A Stability Analysis Method for Power Spring Cascade System Based on Component Connection Method

Abstract

The present invention discloses a method for analyzing the stability of a cascaded system of power springs based on the component connection method, which relates to the technical field of the application of power electronics technology. The present invention includes: controlling a cascaded system of multiple AC power springs to achieve the pure reactive power compensation function of the power springs; dividing the entire cascaded system into separate components, including single-phase power springs, power loads, and line impedances, and respectively establishing linearized state space models for them, and using the component connection method (CCM) to obtain the linearized state space model of the entire cascaded system; solving the eigenvalues of the system state space matrix to obtain the influence of control parameters and circuit elements on the eigenvalue distribution, and thereby analyzing the stability of the system. The method for analyzing the stability of the cascaded system of power springs proposed by the present invention can be used to judge the operating state of the system, obtain the influence of control parameters and circuit elements on the system stability margin, and further propose a design method for controller parameters to ensure the stable operation of the system.

Description

Technical Field

[0001] The present invention relates to the technical field of power electronics technology applications, and specifically to a method for analyzing the stability of a power spring cascade system based on the component connection method. Background Art

[0002] In recent years, renewable energy sources such as wind energy and solar energy have been increasingly used to achieve decarbonized power generation. In response to the problems of grid voltage fluctuations and frequency flicker caused by new energy power generation, the concept of the electric spring (ES) has been proposed. As a new type of management technology for the electricity demand side, the electric spring can effectively solve the problems of voltage amplitude and frequency fluctuations caused by the intermittency and randomness of new energy power generation, and by transferring power fluctuations to non-critical loads (NCLs), achieve the purpose of reducing the configuration capacity of the system energy storage unit and lowering the system operation cost.

[0003] Considering that the configuration capacity of a single electric spring is limited, the joint efforts of multiple electric springs are necessarily required to balance the stability of the entire system. In a microgrid containing multiple electric springs, the coupling effects between electric springs and between electric springs and the grid are likely to cause complex stability problems in the system, affect the power quality of user electricity consumption, and undermine the initial purpose of the electric spring to stabilize the voltage of critical loads. Therefore, in order to ensure the safe and stable operation of the entire system when electric springs are distributed at various nodes of the microgrid, a systematic stability analysis is needed. Summary of the Invention

[0004] The purpose of the present invention is to provide a method for analyzing the stability of a power spring cascade system based on the component connection method, using the component connection method (CCM) to derive the state space model of the power spring cascade system, judge the operating state of the power spring cascade system, ensure the stability of the critical load voltage, and at the same time obtain the influence of control parameters and circuit elements on the system stability margin.

[0005] To achieve the above purpose, the present invention provides the following technical solution: A method for analyzing the stability of a power spring cascade system based on the component connection method, including the following steps:

[0006] (1) Adopt the coordinated control of the power spring cascade system based on droop characteristics to achieve the coordinated operation of multiple electric springs and make the electric springs operate in a pure reactive power compensation mode;

[0007] (2) Divide the entire power spring cascade system into separate components, including single-phase electric springs, electrical loads, and line impedances, and establish linearized state space models for single-phase electric springs, electrical loads, and line impedances respectively;

[0008] (3) According to the established linearized state - space model of the single - phase power spring, power load, and line impedance, obtain the composite model of the power - spring cascaded system, and its expression is as follows:

[0009]

[0010] Among them,

[0011]

[0012] A T = diag(F ES1 ,...,F ESi ,A net ), B T = diag(G ES1 ,...,G ESi ,B net )

[0013] C T = diag(H ES1 ,...,H ESi ,C net ), D T = diag(J ES1 ,...,J ESi ,D net )

[0014] In the formula, i represents the number of cascaded power springs;

[0015] For the cascaded system of multiple power springs, the interconnection relationship between the input and output of component elements is:

[0016]

[0017] Furthermore, the state matrix of the entire cascaded system can be obtained as:

[0018] F T = A T + B T L T1 (I T - D T L T1 ) -1 C T

[0019] Then, perform frequency - domain analysis on the linearized state - space model of multiple power springs to obtain the influence of control parameters and circuit elements on the eigenvalue distribution.

[0020] Furthermore, the coordinated control method of the power - spring cascaded system in step (1) is specifically as follows:

[0021] 1) Collect the key load voltage vpcc , the voltage v of the filter capacitor es , the output current i of the inverter L and the non-critical load current i nc , using the voltage v of the filter capacitor es and the non-critical load current i nc , calculate the reactive power Q output by the power spring. The single-phase phase-locked loop uses the voltage v of the filter capacitor es as the input to obtain the frequency and phase information.

[0022] 2) Multiply the reactive power Q by the droop coefficient n to correct the same reference amplitude v of each critical load voltage * pccmagref , to obtain the corrected reference amplitude v of a single power spring pccmagref . The collected critical load voltage v pcc , after dq transformation, and then calculate the actual amplitude v of the critical load voltage pccmag . Compare the actual value v of the voltage amplitude pccmag with its given value v pccmagref , the difference passes through the outer-loop voltage PI controller, and its output is the q-axis current reference value i Lqref ; at the same time, in order to achieve the pure reactive power compensation function, set the d-axis current reference value i Ldref to 0.

