CN111985066B - Average dynamic phasor model suitable for multi-voltage-class direct-current power distribution system - Google Patents

Average dynamic phasor model suitable for multi-voltage-class direct-current power distribution system Download PDF

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CN111985066B
CN111985066B CN202010659633.9A CN202010659633A CN111985066B CN 111985066 B CN111985066 B CN 111985066B CN 202010659633 A CN202010659633 A CN 202010659633A CN 111985066 B CN111985066 B CN 111985066B
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direct current
transformer
current
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model
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CN111985066A (en
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王振浩
李博文
成龙
李国庆
辛业春
张轶珠
李德鑫
刘座铭
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Electric Power Research Institute of State Grid Jilin Electric Power Co Ltd
Northeast Electric Power University
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Electric Power Research Institute of State Grid Jilin Electric Power Co Ltd
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    • G—PHYSICS
    • G06—COMPUTING; CALCULATING OR COUNTING
    • G06F—ELECTRIC DIGITAL DATA PROCESSING
    • G06F30/00—Computer-aided design [CAD]
    • G06F30/10—Geometric CAD
    • G06F30/18—Network design, e.g. design based on topological or interconnect aspects of utility systems, piping, heating ventilation air conditioning [HVAC] or cabling
    • G—PHYSICS
    • G06—COMPUTING; CALCULATING OR COUNTING
    • G06F—ELECTRIC DIGITAL DATA PROCESSING
    • G06F30/00—Computer-aided design [CAD]
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    • G—PHYSICS
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    • G06F30/367—Design verification, e.g. using simulation, simulation program with integrated circuit emphasis [SPICE], direct methods or relaxation methods
    • H—ELECTRICITY
    • H02—GENERATION; CONVERSION OR DISTRIBUTION OF ELECTRIC POWER
    • H02J—CIRCUIT ARRANGEMENTS OR SYSTEMS FOR SUPPLYING OR DISTRIBUTING ELECTRIC POWER; SYSTEMS FOR STORING ELECTRIC ENERGY
    • H02J1/00—Circuit arrangements for dc mains or dc distribution networks
    • H02J1/08—Three-wire systems; Systems having more than three wires
    • H02J1/082—Plural DC voltage, e.g. DC supply voltage with at least two different DC voltage levels
    • H—ELECTRICITY
    • H02—GENERATION; CONVERSION OR DISTRIBUTION OF ELECTRIC POWER
    • H02M—APPARATUS FOR CONVERSION BETWEEN AC AND AC, BETWEEN AC AND DC, OR BETWEEN DC AND DC, AND FOR USE WITH MAINS OR SIMILAR POWER SUPPLY SYSTEMS; CONVERSION OF DC OR AC INPUT POWER INTO SURGE OUTPUT POWER; CONTROL OR REGULATION THEREOF
    • H02M3/00—Conversion of dc power input into dc power output
    • H02M3/22—Conversion of dc power input into dc power output with intermediate conversion into ac
    • H02M3/24—Conversion of dc power input into dc power output with intermediate conversion into ac by static converters
    • H02M3/28—Conversion of dc power input into dc power output with intermediate conversion into ac by static converters using discharge tubes with control electrode or semiconductor devices with control electrode to produce the intermediate ac
    • H02M3/325—Conversion of dc power input into dc power output with intermediate conversion into ac by static converters using discharge tubes with control electrode or semiconductor devices with control electrode to produce the intermediate ac using devices of a triode or a transistor type requiring continuous application of a control signal
    • H02M3/335—Conversion of dc power input into dc power output with intermediate conversion into ac by static converters using discharge tubes with control electrode or semiconductor devices with control electrode to produce the intermediate ac using devices of a triode or a transistor type requiring continuous application of a control signal using semiconductor devices only
    • H02M3/33569—Conversion of dc power input into dc power output with intermediate conversion into ac by static converters using discharge tubes with control electrode or semiconductor devices with control electrode to produce the intermediate ac using devices of a triode or a transistor type requiring continuous application of a control signal using semiconductor devices only having several active switching elements
    • H02M3/33576—Conversion of dc power input into dc power output with intermediate conversion into ac by static converters using discharge tubes with control electrode or semiconductor devices with control electrode to produce the intermediate ac using devices of a triode or a transistor type requiring continuous application of a control signal using semiconductor devices only having several active switching elements having at least one active switching element at the secondary side of an isolation transformer
    • H02M3/33584—Bidirectional converters
    • G—PHYSICS
    • G06—COMPUTING; CALCULATING OR COUNTING
    • G06F—ELECTRIC DIGITAL DATA PROCESSING
    • G06F2113/00—Details relating to the application field
    • G06F2113/04—Power grid distribution networks

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Abstract

An average dynamic phasor model suitable for a multi-voltage-class direct current power distribution system belongs to the technical field of power system modeling and stability analysis. The invention aims to accurately match the direct current voltage and current which account for the main components in the direct current distribution system, is applied to stability analysis of a large-scale system, can accurately represent the alternating current in the converter through current reconstruction, and is suitable for an average dynamic phasor model which is suitable for a multi-voltage-class direct current distribution system and is suitable for scenes such as component internal current stress analysis and the like. The invention is based on a direct current transformer based on PI control and adopting a single phase shift modulation strategy, an original mathematical model is generated, the dynamic phasors of all variables are calculated, a state space equation set of an average dynamic phasor model of the direct current transformer is calculated, a system-level global model is constructed by a plurality of direct current transformer models through direct current circuit interconnection, and all variables can be written into a sum of steady state values and small signals for obtaining the small signal model. The invention has the advantages of science, rationality, strong practicability, high reliability, simplicity and accuracy.

