with embodiments of the invention; [0035) FIG. 5B is a block diagram of an overall equivalent model of the converter and 12 CA 3018726 2018-09-26 controller of FIG. SA in accordance with embodiments of the invention; 10036) FIG. 6 is a block diagram of a converter and controller with axillary control blocks in accordance with embodiments of the invention;
[0037] FI Gs. 7 A-C arc a series of graphs showing the loci of eigenvalues oflinearized system J versus three controller gains in accordance with embodiments of the invention;
[0038] FIG. 8 is a block diagram the enhanced virtual synchronous machine (eVSM) applied to a two-stage photovoltaic (PV) system in accordance with an embodiment of the invention
[0039] FIG. 9 is a series of graphs illustrating simulation results comparing ine1tial behavior of the eVSM controller with SG (6<t<11.5) in accordance with embodiments of the invention: [0040) FIG. 10 is a series of graphs illustrating simulation results of the cVSM for input power, grid frequency, and grid amplitude jump (t<6);
[0041] 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<l 8); (0042] FIGS. l 2A 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 [0043) (0044] FIG. 13 is a photograph of an experimental set up; 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 (50ms / div), and where the left side of the graph shows the load increased from 150W to 500W and the right side of the graph shows the load decreased;
[0045] FIGS. 15A-15C illustrate the results of the grid frequency jwnp, with the top of 15A 13 CA 3018726 2018-09-26 and I SB showing inverter LN voltages, the middle of l SA and 15B showing inverter currents, and the bottom of ISA and 15B showing Vdc (100 ms / div). Pmpp =450W and fjumps (15A) left: 60Hz to 60.lHz, right: 60.lHz to 60 Hz, (15B) left: 60Hz to 59.9Hz, right: 59.9Hz to 60Hz, and 15C showing pout during these jumps;
[0046] FlGS. l 6A-l 6C illustrate experimental results for the grid amplitude jump, with the top of 16A and 16B showing inve11er LN voltages, the middle of 16A and 16B showing inverter currents, and the bottom of 16A and 16B showing Vdc (l00ms / 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 l 6C showing output active and reactive power; and 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 (50ms / div). DESCRIPTION OF THE INVENTION
[0047] The present invention has utility as an improved method and system for gridsupporting inverters for integration of distributed and renewable energy sources to a power grid. Unlike existing vi11ual 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 de-link element. The e VSM employs an innovative method for enlarging the inertia utilization range which obviates the need for having a large de-link element or a dedicated battery storage system, while still providing the same inertia response of an equivalent synchronous machine. Embodiments of the eVSM inve11er controller use the 14 CA 3018726 2018-09-26 physical de-link capacitor dynamics for emulating the inertial response rather than relying on a dedicated battery storage or large de element. Thus, embodiments of the cVSM implement the virtual inertia outside the controller using only the de 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.
[0048] Embodiments of the c VSM provide stabilizing suppmt to the grid that is similar to a synchronous machine despite the small size of the de-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.
[0049] Embodiments of the invention provide control of an inve1ter. A simple yet effective method of controlling the inverter is provided that is based on the synchronous generator (SG) model, thereby providing dynamic perfo1mance similar to an SG. In this method, the rotor inertia is implemented using the intermediate de-link capacitor, and rotor speed is mimicked with the capacitor voltage. By allowing the de-link voltage to change within an enlarged yet admissible range and mapping the de-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.
[0050] As shown the existing VSM approach with ideal input power source and with proper 15 CA 3018726 2018-09-26 governor and damper models can reproduce a virtual inertia behavior with the moment of inertia J that is implemented by the integrator 2- inside the VSM controller. However, when the existing Js 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 ine11ia should be a single component similar to the rotor mass in SG. Considering and comparing the two equations 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 e VSM 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.
