CN110875600B - Approximate analysis model for dynamic frequency response of two-machine equivalent power system - Google Patents
Approximate analysis model for dynamic frequency response of two-machine equivalent power system Download PDFInfo
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Abstract
A dynamic frequency response approximate analysis model of a two-machine equivalent power system belongs to the field of primary frequency modulation of the power system. The method realizes the approximate analytic solution of the equivalent power system of the two machines by changing the equivalence of the multi-machine power system into two-machine systems (two areas) and simplifying the model by methods of ring solving, approximate substitution and the like, and gives out an approximate analytic expression in the time domain of the model to express the dynamic frequency response process in the two-machine systems. The method can quickly and accurately calculate the dynamic frequency response process of the equivalent power system of the two computers on the basis of considering the time-space distribution characteristics of the frequency, is an expansion form of a single model applied in the actual multi-computer power system, can be directly used for quickly calculating the frequency response characteristics of the power system, can also be directly added with a target equation or constraint conditions to carry out the research related to the optimized operation and protection control of the power system, avoids numerical calculation, and has practical value and wide application prospect.
Description
Technical Field
The invention belongs to the field of primary frequency modulation of an electric power system, and relates to a dynamic frequency response approximate analysis model of a two-machine equivalent electric power system.
Background
The frequency is a key index for reflecting active power balance, and the calculation of the dynamic response process of the frequency is the basis for the research of safety and stability analysis, optimized operation, protection control and the like of the power system. At present, the frequency response capability of a system is remarkably reduced due to the grid connection of a large-capacity unit and high-proportion new energy, and the frequency situation is more severe after disturbance; the construction of a large-scale interconnected power grid makes the dynamic frequency response process after disturbance more complicated; the safety and stability control system based on the wide area measurement technology and the high-speed communication technology realizes real-time decision and real-time control and requires to quickly obtain the frequency situation. Therefore, it is particularly significant to research a fast calculation model capable of reflecting the dynamic frequency response process of the complex power system.
When a high-power loss is caused by faults such as machine dropping, direct current blocking and the like in a complex power system, a primary frequency modulation device supports mutual cross-region in a large range, and the phenomenon of inter-region oscillation occurs, mainly including power angle oscillation, power oscillation on a connecting line, frequency oscillation (also called as the space-time distribution characteristic of frequency) above the system frequency caused by the influence of the power angle oscillation and the power oscillation on the connecting line on a frequency dynamic trajectory, and the actions of protective devices such as system low-frequency load shedding and the like can be influenced in serious cases, even a chain reaction is caused, and the safe and stable operation of the system is threatened. Through a wide-area measurement system, the dynamic state of the global frequency can be visually observed. From the recorded image, the frequencies of different nodes after disturbance, especially the frequencies of the disturbed area and the area far away from the disturbance, show obvious differences. The spatio-temporal distribution characteristics of the frequencies should be taken into account in the calculation.
The single-machine equivalent model ignores the influence of a system network structure and reactive power-voltage, has a simple structure and is quick in calculation, so that the single-machine equivalent model is widely introduced. An Average Frequency response Model (ASFmodel, average System Frequency Model) adopts equivalent rotors, reserves a prime motor-speed regulator of each generator, can describe the Average Frequency dynamic of the System, and still needs simulation calculation. A System Frequency Response Model (SFR Model) further aggregates the equivalent of the prime motor and the speed regulator, realizes the solution of the analysis and lays the foundation of quick calculation. The analytical model has clear physical meaning, can directly calculate the result according to the input, and is suitable for the situation needing rapid calculation, but the single-machine model ignores the network structure and cannot take the space-time distribution characteristics in the dynamic frequency response process into account.
Therefore, the invention provides a dynamic frequency response approximate analysis model of a two-machine equivalent power system, a direct current network equation is added on the basis of a single-machine equivalent model, the frequency response analysis model of the two-machine equivalent power system is established on the basis of a partition theory of the power system, model reduction is carried out by methods of ring solving, approximate substitution and the like, finally, an approximate analysis expression of the dynamic frequency of an equivalent region is solved by adopting a real-mode analysis theory, and the dynamic frequency response process of a node can be rapidly and accurately calculated on the basis of considering the time-space distribution characteristics of the frequency. The model can be directly used for rapidly calculating the frequency response characteristics of the power system, and can also be directly added into a target equation or a constraint condition for relevant research, thereby avoiding numerical calculation and having practical value and wide application prospect.