Description

Average dynamic phasor model suitable for multi-voltage-class direct-current power distribution system
Technical Field
The invention belongs to the technical field of modeling and stability analysis of an electric power system.
Background
With the increase of distributed power sources and the wide application of direct current loads containing frequency conversion devices, the form of the traditional alternating current distribution system is changing, and the limitation of the power supply capacity of the traditional alternating current distribution system is also more remarkable. In the context of the rapid development of power electronics technology, a direct current power distribution network tends to become an important component in the future energy internet. When the current direct current power distribution network is mostly applied to a low-voltage level in a micro-grid mode, along with the investment of a large-scale centralized optical storage station and the requirements of industrial direct current loads, the construction of the multi-voltage-class direct current power distribution network with a medium-voltage level is widely focused.
The DC transformer is a key link of a DC power distribution network with multiple voltage classes. Direct-current transformers based on Dual-Active-Bridge (DAB) converters have strong applicability in power distribution systems due to the advantages of high power density, low element stress, electrical isolation, bidirectional operation and the like. In the analysis of the system level, such as stability evaluation, time domain simulation, etc., it is important to accurately describe the operation state of the DAB converter and its interactions with other converters of the system. Modeling the DAB converter as well as the entire dc system has been a significant challenge due to its high frequency ac commutation phase and ac state variables.
The dynamic phasor model is derived from the Fourier series complex form of the time domain signal, has certain advantages in harmonic characteristic analysis of the AC-DC conversion process, eliminates the discontinuity of the switching function, and is suitable for establishing a system-level global model. The existing mathematical model is mostly applied to analysis of a single converter or a flexible direct current power grid, and for a direct current power distribution network with multiple voltage levels and a direct current transformer, the simplicity and the accuracy of the global model are difficult to consider.
Disclosure of Invention
The invention aims to accurately match the direct current voltage and current which account for the main components in the direct current distribution system, is applied to stability analysis of a large-scale system, can accurately represent the alternating current in the converter through current reconstruction, and is suitable for an average dynamic phasor model which is suitable for a multi-voltage-class direct current distribution system and is suitable for scenes such as component internal current stress analysis and the like.