[0023] 3) The output current i L of the inverter side is transformed by dq to obtain the actual values i Ld and i Lq of the d and q axes of the current. Compare the current reference values i Ldref and i Lqref with them respectively, and the difference passes through the inner-loop current PI controller; the output of the current controller is compensated by feed-forward decoupling to obtain the voltage reference values v md and v mq . v md and v mq obtain the output voltage reference value of the inverter as v ref .

[0024] Furthermore, the single-phase power spring described in step (2) includes AC current control, a phase-locked loop, a critical load voltage droop controller, a critical load voltage controller, and dq transformation.

[0025] Furthermore, the linearized state-space model of the single-phase power spring is specifically established as follows:

[0026] 1) Establish the state-space model of AC current control, and the specific expression is:

[0027]

[0028] Among them, F acc , G acc , H acc , J acc are:

[0029] F acc = A acc + B acc L acc1 (I acc - D acc L acc1 ) -1 C acc , G acc = B acc L acc1 (I acc - D acc L acc1 ) -1 D acc L acc2 + B acc L acc2 , H acc = L acc3 (I acc - D acc L acc1 ) -1 C acc , J acc = L acc3 (I acc - D acc L acc1 ) -1 D acc L acc2 + L acc4

[0030] 2) Respectively use PLL, avc, and vdc to label the variables of the three subsystems of the phase-locked loop, critical load voltage control, and critical load voltage droop control, and establish their state-space models respectively;

[0031] 3) dq transformation

[0032] In the power spring system, the relationship between the critical load voltage in the converter dq coordinate system and the filter capacitor voltage dq coordinate system is:

[0033]

[0034] The relationship between the non-critical load voltage in the converter dq coordinate system and the filter capacitor voltage dq coordinate system is:

[0035]

[0036] The relationship between the filter capacitor voltage in the dq coordinate system and the filter capacitor voltage in the converter dq coordinate system is:

[0037]

[0038] In the above formula, all quantities with subscript "0" represent the values of the static operating point, and Δθ is the phase difference between the filter capacitor voltage dq coordinate system and the converter dq coordinate system;

[0039] 4) According to steps 1)-3), and rearrange these matrices in a diagonal form to obtain the composite model of the power spring system, and its expression is:

[0040]

[0041] In the formula,

[0042]

[0043] A ES = diag(F acc , A PLL , A avc , A vdc ), B ES = diag(G acc , B PLL , B avc , B vdc )

[0044] C ES = diag(H acc , C PLL , C avc , C), D ES = diag(J, D L , D avc , D vdc )

[0045] Define the input and output vectors of the entire power spring system as:

[0046]

[0047] The component interconnection relationship of the power spring system can be described as:

[0048]

[0049] Therefore, the state space model of the entire power spring can be established as:

[0050]

[0051] Among them, F ES , G ES , HES , J ES is obtained according to the following formula.

[0052] F ES = A ES + B ES L ES1 (I ES - D ES L ES1 ) -1 C ES , G ES = B ES L ES1 (I ES - D ES L ES1 ) -1 D ES L ES2 + B ES L ES2 H ES = L ES3 (I ES - D ES L ES1 ) -1 C ES , J ES = L ES3 (I ES - D ES L ES1 ) -1 D ES L ES2 + L ES4

[0053] Furthermore, the AC current control includes a current controller, a control delay, and an LC filter.

[0054] Furthermore, the linearized state - space models of the current controller, the control delay, and the LC filter are specifically as follows:

[0055] 1) The AC current controller adopts a PI controller, and its state - space model is as follows:

[0056]

[0057] In the formula, are the current - controller state, input, and output vectors respectively. Δγ d , Δγ q is the integrator output of G i , and the superscript c indicates that the variable is defined in the dq reference coordinate system of the converter;

[0058] 2) The third-order Pade approximation is applied to make the control delay model have sufficient accuracy within the Nyquist frequency while minimizing the complexity, which is specifically expressed as:

[0059]

[0060] where τ = 1.5T s is the delay time, usually 1.5 times the sampling period T s According to the above formula, the state-space model of the control delay can be expressed as:

[0061]

[0062] where

[0063]

[0064] 3) The state-space model of the LC filter is as follows:

[0065]

[0066] where

[0067] According to steps 1)-3), the composite model of the AC current control is obtained, which is expressed as:

[0068]

[0069] where

[0070]

[0071] X acc = diag(X i , X del , X LC )(X = A, B, C, D)

[0072] Define the input and output vectors of the entire AC current control system as:

[0073]

[0074] where the superscript c indicates that the variable is defined in the dq reference coordinate system of the converter.

[0075] The connection between different components of the AC current control system can be expressed as:

[0076]

[0077] The state-space model of the entire AC current control can be established as:

[0078]

[0079] Among them, F acc , G acc , H acc , J acc are:

[0080] F acc = A acc + B acc L acc1 (I acc - D acc L acc1 ) -1 C acc , G acc = B acc L acc1 (I acc - D acc L acc1 ) -1 D acc L acc2 + B acc L acc2 , H acc = L acc3 (I acc - D acc L acc1 ) -1 C acc , J acc = L acc3 (I acc - D acc L acc1 ) -1 D acc L acc2 + L acc4

[0081] In the formula, I acc is the identity matrix.

[0082] Furthermore, the state - space model of the phase - locked loop has the following specific expression:

[0083]

[0084] In the formula, where Δx1 is the integral output of, and Δθ is the change in the output synchronization angle.