Disclosure of Invention
The invention provides a dynamic frequency response approximate analysis model of a two-machine equivalent power system. The method is characterized in that a multi-machine power system is equivalent to a two-machine system (two regions), model simplification is carried out by methods of ring solving, approximate substitution and the like, approximate analytic solution of the two-machine equivalent power system is achieved, and an approximate analytic expression in a model time domain is given to express a dynamic frequency response process in the two-machine system.
In order to achieve the purpose, the invention adopts the technical scheme that:
a dynamic frequency response approximate analysis model of a two-machine equivalent power system specifically comprises the following steps:
s1: for a multi-machine power system, transient reactance branches in a generator are added into a network model, and a network equation expression based on direct current flow is shown as a formula (1):
in the formula, P g ，P G ，P L Respectively outputting electromagnetic power P for the generator gi The generator terminal bus injection power P Gi Non-generator node injected power P Lj A column vector of components; b GG ，B GL ，B LG ，B LL Sub-arrays of admittance matrix B for DC power flow calculation, B gg ，B gG ，B Gg For admittance between an extended node and an original generator nodeMatrix unit is pu and S B Is a reference capacity; delta. For the preparation of a coating g 、θ G 、θ L Respectively, generator rotor angle delta gi The phase angle theta of the bus voltage at the generator end Gi Non-generator node voltage phase angle theta Lj The unit of the formed column vector is rad; g, G and L are respectively internal potential virtual nodes G of the generator i Generator end bus node G i Load node L j (ii) a m is the number of generator nodes, l is the number of load nodes, i = 1-m, j = m + 1-m + l.
After transient reactance of generator node is added in an expansion mode, generator node G in original load flow calculation i To a non-generator node. When the node operates in a steady state, the injection power of the node is 0, and if the generator is disconnected, the injection power of the corresponding generator terminal bus is negative generator disconnection power. But unlike the load node, the node does not have power frequency regulation capability.
Equation (1) is rewritten and expressed as an incremental equation:
in the formula (I), the compound is shown in the specification,
for generator end node G i And a non-generator node L j Forming an extended non-generator node set, namely an original system node, wherein the node number n = m + l; Δ represents the increment of the corresponding variable; x' di And the transient reactance of the generator node is expressed in pu.
The expression for the generator electromagnetic power increment that can be derived from equation (2) is:
ΔP g ＝B S Δδ g +B L ΔP L (3)
in the formula, B S Is an inter-machine oscillatory matrix, B L A disturbance power distribution matrix, which is respectively:
s2: the network equation is based on a formula (3) in S1, the units all adopt SFR models, a multi-machine system is simplified into a two-machine equivalent system, and the incremental equation is expressed as follows:
in the formula, the parameter with the "'" at the upper left represents a two-machine equivalent system, and the parameter without the "'" represents a multi-machine system; h 'and D' are generator inertia time constants H 'respectively' k Equivalent generator damping D' k Forming a diagonal matrix with the unit of s; b' S Is an inter-machine oscillation matrix with the unit of pu; delta P' d ，ΔP′ M Distributing Power Δ P 'for disturbance' dk Prime mover-governor augmented power Δ P' Mk The unit of the formed column vector is pu; delta delta 'is generator rotor angle delta' k The unit of the formed column vector is rad; delta omega 'is generator rotor angular speed omega' k The unit of the formed column vector is rad/s; omega 0 Is the rated generator angular frequency; s is the complex frequency; k is the equivalent unit number, and k =1,2. Wherein delta P' d ，ΔP′ M The expression of (c) is as follows:
ΔP′ M ＝-K′ m R′ -1 (1+T′ R s) -1 (1+F′ H T′ R s)Δω′ (8)
of formula (II) F' H ，K′ M R ' are respectively a work doing proportion F ' of a high-pressure cylinder of the generator ' Hk Mechanical power gain coefficient K' Mk Reheat time constant R k Formed diagonal matrixNo dimension; t' R Is a difference-adjusting coefficient T' Rk Forming a diagonal matrix with the unit of s;
distributing a matrix for the disturbance power, wherein the unit is pu;
to perturb power
The unit of the formed column vector is pu.