The method comprises the following steps:
Step one: based on a direct current transformer adopting a single phase shift modulation strategy based on PI control, an original mathematical model is generated: u 1 and u 2 are respectively input and output voltages of the direct-current transformer, and i 1 and i 2 respectively represent input and output currents of the direct-current transformer; c 1 and C 2 are respectively the primary side capacitor and the secondary side capacitor of the direct current transformer; the primary side winding current of the internal high-frequency transformer is i t, the leakage inductance is L t, the winding resistance is R t, and the transformation ratio is n 1; s 1、s2 respectively represent the switching functions of the DAB primary side full-bridge and the secondary side full-bridge, the inversion output voltage of the primary side full-bridge is v p, the inversion output voltage of the secondary side full-bridge is converted into v s, Representing a secondary side direct current bus voltage reference value; k p and k i are proportional integral parameters of the PI controller respectively, and gamma u is a PI control integral term;
By controlling the phase shift duty cycle d between v p and v s to achieve the change in the magnitude and direction of the transmission power, the switching function of the primary and secondary full bridges can be expressed as:
Wherein: t=1/2pi f s represents a switching period, f s is a switching frequency, the full-bridge inverter output voltage can be represented as v p＝u1s1,vs＝nu2s2, and then the state space equation of the open loop system is:
the model of the control system is expressed as:
Step two: dynamic phasors for each variable:
for a certain non-sinusoidal periodic time domain signal z (t) of the system, it is spread out in the form of a fourier series:
Wherein: omega s＝2πfs,fs is the inverter switching frequency; the angle bracket term < z > k (T), a dynamic phasor, represents the kth harmonic of the time domain signal, numerically equal to its running average over the switching period T, has:
The differentiation, product and conjugate of the dynamic phasors are:
Only the dc component and the primary dynamic phasors are retained, then z (t) is expressed as:
Wherein: Real and imaginary parts of the primary dynamic phasor, respectively; < z > 0 (t) represents a direct current component;
In the steady state operation, C LV →infinity, obtained from equation (3), i ts2-i2 =0, and the fourier series expansion of the dc transformer output current i LV is expressed as:
Considering only the direct current component (k=0), we get:
The dynamic phasors for the switching function s 2 are:
neglecting high-order dynamic phasors, to ensure that the value of < i 2>0 remains unchanged, the equivalent phase-shifting duty cycle can be used after error correction Instead of d:
The direct current component of the dynamic phasor < i t>0 = 0; assume that:
<ui>R＝<ui>I＝<ii>R＝<ii>I＝0,i＝1,2 (12);
Step three: the state space equation set of the average dynamic phasor model of the direct current transformer is as follows:
Wherein: </SUB >. 0、<·>R and </SUB >. I represent the DC component of the variable, the real and imaginary parts of the primary motion vector, respectively;
the actual phase shift duty cycle d is the output signal of the controller:
Equivalent phase shift duty cycle The correction equation of (2) is:
Wherein: θ=pi R t/(2Xt), where X t＝ωsLt represents transformer leakage reactance;
The large signal model of a single dc transformer, expressed in terms of the average dynamic phasor model, is then:
On the basis of the dynamic phasor modeling of the switching function, a more accurate expression of the transformer current is obtained by reconstructing a complete Fourier series;
the differential equation for the transformer current is:
expanding the full-bridge output voltage into a fourier series:
the full-bridge output voltage contains only odd harmonics, so the coefficients are:
for any k, the laplace transform is performed on equation (17):
Wherein:
The equation (20) is subjected to inverse Laplace transform by partial division expansion, and the result is:
the decay term in equation (21) is ignored, and thus its complete expression is:
Step four: the system-level global model is constructed by a plurality of direct current transformer models through direct current line interconnection: u 0、i0 represents the voltage and current of the equivalent direct current source, and L ij、Rij represents the equivalent inductance and resistance of the direct current line between the nodes i and j (i < j), respectively; the relevant parameters of the mth direct current transformer are indicated by the angle marks m,
The line current between nodes i, j is expressed as:
Wherein: Direct current components of the voltages of the nodes i and j are respectively represented;
when the output end of the DC transformer m is connected with the node n, the output voltage is equal to the node voltage, namely The output current is as follows:
similarly, when the input end of the DC transformer m is connected with the node n, there are The input current is as follows:
Thus, the global system average dynamic phasor large signal model is expressed as:
wherein: x m (m=1, 2 … M), U m is the state phasor, algebraic state variable and input phasor of the mth direct current transformer respectively;
The input phasors of the system are:
step five: to obtain a small signal model, each variable is written as the sum of steady state values and small signals:
Expanding f S (·) taylor to first order, there are:
In the case of a steady-state condition, Then:
similarly, let g S (·) taylor be expanded to first order and consider 0=g S(XS,DS,US) to have:
With respect to Is rewritten as:
Substituting the formula (32) into the formula (30) to obtain a global small signal model, wherein the global small signal model is as follows:
The partial guide matrix in the formula has the following form:
1) Partial guide matrix The subarrays are respectively as follows:
① Sub-array
② Sub-arrayColumn vector and subarray of (a)Is the row vector of (2)
Assuming that the input end and the output end of the direct current transformer m are p and q respectively, the general formula of each element in the formula is:
③ Sub-array Is a diagonal array, the diagonal elements are
2) Partial guide matrixMiddle subarray
3) Partial guide matrixThe subarrays are as follows:
①
Wherein each element is as follows:
② Sub-array
4) Partial guide matrixContaining diagonal elements only
The method is applied to the direct current distribution system with other modulation strategy converters by updating corresponding matrix block elements.
