Multivariate model optimisation during the implementation of flexible interconnection projects for MV distribution networks

Most of the distribution network adopts “closed-loop design, open-loop operation”, using contact switches to form a hand-in-hand or mesh distribution network in each sub-district to realize sectional control and post-accident load transfer. However, as the electrification of urban and rural areas continues to increase, the load on the distribution network also increases, and the daily peak-valley difference gradually expands. At the same time, the penetration rate of distributed power supply and charging pile is gradually increasing, and the problems of new energy consumption, load gap, and power quality are becoming more and more prominent. The traditional distribution network adopts the “closed-loop design, open-loop operation” mode, which cannot realize the trend control, and the low voltage at the end of the line and other problems are increasingly visible.

The flexible interconnection device (FID) based on a power electronic device (PED) can realize the closed-loop interconnection between different AC divisions. By controlling the current of the DC link, the power equalization between different AC sub-divisions can be accomplished without considering the frequency synchronization problem. Compared to distribution network sectionalized switches and contact switches, FID technology makes fault isolation easier, which greatly improves the flexibility and safety of distribution network control. When FID technology is used for feeder interconnection of MV distribution network, asynchronous loop operation between multiple AC distribution feeders can be accomplished without increasing the short-circuit capacity, which can play the roles of trend transferring, load ratio balancing, network loss reduction, fault isolation and power supply restoration, etc., and realize the multi-terminal loop operation of feeders and optimal allocation of adjustable resources.

The flexible interconnection system of power supply sub-districts is shown in Fig. 1, in which two neighboring power supply sub-districts realize real-time interconnection of two regional power supply networks through flexible interconnection devices. The flexible interconnection device consists of two bi-directional AC/DC converters and realizes the flexible control of power supply inter-area currents by means of intelligent fusion terminals. Combined with the distribution master system, it realizes the optimized coordination of source, network, load and storage of the interconnection system, which improves the reliability and flexibility of the power supply of the distribution network system. Similarly, this technology can be applied to low-voltage flexible DC interconnection between two stations to solve operational problems such as difficulty in capacity increase of urban power grid, uneven load ratio of neighboring stations and low terminal voltage.

Figure 1.

The power supply zoning soft interconnection system

The flexible interconnection device is the core equipment of the flexible DC interconnection system for power supply zones, and compared with the conventional contact switch, it has the functions of active scheduling, reactive power support, islanding power supply, load transfer and so on. Flexible interconnection devices have two or more access ports to meet the flexible interconnection of lines of different wiring modes, while DC access ports can be set up to meet the needs of accessing distributed power sources, DC loads and energy storage devices.

According to different distribution grid interconnection patterns, combined with the differences in AC system operation characteristics, as well as the concerns of each application scenario for specific needs such as high power supply reliability, load rate balance control, distributed power consumption and fluctuating power suppression, the functional modes of the distribution flexible interconnection system also differ.

For different application scenarios, the typical functional modes of various flexible interconnection systems can be summarized into the following categories:

Feeder load rate equalization

According to the requirements of monitoring and controlling the transmission power of the FIDVSC connected to the AC feeder, thus realizing the equalization of the load rate of each feeder. The average feeder load factor L_avg is calculated as follows: (1) Lavg=∑i=1nPt..i∑i=1nPT.ii=1,2,⋯,n\[\begin{matrix} {{L}_{\text{avg}}}=\frac{\sum\limits_{i=1}^{n}{{{P}_{\text{t}..i}}}}{\sum\limits_{i=1}^{n}{{{P}_{\text{T}.i}}}} & i=1,2,\cdots ,n \\ \end{matrix}\]

where n is the number of flexible interconnection system ends. P_L.i is the real-time load value accessed by AC feeder i. P_T,i is the rated power value of AC feeder i.

For the purpose of average load factor, the FID-VSC connected to AC feeder i needs to inject the active command value P_VSC,i to the AC feeder as: (2) PVSC,i=P1,i−LavgPT,i\[{{P}_{VSC,i}}={{P}_{1,i}}-{{L}_{avg}}{{P}_{T,i}}\]

The islanding effect in distribution networks can seriously threaten the life safety of distribution network operations and inspection personnel. Most of the existing anti-islanding strategies need to be specially configured for disturbed loads according to different sensitivity requirements, reliability requirements, and capacities, which will not only result in a waste of resources but also consume manpower.