S3: let the power system multi-machine frequency response model parameter be X = { H, D, R -1 }，Y＝{F H ,T R The equivalent parameters of the two-machine equivalent frequency response model are X '= { H', D ', R' -1 The calculation expressions of Y ' = { F ', T ' } are:
of formula (II) to (III)' Nk Rated active power for the kth generator; v k The system is a unit set contained in the kth equivalent unit, and when only one unit is contained in the set, the actual system is a dual-machine model. Wherein λ is i The influence factors of the unit parameters on the equivalent system parameters are as follows:
inter-machine oscillation matrix B 'of two-machine equivalent model' S Perturbation distribution matrix
And disturbance power
The expression of (a) is:
s4: to delta P' M Simplifying the process:
ΔP′ M ＝-R′ -1 F′ H K′ m Δω′-K′ m R′ -1 (1+T′ R s) -1 (1-F′ H )Δω′ (13)
equation (6) can be:
2H′Δδ′s 2 +B′ S Δδ′＝ΔP′ d -ΔP′ D +ΔP′ T (14)
in formula (I), proportional feedback power increment delta P' D And a first-order inertia link feedback power increment delta P' T The expression of (c) is as follows:
ΔP′ D ＝D′ eq Δω′ (15)
ΔP′ T ＝-K′ m R′ -1 (1+T′ R s) -1 (1-F′ H )Δω′ (16)
wherein, D' eq Equivalent damping for analytical model
D′ eq ＝R′ -1 F′ H K′ m +D′ (17)
S5: SFR model parameters X = { H, D, R corresponding to multi-machine frequency response model of power system -1 }，Y＝{F H ,T R Substituting the equation on the left side of the equation (9) to obtain a set V = { V = } 1 ∪V 2 The solution of the corresponding parameter.
The frequency response analytical solution of the SFR model obtained by the ASFR polymerization method has higher approximation degree compared with the frequency response curve of the ASF. Therefore, the frequency difference delta omega 'of the ASF model is subjected to loop release at the feedback position, and the equivalent replacement is carried out by adopting the analytical frequency expression of the SFR, so that the proportional feedback power increment delta P' in the ASF model D And first order inertiaLink feedback power increment delta P ″ T The approximate analytical expression of (a) is as follows:
ΔP″ D ＝(R′ -1 F′ H K′ m +D′)Δω (18)
ΔP″ T ＝-K′ m R′ -1 (1+T′ R s) -1 (1-F′ H )Δω (19)
by nature of SFR, Δ ω is the system center frequency of inertia, which passes through the center of the Δ ω' curve and cannot describe the system frequency oscillation and therefore the error. Proportional feedback directly brings the error into the calculation, while first-order inertia element feedback reduces frequency oscillation accumulation, and as frequency offset accumulation increases, the influence of oscillation error decreases. Thus, Δ P ″' is retained D And use of Δ P ″) T Substitute delta P' T 。
Let Δ P d ＝ΔP′ d1 +ΔP′ d1 The equivalent substitution process using the ASF model and the SFR model is shown in fig. 2.
Therefore, formula (14) is rewritten as:
2H′Δδ′s 2 +D′ eq Δδ′s+B′ S Δδ′＝ΔP′ d +ΔP″ T (20)
the simplified two-machine equivalent frequency response model of equation (20) is shown in fig. 3.
S6: according to the modal superposition method, the undamped free vibration equation of equation (20) is:
2H′Δδ′s 2 +B′ S Δδ′＝0 (21)
two characteristic roots and corresponding characteristic vectors of the equation can be obtained as follows:
a 1 ＝[1,1] T ,a 2 ＝[-2H′ 2 ,2H′ 1 ] T (23)
the modal matrix is then:
introducing a regular transformation:
Δδ′＝Φy (25)
substituting (25) into (20) and multiplying by phi on both sides of the equation T . Assuming that the system is in a steady state before disturbance, performing inverse Laplace transformation, and obtaining a time domain expression of a kinetic equation described by a modal coordinate as follows:
in the formula, M is a main mass matrix, K is a main rigidity matrix, C is a modal damping matrix, F (t) is excitation after transformation, and the expressions are respectively:
M＝Φ T (2H′)Φ＝diag[2H′ 1 +2H′ 2 ,4H′ 1 H′ 2 (2H′ 1 +2H′ 2 )] (27)
K＝Φ T B′ S Φ＝diag[0,k(2H′ 1 +2H′ 2 ) 2 ] (28)
s7: when 2H' 2 D′ eq1 ＝2H′ 1 D′ eq2 When C is classical damping, formula (26) is a decoupling equation, otherwise, the velocity term is coupled, a forced decoupling method is adopted, and the non-diagonal elements of C are ignored, then formula (26) can be expressed as:
wherein, equation (31) is the vibration equation of the system inertia center, and equation (32) is the oscillation equation of the system.