The method has low complexity, is suitable for an average dynamic phasor model of the multi-voltage-class direct current power distribution system, solves the problem of balancing the mathematical modeling accuracy and simplicity of the multi-converter system, and reduces the influence of the discontinuity of the switching function of the converter on the time domain stability analysis of the system; the direct-current voltage and current which are mainly components in the direct-current distribution system can be accurately matched, the direct-current voltage and current matching method is applied to stability analysis of a large-scale system, the alternating current in the converter can be accurately represented through current reconstruction, and the direct-current voltage and current matching method is applicable to scenes such as current stress analysis in elements. The invention considers reducing the complexity of the model and ignores the higher dynamic phasors of each state variable; taking the equivalent phase shift duty ratio of the converter as a model component number state variable to introduce an error correction equation in consideration of improving the accuracy of the model; taking a scene with high requirement on the alternating current accuracy of the converter into consideration, and carrying out complete Fourier series reconstruction on the scene; and (3) considering the applicability of the model, deducing a general formula of the extensible global small signal model partial guide matrix. Has the advantages of science, rationality, strong practicability, high reliability, simplicity and accuracy.
Drawings
Fig. 1 is a diagram of an equivalent circuit of a direct current transformer based on a DAB converter and an explanatory diagram of a control system;
Fig. 2 is a waveform diagram of a DAB converter single phase shift modulation strategy;
FIG. 3 is an illustration of a typical multi-voltage class DC power distribution system topology;
FIG. 4 is a simplified equivalent structural illustration of a multi-voltage class DC power distribution system;
FIG. 5 is a global system feature root schematic;
fig. 6 is a schematic diagram of a system root track when parameters of the converter station 3 are changed; FIG. 6a is A variation graph; FIG. 6b is a graph of f t 3 variation;
fig. 7 is a schematic diagram of a system root track with other converter station parameters changed; fig. 7a is a graph of the converter station 1,2 parameter variation;
fig. 7b is a graph of the change in parameters of the converter station 4;
FIG. 8 is a graph of transformer winding current versus time;
FIG. 9 is a graph of system current variation during a sudden load increase; FIG. 9a is a line current A graph; FIG. 9b is a line currentA graph; FIG. 9c is a line currentA graph; FIG. 9d is a line currentA graph;
FIG. 10 is a graph of system voltage change during a sudden load increase; FIG. 10a is a node voltage A graph; FIG. 10b is a node voltageA graph; FIG. 10c is a node voltageA graph; FIG. 10d is a graph of DC transformer output voltage;
FIG. 11 is a graph of system current voltage variation at the time of a line break fault; FIG. 11a is a line 3 current A graph; FIG. 11b is a line 4 currentA graph; FIG. 11c is an output currentA graph; FIG. 11d is a node voltageA graph; FIG. 11e is a node voltageGraph diagram.
Detailed Description
The method comprises the following steps:
Step 1: generating an original mathematical model of the DC transformer based on PI control by adopting a single phase shift modulation strategy; fig. 1 is an equivalent circuit diagram of a direct current transformer based on a DAB converter and an explanatory diagram of a control system. In fig. 1, u 1 and u 2 are dc transformer input/output voltages, respectively, and i 1 and i 2 represent dc transformer input/output currents, respectively; c 1 and C 2 are respectively the primary side capacitor and the secondary side capacitor of the direct current transformer; the primary side winding current of the internal high-frequency transformer is i t, the leakage inductance is L t, the winding resistance is R t, and the transformation ratio is n 1; s 1、s2 respectively represent the switching functions of the DAB primary side full-bridge and the secondary side full-bridge, the primary side full-bridge inversion output voltage is v p, and the secondary side full-bridge inversion output voltage is converted into the primary side v s. Representing a secondary side direct current bus voltage reference value; k p and k i are proportional integral parameters of the PI controller, and γ u is PI control integral term.
Fig. 2 is a waveform diagram of a single phase shift modulation strategy of the DAB converter. As can be seen from fig. 2, under the single phase shift modulation strategy, the primary side and secondary side full-bridge inverter output voltages v p、vs are two-level square waves with a duty cycle of 0.5. The change in the magnitude and direction of the transmission power is achieved by controlling the phase shift duty cycle d between v p and v s. The switching function of the primary and secondary full bridges can be expressed as:
Wherein: t=1/2pi f s denotes a switching period, and f s is a switching frequency.