[0051] A correspondence is established between the couplet U, <um) and (Cctc, Vctc) using a linear transformation as shown in FIG. SA which is expressed by Eq. 4
[0052] 1t is shown below that this choice results in a moment of inertia equal to J = k(ctc.n)Cctc· As shown in FIG. 58, the de link capacitor dynamics becomes identical to that of the Wn rotor inertia in SG, while the rest of the system is identical to the SG, where Wm is used to generate 16 CA 3018726 2018-09-26 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 de bus capacitor. This is in contrast to the existing VSM methods where inertia is implemented inside the controller and duplicates that of the de link capacitor. (0053] It is further noted that as Vctc 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 Vctc value (not a nominal value) to decouple the de link voltage variations from the smooth and sinusoidal output voltage (23). [0054) 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 I to 4 is intended to include 1-2, 1-3, 2-4, 3-4, and 1-4. [00551 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. [0056) Referring now to the figures, FIG. 6 illustrates a block diagram of an inventive 17 CA 3018726 2018-09-26 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 vittual 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 (0057) In embodiments of the inventive eVSM controller, the SG damping tenn in the swing equation (2), Pdmp = D(mm - Wg) may also be added. Unlike the VSM methods, however, the point of applying Pdmp is not available in embodiments of the inventive controller because the inertiagenerating integrator is not inside the controller, but rather in the actual system. Therefore, Pctmp has to be applied differently. As shown in FIG. 6, the damping scheme Pdmp is translated into voltage amplitude, namely Vimr, and is implemented at the output of the QC block. A shown in FIG. 6 Vctmp = -f D :t (c6s0 • vg) = -DVg:t [sin(0 - 0g)] ""DVg(wg - wm)cos(0 - 0g)- Eq. 5 This introduces a damping effect in the voltage magnitude E which in turn translates into Pdmp. It is worth noting that in embodiments of the inventive method the term ( Wm - Wg) is calculated instantaneously with no dynamics without directly measuring or calculating wg. (0058] 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 docs. The adopted 18 CA 3018726 2018-09-26 voltage droop controller is given by Eq. 6 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 Q = -£ cos 0 • i = -Ecos0ia - Ecos(0 - 2 ;)ib - Ecos(0 + 2 3 n)ic and the voltage amplitude Vg is calculated according to VB, -- 4 - 3(VaVb + Vb Ve+ Ve Va) or any other methods to estimate reactive power and voltage magnitude.
[0059] 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), (I 0) is modified and used in embodiments of the inventive method as a cross product of the grid voltage and inverter voltage is added as shown in FIG. 6 where a and fJ indices denote the variables transformed to afJ domain by the Clarke transformation, Vo is the output voltage amplitude, and c / Jg and ¢0 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 Vo to grid voltage Vg prior to connecting to the grid. Prior to the grid connection, due to the c01Tespondence between the active power and Wm, 19 CA 3018726 2018-09-26 the signal Sync may be adjusted so that ¢g and ¢0 become equal. At this time, a grid connection switch can be closed. Subsequently, v0 = Vg and the signal Sync will reset to zero.
[0060] ln inventive embodiments of the eVSM controller, the internal frequency and the delink voltage arc coupled through equation 4. There are two key points to this relationship: (i) the de-link voltage is shaped to follow the grid frequency because the internal frequency has to closely follow the grid frequency, and (ii) the de-link voltage will satisfy ii de = k Ct>m which can be shown to amp I if y the inertia utilization. To clarify the second point, assume that the center delink voltage is Vctc,n = k0wn, Then, the inertia power of the proposed PY system is equal to [00611 where Eq. 9 Equation (9) signifies an ineraa utilization amplification with gain K = k0k. The first gain, k 0, depends on the value of the center de-link voltage Vctc,n· For a 500 V de-link voltage in a 60 Hz system, for instance, k0 is equal to :~~ = 1.33. The second gain, k, depends on the allowable range of swings of the de-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-phasc grid and a center de-link voltage of Vctc,n = 500 V, the physical limitations of the VSC require that the grid peak voltage 120\1"2:::::: 170 V remain below Vdc,n = 250 V. Even if 2 a margin of about 20 V is left for this limit, the de-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 20 CA 3018726 2018-09-26 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.
[0062] 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 pat1ially provided from the input source. Assume that the desired total emulated moment of inertia is ltotal and the amount of the moment of inertia provided by bus capacitor in the proposed controller is J = KC. Therefore, .!total = J + . / in, where .!in is the moment of inertia supply by the input stage. In this case, the desired by adding -Jin mm mm to the reference of the input power.
[0063] Design of synchronization block gains, μ1 and μ2, 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 μ1 and μ2 may be derived as Examples Example I CA 3018726 2018-09-26 Eq. 10 21 [0064) 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 uncc11ainty 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 dp to regulate the PV system dv to ( dp )" + krv ( cvn - cvm). This mechanism will operate the PV at a point slightly on the right side dv of the MPP determined by (dp)n as long as the grid frequency is close to the nominal value. dv Similar to the governor of an SG, this loop will adjust the input power according to a droop characteristic (kpv) with respect to the internal grid frequency variable ( rom), A first PI, with a slow time constant is used to emulate the slow behavior of an SG governor.
