Inertia sharing control method based on hybrid five-port converter

Abstract

The invention belongs to the technical field of multi-port converter control, and relates to an inertia sharing control method based on a hybrid five-port converter, which realizes overall and inter-phase energy balance through overall and inter-phase inertia sharing control. Inertia support of a sub-module capacitor and a DAB output capacitor is realized through inertia sharing control of an LVDC port. The inertia support of the DAB output capacitor is realized through the inertia sharing control of the sub-module capacitor; the performance of the inertia support is improved through inertia sharing control of DAB output capacitance based on model prediction; power supply to a passive port is realized through voltage and current double-closed-loop control of an MVAC port; according to the method, four inertia sharing controllers and three balance controllers are provided, multi-stage joint control is fully scheduled, the virtual inertia control coefficient is increased, the fluctuation quantity of capacitor voltage is reduced, the system is helped to rapidly transit the transient state, the transient state performance of the system is optimized, and the dynamic performance and the operation boundary of the system are improved.

Description

Technical Field

[0001] The invention belongs to the technical field of multi-port converter control, and relates to an inertia sharing control method based on a hybrid five-port converter. Background Art

[0002] AC and DC power distribution systems under high power background, with the introduction of different types of loads and energy storage subsystems, the continuous complexity of actual scene conditions, the continuous improvement and enrichment of voltage and power levels, and the continuous expansion of the number of ports, when routing energy at different ports, it involves the operation of multiple working conditions. The ports are faced with the problem of flexible power scheduling, and seamless working condition switching is a major difficulty. During the transient transition period, the system will still have large fluctuations and long recovery time, which will maximize the stress of power electronic devices and easily cause system instability. There are problems such as complex working conditions and continuous load switching that have a large disturbance effect on the system. In addition, there are voltage and power imbalances between the modules inside the port. Therefore, improving the ability of the converter to resist transient shocks, realizing system switching and balancing, and designing high dynamic performance control strategies have become the focus of research. There are three mainstream multi-port converter control schemes: 1. Energy expansion control framework; 2. Fluctuating power transfer control framework; 3. Enhanced LVDC port performance control framework.

[0003] The traditional control scheme is to achieve energy balance through overall and bridge arm energy control, and capacitor voltage balance through PI control. PI control based on double frequency circulating current suppression control is used to achieve circulating current suppression control, and PI decoupling control is used to achieve AC port current control. At the LVDC port, the outer loop uses PI control to control the bus voltage, and the inner loop uses master-slave control to achieve current sharing between modules.

[0004] The traditional energy expansion control scheme achieves energy balance through overall, phase-to-phase and bridge-arm energy control. The output of the three controllers is nested together as the reference value of the PIR circulating current suppression controller, and PI decoupling control is used to achieve AC port current control. At the LVDC port, only a single PI control loop is used to control the bus voltage.

[0005] The traditional fluctuating power transfer control scheme uses the same PI control as the traditional control scheme to achieve capacitor voltage balance, and PI decoupling control to achieve AC port current control. The highlight is that at the LVDC port, the outer loop uses PI control to control the bus voltage, and superimposes the phase shift value generated by the fluctuating power of the sub-module capacitor to reduce the capacitor voltage fluctuation value.

[0006] The traditional enhanced LVDC port performance solution achieves energy balance through phase-to-phase and bridge-arm energy control. The two controllers are nested together and the output is used as the reference value of the PI circulating current suppression controller. The outer loop of the MVAC port adopts power control to generate a reference value as the inner loop PI decoupling control to achieve AC port current control. The highlight is that at the LVDC port, dual-loop PIR control is used to achieve bus voltage control and module current sharing.

[0007] However, in the traditional control scheme, the MVAC port is used as the source end, while in the actual high-power converter, the MVDC port is used as the source end. The three-layer energy control scheme will be nested, the control parameters are difficult to adjust, and the dynamic response speed will be greatly reduced. During the transient transition, the system will have large fluctuations and a long recovery time. At the same time, the DAB parallel port uses master-slave current sharing control, and the hardware cost is relatively high. At the same time, this solution is only applicable to the case of one LVDC port and cannot meet the power distribution needs of multiple ports.

[0008] Although the traditional energy expansion control scheme can be applied to multiple LVDC ports, it is similar to the traditional control scheme in that it has nested control and difficult parameter adjustment. Moreover, the scheme does not propose a control strategy for submodule capacitor voltage balancing and DAB module current balancing.

[0009] The traditional fluctuating power transfer control scheme lacks circulating current suppression control and overall and phase-to-phase energy balance control, and cannot achieve energy balance. It is also only applicable to the case of one LVDC port.

[0010] In the traditional solution for enhancing the performance of LVDC ports, the energy controllers between phases and bridge arms are nested, which makes the control complex. The master-slave current sharing control is sampled at the same time, and the number of hardware is large. At the same time, none of the above solutions analyzes and controls the transient state of the system. Multi-port converters still have problems such as serious mutual influence between ports, large fluctuations in the operation of dynamic switching systems, long transient recovery time, and poor dynamic performance of the system.

[0011] Therefore, a control method that can achieve energy coordination, transient switching and sub-module balancing is needed to solve this technical problem. Summary of the invention

[0012] In order to solve the above technical problems, the present invention is specifically implemented through the following technical solutions.

[0013] An inertia sharing control method based on a hybrid five-port converter is proposed. The inertia sharing characteristics between the internal modules and the external ports of the converter are proposed. The control quantity and degree of freedom of the inertia sharing characteristics are analyzed to realize the control of the five-port converter. The control method realizes the energy balance between the overall and inter-phase inertia sharing control, and realizes the transmission and regulation of inertia sharing by circulating current control.

[0014] The inertial support of the submodule capacitor and DAB output capacitor is achieved through the inertial sharing control of the LVDC port;

[0015] The inertial support of the DAB output capacitor is achieved through the inertial sharing control of the sub-module capacitor;

[0016] Improve the performance of inertial support through inertial sharing control of DAB output capacitors based on model prediction, and embed a current sharing controller so that each DAB module transmits the same power to avoid power imbalance;

[0017] Power is supplied to the passive port through dual closed-loop control of the MVAC port voltage and current, and a submodule capacitor voltage balancing method based on sorting is used to reduce the embedding of the PI controller and achieve the balancing of the submodule capacitor voltage;

[0018] The circulating current suppression is realized by combining the double frequency circulating current suppression controller with the PIR controller.