[0085] Furthermore, the steps for specifically establishing the state - space model of the critical load voltage control are as follows:

[0086] The critical load voltage (i.e., the voltage at the PCC) v pcc is expressed in its dq components as:

[0087]

[0088] After linearizing the above formula, we can get:

[0089]

[0090] In the formula, V pccd0 , V pccq0 respectively represent the values of the dq components of the static operating point of the voltage v pcc .

[0091] The state - space model of the voltage control loop is as follows:

[0092]

[0093] In the formula, Among them, the state variable Δx2 is the integral output of Δv pccmag -Δv pccmagref .

[0094] Furthermore, the specific establishment steps of the state - space model of the critical load voltage droop control are as follows:

[0095] 1) During the control process, adjust the reference voltage amplitude, and the adjustment method is as follows:

[0096]

[0097] In the formula, v * pccmagref represents the rated value of the critical load voltage amplitude corresponding to zero reactive power provided by the power spring; Q represents reactive power; n represents the droop coefficient;

[0098] 2) The instantaneous reactive power output by the power spring is:

[0099]

[0100] According to the obtained instantaneous reactive power q, after being processed by a low - pass filter, the input of the voltage droop controller, that is, the reactive power Q, is obtained as follows:

[0101]

[0102] In the formula, ω f is the cut - off frequency of the low - pass filter;

[0103] 3) Linearize the expressions in steps 1) - 2), then the small - signal model corresponding to the voltage droop control link is:

[0104]

[0105] In the formula,

[0106] The present invention has at least the following beneficial effects:

[0107] By using the Component Connection Method (CCM) to derive the state - space model of the power - spring cascaded system, and then performing small - signal stability analysis on the system. The stability analysis method for the power - spring cascaded system proposed by the present invention is used to judge the operating state of the cascaded system, ensure the stability of the critical load voltage, and at the same time obtain the influence of control parameters and circuit elements on the system stability margin. Furthermore, a design method for the power - spring controller parameters is proposed, laying a theoretical foundation for the layout of power springs in the micro - grid.

[0108] Of course, it is not necessary for any product implementing the present invention to achieve all the above - mentioned advantages simultaneously. Brief Description of the Drawings

[0109] Figure 1 is the overall structure diagram of the present invention;

[0110] Figure 2 is the droop - control block diagram of the power spring of the present invention;

[0111] Figure 3 is the current - control block diagram of a single power spring of the present invention in the dq coordinate system;

[0112] Figure 4 is the Synchronous Reference Frame Phase - Locked Loop (SRF - PLL) block diagram;

[0113] Figure 5 is the schematic diagram of the phase difference between the dq coordinate system of the filter - capacitor voltage and the dq coordinate system of the phase - locked loop;

[0114] Figure 6 is the eigenvalue distribution diagram of the cascaded - system state matrix. Detailed Embodiment

[0115] Next, the technical solutions in the embodiments of the present disclosure will be clearly and completely described in conjunction with the accompanying drawings in the embodiments of the present disclosure. Obviously, the described embodiments are only a part of the embodiments of the present disclosure, rather than all the embodiments. Based on the embodiments in the present disclosure, all other embodiments obtained by those of ordinary skill in the art without creative efforts belong to the scope of protection of the present disclosure.

[0116] Please refer to Figure 1-6 , the present invention provides a technical solution: a method for analyzing the stability of a power - spring cascaded system based on the Component Connection Method, including the following steps:

[0117] (1) Coordinate control of the power spring cascade system based on droop characteristics is adopted to achieve the coordinated operation of multiple power springs and make the power springs work in the pure reactive power compensation mode;

[0118] The control structure diagram of the power spring is as Figure 2 shown, where L f is the inductor of the low-pass filter, C f is the capacitor of the low-pass filter, R c is the critical load, R nc is the non-critical load, R1 and L1 are the impedances of the transmission line, v es is the voltage of the filter capacitor, v pcc is the voltage of the critical load, v G is the grid voltage, i L is the output current of the inverter, i nc is the current of the non-critical load, Figure 1 is the system structure diagram including 3 ESs, Figure 2 The specific ES control method represented is as follows:

[0119] 1) Collect the critical load voltage v pcc , the voltage of the filter capacitor v es , the output current of the inverter i L and the current of the non-critical load i nc . Use the voltage of the filter capacitor v es and the current of the non-critical load i nc to calculate the reactive power Q output by the power spring. The single-phase phase-locked loop uses the voltage of the filter capacitor v es as the input to obtain the frequency and phase information.

[0120] 2) Multiply the reactive power Q by the droop coefficient n to correct the same reference amplitude v * pccmagref of each critical load voltage to obtain the corrected reference amplitude v pccmagref of a single power spring; the collected critical load voltage v pcc , after dq transformation, calculate the amplitude v pccmag of the critical load voltage; compare the obtained actual voltage amplitude v pccmag with the given voltage amplitude v pccmagref . The difference passes through the outer-loop voltage PI controller, and its output is the q-axis current reference value i Lqref ; at the same time, to achieve the pure reactive power compensation function, set the d-axis current reference value i Ldref to 0;

[0121] 3) The output current i L of the inverter side is transformed by dq to obtain the actual values i Ld , i Lq, the current reference values \(i\) Ldref and \(i\) Lqref are respectively compared with it, and the difference passes through the inner-loop current PI controller; the output of the current controller is compensated by feed-forward decoupling to obtain the voltage reference values \(v\) md and \(v\) mq , \(v\) md and \(v\) mq The output voltage reference value of the inverter is obtained by dq / αβ transformation as \(v\) ref .