[0065] As observed in Pig. 8, the optional input ine1iia term -JinmmdJm can be added to the input power reference Pt" and a fast P 12 controller is used to allow tracking of a power reference up to the desired frequency band of the inertia. Example 2
[0066] A stability analysis of the embodiment of the inventive control system shown in FIG. 6 is performed to obtain system parameters.
[0067] Differential equations of the system shown in PIG. 6 are summarized as follows: 22 CA 3018726 2018-09-26 Eq. 11 where Wac = ~ Cctc VJc, the variables are transformed to the dq-frame using the internal angle 0, 2 and it is assumed that the virtual impedance is equal to R. The damping term is calculated grid voltage dq components are Vi:d = Vg cos b', Vgq = Vg sin o. It is also noted that ~c = CJ / ;i / ~c = Ck[J / ;ic n + k( Wm - (Un )]wm. and substituting in equation 10 gives, Wm = - 1 -[Pin -Eid+ DVgiAwg -wm)coso] t[v.ic n + k(cvm -cvn)l Ck ' ft;= k / Q' -Q) = kq[Qn + kv(Vn -Vg) + 1',\] t =cvg -cvm i = - R i +_!__[E- DV (cv -cv )cos<5]-_!__V cos<5 +cv i d L d L g g m Lg mq d. R. lV.£. ~z =--1 -- sinu-OJ 1. d{ 'I L q L g m d Eq. 12
[0068] Equation set 12 represents a fifth order nonlinear system with state variables ( Wm, E, (), ia, iq). For every given set of values of ( / Jin, Qn), the equilibrium point of the equations may be derived and used in linear stability analysis using the Jacobian lincarization method. Assuming that the grid frequency / voltage is at its rated, i.e., cog= con, Vg = Vn, and the equilibrium point is given by (cog, Ea, Jo, iao, iqo), the Jacobian matrix will be given by 23 CA 3018726 2018-09-26 DV / ,1,, _ ido 0 Eo ('kV COS(),, 0 dc.o Ckv:ic.n CkVdc,n () k, / "o 0 0 k"E0 -I 0 0 0 0 DV V R iqo +--g cos5 L 2-sinoo L O}g L L -ido 0 vg R --cos5 -O}g L o L To obtain the equilibrium point, it is noted that the first equation implies E0ia0 = Pin, the second one implies E0 iqo = -Q11 , the fourth and fifth equations imply E0 - Ria0 + Lwgiqo = Vgcoso0 which leads to the second-order polynomial versus £5. Thus, E0 is calculated according to When Ea is calculated, other variables are also easily calculated from the above relationships.
[0069] Three controller parameters arc R, D and kq- Let us define R = f31X and kq = / 32 wg X. These definitions normalize the three controller parameters and facilitates a more general 3Vg design stage. With this definition, it was shown that 0.25 < {31 < 1.25 and 0.025 < / 3z < 0.125 result in desirable perfonnances. The damping gain D is already nonnalized. Here a value between 2 to 5%, i.e., 0.02 < D < 0.05, is a desired selection. FIG. 7A shows the loci of eigenvalues of] when / 31 = 0.75, / Jz = 0.075 and D is varied from O to 0.05. The strong stabilizing effect of D is clearly observed from the graph. FIG. 7B shows the case where {31 = 24 CA 3018726 2018-09-26 0.75, D = 0.025 and / 32 is varied from 0.025 to 0.125. This parameter mainly shifts the automatic voltage regulator (A VR) 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 {32 = 0.075, D = 0.025 and {31 is 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 / 31 = 1, (]2 = 0.1, D = 0.03 that results in the location of poles at- 340 ± j418, -20 ± j 19, - 45. Example 3
[0070] 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: 25 CA 3018726 2018-09-26