[0019] The inertia sharing characteristics are: 1) the sub-module capacitor can share inertia for the DAB output capacitor; 2) the DAB output capacitor can share inertia for the sub-module capacitor; 3) the MVDC port can share inertia for the sub-module capacitor; 4) the MVDC port can share inertia for the DAB output capacitor; 5) the LVDC port can share inertia for the DAB output capacitor; 6) the LVDC port can share inertia for the sub-module capacitor.

[0020] Preferably, the controller of the overall inertia sharing control in the overall and inter-phase inertia sharing control is:

[0021]

[0022] In formula (13), is the current reference value on the MVDC side, is the average value of the capacitor voltage of all submodules in the multi-port converter, U MVDC / N is the submodule capacitor voltage rating, k p With k i is the control parameter of the overall inertia sharing controller, and s is the integral link.

[0023] More preferably, the controller of the phase-to-phase inertia sharing control in the overall and phase-to-phase inertia sharing control is:

[0024]

[0025] In formula (14), is the reference value of the p-phase circulating current instruction, is the average value of the capacitor voltage of all submodules in the p-phase of the MMC, kp With k i is the control parameter of the phase inertia sharing controller, and s is the integral link.

[0026] Preferably, the controller for implementing the inertia sharing control of the submodule capacitors in the inertial support of the submodule capacitors and the DAB output capacitor is:

[0027]

[0028] In formula (15), Δd pn is the phase shift adjustment value of DAB obtained after passing through the PI controller, u cpn is the capacitor voltage of the nth submodule of phase p, k p With k i is the control parameter of the inertia sharing controller of the submodule capacitor, and s is the integral link.

[0029] Preferably, the controller of the inertia sharing control of the LVDC port is configured in steps as follows:

[0030]

[0031] In formula (16), is the overall reference value of energy storage power, is the main reference value of energy storage power, k p With k i is the control parameter of the inertia sharing controller of the LVDC port, s is the integral link;

[0032] In formula (17), i LVDCp is the LVDC port current, i LVDCpl is the current of the LVDC port load, U LVDCp is the voltage of the LVDC port, k p With k i is the control parameter of the inertia sharing controller of the LVDC port, and s is the integral link.

[0033] Preferably, the inertia sharing control of the DAB output capacitor based on model prediction and the current sharing control in the embedded current sharing controller are:

[0034]

[0035] in

[0036]

[0037] In formula (30), L1 and n DAB1 are DAB leakage inductance and transformer turns ratio, C opn is the DAB output capacitance, f sDABis the switching frequency of DAB, u cpn is the submodule capacitor voltage;

[0038] In formula (31), i LVDCp [k] is the output current of the p-phase LVDC port at time k, is the voltage reference value of the p-phase LVDC port, u opn [k] is the output voltage of the nth DAB of phase p at time k, and N is the number of sub-modules in each bridge arm.

[0039] Preferably, in the scheme of combining a double frequency circulating current suppression controller with a PIR controller to realize circulating current suppression, the double frequency circulating current suppression controller is:

[0040]

[0041] In formula (33), and are the reference values ​​of the d-axis and q-axis components of the circulation respectively; i cird with i cirq are the d-axis and q-axis components of the circulation, respectively, and K p3 , K p4 , K i3 , K i4 is the parameter of the double frequency circulating current suppression PI controller, s is the integral link, ω is the passive network frequency on the AC side of the MVAC port, L m is the bridge arm inductance.

[0042] The PIR controller is:

[0043]

[0044] In formula (34), with i cirp are the p-phase circulating current reference value and p-phase circulating current, K p5 , K i5 is the PIR controller parameter, s is the integral link, K r is the resonant gain, ω c is the bandwidth, ω r is the selected resonant angular frequency.

[0045] Preferably, in the MVAC port voltage and current dual closed-loop control method, the outer loop control adopts voltage feedback control, and the output of the voltage controller is adjusted by comparing the actual voltage with the reference voltage to achieve accurate voltage tracking; at the same time, the i obtained by the outer loop voltage control is vd and i vq As a reference value for current inner loop control and The inner loop control adopts current feedback control. By comparing the actual current with the reference current, the output of the current controller is adjusted to achieve accurate current tracking. The variables are coupled with each other. In order to eliminate the coupling in the dq axis, the controller is designed using feedforward decoupling. The controller is:

[0046]

[0047] In formula (29), and are the reference values ​​of the d-axis and q-axis components of the AC side output current respectively; i d and i q are the d-axis and q-axis components of the AC side output current, respectively, K p1 , K p2 , K i1 , K i2 are the PI controller parameters of the current inner loop, s is the integral link, L eq is the equivalent inductance.

[0048] The beneficial effects of the present invention are:

[0049] 1. The present invention proposes a global inertia sharing control method, designs four inertia sharing controllers and three balancing controllers, fully dispatches multi-level joint control, increases the virtual inertia control coefficient, reduces the fluctuation of capacitor voltage, helps the system to quickly transition to transient state, optimizes the transient performance of the system, and improves the system dynamic performance and operation boundary. At the same time, it reduces the mutual influence of power coupling between ports and circuit instability under unbalanced conditions. This control scheme realizes energy scheduling, transient control and module balancing.

[0050] 2. The inertia sharing control of submodule capacitors in the global inertia sharing control proposed by the present invention clearly analyzes the coupling relationship of energy between modules, reduces capacitor voltage fluctuations, and simplifies or deletes redundant controllers. At the same time, reasonable module balance control is adopted to reduce the number of control parameter adjustments and the number of hardware circuits such as control and sampling.