[0122] (2) Divide the entire cascaded system into separate components, including a single-phase power spring (ES), a power load, and a line impedance. Establish a linearized state-space model for each of them, and based on the established linearized state-space models of the single-phase power spring, the power load, and the line impedance, obtain the overall linearized state-space model of the power spring cascaded system. The single-phase power spring consists of subsystems such as AC current control, a phase-locked loop, a critical load voltage droop controller, a critical load voltage controller, and dq transformation. The specific method for establishing the overall linearized state-space model of the power spring cascaded system is as follows:

[0123] 1) AC current control

[0124] The block diagram of AC current control is as Figure 3 shown. It can be seen from the figure that the AC current control link includes three subsystems: a current controller \(G\) i , a control delay \(G\) del and an LC filter \(G\) LC .

[0125] The current controller \(G\) i adopts a PI controller, and its state-space model is as follows:

[0126]

[0127] where are respectively the vectors of the state, input, and output variables of the current controller, and Δγ d , Δγ q is the output of the integrator of \(G\) i . The superscript c indicates that the variable is defined in the dq reference coordinate system of the converter; \(A\) i , \(B\) i , \(C\) i , \(D\) i are the system matrices of the single-phase power spring;

[0128] In digital control, calculations and pulse-width modulation (PWM) introduce control delays. Applying a third-order Pade approximation enables the delay-link model to have sufficient accuracy within the Nyquist frequency (i.e., half of the sampling frequency), while minimizing complexity, specifically expressed as:

[0129]

[0130] where e -τ·s is the model of the delay link, and τ = 1.5T s is the delay time, usually 1.5 times the sampling period T s . According to the above equation, the state-space model of the control delay can be expressed as:

[0131]

[0132] where are the vectors of the states, inputs, and outputs of the control delay link, respectively, and is the state variable of the control delay link;

[0133] The state-space model of the LC filter is as follows:

[0134]

[0135] where are the vectors of the states, inputs, and outputs of the LC filter link, respectively, and

[0136] To derive the state-space model of the inner current control loop using the component connection method (CCM), a composite model for AC current control is obtained by rearranging these matrices in diagonal form, which is expressed as:

[0137]

[0138] where are the combined vectors of the states, inputs, and outputs of the components in the AC current control link, respectively. A acc , B acc , C acc , D acc are the diagonal matrices formed by the system matrices of the components in the AC current control link; among them,

[0139]

[0140] X acc = diag(X i , X del , X LC )(X = A, B, C, D)

[0141] Define the input and output vectors of the entire AC current control system as:

[0142]

[0143] where the superscript c indicates that the variable is defined in the converter dq reference frame;

[0144] The connections between different components of the AC current control system can be expressed as:

[0145]

[0146] where L acc1 , L acc2 , L acc3 , L acc4 are matrices reflecting the interconnection relationships between the components of the AC current control;

[0147] The state - space model of the entire AC current control can be established as:

[0148]

[0149] where F acc , G acc , H acc , J acc are:

[0150] F acc = A acc + B acc L acc1 (I acc - D acc L acc1 ) -1 C acc , G acc = B acc L acc1 (I acc - D acc L acc1 ) -1 D acc L acc2 + B acc L acc2 , H acc = L acc3 (I acc - D acc L acc1 ) -1 C acc , J acc = L acc3 (I acc - D acc L acc1) -1 D acc L acc2 +L acc4 (9)

[0151] Wherein, A acc , B acc , C acc , D acc is a diagonal matrix formed by the component system matrices of each component in the AC current control link; L acc1 , L acc2 , L acc3 , L acc4 is a matrix representing the interconnection relationship between the components of each component in the AC current control link; I acc is the identity matrix;

[0152] 2) Phase-locked loop

[0153] The block diagram of the synchronous reference frame phase-locked loop (SRF-PLL) is as shown in Figure 4 . In the figure, where v esα and v esβ are the filtered capacitor voltage components in the αβ coordinate system; G PLL is the PI controller of the phase-locked loop. According to Figure 4 , the state space model of the phase-locked loop can be obtained as follows:

[0154]

[0155] Wherein, are the vectors of the state, input, and output variables of the phase-locked loop respectively, and wherein, Δx1 is the integral output of , and Δθ is the change in the output synchronous angle.

[0156] 3) Critical load voltage control

[0157] The critical load voltage (i.e., the voltage at the PCC) v pcc is expressed in its dq components as:

[0158]

[0159] After linearizing the above formula, we can get:

[0160]

[0161] Wherein, V pccd0 , V pccq0 respectively represent the values of the dq component static operating points of the voltage v pcc .