[0071] l Pin -Eid-'\ Jmpid C0_.J.. \'uc.n f / ,·(Wrn-Wn) • 1 / = kq(Q* - Q) = kq [Qn + kv(\r~ - \lg)+ Eiq], t5 = w'g -Wm1 •• u • 1 (E'' \ ~ ) 1 + · ~ti= -T / J T, '· + 'dlllp - 1,Vi:d Wmlq 1 • H · I • 'l4=- 1 ty - 1 rn1 - ~'mlJ. i\!d = -1 1 , (lhl -\~cui-; d)~(.(,•m·ii;q• <... ~ d '- L,1 = -t, (un1 - \·~:--in<i)--..v'mi,.,,, C" ., u ..,. 5u Vn! = ,\ Cid - ·igu) + Wml'cq• 1\q = / . (iq - igy) - u.JmL\d, \ ·•.,J mp -- - rI u~ Jmp - -l; ; 1r. )l1, g' (,:...,v g - w m ) cos· uJ: . Eq. 13 The capacitor voltage of LCL filter is Ve and its capacitance is C. The grid side current is ig and its inductance is Lg. Equation set (13) represents a tenth order nonlinear system with state variables (wm,E,8,id,'lq: 1·gd,igq,Vcd,Vc{1iVctmp)_ 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 (wg, Eo, 8n~ ido, ·iqo, igdo, ·igqo, l'rdn, Vcqo, 0). To obtain the equilibrium point, the equations imply: 26 CA 3018726 2018-09-26 (0072] Comparing with the similar equation derived for L filter in Example 2 above and its solution, here the solution is where [0073) By calculating Eo, other variables are obtained from the above relationship. The Jacobian matrix (f) is given by: 27 CA 3018726 2018-09-26 r 0 t.;.x 0 Ea 0 0 0 0 0 to, (',~U:t, (. ~~ ;, i ·:t,, CJc / .:\-., n () kqiqo I) 0 ~~Eo 0 0 0 0 0 l 0 I.I 0 0 0 0 0 {) 0 lqo I (I fl 0 0 l 0 I L --y: "''g -7: L -J<lo 0 u --(<-'g R 0 0 0 1 0 I -L --y; 0 \ I • - 0 0 0 I lgqo '[";Htl,\. Wg l,., 0 0 - lgcJo 0 l\ ,J I 0'>. c\. , 0 0 -(.tlg 0 0 / ,I. 0 t'.c{JO 0 ll VI 0 -v1 0 0 ~~£ 0 --- l'cd,n 0 0 0 J_ 0 l -Wg 0 0 C} ~c rJ) \·rg COSr5 0 (j 0 0 0 0 0 0 1 0 r
[0074] Figure 10 shows the loci of system eigenvalues with LCL filter versus changes in the grid inductance. Jn this figure, Lg changes from 1 ml-I to 0.1 mH and from I mH to 5 mH. It is observed that although Lg 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
[0075] A simulation was performed to determine the performance of an embodiment of the eVSM controller. System parameter are presented in Table 1. [00761 Table 1 System parameters De link nominal voltage De link capacitor 28 CA 3018726 2018-09-26 v:icn 430V cdc 880μF Grid LN voltage (peak) vg 120✓2 V Inverter power rating s lkVA Grid frequency I 60Hz Switching frequency Is lOkHz L; 5mH Pilter Parameters C 2μF -
[0077] The simulation of the embodiment of the eVSM controller was conducted under the following conditions: • 0<t<l: MPPT operation(::= 0, P = Pmpp = 1000W) • t=l: (::r jump 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 of2.5% • 6<t<9: slow grid frequency swing (0.5 Hz frequency, 0.2 Hz peak) • 9.5<t<l 1.5: fast grid frequency swing (5 Hz frequency, 0.2 Hz peak) • 1 0<t< 11: input inertia command enabled • t=12: islanding (initial resistive load of700W) • t=13: l0OW load shedding • t=14: 255W / 255Var load addition 29 CA 3018726 2018-09-26 • t= 16: pre-synchronization enabled • t=l 8: reconnection to grid [0078) 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 com and Vic 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 Vctc and Wm 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
[0079] 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 de link capacitor. Controllers for both stage I and the proposed eVSM were implemented on a float point Texas Instrument TMS320F28335 DSP. A de link center voltage was designed to be at 250V, and the rest of the system parameters arc as presented in TABLE I. A view of the experimental set up is shown in FIG. 13.
[0080] F1Gs.12A and 12B illustrate the results of the performance of the proposed controller during SJ\, GC, and transition between these two modes of operation. Channels Ch. l and Ch.2 30 CA 3018726 2018-09-26 are inverter currents ia and i& (2A / div), Ch.3 and Ch.4 are inve11er phase voltages Voa and Vob (lO0V / div). In FIG. 12A, the inverter is initially supplying local loads in the SA mode, and then the inverter is co1mccted 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 12D both show the stable operation of the inventive inve11er during both SA and GC and also the smooth transition of the inverter from SA to GC and vice versa, respectively.