The full bridge inverter output voltage may be represented as v p＝u1s1,vs＝nu2s2. The state space equation for the open loop system is:
the mathematical model of the control system can be expressed as:
Step 2: calculating the dynamic phasors of all variables, and averaging the switching function and all the variables by neglecting the higher dynamic phasors with lower influence on the direct current component, only keeping the direct current component and +/-1 dynamic phasors and considering the error correction of the phase shift duty ratio; for a certain non-sinusoidal periodic time domain signal z (t) of the system, it can be expanded into the form of a fourier series:
Wherein: omega s＝2πfs,fs is the inverter switching frequency; the angle bracket term < z > k (T), a dynamic phasor, represents the kth harmonic of the time domain signal, numerically equal to its running average over the switching period T, has:
The average dynamic phasor model takes the dynamic phasor < z > k (t) as a state variable of the model and averages the opening Guan Hanshu, so that the obtained model is continuous in the time domain, and is beneficial to time domain analysis of the system. By approximately omitting some unimportant components in the dynamic phasors, the complexity of the model can be reduced, and in fact, the conventional state space averaging method is a special case in which only the direct current component (k=0) is retained.
The differentiation, product and conjugate of the dynamic phasors are:
only the dc component and the primary dynamic phasors are preserved, then z (t) can be expressed as:
Wherein: Real and imaginary parts of the primary dynamic phasor, respectively; < z > 0 (t) represents a direct current component.
This approximation condition, by ignoring higher order dynamic phasors, reduces the complexity of the final model, but affects the accuracy of the system. Therefore, in modeling a direct current transformer based on DAB, certain algebraic constraints are required to correct errors. In a closed loop system, the controller stably controls the low voltage dc bus voltage at a reference value, so that an error is usually generated on the phase shift duty cycle d of the system input signal.
In the steady state operation, C LV →infinity, as can be obtained from equation (3), i ts2-i2 =0, and fourier series expansion of the dc transformer output current i LV can be expressed as:
Considering only the direct current component (k=0), we get:
The dynamic phasors for the switching function s 2 are:
neglecting high-order dynamic phasors, to ensure that the value of < i 2>0 remains unchanged, the equivalent phase-shifting duty cycle can be used after error correction Instead of d:
The dc transformer winding current can be considered to be purely ac, so that the dc component of its dynamic phasor < i t>0 = 0. The input/output variable of the dc transformer is dc, and its dynamic change is far slower than the ac component in the DAB module, then it can be assumed that:
<ui>R＝<ui>I＝<ii>R＝<ii>I＝0,i＝1,2 (12)
step 3: in summary, the state space equation set of the average dynamic phasor model of the direct current transformer is as follows:
Wherein: </SUB >. 0、<·>R and </SUB >. I represent the DC component of the variable, the real and imaginary parts of the primary motion vector, respectively.
The actual phase shift duty cycle d is the output signal of the controller:
Equivalent phase shift duty cycle The correction equation of (2) is:
Wherein: θ=pi R t/(2Xt), where X t＝ωsLt represents transformer leakage reactance.
The large signal model of a single dc transformer, expressed in terms of the average dynamic phasor model, is then:
The dynamic phasor average modeling corrects errors generated by neglecting higher harmonics through constraint equations, and can better represent direct current variables in a system in a situation of small influence of voltage ripple such as a well-designed closed-loop system or large-scale system simulation. However, ac variables in the system, such as transformer current, are sinusoidal approximations calculated by < i t>R and < i t>I in equation (13), and are not suitable for evaluating situations where current accuracy requirements such as component current stress are high. On the basis of the dynamic phasor modeling of the switching function, a more accurate expression of the transformer current is obtained by reconstructing the complete Fourier series.