In order to solve the above problems, this section proposes an anti-islanding strategy with self-adaptive capability for distribution networks based on back-to-back voltage-source converter (B2B-VSC) based on flexible interconnection engineering. The structure of the B2B VSC is shown in Fig. 2, and U_DC is the DC intermediate level voltage of the B2B-VSC. C is the DC intermediate capacitance, L_t is the filter inductance, C_t is the filter capacitance, and R_d−ini is the initial disturbance resistance of the anti-islanding device. I˙0\[{{\dot{I}}_{0}}\] is the output current of the B2B-VSC, U˙o${{\dot{U}}_{o}}$ is the output voltage of the B2B-VSC. U˙ d${{\dot{U}}_{~d~}}$ is the voltage across the initial perturbation resistor. U˙ s${{\dot{U}}_{~s~}}$ is the overall two-terminal voltage of B2B-VSC and R_d−ini.

Figure 2.

Basic structure of adaptive low-voltage anti-islanding strategy

The B2B-VSC can output a specified voltage phase U˙ d${{\dot{U}}_{~d~}}$ according to the control strategy of the inverter stage, thus changing the amplitude and phase angle of U˙ d${{\dot{U}}_{~d~}}$. As a result, the power value of the disturbed load unit, which is composed of B2B-VSC together with R_d−ini, will be controllable, and it can be presented as either purely resistive or resistive-inductive and resistive-capacitive.

If the undervoltage protection alone is used to realize the low-voltage anti-islanding strategy, the perturbation load unit needs to exhibit a purely resistive characteristic with a phase relationship that indicates the equivalent load value of the perturbation load unit R_d-eq < R_d-ini, for higher DG capacity or lower sensitivity requirements. R_d-eq > R_d-ini, for lower DG capacity or higher sensitivity requirements.

In the conventional low-voltage anti-islanding strategy, if undervoltage protection is used, the perturbed load configuration is modeled as: (9) Rd=Un2Pinv(UnUn−U−1)\[{{R}_{d}}=\frac{U_{n}^{2}}{{{P}_{inv}}}(\frac{{{U}_{n}}}{{{U}_{n}}-U}-1)\]

Where, R_d is the desired configuration of the disturbed load, U_n is the rated voltage of the bus, P_inv is the DG capacity, and U is the set undervoltage threshold.

The equivalent load value of the perturbation load unit if the B2B-VSC based adaptive anti-islanding strategy is used: (10) Rd-eq=(220V)2|U˙d|×|I˙0|=(220V)2×Rd-ini|U˙s−U˙o|2\[{{R}_{d-eq}}=\frac{{{(220V)}^{2}}}{|{{{\dot{U}}}_{d}}|\times |{{{\dot{I}}}_{0}}|}=\frac{{{(220V)}^{2}}\times {{R}_{d-ini}}}{|{{{\dot{U}}}_{s}}-{{{\dot{U}}}_{o}}{{|}^{2}}}\]

It should be noted that since the active power consumed/emitted at the output of the B2B-VSC is equal to the power emitted/consumed at the input, the active power of the B2B-VSC is constant at 0, and the active power of the perturbed load cell includes only the power dissipation of R_d-ini.

According to Eq. It can be seen that the resistance value of R_d-eq can be controlled by adjusting the amplitude of U˙o${{\dot{U}}_{o}}$, thus adapting to different anti-islanding under-voltage protection sensitivity requirements (different under-voltage thresholds), as well as different DG capacities. Therefore, with the use of the B2B-VSC-based adaptive anti-islanding strategy, the disturbed load no longer needs to be reconfigured when the sensitivity requirement or the DG capacity changes.

If, in order to improve reliability, Undervoltage protection and over/underfrequency protection are used to realize low-voltage anti-islanding, the disturbing load unit needs to present resistive-inductive (dual protection of undervoltage and overfrequency) or resistive-capacitive characteristics (dual protection of undervoltage and underfrequency). It should be noted that, due to the external load quality factor Q_f has a certain impact on the reliability of over/under-frequency protection, the larger the Q_f over 1 under-frequency protection, the lower the reliability. Therefore, in this section, under-voltage protection is selected as the basic protection, over 1 under-frequency protection as additional protection, so as to effectively improve the reliability of the anti-islanding. Where θ is the phase of U˙o${{\dot{U}}_{o}}$ over I˙o${{\dot{I}}_{o}}$.