[0081] Load jump scenarios are presented in FIG. 14, where the load in SA mode jumps from 150W to 500W and vice versa. The de link voltage variations in response to the frequeny variations are also shown in FIG. 14.
[0082] The results shown in FIGS. 15a-c show the invcrtor working in GC mode, where the grid frequency jumps in the following sequence: from 60Hz to 60.1 Hz to 59.9Hz and back to 60Hz. It can be observed that the controller reacts to frequency jumps and the active power as well as the de link voltage change accordingly.
[0083] Grid voltage amplitude is also changed by + / -2.5% and results are presented in FIGS. l 6a-c. Change in the output reactive power can be observed in these results. (0084] The controller also provides a seamless transient between SA and GC modes of operation. Results in FIG. 17 show the inve11er voltage and current during transition from GC to SA mode, where the voltage remains stable and the inverter continues supplying local loads.
[0085] 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 31 CA 3018726 2018-09-26 CA 3018726 Date reçue / Received date 2025-04-11 Power Systems, IEEE Transactions on, vol. Digital Object Identifier 10.1109 / TPWRS.2014.2306016. (8) F. Gao and M. R. Iravani, "A control strategy for a distributed generation unit in gridconnected and autonomous modes of operation," IEEE Transactions on power delivery, vol. 23, no. 2,pp. 850-859, 2008. (9) M. Karimi-Ghartemani, "Universal integrated synchronization and control for single-phase DC / AC converters," IEEE Transactions on Power Electronics, vol. 30, no. 3, pp. 1544- 1557, 2015. (10) M. Karimi-Ghartemani, S. A. Khajehoddin, P. Piya, and M. Ebrahimi, "Universal controller for three-phase invetters in a microgrid," IEEE Journal of Emerging and Selected Topics in Power Electronics, vol. 4, no. 4, pp. 1342-1353, 2016. (11) M. Ashabani and Y. Abdel-Rady I. Mohamed, "Novel comprehensive control framework for incorporating vscs to smart power grids using bidirectional synchronous-vsc," Power Systems, IEEE Transactions on, vol. 29, no. 2, pp. 943-957, 2014. (12) S. B. Kjaer, J. K. Pedersen, and F. Blaabjerg, "A review of single-phase grid-connected inverters for photovoltaic modules,'' Industry Applications, IEEE 7,-ansactions on, vol. 41, no. 5, pp. 1292-1306, 2005. (13) F. Blaabjerg, R. Teodorcscu, M. Liserre, and A. Timbus, "Overview of control and grid synchronization for distributed power generation systems," Industrial Electronics, IEEE Transactions on, vol. 53, no. 5, pp. 1398-1409, Oct. 2006. (14) M. Ebrahimi, S. A. Khajehoddin, and M. Karimi-Ghartemani, "Fast and robust singlephase dq current controller for smart inverter applications," IEEE Transactions on PmFer Electronics, vol. 31, no. 5, pp. 3968-3976, 2016. (15) S. Khajchoddin, M. Karimi-Gharteman, A. Bakhshai, and P. Jain, "High quality output cunent control for single phase grid-connected inverters," in Applied Power Electronics Conference and Exposition (AP EC), 2014 Twenty-Ninth Annual IEEE. IEEE, 2014, pp. 1807-1814. (16) S. D'Arco and J. A. Suul, "Equivalence of virtual synchronous machines and frequency-droops for converter-based microgrids," IEEE Transactions on Smart Grid, vol. 5, no. 1, pp. 394-395, 2014. (17) -~, ;'Virtual synchronous machines classification of implementations and analysis of equivalence to droop controllers for microgrids," in PowerTech (POWERTECH), 2013 IEEE Grenoble. IEEE, 2013, pp. 1-7. (18) H. Alatrash, A. Mensah, E. Mark, G. Haddad, and .T. Enslin, "Generator emulation controls for photovoltaic inverters," IEEE Transactions on Smart Grid, vol. 3, no. 2, pp. 996-1011, 2012. ( 19) Y. Chen, R. Hesse, D. Turschner, and H.-P. Beck, "Comparison of methods for implementing virtual synchronous machine on inverters," in International Co71ference on Renewable Energies and Power Quality-lCREPQ12, 2012. 33 CA 3018726 2018-09-26 (20) J. A. Suul, S. D' Arco, and G. Guidi, "Virtual synchronous machine-based control of a single-phase bi-directional battery charger for providing vehicle-to-grid services," IEEE Transactions on Industry Applications, vol. 52, no. 4, pp. 3234--3244, 2016. (21) S. Dong and Y. C. Chen, "Adjusting synchronverter dynamic response speed via damping correction loop," IEEE Transactions on Energy Conversion, 2016. (22) D. Chen, Y. Xu, and A. Q. Huang, "Integration of de microgrids as virtual synchronous machines into the ac grid," IEEE Transactions on Industrial Electronics, vol. PP, no. 99, pp 1-1, 2017. (23) S. J\. Khajehoddin, M. Karimi-Ghartemani, P. K. Jain, and A. Bakhshai, "De-bus design and control for a single-phase grid-connected renewable converter with a small energy storage component," IEEE Transactions on Power Electronics, vol. 28, no. 7, pp. 3245- 3254, 2013. 34 CA 3018726 2018-09-26