[0051] 3. The inertia sharing control of the LVDC port in the global inertia sharing control proposed by the present invention increases the freedom of energy control by introducing distributed energy storage, enriches the energy flow mode of the converter, expands the energy operation range of the system, and realizes adaptive smooth switching between various working conditions. At the same time, mode switching does not require reconstruction of the control algorithm, which reduces the control complexity. BRIEF DESCRIPTION OF THE DRAWINGS

[0052] Figure 1 A control block diagram of a global inertia sharing control method in an inertia sharing control method based on a hybrid five-port converter of the present invention;

[0053] Figure 2 It is the topological diagram of the hybrid five-port converter;

[0054] Figure 3 It is a submodule topology diagram of the hybrid five-port converter;

[0055] Figure 4 is the equivalent circuit diagram of inertia sharing mode 1;

[0056] Figure 5 is the equivalent circuit diagram of inertia sharing mode 2;

[0057] Figure 6 is the equivalent circuit diagram of inertia sharing mode 3;

[0058] Figure 7 is the equivalent circuit diagram of inertia sharing mode 4;

[0059] Figure 8 is the equivalent circuit diagram of inertia sharing mode 5;

[0060] Fig. 9 is the equivalent circuit diagram of inertia sharing mode 6;

[0061] Fig.10 It is the inertia sharing superposition equivalent circuit diagram;

[0062] Fig.11 is a schematic diagram of energy transfer of a hybrid five-port converter;

[0063] Fig.12 It is the flow chart of the submodule capacitor voltage balancing algorithm;

[0064] Fig.13 is the power waveform of the MVDC port before and after the inertia sharing control of the LVDC port is applied;

[0065] Fig.14 is the voltage waveform of the a-phase LVDC port before and after the inertia sharing control of the LVDC port is applied;

[0066] Fig.15 It is the capacitor voltage waveform of the a-phase submodule before and after the inertia sharing control of the LVDC port is applied;

[0067] Fig.16 is the capacitor voltage waveform of the b-phase submodule before and after the inertia sharing control of the LVDC port is applied;

[0068] Fig.17 It is the capacitor voltage waveform of the c-phase submodule before and after the inertia sharing control of the LVDC port is applied;

[0069] Fig.18 It is the waveform of system circulating current before and after applying inertia sharing control of LVDC port;

[0070] Fig.19 It is a graph of the THD value of the MVAC port voltage before and after applying the inertia sharing control of the LVDC port;

[0071] Fig. 20 It is a comparison waveform of the capacitor voltage of the a-phase submodule under the submodule capacitor inertia sharing control and the traditional bridge arm energy balance control;

[0072] Fig.21 It is a comparison waveform of the capacitor voltage of the b-phase submodule under the submodule capacitor inertia sharing control and the traditional bridge arm energy balance control;

[0073] Fig. 22 It is a comparison waveform of the capacitor voltage of the c-phase submodule under the submodule capacitor inertia sharing control and the traditional bridge arm energy balance control;

[0074] Fig.23 It is the MVAC port voltage THD value diagram of sub-module capacitor inertia sharing control and traditional bridge arm energy balance control;

[0075] Fig.24 It is a comparison waveform of the voltage transient state of the a-phase LVDC port when DAB adopts three phase-shift control strategies. DETAILED DESCRIPTION

[0076] The following will be combined with the drawings in the embodiments of the present invention to clearly and completely describe the relevant technologies in the present invention. Obviously, the described embodiments are only part of the embodiments of the present invention, not all of the embodiments. Based on the embodiments of the present invention, all other embodiments obtained by ordinary technicians in this field without creative work are within the scope of protection of the present invention.

[0077] The control strategy designed in the present invention is a global inertia sharing control scheme for a hybrid five-port converter based on MMC (Modular Multilevel Converter), DAB (Dual Active Bridge) and energy storage subsystem. First, based on the characteristics of the capacitor, the scheme proposes the inertia sharing characteristics between the internal modules and the external ports of the converter, and analyzes the control quantity and degree of freedom of the inertia sharing characteristics. Then, based on the mathematical model of the converter, four inertia sharing controllers and three balancing controllers are designed for the global inertia sharing control scheme to achieve coordinated control of energy, optimization of the transient performance of the system and balance between sub-modules.

[0078] Based on the analyzed variables for adjusting shared inertia, the present invention proposes a control strategy for global inertia sharing. The control framework is as follows: Figure 1As shown. The overall and inter-phase inertia sharing control is used to achieve overall and inter-phase energy balance, and the circulating current control is used to achieve the transmission and regulation of inertia sharing. The inertia sharing control of the LVDC port is used to achieve inertia support for the sub-module capacitor and the DAB output capacitor. The inertia sharing control of the sub-module capacitor is used to achieve inertia support for the DAB output capacitor. The inertia sharing control of the DAB output capacitor based on model prediction is used to improve the performance of the inertia support, and the current sharing controller is embedded so that each DAB module transmits the same power to avoid power imbalance. The MVAC port voltage and current dual closed-loop control is used to realize power supply to the passive port, and the sorting-based sub-module capacitor voltage balancing method is used to reduce the embedding of the PI controller and realize the balance of the sub-module capacitor voltage. The solution of combining the double frequency circulating current suppression controller with the PIR controller is adopted to achieve circulating current suppression. Figure 1 The markings are as follows: Icons: A-overall and phase-to-phase inertia sharing controller; B-circulating current suppression controller; C-MVAC port voltage and current controller; D-LVDC port inertia sharing controller; E-submodule capacitor inertia sharing control; F-submodule capacitor voltage equalizing controller and PWM generator; G-sequencing-based submodule voltage equalizing control link; H-energy collaborative management controller; I-DAB output current balancing controller; J-DAB output capacitor inertia sharing controller based on predictive control.

[0079] The control strategy designed by the present invention applies the hybrid five-port converter topology as follows: Figure 2 The converter has five ports, namely, MVDC port, MVAC port and three LVDC ports. MVDC 、i MVDC are the voltage and current of the MVDC port respectively; u p 、i p are the phase voltage and phase current of the passive network MVAC port, L T is the filter inductor; U LVDCp 、i LVDCp are the voltage and current of the LVDCp (p = a, b, c) port, i LVDCpl is the load current of the LVDCp port, i bp is the output current of each energy storage port. Each port contains three-phase six bridge arms, i px (x=u,l) is the current of the upper and lower bridge arms, R m , L m They are the bridge arm resistance and bridge arm inductance respectively. The middle of the upper and lower bridge arms is used as the output terminal of the MVAC port to supply power to the passive network. There are 6N submodules SM between each bridge arm. pn (n=1,…,2N), each submodule consists of an MMC half-bridge submodule and a DAB, and the topology is as follows Figure 3 As shown. Among them, Cmpn is the submodule capacitance, u cpn is the submodule capacitor voltage, i cipn is the submodule capacitor current, u Hpn is the equivalent bridge arm voltage of each submodule, i ipn is the equivalent input current of the submodule. MMC and DAB are power coupled through the submodule capacitors. The LVDC port is connected to the MVDC and MVAC ports through DAB as a power channel. The output sides of 2N DAB submodules are connected in parallel to form an input independent output parallel (IIOP) connection mode. DAB1 are DAB leakage inductance and transformer turns ratio, C opn is the output filter capacitor, i copn is the capacitor current, u opn with i opn are the module output voltage and current, i pn1 is the DAB primary input current, i pn2 is the DAB secondary side output current. Figure 2 and Figure 3 The hybrid five-port converter on display compactly links MVDC, MVAC, three LVDC ports with unbalanced power levels, and distributed energy storage. At the same time, it combines the advantages of series and parallel converters to achieve flexible energy flow between ports, improve the operating capacity under unbalanced energy conditions, meet the power requirements of different working conditions, and ensure fast switching of working modes under uninterruptible power supply.