In the traditional low-voltage anti-islanding strategy, if Undervoltage and over-frequency double protection are used, the resistance configuration model of the disturbed load is the same as equation (1), and the inductive resistance configuration model of the disturbed load is (11) XL=U02f02πQL(fH2−f02)\[{{X}_{L}}=\frac{U_{0}^{2}{{f}_{0}}}{2\pi {{Q}_{L}}(f_{H}^{2}-f_{0}^{2})}\]

Where X_L is the inductive reactance value of the disturbed load to be configured, U₀ and f₀ are the bus voltage and the output frequency of the PV system during islanding operation, respectively. Q_L is the inductive reactive power of the local load. f_H is the set over-frequency threshold.

In the traditional low-voltage anti-islanding strategy, if undervoltage and underfrequency double protection are used, the perturbed load resistance configuration model is the same as above, and the perturbed load capacitive reactance configuration model is: (12) Xc=f0Qc2πfL2U02−Qc2πf02U02\[{{X}_{c}}=\frac{{{f}_{0}}{{Q}_{c}}}{2\pi f_{L}^{2}U_{0}^{2}}-\frac{{{Q}_{c}}}{2\pi f_{0}^{2}U_{0}^{2}}\]

Where, X_c is the value of perturbed load capacitive reactance to be configured, Q_c is the local load capacitive reactive power, and f_L is the set underfrequency threshold.

The equivalent load impedance of the perturbation load unit if the B2B-VSC based adaptive anti-islanding strategy is used: (13) Zdeq=(220V)2×Rd-ini|U˙s−U˙0|2+j(220V)2×Rd-ini|U˙s−U˙o|2×sinθ\[{{Z}_{deq}}=\frac{{{(220V)}^{2}}\times {{R}_{d-ini}}}{|{{{\dot{U}}}_{s}}-{{{\dot{U}}}_{0}}{{|}^{2}}}+j\frac{{{(220V)}^{2}}\times {{R}_{d-ini}}}{|{{{\dot{U}}}_{s}}-{{{\dot{U}}}_{o}}{{|}^{2}}\times \sin \theta }\]

According to Eq. 1, the value of Z_d-eq can be controlled by adjusting the amplitude of U˙o${{\dot{U}}_{o}}$, thus adapting to different requirements for undervoltage and over1 underfrequency protection sensitivity of the ASI (different undervoltage and over/underfrequency thresholds), as well as different DG capacities and different reactive power sizes of the local load. When the reliability requirements for LV anti-islanding change (e.g., when it is necessary to switch from a single protection mode to multiple protection modes), it is sufficient to change the phase of U˙o${{\dot{U}}_{o}}$ (Adjustment θ).

Therefore, with the B2B-VSC-based adaptive anti-islanding strategy, the disturbed loads no longer need to be reconfigured when any one of the sensitivity requirements, reliability requirements, or DG capacity changes.

Since the traditional distribution network control method relying on “hard” switches cannot meet the demand for flexible and active trend regulation, this study introduces the intelligent soft on/off switch (SOP) to adapt to the needs of the smart grid construction, and the MMC B2B has the ability of flexible trend regulation and control, which can realize real-time, fast and smooth power control, and guarantee inter-area trend mutual aid. It can achieve real-time, fast, and smooth power control, guarantee mutual aid in inter-area trends, and greatly improve the reliability and flexibility of the distribution network.

The MMC B2B is based on a new type of power electronic device, which adjusts the power output value of the converter in real time by controlling the command value, so as to change the system’s current distribution. In this paper, a flexible interconnected substation system architecture using MMC B2B to replace traditional circuit breakers and sectionalized switches is proposed for low-voltage distribution networks, and the flexible interconnected substation system architecture is shown in Fig. 3. B2B-MMC consists of MMMC2, with O₁ and O₂ being the capacities of MMC1 and MMC2, respectively, P₁ + J_Q1 and P₂ + J_Q2 being the equivalent loads of the 10 kV buses, respectively, and j being the dummy part, and QC and QF being the network contact switches and circuit breakers, respectively. Contact switches and circuit breakers, respectively. The architecture realizes the following four functions:

Bus interconnection. It realizes inter-regional energy intercommunication and reactive power compensation functions and provides mutual support for the busbar voltage.