[0162] The state space model of the voltage control loop is as follows:

[0163]

[0164] In the formula, are respectively the vectors of the key load voltage control state, input, and output variables, and Among them, the state variable Δx2 is Δv pccmag -Δv pccmagref The integral output of

[0165] 4) Key load voltage droop control

[0166] The method of adjusting the reference voltage amplitude during the control process is as follows:

[0167]

[0168] In the formula, v * pccmagref represents the rated value of the key load voltage amplitude corresponding to when the reactive power provided by the power spring is zero; Q represents the reactive power; n represents the droop coefficient;

[0169] The instantaneous reactive power output by the power spring is:

[0170]

[0171] In order to reduce the influence of harmonics on the system, the input of the droop controller, that is, the reactive power Q, can be obtained by processing the instantaneous reactive power q through a low-pass filter:

[0172]

[0173] In the formula, ω f is the cut-off frequency of the low-pass filter, and s is the Laplace transform factor;

[0174] Performing linearization processing on equations (14) - (16), the small-signal model corresponding to the key load voltage droop control link can be obtained as:

[0175]

[0176] In the formula, are respectively the vectors of the state, input, and output variables of the key load voltage droop control, and

[0177] 5) dq transformation

[0178] The SRF-PLL affects the characteristics of variables through the Park transformation and its inverse transformation. This is due to the small-signal perturbation of the filter capacitor voltage, which propagates to the PLL phase angle, resulting in two dq coordinate systems: one is the filter capacitor voltage dq coordinate system, which is composed of the filter capacitor voltage v esThe phase angle definition; the other is the phase-locked loop dq coordinate system, which is defined by the phase angle obtained from the SRF-PLL.

[0179] The phase difference Δθ between the filter capacitor voltage dq coordinate system and the phase-locked loop dq coordinate system (i.e., the converter dq coordinate system) is expressed as Figure 5 shown by Figure 5 It can be seen that in the power spring system, the relationship between the key load voltages in the converter dq coordinate system and the filter capacitor voltage dq coordinate system:

[0180]

[0181] In the formula, are the key load voltages in the converter dq coordinate system and the filter capacitor voltage dq coordinate system respectively; Δθ is the phase difference between the filter capacitor voltage dq coordinate system and the converter dq coordinate system; V pccd0 , V pccq0 represent the dq components of the static operating point of the key load voltage.

[0182] The relationship between the non-key load voltages in the converter dq coordinate system and the filter capacitor voltage dq coordinate system:

[0183]

[0184] In the formula, are the non-key load voltages in the converter dq coordinate system and the filter capacitor voltage dq coordinate system respectively; V ncd0 , V ncq0 represent the dq components of the static operating point of the non-key load voltage.

[0185] Let V es = V es0 +j0 be the steady-state vector of the filter capacitor voltage in the filter capacitor voltage dq coordinate system. Then, in the power spring system, the relationship between the filter capacitor voltages in the filter capacitor voltage dq coordinate system and the converter dq coordinate system:

[0186]

[0187] 6) State-space model of the power spring

[0188] Considering the state-space models of the alternating current control ACC, phase-locked loop PLL, key load voltage control AVC, and key load voltage droop control VDC, the composite system model of the power spring is as follows:

[0189]

[0190] In the formula, are the combined vectors of the states, inputs, and outputs of each component of the power spring respectively, and AES , B ES , C ES , D ES is the diagonal matrix formed by the system matrices of each component of the power spring; where,

[0191]

[0192] A ES = diag(F acc , A PLL , A avc , A vdc ), B ES = diag(G acc , B PLL , B avc , B vdc )

[0193] C ES = diag(H acc , C PLL , C avc , C), D ES = diag(J, D L , D avc , D vdc )

[0194] In the formula, are the vectors of the state, input, and output variables of the phase-locked loop respectively, and A PLL , B PLL , C PLL , D PLL are the system matrices of the phase-locked loop; are the vectors of the state, input, and output variables of the critical load voltage control link respectively, and A avc , B avc , C avc , D avc are the system matrices of the critical load voltage control link; are the vectors of the state, input, and output variables of the critical load voltage droop control link respectively, and A vdc , B vdc , C vdc , D vdc are the system matrices of the critical load voltage droop control link;

[0195] Define the input and output vectors of the entire power spring control system as:

[0196]

[0197] The component interconnection relationship of the power spring system can be described as:

[0198]

[0199] In the formula, L ES1 , L ES2 , L ES3 , L ES4 is a matrix representing the interconnection relationship between the components of the single-phase power spring;

[0200] Therefore, the state-space model of the entire power spring can be established as:

[0201]

[0202] In the formula, are the vectors of the state, input, and output variables of the single-phase power spring respectively, F ES , G ES , H ES , J ES are the system matrices of the single-phase power spring.

[0203] 7) State-space model of the power load and line impedance

[0204] The structure diagram of the three power spring cascaded system is as shown in Figure 1 . The state-space model of the power load and line impedance part is as follows:

[0205]

[0206] In the formula, are the vectors of the state, input, and output variables of the power load and line impedance components respectively, A net , B net , C net , D net are the system matrices of the power load and line impedance components; i represents the number of cascaded power springs; among them,

[0207]

[0208]

[0209]

[0210] In the formula, i L1d , i L1q , i L2d , i L2q , i L3d , i L3q are the dq components of the line current, v Gd , v Gq are the dq components of the grid voltage.

[0211] Use the Component Connection Method (CCM) to obtain the composite model of the power spring cascade system as follows:

[0212]

[0213] Wherein, are the combined vectors of the states, inputs, and outputs of each component of the cascade system, respectively. A T , B T , C T , D T is the diagonal matrix formed by the system matrices of each component of the cascade system; among them,

[0214]

[0215] A T = diag(F ES1 ,..., F ESi , A net ), B T = diag(G ES1 ,..., G ESi , B net )

[0216] C T = diag(H ES1 ,..., H ESi , C net ), D T = diag(J ES1 ,..., J ESi , D net )