[0080] The present invention proposes the characteristics of circuit inertia sharing for a five-port converter, and constructs a control scheme based on this. When the port power of the multi-port converter changes, since the switching frequency in the half-bridge submodule is much lower than the switching frequency in the DAB, and the port energy mutation is drastic, the input current and output current on both sides of the capacitor three-port network will be inconsistent, and the capacitor will spontaneously output or absorb current to maintain the node energy balance, acting as the inertial support of the system. After designing the circuit parameters, the parameters of the capacitor are determined and unchanged. After the system port power mutation, only the capacitor voltage change amount provides inertial support, and transient power mismatch will occur, which will cause the bus voltage to change drastically.

[0081] According to Kirchhoff's current law, i cipn andi copn Can be expressed as

[0082]

[0083] The natural inertia provided by the submodule capacitor and the DAB output capacitor can be expressed as

[0084]

[0085] Among them, P ipn is the input power of the submodule capacitor three-port network, P pn is the power of DAB transmission, P opn is the power output of DAB.

[0086] It can be seen from formula (2) that the submodule capacitor inside the port and the DAB output capacitor can share inertia. Assuming that the submodule capacitor provides ΔP to the DAB output capacitor pn1 The inertial power support can be expressed as

[0087]

[0088] Among them, C vir1 is the virtual capacitance shared by the submodule capacitance and the DAB output capacitance, u' opn is the output voltage of the nth DAB module of the p-phase with enhanced inertia. From the above formula, it can be analyzed that ΔP pn1 The virtual inertia control coefficient can be increased to reduce the impact of power mutation on the LVDC side, which can be expressed as

[0089]

[0090] Combining formula (3) and formula (4), the inertia sharing of the submodule capacitor to the DAB output capacitor can be intuitively reflected by the equivalent circuit, as follows: Figure 4 shown.

[0091] The inertia of the DAB output capacitor shared with the submodule capacitor can be expressed as

[0092]

[0093] Among them, ΔP pn2 The inertial power support provided by the DAB output capacitor to the submodule capacitor, C vir2 is the virtual capacitance shared by the DAB output capacitor to the submodule capacitor, u' cpn is the capacitor voltage of the nth submodule of the p-phase with enhanced inertia, and the equivalent circuit is as follows: Figure 5 shown.

[0094] In addition, inertial support can be provided between ports to achieve inertial sharing across the entire system. MVDC ports can share inertia to enhance the submodule capacitance inertia, which can be expressed as

[0095]

[0096] Among them, ΔP pn3The inertial power support provided by the MVDC port to the submodule capacitor, C vir3 is the virtual capacitor shared by the MVDC port to the submodule capacitor, d Hpn is the duty cycle of the average model of the submodule. The equivalent circuit is as follows Figure 6 shown.

[0097] The MVDC port can share inertia to enhance the DAB output capacitor inertia, which can be expressed as

[0098]

[0099] Among them, ΔP pn4 The inertial power support provided by the MVDC port to the DAB output capacitor, C vir4 It is the virtual capacitor shared by the MVDC port and the DAB output capacitor. The equivalent circuit is as follows: Figure 7 shown.

[0100] The LVDC port can share inertia to enhance the DAB output capacitor inertia, which can be expressed as

[0101]

[0102] Among them, ΔP pn5 The inertial power support provided by the LVDC port to the DAB output capacitor, C vir5 It is a virtual capacitor shared by the LVDC port and the DAB output capacitor. The equivalent circuit is as follows: Figure 8 shown.

[0103] The LVDC ports can share inertia to enhance the submodule capacitance inertia, which can be expressed as

[0104]

[0105] Among them, ΔP pn6 The inertial power support provided by the LVDC port to the submodule capacitor, C vir6 It is the virtual capacitor shared by the LVDC port to the sub-module capacitor. The equivalent circuit is as follows: Fig. 9 shown.

[0106] According to formulas (3)-(9) and Figure 4-9 It can be seen that the capacitors between ports and inside ports can share a part of the inertia with each other, thereby increasing the virtual inertia control coefficient, reducing the fluctuation of capacitor voltage, and increasing the inertia of the system, such as Fig.10 As shown in the figure, the global inertia sharing can be realized within the circuit, and the transient power can be distributed more efficiently, which is beneficial to improving the transient response capability.

[0107] According to the above analysis, in addition to the inertia sharing provided by the internal capacitor, the MVDC and LVDC ports can provide inertia support for the system transient state. The inertia support corresponds to the converter steady-state and transient energy balance. Therefore, analyzing the inertia sharing control quantity is equivalent to analyzing the energy balance control quantity. Fig.11 As shown in Figure 2, the power absorbed by the multi-port converter is the difference between the power on the MVDC side, the MVAC side, and the LVDC side, which can be expressed as

[0108]

[0109] in, is the amplitude of the AC phase voltage on the MVAC side, is the amplitude of the AC phase current on the MVAC side, It is the phase angle difference between the AC phase voltage and phase current on the MVAC side.

[0110] The power absorbed by the P phase in steady state is the difference between the power on the MVDC side, the p-phase MVAC side, and the p-phase LVDC side, which is

[0111]

[0112] Among them, I cirp,0 is the DC component in the p-phase circulating current. According to formulas (3)-(11), it can be observed that ΔP PET By adjusting U MVDC 、i MVDC , U LVDCp 、i LVDCp To change its size, ΔP p By adjusting U MVDC ,I cirp,0 , U LVDCp 、i LVDCp To change its size. In a multi-port converter system, energy is transferred between any ports. MVDC The value of is constant at the set value. The amplitude of the AC phase voltage on the MVAC side and the amplitude of the AC phase current The LVDC bus voltage of each phase is controlled at a fixed value, and the LVDC current of each phase can be adjusted by controlling the energy storage connected to the bus of each phase LVDC, thereby adjusting the energy. Therefore, it is suitable for adjusting ΔP PET and the variable of shared inertia is i MVDC and i LVDCp , suitable for adjusting ΔP p and the variable of shared inertia is I cirp,0 and i LVDCp .