The differential equation for the transformer current is:
expanding the full-bridge output voltage into a fourier series:
The dc transformer input-output voltage in a single period can be regarded as a constant, and the full-bridge output voltage contains only odd harmonics, so the coefficients are:
for any k, the laplace transform is performed on equation (17):
Wherein:
The equation (20) is subjected to inverse Laplace transform by partial division expansion, and the result is:
considering the periodicity of the transformer current, the decay term in equation (21) can be ignored, and thus its complete expression is:
Step 4: the system-level global model is constructed by interconnecting a plurality of direct-current transformer models through direct-current lines and generally consists of N nodes and M direct-current transformers. FIG. 3 is a schematic diagram of a typical multi-voltage class DC power distribution system topology, in which a medium voltage DC bus is interconnected with an AC power distribution system via a rectifier or an inverter on the one hand, and connected with medium voltage DC loads of various voltage classes, a centralized photovoltaic energy storage station, and low voltage DC buses of a DC micro-grid via DC transformers of different transformation ratios on the other hand; the low-voltage direct current bus is also connected with low-voltage loads or distributed photovoltaics with different voltage grades.
Fig. 4 is a simplified structure of a six-node four-dc transformer multi-voltage class dc distribution system. In fig. 4, u 0、i0 represents the voltage and current of the equivalent dc source, respectively, which are equal in value to the output voltage and output current of the ac distribution network through the rectifier; l ij、Rij respectively represents the equivalent inductance and resistance of the direct current circuit between the nodes i and j (i < j); the relevant parameter of the mth DC transformer is indicated by the angle mark m, e.gIs the output current DC component of the mth DC transformer.
The line current between nodes i, j can be expressed as:
Wherein: the dc components of the voltages at nodes i, j are shown, respectively.
In order to construct the system-level model, the junction points of the direct current transformer and other parts of the system should be represented in the form of state variables. When the output end of the DC transformer m is connected with the node n, the output voltage is equal to the node voltage, namelyThe output current is as follows:
similarly, when the input end of the DC transformer m is connected with the node n, there are The input current is as follows:
Thus, the global system average dynamic phasor large signal model can be expressed as:
wherein: x m (m=1, 2 … M), U m is the state phasor, algebraic state variable and input phasor of the mth dc transformer, respectively.
The input phasors comprise all reference voltages of the dc transformer control system and current variables not described by the current equation, preferably the input phasors of the example system of fig. 4 are:
step 5: to obtain a small signal model, variables can be written as the sum of steady state values and small signals:
Expanding f S (·) taylor to first order, there are:
In the case of a steady-state condition, Then:
similarly, let g S (·) taylor be expanded to first order and consider 0=g S(XS,DS,US) to have:
With respect to The expression of (c) can be rewritten as:
Substituting the formula (32) into the formula (30) to obtain a global small signal model, wherein the global small signal model is as follows:
The partial guide matrix in the formula has the following form:
1) Partial guide matrix The subarrays are respectively as follows:
① Sub-array
② Sub-arrayColumn vector and subarray of (a)Is the row vector of (2)
Assuming that the input end and the output end of the direct current transformer m are p and q respectively, the general formula of each element in the formula is:
③ Sub-array Is a diagonal array, the diagonal elements are
2) Partial guide matrixMiddle subarray
3) Partial guide matrixThe subarrays are as follows:
①
Wherein each element is as follows:
② Sub-array
4) Partial guide matrixContaining diagonal elements onlyEach matrix block of the partial guide matrix in the global average dynamic phasor small signal model corresponds to the relevant converter and the circuit, so that the method has expansibility, and can be applied to a direct current distribution system with other modulation strategy converters by updating corresponding matrix block elements.
Fig. 5-11 illustrate an embodiment of the present invention in a Matlab/Simulink platform, which is specifically described below:
the average dynamic phasor model of the 6-node simplified multi-voltage-class direct current power distribution system shown in fig. 4 is built, and specific parameters are shown in table 1.
TABLE 1 System parameter Table
Fig. 5 is a schematic diagram of a feature root of the global system, it can be seen that the real part of the feature root is negative, the system is stable, and there are 9 oscillation modes in total, wherein mode 1 to mode 4 are dominant modes, and respectively correspond to 4 converter stations, and the analysis of corresponding feature values is shown in table 2. Fig. 6-7 are schematic diagrams of the system characteristic root trajectories of the converter station 3 and other converter station dc transformers when the leakage inductance and frequency are changed. As can be seen from the figure, when the leakage inductance of the direct current transformer of the converter station increases, the corresponding mode moves rightwards, and the system is kept stable; when the frequency of the transformer is increased, the corresponding mode moves in a direction away from the real axis, the damping is reduced, the resonance frequency is correspondingly increased, the stability of the system is weakened, and the system can still be kept stable; other modalities are not substantially affected. The method can be well applied to stability analysis of the multi-voltage-class direct current power distribution system.