[0217] Wherein, are the combined vectors of the states, inputs, and outputs of each component of the cascade system, respectively. A T , B T , C T , D T is the diagonal matrix formed by the system matrices of each component of the cascade system; are the vectors of the states, inputs, and outputs of the i-th power spring, respectively. F ESi , G ESi , H ESi , J ESi is the system matrix of the i-th power spring; are the vectors of the states, inputs, and outputs of the power load and line impedance components, respectively. A net , B net , C net , D net is the system matrix of the power load and line impedance components; i represents the number of cascaded power springs;

[0218] For multiple power springs, the interconnection relationship between the component inputs and outputs is as follows:

[0219]

[0220] where is the vector of the input and output variables of the entire cascaded system; L T1 , L T2 , L T3 , L T4 are the matrices reflecting the interconnection relationships between the components of the cascaded system;

[0221] Furthermore, the state matrix of the entire cascaded system can be obtained as F T = A T + B T L T1 (I T - D T L T1 ) -1 C T ;

[0222] The eigenvalues of the system state matrix are obtained through MATLAB programming, and then the frequency-domain analysis of the linearized state-space model of multiple power springs is carried out to obtain the influence of control parameters and circuit elements on the system stability margin.

[0223] Example:

[0224] The DC voltage V dc is 400V; the filter inductor L f on the inverter side is 3.5mH, the filter capacitor C f is 50uF; the line resistance R1 is 0.5Ω, the line inductor L1 is 10mH, the line resistances R2 and R2 are 0.3Ω, and the line inductors L2 and L3 are 1.1mH; the critical load R c is 180Ω, the non-critical load R nc is 30Ω, the switching frequency is 10kHz; the proportional coefficient and integral coefficient of the voltage controller are 0.1 and 15.5 respectively; the proportional coefficient and integral coefficient of the current controller are 0.25 and 0.75 respectively; the voltage droop coefficient is 0.00005, and the eigenvalue distribution of the obtained cascaded system state matrix is as shown in Figure 6 . As can be seen from the figure, the eigenvalues of the state matrix are all distributed on the left side of the imaginary axis. It can be seen that the power spring cascaded system is stable under the above parameter selection.

[0225] It should be noted that, in this document, relational terms such as first and second are only used to distinguish one entity or operation from another entity or operation, and do not necessarily require or imply any actual relationship or order between these entities or operations. Moreover, the terms "comprising", "including" or any other variant thereof are intended to cover non-exclusive inclusion, such that a process, method, article or apparatus comprising a series of elements includes not only those elements but also other elements not expressly listed, or elements inherent to such process, method, article or apparatus.

[0226] For those of ordinary skill in the art, the specific meanings of the above terms in the present invention can be understood according to specific circumstances. When an element is referred to as being "assembled on", "mounted on", "fixed to" or "disposed on" another element, it can be directly on the other element or there may also be an intermediate element. When an element is considered to be "connected" to another element, it can be directly connected to the other element or there may be an intermediate element at the same time. The terms "vertical", "horizontal", "upper", "lower", "left", "right" and similar expressions used herein are for illustrative purposes only and do not represent the only implementation.

[0227] Although embodiments of the present invention have been shown and described, for those of ordinary skill in the art, it can be understood that various changes, modifications, substitutions and variations can be made to these embodiments without departing from the principles and spirit of the present invention, and the scope of the present invention is defined by the appended claims and their equivalents.

[0228] In the description of this specification, the description with reference to terms such as "one embodiment", "example", "specific example", etc. means that the specific features, structures, materials or characteristics described in connection with the embodiment or example are included in at least one embodiment or example of the present disclosure. In this specification, the schematic representations of the above terms do not necessarily refer to the same embodiment or example. Moreover, the specific features, structures, materials or characteristics described can be combined in a suitable manner in any one or more embodiments or examples.

Claims

1. A method for analyzing the stability of a cascaded system of power springs based on the component connection method, characterized in that It includes the following steps: (1) Adopt the coordinated control of the power spring cascade system based on droop characteristics to achieve the coordinated operation of multiple power springs and make the power springs operate in the pure reactive power compensation mode; (2) Divide the entire power spring cascade system into separate components, including single-phase power springs, power loads, and line impedances, and establish linearized state space models for them respectively; (3) According to the established linearized state space models of single-phase power springs, power loads, and line impedances, obtain the composite model of the power spring cascade system, and its expression is specifically as follows: Wherein, A T = diag(F ES1 ,..., F ESi , A net ), B T = diag(G ES1 ,..., G ESi , B net ) C T = diag(H ES1 ,..., H ESi , C net ), D T = diag(J ES1 ,..., J ESi , D net ) wherein, are respectively the combined vectors of the states, inputs, and outputs of the components of the cascaded system, A T , B T , C T , D T is the diagonal matrix formed by the system matrices of the components of the cascaded system; are respectively the vectors of the states, inputs, and outputs of the i-th power spring, F ESi , G ESi , H ESi , J ESi is the system matrix of the i-th power spring; are respectively the vectors of the states, inputs, and outputs of the power load and line impedance components, A net , B net , C net , D net is the system matrix of the power load and line impedance components; i represents the number of cascaded power springs; For the multi-power spring cascade system, the interconnection relationship between the inputs and outputs of each component is: wherein, is the vector of the input and output variables of the entire cascaded system; L T1 , L T2 , L T3 , L T4 are the matrices reflecting the interconnection relationships among the components of the cascaded system; Furthermore, the state matrix of the entire cascade system can be obtained as: F T = A T + B T L T1 (I T - D T L T1 ) -1 C T Solve the eigenvalues of the system state matrix, and then conduct frequency domain analysis on the linearized state space models of multiple power springs to obtain the influence of control parameters and circuit elements on the eigenvalue distribution.