[0113] At the same time, according to formula (1), it can be observed that the characterization quantity of the shared inertia of the primary and secondary sides of DAB is i pn1 with i pn2 , both can be expressed as

[0114]

[0115] It can be seen from formula (12) that d in DAB control pn This will affect the inertia sharing performance, and thus affect the transient characteristics of the system. Therefore, the variables that the system can adjust the shared inertia can be i MVDC ,I cirp,0 、i LVDCp d pn , the degrees of freedom of transient energy control reach 6N+7.

[0116] The control of the global inertia sharing control scheme designed by the present invention is described in detail below.

[0117] The overall inertia sharing control mainly describes the use of i MVDC To adjust the inertia sharing, since i MVDC The feedback is set to the average value of the capacitor voltage of all submodules in the multi-port converter. The control target is to control the average value to the submodule capacitor voltage rated value U MVDC / N, so U MVDC / N is set as the reference value, and the error between the two is passed through the PI controller to obtain the MVDC side current reference value In this way, the overall inertia sharing of the multi-port converter is adjusted to achieve overall energy balance. Based on this rule, the controller of the overall inertia sharing control is

[0118]

[0119] Phase inertia sharing control mainly describes the use of I cirp,0 To adjust the inertia sharing amount, the PI control used achieves a tracking effect without static error. The feedback is set to the average value of the capacitor voltage of all submodules in the p phase of the MMC. The control target is to control the average value to The error between the two is added after the PI controller One third of the p-phase circulating current command In this way, the inter-phase inertia sharing and energy balance of the multi-port converter are regulated. Based on this rule, the controller of inter-phase inertia sharing control is

[0120]

[0121] Energy coupling is carried out between MMC and DAB through capacitors. The voltage of submodule capacitors has low-frequency fluctuations. Using DAB as a power channel to transmit part of the energy to the LVDC side can effectively reduce the fluctuation of submodule capacitor voltage. When the power on the LVDC side changes suddenly, this part of energy can provide inertial support to ensure the stability of the voltage at the LVDC end. The given and feedback of the inertial sharing control of the submodule capacitor are u cpn and The error is passed through the PI controller to obtain the phase shift adjustment value Δd of the DAB phase shift value. pn , thereby achieving partial low-frequency power transmission and inertia sharing of submodule capacitors. Based on this rule, the controller for inertia sharing control of submodule capacitors is

[0122]

[0123] The variables applicable to adjusting the inertia sharing of multi-port converters are as follows: MVDC and I cirp,0 In addition, after the introduction of energy storage system, i LVDCp It can also be used as an adjustment variable for inertia sharing control. The LVDC port will change from passive to active participation in inertia sharing. The voltage on the LVDC bus is controlled by the DAB connected to the IIOP. The constant power control of the energy storage system is equivalent to constant current control. Therefore, in the mode where the energy storage power is required, different energy storage power reference values ​​are set. The energy storage output current will be controlled according to the reference value, thereby changing i LVDCp The value of can realize the control of energy and inertia sharing. In the mode where there is no demand for energy storage power, set The outer loop adopts PI control, according to U MVDC / N and The difference between the inertia sharing current adjustment and the virtual control coefficient of the inertia sharing are dynamically adjusted to achieve transient fast control. Based on this rule, the controller of the inertia sharing control of the LVDC port can be set in steps as

[0124]

[0125]

[0126] Different DAB controllers pnDifferent from the steady-state transient control effect, the high-performance control method can quickly adjust the inertial support of the remaining circuit parts for the DAB output capacitor. Model predictive control is a control method based on the mathematical model of the converter. It calculates the control input by predicting the future state of the system and updates the control input at each sampling moment. Specifically, model predictive control models the system as a discrete-time dynamic model and uses the model to predict the future system state. Then, based on these prediction results, the optimal control input is determined by solving an optimization problem, thereby achieving the optimal use of Adjust the optimal amount of inertia sharing.

[0127] Let the equivalent output load of each DAB module be R opn According to Kirchhoff's voltage law, the dynamic average model of DAB can be expressed as

[0128]

[0129] For three-phase 2N DABs, 6N dynamic average models can be established. By discretizing the converter dynamic average model using the Euler forward approximation method, the discrete mathematical model of DAB can be obtained, namely:

[0130]

[0131] Among them, u opn [k] represents the output voltage of the nth DAB submodule of the p-phase at time k, i.e., the voltage of the p-phase LVDC port; u opn [k+1] indicates the predicted output voltage of the nth DAB submodule of phase p at time k+1. In the multi-port conversion system, the low-voltage DC port power switching is frequent, and the load parameter changes vary greatly. The equivalent output load included in the discrete model will make the model very inaccurate, and the voltage deviation of the port output voltage will deviate from the reference value. By equating the voltage and load to the output current of each module port, the prediction model can be improved as follows:

[0132]

[0133] In order to control the stability of the low-voltage DC bus voltage, the cost function is considered as

[0134] cost = α1G1 (21)

[0135] in, When the optimal phase shift value is selected, the optimal problem can be solved to minimize the cost function, that is,

[0136]

[0137] The optimal phase shift value can be obtained as

[0138]

[0139] in

[0140]

[0141] MVAC ports often supply power to passive receiving networks in the context of high-power AC / DC hybrid power supply systems, and the load will change according to different working conditions. Since the receiving network frequency is a given value, the rotation speed of the dq conversion shaft is fixed, so there is no need for a phase-locked loop. When the load changes, although the rotation speed of the dq conversion shaft is fixed, u p The angle between the amplitude and the d-axis will change, so it is necessary to control u p The amplitude remains unchanged, so the power at the MVAC port is determined by the load requirement.

[0142] The average model of the MMC submodules is

[0143]

[0144] According to the above formula, U MVDC The middle point of is regarded as the neutral point, and Kirchhoff voltage equations are written for the upper and lower bridge arms respectively:

[0145]

[0146] Among them, u dp is the differential voltage of the upper and lower bridge arms of the p-phase, u cp is the common mode voltage of the upper and lower bridge arms of the p phase, i cirp is the p-phase circulating current, which can be derived from the following formulas:

[0147]

[0148] Let L eq =L T +1 / 2L m and R eq =1 / 2R m Applying abc→dq coordinate transformation to the first equation of formula (26), we can obtain the mathematical model in the dq coordinate system:

[0149]

[0150] Among them, i d with i q are the reference values ​​of the d-axis and q-axis components of the output current on the MVAC side, and u dd with u dq u dp The d and q axis components, u d with u qare the reference values ​​of the d-axis and q-axis components of the output voltage on the MVAC side, and ω is the passive network frequency on the AC side of the MVAC port.