TABLE 2 eigenvalue analysis
And (3) constructing an electromagnetic transient model according to the parameters of the table 1, and comparing and analyzing the average dynamic phasor model with the electromagnetic transient model. Under steady-state conditions, the bus voltage at the output ends of the direct current transformers 1 and 2 is 400V, and the output rated voltages of the direct current transformers 3 and 4 are 48V and 110V respectively. Fig. 8 to 10 are schematic diagrams of the current-voltage change of the system when the end load of the converter station 3 suddenly increases and the front-end line 4 of the converter station 4 breaks down.
① The load of the converter station 3 suddenly increases; when t=4ms, the resistive load power carried by the tail end of the converter station 3 rises from 240W to 480W, the output voltage of each direct current transformer is restored to a stable state under the action of a control system, and the change condition of the current and the voltage of the system is shown in fig. 8 and 9. The output current of the DC transformer 3 rises from 5A to 10A, and the current of the input terminal line 3 rises from 0.76A to 1.49A. Since line 4 is connected to node 4 along line 3, but the line end load is unchanged, the line current returns to 1.6A after a brief disturbance. The voltage of the node 4, namely the output voltage of the direct current transformer 2, can still be stabilized at 400V after disturbance under the action of a control system; as the load suddenly increases, the line 3 loss increases with current, and the voltage at node 5 decreases from 399.52V to 399.04V; likewise, the voltage at node 6 can be maintained at around 399.50V after the perturbation. The disturbance of the input voltage of the transformer has little influence on the output voltage, and can be stabilized at a set reference value.
② Line 4 break fault; in the initial condition, dc transformers 3 and 4 are loaded with 320W and 550W of resistive load, respectively. When t=0.2 s, the line 4 breaks, the dc transformer 4 is taken out of operation, and the system current and voltage change condition is shown in fig. 10. The current of the line 4 is reduced from 1.6A to 0 after the occurrence of the disconnection fault; the line 3 current returns to 0.99A after the disturbance. The node voltage can be stabilized at the original amplitude after a short steep rise.
From the simulation results of the two situations, it can be seen that in the aspect of transient process, the electromagnetic transient model contains more power electronic devices, and the modulation triggering unit is far more complex than the dynamic phasor average model, so that the disturbance is larger; in a stable state, the provided model can be accurately matched with the amplitude of each variable, and is more suitable for analysis under a large time scale. The average error rates of the proposed model and the electromagnetic transient model are shown in table 3. From the values, the simulation error of the voltage is almost zero, and the error of the current is slightly larger than the voltage, which shows that the accuracy of the invention is higher.
Table 3 simulation error Rate of System (%)
The dynamic phasor average model, the electromagnetic transient model and the current waveform of the winding of the direct-current transformer adopting the current reconstruction method are shown in fig. 11. As can be seen from the figure, the transformer current in the dynamic phasor average model is only an approximate sine waveform compared with the actual value; by means of the current reconstruction method, the current of the transformer winding can be accurately represented, and the method can accurately restore the AC current in the converter.

Claims (1)

1. An average dynamic phasor modeling method suitable for a multi-voltage-class direct current power distribution system is characterized by comprising the following steps of: the method comprises the following steps:
Step one: based on a direct current transformer adopting a single phase shift modulation strategy based on PI control, an original mathematical model is generated: u 1 and u 2 are respectively input and output voltages of the direct-current transformer, and i 1 and i 2 respectively represent input and output currents of the direct-current transformer; c 1 and C 2 are respectively the primary side capacitor and the secondary side capacitor of the direct current transformer; the primary side winding current of the internal high-frequency transformer is i t, the leakage inductance is L t, the winding resistance is R t, and the transformation ratio is n 1; s 1、s2 respectively represent the switching functions of the DAB primary side full-bridge and the secondary side full-bridge, the inversion output voltage of the primary side full-bridge is v p, the inversion output voltage of the secondary side full-bridge is converted into v s, Representing a secondary side direct current bus voltage reference value; k p and k i are proportional integral parameters of the PI controller respectively, and gamma u is a PI control integral term;
By controlling the phase shift duty cycle d between v p and v s to achieve the change in the magnitude and direction of the transmission power, the switching function of the primary and secondary full bridges can be expressed as:
Wherein: t=1/2pi f s represents a switching period, f s is a switching frequency, the full-bridge inverter output voltage can be represented as v p＝u1s1,vs＝nu2s2, and then the state space equation of the open loop system is:
the model of the control system is expressed as:
Step two: dynamic phasors for each variable:
for a certain non-sinusoidal periodic time domain signal z (t) of the system, it is spread out in the form of a fourier series:
Wherein: omega s＝2πfs,fs is the inverter switching frequency; the angle bracket term < z > k (T), a dynamic phasor, represents the kth harmonic of the time domain signal, numerically equal to its running average over the switching period T, has:
The differentiation, product and conjugate of the dynamic phasors are:
Only the dc component and the primary dynamic phasors are retained, then z (t) is expressed as:
Wherein: Real and imaginary parts of the primary dynamic phasor, respectively; < z > 0 (t) represents a direct current component;
In the steady state operation, C 2 →infinity, obtained from equation (2), i ts2-i2 =0, and the fourier series expansion of the dc transformer output current i 2 is expressed as:
Considering only the direct current component, i.e. k=0, we get:
The dynamic phasors for the switching function s 2 are:
neglecting high-order dynamic phasors, to ensure that the value of < i 2>0 remains unchanged, the equivalent phase-shifting duty cycle can be used after error correction Instead of d:
The direct current component of the dynamic phasor < i t>0 = 0; assume that:
<ui>R＝<ui>I＝<ii>R＝<ii>I＝0,i＝1,2 (12);
Step three: the state space equation set of the average dynamic phasor model of the direct current transformer is as follows:
Wherein: </SUB >. 0、<·>R and </SUB >. I represent the DC component of the variable, the real and imaginary parts of the primary motion vector, respectively;
the actual phase shift duty cycle d is the output signal of the controller:
Equivalent phase shift duty cycle The correction equation of (2) is:
Wherein: θ=pi R t/(2Xt), where X t＝ωsLt represents transformer leakage reactance;
The large signal model of a single dc transformer, expressed in terms of the average dynamic phasor model, is then:
On the basis of the dynamic phasor modeling of the switching function, a more accurate expression of the transformer current is obtained by reconstructing a complete Fourier series;
the differential equation for the transformer current is:
expanding the full-bridge output voltage into a fourier series:
the full-bridge output voltage contains only odd harmonics, so the coefficients are:
for any k, the laplace transform is performed on equation (17):
Wherein:
The equation (20) is subjected to inverse Laplace transform by partial division expansion, and the result is:
the decay term in equation (21) is ignored, and thus its complete expression is:
Step four: the system-level global model is constructed by a plurality of direct current transformer models through direct current line interconnection: u 0、i0 represents the voltage and current of the equivalent direct current source, and L ij、Rij represents the equivalent inductance and resistance of the direct current line between the nodes i and j, respectively, wherein i is less than j; the relevant parameters of the mth direct current transformer are indicated by the angle marks m,
The line current between nodes i, j is expressed as:
Wherein: Direct current components of the voltages of the nodes i and j are respectively represented;
when the output end of the DC transformer m is connected with the node n, the output voltage is equal to the node voltage, namely The output current is as follows:
similarly, when the input end of the DC transformer m is connected with the node n, there are The input current is as follows:
Thus, the global system average dynamic phasor large signal model is expressed as:
wherein: x m (m=1, 2 … M), U m is the state phasor, algebraic state variable and input phasor of the mth direct current transformer respectively;
The input phasors of the system are:
step five: to obtain a small signal model, each variable is written as the sum of steady state values and small signals:
Expanding f S (·) taylor to first order, there are:
In the case of a steady-state condition, Then:
similarly, let g S (·) taylor be expanded to first order and consider 0=g S(XS,DS,US) to have:
With respect to Is rewritten as:
Substituting the formula (32) into the formula (30) to obtain a global small signal model, wherein the global small signal model is as follows:
The partial guide matrix in the formula has the following form:
1) Partial guide matrix The subarrays are respectively as follows:
① Sub-array
② Sub-arrayColumn vector and subarray of (a)Is the row vector of (2)
Assuming that the input end and the output end of the direct current transformer m are p and q respectively, the general formula of each element in the formula is:
③ Sub-array Is a diagonal array, the diagonal elements are
2) Partial guide matrixMiddle subarray
3) Partial guide matrixThe subarrays are as follows:
① Wherein each element is as follows:
② Sub-array
4) Partial guide matrixContaining diagonal elements onlyThe method is applied to the direct current distribution system with other modulation strategy converters by updating corresponding matrix block elements.
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