Solving equation (31) yields:
in the formula:
because, in formula (32),. DELTA.P ″ T (t) the resulting response is much less than Δ P' d The resulting response, ignoring Δ P "in the equation T (t), solving the equation to obtain:
in the formula:
s8: the frequency difference approximate analysis expression of the two equivalent model nodes can be obtained by the formulas (6), (25), (33) and (35):
the invention has the beneficial effects that: the invention provides a dynamic frequency response approximate analysis model of a two-machine equivalent power system. The electric power system is equivalent to a two-machine system, model simplification is carried out by methods of ring solving, approximate substitution and the like, approximate analytic solution of the two-machine equivalent electric power system is achieved, and an approximate analytic expression in a model time domain is given. The model can quickly and accurately calculate the dynamic frequency response process of the equivalent power system of two machines on the basis of considering the time-space distribution characteristics of the frequency, is an expansion form of a single-machine model applied to an actual multi-machine power system, can be directly used for quickly calculating the frequency response characteristics of the power system, can also be directly added into a target equation or a constraint condition to carry out the research related to the optimized operation and protection control of the power system, avoids numerical calculation, and has practical value and wide application prospect.
Drawings
FIG. 1 is a frequency response model of a two-machine equivalent power system.
Fig. 2 is an equivalent substitution process using an ASF model and an SFR model.
FIG. 3 is a simplified two-machine equivalent frequency response model.
FIG. 4 is a diagram of an IEEE four-machine two-zone standard test system.
FIG. 5 is a dynamic frequency response process of the two equivalent units after disturbance.
Detailed Description
The invention provides a dynamic frequency response approximate analysis model of a two-machine equivalent power system. The electric power system is equivalent to a two-machine system, model simplification is carried out by methods of ring solving, approximate substitution and the like, approximate analytic solution of the two-machine equivalent electric power system is achieved, and an approximate analytic expression in a model time domain is given.
The calculation process of the invention is further explained below by taking an IEEE standard two-zone four-machine power system as an example, and the accuracy of the invention is verified by comparing with a simulation method. The test system is shown in fig. 4.
In an IEEE standard two-area four-machine power system, the number m of generator nodes is 4, the number l of load nodes is 7, and the system capacity S B =100MVA, and the power generation load node power data is shown in table 1.
TABLE 1 Generator load node Power parameters
Substituting the data and the line parameters and the transformer parameters in the IEEE two-zone four-machine standard calculation example into the step S1, the equation after the non-generator node is eliminated is as follows:
according to step S2, let v 1 V is a set containing nodes 1 and 2 2 To contain the set of nodes 3 and 4, the system is divided into v 1 And v 2 The incremental equation of the represented two-machine equivalent system is expressed as follows:
in the formula:
ΔP′ M ＝-K′ m R′ -1 (1+T′ R s) -1 (1+F′ H T′ R s)Δω′
wherein the parameters can be determined by the following equations (9) to (12):
in the unit model, P N ＝diag(900,900,900,900)，K M ＝diag(1,1,1,1)，D M ＝diag(0,0,0,0)，ω 0 And the other important parameters are shown in the table 2.