Claims
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 ( rom) to convert a de input to an ac output, said method compnsmg: implementing the virtual rotor to emulate an inertia (J) of an actual rotor using an intermediate de-link capacitor (Cdc) of the inverter, the implementing comprising: controlling a voltage of the intermediate de-link capacitor (Cdc); simulating a rotor speed of the virtual rotor based on the voltage of the intermediate de-link capacitor (Cdc), while allowing the voltage applied to change in a defined range; and mapping a changing voltage of the intermediate de-link capacitor (Cdc) into the inverter as the internal frequency (rom), where the controller for the inverter does not duplicate the emulated the 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. 35 Date rei,me / Date received 2023-09-26 4. The method of claim 1 further comprising applying a power dampening term Pdmp as a voltage amplitude Vimp of a grid voltage Vg, where that introduces a damping effect into an internal voltage magnitude E which in tum translates into Pdmp.
5. The method of claim 4 wherein a calculated reference Q* is used to derive the internal voltage amplitude E given by sync where the reactive power Q is given by Q = -E cos 0 • i = -Ecos0ia - Ecos(0 - 2n)ib - Ecos(0 + 2n)ic. 3 3 6. The method of claim 1 wherein the method of self-synchronization comprises: calculating a cross product of a grid voltage Vg and an inverter voltage Vo as a signal where a and fJ indices denote variables transformed to a / J domain by a Clarke transformation and Vo is an output voltage amplitude and < / Jg and ¢0 are grid and output voltage phase angles.
7. The method of claim 6 further comprising adjusting the signal Sync to make < / Jg and ¢0 to be equal prior to closing a grid connection switch. 36 Date rei,me / Date received 2023-09-26 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 ( rom) to convert a de input to an ac output, said method comprising: implementing a virtual rotor to emulate an inertia (J) of an actual rotor using an intermediate de-link capacitor (Cdc) of the inverter, the implementing comprising: controlling a voltage of the intermediate de-link capacitor (Cdc); simulating the rotor speed of the virtual rotor based on the measured voltage of the intermediate de-link capacitor (Cdc), while allowing the voltage applied to change in a defined range; and mapping the changing capacitor voltage of the intermediate de-link capacitor (Cdc) into the inverter as the internal frequency ( rom), where the controller for the inverter does not duplicate the emulated the 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. 37 Date rei,me / Date received 2023-09-26 10. The system of claim 8 further comprising applying a power dampening term Pdmp as a voltage amplitude Vimp of a grid voltage Vg, where that introduces a damping effect in to an internal voltage magnitude E which in tum translates into Pdmp- 11. The system of claim 10 further comprising a calculated reference Q* that is used to derive the internal voltage amplitude E given by where the reactive power Q is given by Q = -E cos 0 • i = -Ecos0ia - Ecos(0 - 2n)ib - Ecos(0 + 2n)ic. 3 3 12. The system of claim 8 further comprising providing self-synchronization prior to connecting the inverter to a grid. signal 13. The system of claim 12 wherein the self-synchronization further comprises: calculating a cross product of a grid voltage Vg and an inverter voltage Vo as a sync where a and / J indices denote variables transformed to a / J domain by a Clarke transformation and Vo is an output voltage amplitude and < / Jg and ¢0 are grid and output voltage phase angles. 38 Date rei,me / Date received 2023-09-26 14. The system of claim 13 wherein the self-synchronization further compnses adjusting the signal Sync to make < / >g and < / >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. 39 Date rei,me / Date received 2023-09-26