[0151] According to the above analysis, the outer loop control adopts voltage feedback control, which achieves accurate voltage tracking by comparing the actual voltage with the reference voltage and adjusting the output of the voltage controller. vd and i vq As a reference value for current inner loop control and The inner loop control uses current feedback control, which achieves accurate current tracking by comparing the actual current with the reference current and adjusting the output of the current controller. The variables are coupled to each other. In order to eliminate the coupling in the dq axis, a feedforward decoupling controller is used to design the controller. The controller is

[0152]

[0153] in, and are the reference values ​​of the d-axis and q-axis components of the AC side output current respectively; K p1 , K p2 , K i1 , K i2 are the PI controller parameters of the inner current loop.

[0154] For DABs with 2N IIOP connections, under normal operating conditions, the output current of each DAB module should remain equal to achieve equal power sharing. In practical applications, in order to reduce system costs and enhance the flexibility of control algorithms, it is necessary to study control strategies using fewer sensors. For the output current value used in the model, the low-voltage DC bus current value is divided by the number of modules to replace the output current of each module, thereby achieving 2N degrees of freedom control and current sharing control at the same time, that is,

[0155]

[0156] in

[0157]

[0158] According to the control requirements, it is necessary to detect the input voltage, output voltage and parallel output low-voltage DC bus current of each module, so 1 current sensor is required, and 3 current sensors are required for three phases. Compared with master-slave control, the number of current sampling hardware is greatly reduced.

[0159] Since the capacitors of each submodule in the MMC are independent, the differences in the switching time of each submodule, the differences in communication delays, and the differences in component parameters in actual applications will cause inconsistent charging and discharging of the capacitor. If not controlled, the voltage of the submodule capacitor will diverge, making it more difficult to control the DAB converter connected to the subsequent IIOP, distorting the output voltage and current on the MVAC side, and even endangering the safety and service life of the equipment. Therefore, capacitor voltage balancing control is the basis for the stable operation of the multi-port converter system, and its specific implementation process is closely related to the PWM strategy adopted, that is, according to the current submodule capacitor voltage level and combined with the corresponding PWM strategy to change the submodule's input / removal state, thereby adjusting its charging and discharging power to ensure the stable balance of the capacitor voltage.

[0160] The present invention combines this idea with the carrier phase shift modulation strategy to provide a submodule capacitor expansion balancing method based on sorting. First, the capacitor voltages of all submodules in each phase are detected, and the capacitor voltages of each submodule are compared and arranged in descending order by the bubble method. The number of submodules required for the bridge arm and the direction of the bridge arm current are obtained according to the carrier phase shift PWM generator at the current moment, and the submodules required for the bridge arm are selected. The specific selection rules are as follows: Fig.12 As shown. The purpose is to make the submodule with low capacitor voltage put into operation for a longer time when the bridge arm current is positive, so that the capacitor can be charged; conversely, when the bridge arm current is negative, the submodule with high capacitor voltage is put into operation for a longer time. After using phase shift modulation to calculate the number of submodules required, the capacitor voltage balance method in the nearest level approximation is used to select the specific submodules to be put into operation and removed in order to maintain the capacitor voltage balance. Compared with the method of introducing voltage regulation in carrier phase shifting, this method reduces the number of embedded PI controllers and simplifies the difficulty of parameter adjustment.

[0161] Apply abc→d to the second equation of formula (26) -2 q -2 Coordinate transformation, we can get the expression of this formula in the rotating coordinate system:

[0162]

[0163] Among them, i cird with i cirq are the reference values ​​of the d-axis and q-axis components of the circulation, u cd with u cq u cp The d and q axis components.

[0164] The detected three-phase circulating current i cirp Through Park transformation as feedback, Perform coordinate transformation to obtain the given signal The errors of the two are controlled by the PI controller, and the cross-coupling terms of the dq axis current are feedforward compensated. Finally, the circulating current suppression voltage command of each phase of the MMC is obtained by Park inverse transformation. The mathematical model of the circulating current shows that the variables are mutually coupled. In order to eliminate the coupling in the dq axis, the controller is designed using feedforward decoupling. The double frequency circulating current suppression controller is

[0165]

[0166] in, and are the reference values ​​of the d-axis and q-axis components of the circulation respectively; K p3 , K p4 , K i3 , K i4 are the circulating current suppression PI controller parameters.

[0167] The resonant controller is a second-order generalized integrator that can achieve infinite gain at the selected frequency, so it can selectively control a certain frequency harmonic. The PIR controller is:

[0168]

[0169] In formula (34), with i cirp are the p-phase circulating current reference value and p-phase circulating current, K p5 , K i5 is the PIR controller parameter, s is the integral link, K r is the resonant gain, ω c is the bandwidth, ω r is the selected resonant angular frequency.

[0170] In view of the deficiency that the traditional PI control method cannot fully track the periodic AC harmonic components, it is only effective for the double frequency harmonics and cannot suppress other harmonics such as the quadruple frequency and sextuple frequency in the circulating current. The resonant controller needs to set a series of resonant controller parameters, and the debugging is complicated. The present invention combines the traditional PI control method with the resonant controller, uses the PI control as the main controller to generate the most core common mode component, and uses the most basic resonant controller that suppresses the quadruple frequency in the circulating current as the slave controller to generate fine-tuning common mode components, so as to suppress the quadruple frequency harmonics, thereby realizing the zero-static tracking of the DC component in the circulating current.

[0171] In order to verify the effectiveness of the proposed control strategy, the circuit and controller of the hybrid five-port converter were built based on MATLAB / SIMULINK. The specific circuit parameters are shown in Table 1. In a multi-port converter, there is a situation of three-phase energy imbalance, especially when a phase performs rapid load switching, the power, voltage and current characteristics of each port, as well as the voltage of each module and the circulating current in the system will be greatly affected. During the transient period, the huge load switching will have a significant impact on both the classical controller and the controller proposed in this article. The present invention selects the working condition where the a-phase LVDC load suddenly increases from 100kW to 400kW at 0.2s as a comparison to verify the superiority of the proposed control strategy. The specific working condition changes are shown in Table 2.