2. The stability analysis method of a power spring cascaded system based on a component connection method according to claim 1, wherein: The coordinated control method of the power spring cascade system in step (1) is specifically as follows: 1) Collect the key load voltage v pcc , the filter capacitor voltage v es , the inverter output current i L and the non-critical load current i nc . Using the filter capacitor voltage v es and the non-critical load current i nc , calculate the reactive power Q output by the power spring. The single-phase phase-locked loop uses the filter capacitor voltage v es as the input to obtain the frequency and phase information; 2) Modify the same reference amplitude v of each key load voltage by multiplying the reactive power Q by the droop coefficient n * pccmagref to obtain the corrected reference amplitude v of a single power spring pccmagref ; The key load voltage v collected pcc , after dq transformation, then calculate the amplitude v of the key load voltage pccmag ; Compare the obtained actual voltage amplitude v pccmag with the given voltage amplitude v pccmagref , and the difference passes through the outer-loop voltage PI controller, and its output is the q-axis current reference value i Lqref ; At the same time, in order to achieve the pure reactive power compensation function, set the d-axis current reference value i Ldref to 0; 3) Inverter-side output current i L The actual values of the d-axis and q-axis currents i Ld , i Lq are obtained through dq transformation. The current reference values i Ldref and i Lqref are respectively compared with them. The difference is passed through the inner-loop current PI controller; the output of the current controller is compensated by feedforward decoupling to obtain the voltage reference values v md and v mq . v md and v mq obtain the output voltage reference value of the inverter as v ref .

3. A method for analyzing the stability of a cascaded system of power springs based on a component connection method according to claim 1, characterized in that The single-phase power spring described in step (2) includes alternating current control, phase-locked loop, critical load voltage (i.e., the voltage at the PCC) control, critical load voltage droop control, and dq transformation.

4. A method for analyzing the stability of a power spring cascading system based on a component connection method according to claim 3, characterized in that The specific steps for establishing the linearized state space model of the single-phase power spring are as follows: 1) Establish the state space model of alternating current control, and its specific expression is: wherein, are respectively the vectors of the states, inputs, and outputs of the AC current control loop, F acc , G acc , H acc , J acc is the system matrix of the AC current control loop; Among them, F acc , G acc , H acc , J acc are as follows: F acc = A acc + B acc L acc1 (I acc - D acc L acc1 ) -1 C acc , G acc = B acc L acc1 (I acc - D acc L acc1 ) -1 D acc L acc2 + B acc L acc2 , H acc = L acc3 (I acc - D acc L acc1 ) -1 C acc ,J acc = L acc3 (I acc - D acc L acc1 ) -1 D acc L acc2 + L acc4 Where, A acc , B acc , C acc , D acc is a diagonal matrix formed by the system matrices of the components in the AC current control link; L acc1 , L acc2 , L acc3 , L acc4 is a matrix representing the interconnection relationship between the components in the AC current control link; I acc is the identity matrix; 2) Use PLL, avc, and vdc to mark the variables of the three subsystems of the phase-locked loop, critical load voltage control, and critical load voltage droop control respectively, and establish their state space models; 3) dq transformation In the power spring system, the relationship between the critical load voltages in the dq coordinate system of the converter and the dq coordinate system of the filter capacitor voltage is: In the formula, v pccd are the key load voltages in the dq coordinate system of the converter and the dq coordinate system of the filter capacitor voltage respectively; Δθ is the phase difference between the dq coordinate system of the filter capacitor voltage and the dq coordinate system of the converter; V pccd0 , V pccq0 represent the dq components of the static operating point of the key load voltage; The relationship between the non-critical load voltages in the dq coordinate system of the converter and the dq coordinate system of the filter capacitor voltage is: wherein, v ncd are the non-critical load voltages in the dq coordinate system of the converter and the dq coordinate system of the filter capacitor voltage respectively; V ncd0 , V ncq0 represent the dq components of the static operating point of the non-critical load voltage; The relationship between the filter capacitor voltages in the dq coordinate system of the filter capacitor voltage and the dq coordinate system of the converter is: In the formula, v esd are the filter capacitor voltages in the dq coordinate system of the converter and the dq coordinate system of the filter capacitor voltage respectively; V esd0 , V esq0 represent the dq components of the static operating point of the filter capacitor voltage. 4) According to steps 1)-3), use the component connection method to deduce the composite system model of the power spring as: In the formula, are respectively the combined vectors of the states, input, and output variables of each component of the electric spring, A ES , B ES , C ES , D ES is the diagonal matrix formed by the system matrices of each component of the electric spring; among them, A ES = diag(F acc , A PLL , A avc , A vdc ), B ES = diag(G acc , B PLL , B avc , B vdc ) C ES = diag(H acc , C PLL , C avc , C), D ES = diag(J, D L , D avc , D vdc ) In the formula, are the vectors of the state, input, and output variables of the phase-locked loop, respectively, A PLL , B PLL , C PLL , D PLL is the system matrix of the phase-locked loop; are the vectors of the state, input, and output variables of the critical load voltage control link, respectively, A avc , B avc , C avc , D avc is the system matrix of the critical load voltage control link; are the vectors of the state, input, and output variables of the critical load voltage droop control link, respectively, A vdc , B vdc , C vdc , D vdc is the system matrix of the critical load voltage droop control link; Define the input and output vectors of the entire power spring system as: The component interconnection relationship of the power spring system can be described as: Wherein, L ES1 , L ES2 , L ES3 , L ES4 is a matrix representing the interconnection relationship between the components of the single-phase power spring; Therefore, the state space model of the entire power spring can be established as: wherein, are respectively the vectors of the state, input, and output variables of the single-phase power spring, F ES , G ES , H ES , J ES are the system matrices of the single-phase power spring; wherein, F ES , G ES , H ES , J ES are obtained according to the following formula: F ES = A ES + B ES L ES1 (I ES - D ES L ES1 ) -1 C ES , G ES = B ES L ES1 (I ES - D ES L ES1 ) -1 D ES L ES2 + B ES L ES2 H ES = L ES3 (I ES - D ES L ES1 ) -1 C ES ,J ES = L ES3 (I ES - D ES L ES1 ) -1 D ES L ES2 + L ES4 where I ES is the identity matrix.