Figure 3.

Flexible interconnection substation system architecture

When the flexibly interconnected substation with B2B-MMC is put into operation, QF and QC are in the disconnected state.

In the flexible power distribution system based on B2B-MMC, the communication state between converter stations is categorized into communication control and no-communication control according to different operation scenarios. Master-slave control is used in the communication control state, and voltage sag control is used in the no-communication control state.

For the multi-terminal flexible interconnection system, it is set that the system has n DC node, then the corresponding model of the master-slave control strategy of the system is: (14) ΔP1i=(Prefi+PGi−PDi)−Udci∑j=1n(Udci−Udcj)/Rijg,i=2,3,⋯n\[\Delta {{P}_{1i}}=({{P}_{refi}}+{{P}_{Gi}}-{{P}_{Di}})-{{U}_{dci}}\sum\limits_{j=1}^{n}{({{U}_{dci}}-{{U}_{dcj}})}/{{R}_{ijg}},i=2,3,\cdots n\]

Where ΔP_1i is the mismatch power of node i in master-slave control mode, P_ref is the initial active power set by the converter. P_Gi,P_Di is the active power issued by the generator and the active power absorbed by the load at node i, U_dci,U_dcj is the DC voltage at node i,j, and R_ij is the line resistance between node i,j.

DC voltage sag control means that the active class control of multiple converter stations is DC voltage sag control. In order to ensure the stable operation of the DC node voltage of multiple converter stations, it is necessary to increase the injected active power according to the sag coefficient e_i in its own characteristic curve. The operating characteristics at node i are: (15) Prefi−Pi=−(Uderef−Uref)/ei\[{{P}_{refi}}-{{P}_{i}}=-({{U}_{deref}}-{{U}_{ref}})/{{e}_{i}}\]

Where P_i is the actual output active power of the converter, U_dcref is the set voltage value of the DC node, and U_refi is the actual voltage value of the DC node.

For the multi-terminal flexible interconnection system, node 1~w is set as DC voltage sag control and node w+1~k is fixed active power control, then the corresponding model of DC voltage sag control strategy of the system is: (16) ΔP2i={ [Pref−Uderefilei+Udei/ei+PGi−PDi]−Uderefi∑j=1n(Uderefi−Udej)/Rij,i=1,2,⋯,w(Pref+PGi−PDi)−Uderefi∑j=1n(Uderefi−Udej)/Rij,i=w+1,w+2,⋯,k\[\Delta {{P}_{2i}}=\left\{ \begin{array}{*{35}{l}} [{{P}_{ref}}-{{U}_{derefi}}l{{e}_{i}}+{{U}_{dei}}/{{e}_{i}}+{{P}_{Gi}}-{{P}_{Di}}]- \\ {{U}_{derefi}}\sum\limits_{j=1}^{n}{({{U}_{derefi}}-{{U}_{dej}})}/{{R}_{ij}},i=1,2,\cdots ,w \\ ({{P}_{ref}}+{{P}_{Gi}}-{{P}_{Di}})-{{U}_{derefi}}\sum\limits_{j=1}^{n}{({{U}_{derefi}}-{{U}_{dej}})}/{{R}_{ij}}, \\ i=w+1,w+2,\cdots ,k \\ \end{array} \right.\]

Where ΔP_2i is the mismatch power of node i in the DC voltage sag control mode.

Through the analysis of the B2B-MMC converter control mode, it can be seen that its model structure is different in different control modes, and the formation of the solution about the state variables is not consistent. Master-slave control can realize the accurate control of converter output power. The strategy is simple, clear, and easy to realize; accordingly, this paper adopts the master-slave control mode for solving.

According to the power characteristics of the modular multilevel converter, the AC side current equation and the AC measurement flow into the converter current equation are obtained.