[0172] Table 1 Main parameters of five-port converter

[0173]

[0174]

[0175] Table 2 Rated power parameters under working conditions

[0176]

[0177] Fig.13 The power waveform of the MVDC port before and after the inertia sharing control of the LVDC port is applied is shown. It can be clearly seen from the figure that after the inertia sharing control of the LVDC port is applied, the transient fluctuation of the MVDC port power is significantly reduced. Fig.14 The LVDC port voltage waveform of phase a before and after the inertia sharing control of the LVDC port is applied is shown. Without the inertia sharing control of the LVDC port, the voltage drops to 980V at 0.2s. Due to the fluctuation of the MVDC port and the sudden change of the submodule capacitor voltage, the voltage continues to drop and oscillate until it stabilizes after 0.08s. After the inertia sharing control of the LVDC port is applied, the voltage drops to 985V at 0.2s, but there is no continuous drop and oscillation, and it enters a steady state after 0.025s.

[0178] Fig.15 The voltage waveform of the a-phase submodule capacitor before and after the inertia sharing control of the LVDC port is applied is shown. When the inertia sharing control of the LVDC port is not applied, the voltage drops to a maximum of 789V, and an overshoot of 1109V occurs when it recovers, and it stabilizes after 0.15s. After the inertia sharing control of the LVDC port is applied, the voltage drops to a maximum of 868V, without overshoot, and stabilizes after 0.12s. Fig.16 and Fig.17The capacitor voltage waveforms of the b-phase and c-phase sub-modules before and after the inertia sharing control of the LVDC port is applied are shown. The maximum sudden increase in the capacitor voltage after the inertia sharing control of the LVDC port is applied is reduced by 42V and 49V respectively.

[0179] Fig.18 The system circulating current waveforms before and after the inertia sharing control of the LVDC port is applied are shown. It can be clearly seen that after the inertia sharing control of the LVDC port is introduced, the fluctuation and oscillation of the circulating current in the transient process are significantly reduced. Fig.19 The THD values ​​of the MVAC port voltage before and after the inertia sharing control of the LVDC port are shown. The maximum THD values ​​before and after the inertia sharing control of the LVDC port are 0.193% and 0.149%, respectively, indicating that when the LVDC port power fluctuates greatly, the introduction of the inertia sharing control of the LVDC port can reduce the impact on the MVAC port. According to the above results, it can be concluded that when the LVDC port power is switched, the impact on the MVDC and MVAC ports is significantly reduced after the inertia sharing control of the LVDC port is applied, which is reflected in the smaller MVDC power fluctuation and lower AC voltage THD value. At the same time, during the transient period, the voltage value mutation of the capacitor that provides natural inertia inside the system is also significantly reduced, the response speed is slightly faster, and the system's ability to withstand transient overshoot is improved, thereby verifying the effectiveness of the proposed transient controller, namely the inertia sharing control of the LVDC port.

[0180] Submodule capacitor inertia sharing control can effectively reduce the fluctuation of submodule capacitor voltage, so that the system has a smaller voltage surge and decrease value during the transient process, and can quickly reach a steady state. At the same time, submodule capacitor inertia sharing control can indirectly balance the power between bridge arms, so that the energy of the upper and lower bridge arms is kept as equal as possible.

[0181] Fig. 20 The comparison waveform of the capacitor voltage of the a-phase submodule under the submodule capacitor inertia sharing control and the traditional bridge arm energy balance control is shown. Using the traditional bridge arm energy balance control, the capacitor voltage fluctuation value is 168V under normal conditions, and the transient voltage drops to a maximum of 778V until it stabilizes after 0.2s. Using the submodule capacitor inertia sharing control, the capacitor voltage fluctuation value is 116V under normal conditions, and the transient voltage drops to a maximum of 868V, and stabilizes after 0.12s.

[0182] Fig.21The comparison waveform of the capacitor voltage of the b-phase submodule under the submodule capacitor inertia sharing control and the traditional bridge arm energy balance control is shown. When using the traditional bridge arm energy balance control, the transient voltage drops to 873V at most and increases to 1159V at most. The upper and lower bridge arm voltages need to undergo a long period of energy modulation before they can enter a steady state. When using the submodule capacitor inertia sharing control, the transient voltage increases to 1079V at most and stabilizes after 0.06s.

[0183] Fig. 22 The comparison waveform of the sub-module capacitor voltage of the phase C under the sub-module capacitor inertia sharing control and the traditional bridge arm energy balance control is shown. When using the traditional bridge arm energy balance control, the maximum transient voltage surges to 1149V, and the upper and lower bridge arm voltages take 0.1s to enter a steady state. When using the sub-module capacitor inertia sharing control, the maximum transient voltage surges to 1111V and stabilizes after 0.07s.

[0184] Fig.23 The MVAC port voltage THD values ​​of submodule capacitor inertia sharing control and traditional bridge arm energy balance control are shown. When using traditional bridge arm energy balance control, the maximum THD value is 0.195%, and when using submodule capacitor inertia sharing control, the maximum THD value is 0.149%, indicating that when the LVDC port power fluctuates greatly, the submodule capacitor inertia sharing control can better reduce the impact on the MVAC port. According to the above results, it can be concluded that under normal working conditions, using submodule capacitor inertia sharing control, the submodule capacitor has a smaller fluctuation value. When the LVDC port power is switched, using submodule capacitor inertia sharing control, during the transient period, the voltage value mutation of the capacitor is significantly reduced, the recovery speed is greatly accelerated, and the system's ability to withstand transient overshoot is improved, thereby verifying the effectiveness of the proposed submodule capacitor inertia sharing control.

[0185] In LVDC, the DAB output capacitor is a key component of shared inertia, and its energy and voltage stability will seriously affect the operation of the entire system. Fig.24The comparative waveforms of the voltage transient of the a-phase LVDC port are shown when the DAB adopts three phase-shift control strategies. It can be observed from the figure that the voltage drop value of the DAB output capacitor inertia sharing control based on master-slave control is the largest, which is 78.9V, and the transient response time is the longest, reaching 35.1ms, and the voltage ripple is the most obvious. The control effect of the DAB output capacitor inertia sharing control based on direct power control is second, with a voltage drop of 44.5V and a transient response time of 24.6ms. The control effect of the DAB output capacitor inertia sharing control based on model predictive control is the best, with a voltage drop of only 18.7V and a transient response time of 13.6ms. According to the above results, it can be verified that the DAB output capacitor inertia sharing control based on model predictive control is more superior in the case of load mutation.