5. A method for analyzing the stability of a power spring cascade system based on the component connection method according to claim 3, characterized in that The alternating current control includes a current controller, control delay, and LC filter.

6. The stability analysis method of the power spring cascade system based on the component connection method according to claim 5, characterized in that The linearized state space models of the current controller, control delay, and LC filter are specifically as follows: 1) The current controller adopts a PI controller, and its state space model is as follows: wherein, are the vectors of the state, input, and output variables of the current controller, respectively, and Δγ d , Δγ q is the output of the integrator of G i . The superscript c indicates that the variable is defined in the dq reference coordinate system of the converter; A i , B i , C i , D i are the system matrices of the single-phase power spring; 2) Apply the third-order Pade approximation to make the control delay model have sufficient accuracy within the Nyquist frequency (i.e., half of the sampling frequency), and at the same time minimize the complexity, which is specifically expressed as: where, e -τ·s is the model of the delay link, τ = 1.5T s is the delay time, usually 1.5 times the sampling period T s According to the above equation, the state space model for controlling the delay can be expressed as: wherein are respectively the vectors of the state, input, and output variables of the control delay link, and Δx del_d1 , Δx del_d2 , Δx del_d3 , Δx del_q1 , Δx del_q2 , Δx del_q3 are the state variables of the control delay link; 3) The state space model of the LC filter is as follows: wherein, is the vector of the states, inputs, and outputs of the LC filtering section, and According to steps 1)-3), deduce the composite model of the alternating current control, and its expression is: In the formula, is the combined vector of the states, inputs, and outputs of the components in the AC current control section, A acc , B acc , C acc , D acc is the diagonal matrix formed by the system matrices of the components in the AC current control section; where X acc = diag(X i , X del , X LC )(X = A, B, C, D) Define the input and output vectors of the entire alternating current control system as: In the formula, the superscript c indicates that the variable is defined in the dq reference coordinate system of the converter, and the connection between different components of the alternating current control system can be expressed as: where L acc1 , L acc2 , L acc3 , L acc4 is a matrix reflecting the interconnection relationships among the components of the AC current control system; the state-space model of the entire AC current control can be established as: Among them, F acc , G acc , H acc , J acc are as follows: F acc = A acc + B acc L acc1 (I acc - D acc L acc1 ) -1 C acc ,G acc = B acc L acc1 (I acc - D acc L acc1 ) -1 D acc L acc2 + B acc L acc2 , H acc = L acc3 (I acc - D acc L acc1 ) -1 C acc , J acc = L acc3 (I acc - D acc L acc1 ) -1 D acc L acc2 + L acc4 。 7. A method for analyzing the stability of a cascaded system of power springs based on the component connection method according to claim 3, characterized in that: The state - space model of the phase - locked loop has the following specific expression: wherein, are respectively the vectors of the states, inputs, and output variables of the phase-locked loop, and where Δx1 is the integral output of, and Δθ is the change in the output synchronization angle.

8. A method for analyzing the stability of a cascaded system of power springs based on the component connection method according to claim 3, characterized in that: The state - space model of the critical load voltage control is established as follows: The critical load voltage (i.e., the voltage at the PCC) v pcc Expressed in its dq components as: After linearizing the above formula, we can get: Where, V pccd0 and V pccq0 respectively represent the values of the dq components of the static operating point of the voltage v pcc ; The state - space model of the critical load voltage control is as follows: In the formula, are respectively the vectors of the state, input, and output variables for the critical load voltage control, and where the state variable Δx2 is the integral output of Δv pccmag -Δv pccmagref .

9. A method for analyzing the stability of a power spring cascade system based on a component connection method according to claim 3, characterized in that: The state - space model of the critical load voltage droop control is established as follows: 1) Adjust the reference voltage amplitude during the control process, and the specific method is as follows: where, v * pccmagref represents the rated value of the critical load voltage amplitude corresponding to the zero reactive power provided by the power spring; Q represents reactive power; n represents the droop coefficient; 2) The instantaneous reactive power output by the power spring is: According to the obtained instantaneous reactive power q, after being processed by a low - pass filter, the input of the droop controller, that is, the reactive power Q, is obtained as follows: where ω f is the cut-off frequency of the low-pass filter, and s is the Laplace transform factor; 3) Linearize the expressions in steps 1) - 2), and the state - space model of the critical load voltage droop control is obtained as: wherein, are respectively the vectors of the state, input, and output variables of the critical load voltage droop control, and .

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