The AC side current equation is: (17) { Psi=−Usi2Gi+UsiUci[Gicos(δsi−δci)+Bisin(δsi−δci)]Qsi=Usi2Bi+UsiUci[Gisin(δsi−δci)−Bicos(δsi−δci)]\[\left\{ \begin{matrix} {{P}_{si}}=-U_{si}^{2}{{G}_{i}}+{{U}_{si}}{{U}_{ci}}[{{G}_{i}}\cos ({{\delta }_{si}}-{{\delta }_{ci}})+{{B}_{i}}\sin ({{\delta }_{si}}-{{\delta }_{ci}})] \\ {{Q}_{si}}=U_{si}^{2}{{B}_{i}}+{{U}_{si}}{{U}_{ci}}[{{G}_{i}}\sin ({{\delta }_{si}}-{{\delta }_{ci}})-{{B}_{i}}\cos ({{\delta }_{si}}-{{\delta }_{ci}})] \\ \end{matrix} \right.\]

Where G_i,B_i is the AC system line equivalent conductance and conductance, respectively.

The current equation of the AC current meter converter is: (18) { Pci=Uci2Gi−UsiUci[Gicos(δsi−δci)−Bisin(δsi−δci)]Qci=−Uci2Bi+UsiUci[Gisin(δsi−δci)+Bicos(δsi−δci)]\[\left\{ \begin{array}{*{35}{l}} {{P}_{ci}}=U_{ci}^{2}{{G}_{i}}-{{U}_{si}}{{U}_{ci}}[{{G}_{i}}\cos ({{\delta }_{si}}-{{\delta }_{ci}})- \\ {{B}_{i}}\sin ({{\delta }_{si}}-{{\delta }_{ci}})] \\ {{Q}_{ci}}=-U_{ci}^{2}{{B}_{i}}+{{U}_{si}}{{U}_{ci}}[{{G}_{i}}\sin ({{\delta }_{si}}-{{\delta }_{ci}})+ \\ {{B}_{i}}\cos ({{\delta }_{si}}-{{\delta }_{ci}})] \\ \end{array} \right.\]

MMC power constraints, for MMC1 and MMC2, the power output and capacity of the converter station in master-slave control mode need to satisfy Eq: (19) { PMMC1+PMMC2=0PMMC12+QMMC12≤O1PMMC22+QMMC22≤O2\[\left\{ \begin{array}{*{35}{l}} {{P}_{MMC1}}+{{P}_{MMC2}}=0 \\ \sqrt{P_{MMC1}^{2}+Q_{MMC1}^{2}}\le {{O}_{1}} \\ \sqrt{P_{MMC2}^{2}+Q_{MMC2}^{2}}\le {{O}_{2}} \\ \end{array} \right.\]

Where P_MMC1 and P_MMC2 are the active power injected by the converter, and Q_MMC1 and Q_MMC2 are the reactive power injected by the converter, respectively.

The main transformer constraints, in order to prevent the equivalent load power plus the B2B-MMC output power from exceeding the transformer carrying capacity, are: (20) PMMC2+QMMC2+PDi2+QDi2≤T\[\sqrt{P_{MMC}^{2}+Q_{MMC}^{2}}+\sqrt{P_{Di}^{2}+Q_{Di}^{2}}\le T\]

Where P_MMC and Q_MMC are the active and reactive power injected by the converter, P_Di and Q_Di are the equivalent active and reactive power loads at the connection point with the converter, and T is the transformer capacity.

Based on the optimal scheduling of flexibly interconnected substations with B2B-MMC as a bridge between the control objectives and the control means, the genetic algorithm (GA) is used to assign values to the converter power setting value during the distribution network operation process, and the optimal results calculated according to the improved alternating iteration method can continuously correct the converter power setting value to realize the system network loss minimization. In this paper, the objective function is set to minimize the network loss, in which the converter active loss is about 1% of the injected power, then the high-voltage grid network loss power is: (21) Ploss=∑i=1mUi∑j=1mUj(Gijcosθij+Bijsinθij)+PMMCloss\[{{P}_{loss}}=\sum\limits_{i=1}^{m}{{{U}_{i}}}\sum\limits_{j=1}^{m}{{{U}_{j}}}({{G}_{ij}}\cos {{\theta }_{ij}}+{{B}_{ij}}\sin {{\theta }_{ij}})+{{P}_{MM{{C}_{l}}oss}}\]

Where P_loss is the power lost in the HV grid network, U_i and U_j are the voltage amplitude at node i,j, and G_ij,B_ij,θ_ij is the conductance, conductivity and phase angle difference between node i,j, respectively. m is the number of AC system nodes, and P_{MMC_loss} is the converter loss value.