[0186] In summary, the present invention uses four inertia sharing controllers and three balancing controllers to fully dispatch multiple levels for joint control, increase the virtual inertia control coefficient, reduce the fluctuation of capacitor voltage, help the system to quickly transition to a transient state, optimize the transient performance of the system, and improve the system dynamic performance and operating boundaries. At the same time, it reduces the mutual influence of power coupling between ports and circuit instability under unbalanced conditions. This control scheme realizes energy scheduling, transient control and module balancing, so the present invention has broad application prospects.

[0187] It should be emphasized that the above are only preferred embodiments of the present invention and do not limit the present invention in any form. Any simple modifications, equivalent changes and modifications made to the above embodiments based on the technical essence of the present invention are still within the scope of the technical solution of the present invention.

Claims

1. An inertia sharing control method based on a hybrid five-port converter, Features: The overall and inter-phase inertia sharing control is used to achieve overall and inter-phase energy balance, and the transmission and regulation of inertia sharing is achieved through circulation control; The inertial support of the submodule capacitor and DAB output capacitor is achieved through the inertial sharing control of the LVDC port; The inertial support of the DAB output capacitor is achieved through the inertial sharing control of the sub-module capacitor; Improve the performance of inertial support through inertial sharing control of DAB output capacitors based on model prediction, and embed a current sharing controller so that each DAB module transmits the same power to avoid power imbalance; Power is supplied to the passive port through dual closed-loop control of the MVAC port voltage and current, and a submodule capacitor voltage balancing method based on sorting is used to reduce the embedding of the PI controller and achieve the balancing of the submodule capacitor voltage; The circulating current suppression is achieved by combining a double frequency circulating current suppression controller with a PIR controller.

2. An inertia sharing control method based on a hybrid five-port converter, It is characterized in that The controller of the overall inertia sharing control in the overall and phase inertia sharing control is: In formula (13), is the current reference value on the MVDC side, is the average value of the capacitor voltage of all submodules in the multi-port converter, U MVDC / N is the submodule capacitor voltage rating, k p With k i is the control parameter of the overall inertia sharing controller, and s is the integral link.

3. The inertia sharing control method based on a hybrid five-port converter according to claim 2, It is characterized in that The controller of the phase-to-phase inertia sharing control in the overall and phase-to-phase inertia sharing control is: In formula (14), is the reference value of the p-phase circulating current instruction, is the average value of the capacitor voltage of all submodules in the p-phase of the MMC, k p With k i is the control parameter of the phase inertia sharing controller, and s is the integral link.

4. The inertia sharing control method based on a hybrid five-port converter according to claim 1, It is characterized in that The controller for implementing the inertia sharing control of the submodule capacitors in the inertial support of the submodule capacitors and the DAB output capacitors is: In formula (15), Δd pn is the phase shift adjustment value of DAB obtained after passing through the PI controller, u cpn is the capacitor voltage of the nth submodule of phase p, k p With k i is the control parameter of the inertia sharing controller of the submodule capacitor, and s is the integral link.

5. The inertia sharing control method based on a hybrid five-port converter according to claim 1, It is characterized in that The controller for the inertia sharing control of the LVDC port is set up step by step as follows: In formula (16), is the overall reference value of energy storage power, is the main reference value of energy storage power, k p With k i is the control parameter of the inertia sharing controller of the LVDC port, s is the integral link; In formula (17), i LVDCp is the LVDC port current, i LVDCpl is the current of the LVDC port load, U LVDCp is the voltage of the LVDC port, k p With k i is the control parameter of the inertia sharing controller of the LVDC port, and s is the integral link.

6. The inertia sharing control method based on a hybrid five-port converter according to claim 1, It is characterized in that The inertia sharing control of the DAB output capacitor based on model prediction and the current sharing control in the embedded current sharing controller are: in In formula (30), L 1 With n DAB1 are DAB leakage inductance and transformer turns ratio, C opn is the DAB output capacitance, f sDAB is the switching frequency of DAB, u cpn is the submodule capacitor voltage; In formula (31), i LVDCp [k] is the output current of the p-phase LVDC port at time k, is the voltage reference value of the p-phase LVDC port, u opn [k] is the output voltage of the nth DAB of phase p at time k, and N is the number of sub-modules in each bridge arm.

7. The inertia sharing control method based on a hybrid five-port converter according to claim 1, It is characterized in that In the scheme of combining the double frequency circulating current suppression controller with the PIR controller to realize circulating current suppression, the double frequency circulating current suppression controller is: In formula (33), and are the reference values ​​of the d-axis and q-axis components of the circulation respectively; i cird with i cirq are the d-axis and q-axis components of the circulation, respectively, and K p3 , K p4 , K i3 , K i4 is the parameter of the double frequency circulating current suppression PI controller, s is the integral link, ω is the passive network frequency on the AC side of the MVAC port, L m is the bridge arm inductance; The PIR controller is: In formula (34), with i cirp are the p-phase circulating current reference value and p-phase circulating current, K p5 , K i5 is the PIR controller parameter, s is the integral link, K r is the resonant gain, ω c is the bandwidth, ω r is the selected resonant angular frequency.

8. The inertia sharing control method based on a hybrid five-port converter according to claim 1, It is characterized in that In the voltage and current dual closed-loop control method of the MVAC port, the outer loop control adopts voltage feedback control, and the output of the voltage controller is adjusted to achieve accurate voltage tracking by comparing the actual voltage with the reference voltage; at the same time, the i obtained by the outer loop voltage control is vd and i vq As a reference value for current inner loop control and The inner loop control adopts current feedback control. By comparing the actual current with the reference current, the output of the current controller is adjusted to achieve accurate current tracking. The variables are coupled with each other. In order to eliminate the coupling in the dq axis, the controller is designed using feedforward decoupling. The controller is: In formula (29), and are the reference values ​​of the d-axis and q-axis components of the AC side output current respectively; i d and i q are the d-axis and q-axis components of the AC side output current, respectively, K p1 , K p2 , K i1 , K i2 are the PI controller parameters of the current inner loop, s is the integral link, L eq is the equivalent inductance.

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