Considering the voltage margin constraints of distribution nodes, the penalty function is added to the objective function, and the objective function considering the system reliability constraints is established as: (22) Ude=∑i=1mmax(Umin−Ui)2+∑i=1mmax(Umax−Ui)2\[{{U}_{de}}=\sum\limits_{i=1}^{{{m}_{\max }}}{{{({{U}_{\min }}-{{U}_{i}})}^{2}}}+\sum\limits_{i=1}^{{{m}_{\max }}}{{{({{U}_{\max }}-{{U}_{i}})}^{2}}}\] (23) *minPallloss=Ploss+MUde\[*\min {{P}_{al{{l}_{l}}oss}}={{P}_{loss}}+M{{U}_{de}}\]

Where U_de is the grid node voltage margin constraint, m_max and m_min are the number of exceedances above and below the upper and lower voltage limits, respectively, and U_max and U_min are the upper and lower node voltage margin limits, respectively. minP_{all_loss} is the loss minimum and M is the penalty factor.

In the distribution network, assuming that the line current is not controlled, then the distribution power imbalance is bound to occur, resulting in some distribution lines being overloaded while some other lines are far from the upper limit of their current-carrying capacity, which will seriously affect the stable operation of the system. Therefore, the future development trend of the distribution network and the necessity of controlling the distribution system’s trend determine that it is extremely important and indispensable to regulate the line trend in the distribution network.

D-UPFC technology is the application of UPFC in a distribution network, which is a derivative of UPFC. The concept of UPFC is a kind of hybrid flexible AC transmission system (FACTS) equipment with an integrated series-parallel voltage source converter (VSC), which is composed of a static synchronous compensator (STATCOM) on the shunt side and static synchronous series compensator (SSSC) on the series side, and the function and structure of D-UPFC and UPFC are the same, and the working principle is also similar. D-UPFC and UPFC have the same function and structure, and the working principle is also similar. Mainly series compensation, parallel compensation, phase shift adjustment, and other functions. The difference between the two is that UPFC is used for equipment in transmission lines, while D-UPFC is suitable for equipment in the distribution network. The structure of D-UPFC can be seen in Fig. 4, and its main structure is composed of two voltage-source converters, VSC1 and VSC2, and a DC capacitor. If the current VSC1 flows to VSC2, then the process is equivalent to going through a rectification and then inversion process.

Figure 4.

The structure of the D-UPFC

The realization of the D-UPFC function must depend on a good control strategy. Regarding the control strategy of D-UPFC, experts and scholars have done much research since the original amplitude-phase control to the development of fuzzy PID control and intelligent control in recent years. Among the many control schemes of D-UPFC, cross-decoupling, cross-coupling, multi-objective coordinated control, etc., have been verified and applied through simulation and experiment. In this paper, according to the needs of the implementation of the flexible interconnection project of a medium voltage distribution network, the series-type D-UPFC is applied to the control system to realize the fast MPC control of D-UPFC based on current feedback.

The mathematical model of the D-UPFC series side is constructed as follows:

The D-UPFC series side transformer is essentially a PWM converter that can operate in four quadrants. By equating it to a controllable voltage source, the line current can be dispatched by simply adjusting the voltage amplitude and phase angle.

The principle of the converter is shown in Fig. 5, where V_1a,V_1b,V_1c shows the three-phase voltages at the transmitting side, V_2a,V_2b,V_2c shows the three-phase voltages at the receiving side, i_a,i_b,i_c shows the line currents, R shows the line resistances, and L shows the line inductances, respectively. V_pqa,V_pqb,V_pqc represents the three-phase voltage added to the system line at the series side. The circuit equation for the series side in the abc coordinate system is: (24) { V1a=−Vpqa+Ria+Ldiadt+V2aV1b=−Vpqb+Rib+Ldibdt+V2bV1c=−Vpqc+Ric+Ldicdt+V2c\[\left\{ \begin{matrix} {{V}_{1a}}=-{{V}_{pqa}}+R{{i}_{a}}+L\frac{d{{i}_{a}}}{dt}+{{V}_{2a}} \\ {{V}_{1b}}=-{{V}_{pqb}}+R{{i}_{b}}+L\frac{d{{i}_{b}}}{dt}+{{V}_{2b}} \\ {{V}_{1c}}=-{{V}_{pqc}}+R{{i}_{c}}+L\frac{d{{i}_{c}}}{dt}+{{V}_{2c}} \\ \end{matrix} \right.\]

The mathematical model in the transformed dq-coordinate system is: (25) Lddt[ idiq ]=L[ 0ω−ω0 ][ idiq ]−R[ idiq ]+[ V1d−V2d+VpqdV1q−V2q+Vpqq ]\[L\frac{d}{dt}\left[ \begin{matrix} {{i}_{d}} \\ {{i}_{q}} \\ \end{matrix} \right]=L\left[ \begin{matrix} 0 & \omega \\ -\omega & 0 \\ \end{matrix} \right]\left[ \begin{matrix} {{i}_{d}} \\ {{i}_{q}} \\ \end{matrix} \right]-R\left[ \begin{matrix} {{i}_{d}} \\ {{i}_{q}} \\ \end{matrix} \right]+\left[ \begin{matrix} {{V}_{1d}}-{{V}_{2d}}+{{V}_{pqd}} \\ {{V}_{1q}}-{{V}_{2q}}+{{V}_{pqq}} \\ \end{matrix} \right]\] (26) Vdcidc=32(Vpqdid+Vpqqiq)\[{{V}_{dc}}{{i}_{dc}}=\frac{3}{2}({{V}_{pqd}}{{i}_{d}}+{{V}_{pqq}}{{i}_{q}})\]

Let V₁ coincide with axis d, then V_1q = 0, there: (27) [ pq ]=[ V1d00−V1d ][ idiq ]\[\left[ \begin{matrix} p \\ q \\ \end{matrix} \right]=\left[ \begin{matrix} {{V}_{1d}} & 0 \\ 0 & -{{V}_{1d}} \\ \end{matrix} \right]\left[ \begin{matrix} {{i}_{d}} \\ {{i}_{q}} \\ \end{matrix} \right]\]

In a real circuit, ωL ≫ R, the effect of R can be neglected, and at steady state, then there is (28) id=1ωL(Vpqq−V2q)\[{{i}_{d}}=\frac{1}{\omega L}({{V}_{pqq}}-{{V}_{2q}})\] (29) iq=−1ωL(V1d−V2d+Vpqd)\[{{i}_{q}}=-\frac{1}{\omega L}({{V}_{1d}}-{{V}_{2d}}+{{V}_{pqd}})\]

From there, there is: (30) p=V1d*1ωL(Vpqq−V2q)\[p={{V}_{1d}}*\frac{1}{\omega L}({{V}_{pqq}}-{{V}_{2q}})\] (31) q=V1d*1ωL(V1d−V2d+Vpqd)\[q={{V}_{1d}}*\frac{1}{\omega L}({{V}_{1d}}-{{V}_{2d}}+{{V}_{pqd}})\]

Figure 5.

Schematic diagram of the series converter

From Eq. 2, when line resistance R is neglected and the system is in steady state, P is related to V_pqq only and Q is related to V_pqp only. The instantaneous active and instantaneous reactive power can be controlled by adjusting V_pq. Since a constant amplitude transformation is used for the dq transformation, a correction factor of 1.5 needs to be added to the calculation. The instantaneous power is calculated as: (32) { p=32V1d*1ωL(Vpqq−V2q)q=32V1d*1ωL(V1d−V2d+Vpqd)\[\left\{ \begin{array}{*{35}{l}} p=\frac{3}{2}{{V}_{1d}}*\frac{1}{\omega L}({{V}_{pqq}}-{{V}_{2q}}) \\ q=\frac{3}{2}{{V}_{1d}}*\frac{1}{\omega L}({{V}_{1d}}-{{V}_{2d}}+{{V}_{pqd}}) \\ \end{array} \right.\]