TABLE 2 two-zone four-machine model Generator parameters
From equation (13):
ΔP′ M ＝-R′ -1 F′ H K′ m Δω′-K′ m R′ -1 (1+T′ R s) -1 (1-F′ H )Δω′
the formula of (6) is:
2H′Δδ′s 2 +B′ S Δδ′＝ΔP′ d -ΔP′ D +ΔP′ T
in formula, proportional feedback Power increment Δ P' D Feeding back power increment delta P 'with first-order inertia link' T The expression of (a) is as follows:
ΔP′ D ＝D′ eq Δω′
ΔP′ T ＝-K′ m R′ -1 (1+T′ R s) -1 (1-F′ H )Δω′
wherein, D' eq For equivalent damping of analytical model
D′ eq ＝R′ -1 F′ H K′ m +D′
By step S4, let V = { V 1 ∪V 2 The corresponding SFR model parameters X = { H from equations (9) - (10),D,R -1 }， Y＝{F H ,T R obtaining a first-order inertia link feedback power increment delta P ″' according to a formula (19) T The approximate analytical expression of (a) is as follows:
ΔP″ T ＝-K′ m R′ -1 (1+T′ R s) -1 (1-F′ H )Δω
and (5) obtaining a simplified equivalent frequency response model expression of the two machines according to the formula (20):
2H′Δδ′s 2 +D′ eq Δδ′s+B′ S Δδ′＝ΔP′ d +ΔP″ T
a simplified two-machine equivalent frequency response model is shown in fig. 3.
Through step S5, according to the mode superposition method, the undamped free vibration equation of equation (20) is:
2H′Δδ′s 2 +B′ S Δδ′＝0
the two characteristic roots and the corresponding characteristic vectors of the equation are respectively:
a 1 ＝[1,1] T ,a 2 ＝[-2H′ 2 ,2H′ 1 ] T
the mode matrix is:
introducing a regular transformation:
Δδ′＝Φy
the time domain expression of the kinetic equation described by the available modal coordinates of equation (26) is:
in the formula, the parameter expression is as follows:
M＝Φ T (2H′)Φ＝diag[2H′ 1 +2H′ 2 ,4H′ 1 H′ 2 (2H′ 1 +2H′ 2 )]
K＝Φ T B′ S Φ＝diag[0,k(2H′ 1 +2H′ 2 ) 2 ]
through step S6, when 2H' 2 D′ eq1 ＝2H′ 1 D′ eq2 When C is classical damping, the above formula is a decoupling equation, otherwise, the velocity term is coupled, and according to the formulas (31) to (32), the following results are obtained:
from (34) to (37), solving the expression of the equation, respectively, can obtain:
in the formula:
in the formula:
through step S7, a frequency difference approximate analysis expression of the two-machine equivalent model node can be obtained from equation (37):
if it is assumed that t =1s, L7 has a sudden load increase of 300MW, then
The disturbance distribution power is as follows:
thus:
ΔP d ＝-3
the above formula is substituted into a frequency difference approximate expression of two-machine equivalence, a dynamic frequency response curve can be obtained, and the response curve of 0-25 s is shown in figure 5. As can be seen from the figure, the method provided by the invention has small simulation curve error and high precision compared with the existing numerical simulation method.
The above-mentioned embodiments only represent the embodiments of the present invention, but they should not be understood as the limitation of the scope of the present invention, and it should be noted that those skilled in the art can make several variations and modifications without departing from the spirit of the present invention, and these all fall into the protection scope of the present invention.
Claims (1)
1. A dynamic frequency response approximate analysis model of a two-machine equivalent power system is characterized by comprising the following steps:
s1: for a multi-machine power system, transient reactance branches in a generator are added into a network model, and a network equation expression based on direct current flow is shown as a formula (1):
in the formula, P g ，P G ，P L Respectively outputting electromagnetic power P for the generator gi The generator terminal bus injection power P Gi Load node injected power P Lj A column vector of components; b GG ，B GL ，B LG ，B LL For sub-arrays of admittance matrix B in DC power flow calculation, B gg ，B gG ，B Gg Is an admittance array between an expansion node and an original generator node, the unit is pu and is S B Is a reference capacity; delta g 、θ G 、θ L Respectively generator rotor angle delta gi The phase angle theta of the bus voltage at the generator end Gi Load node voltage phase angle θ Lj The unit of the formed column vector is rad; g, G and L are potential virtual nodes G in the generator respectively i Generator node G i Load node L j (ii) a m is the number of generator nodes, l is the number of load nodes, i = 1-m, j = m + 1-m + l;
after transient reactance of generator node is added in an expansion mode, generator node G in original load flow calculation i Converting into a load node; when the node operates in a steady state, the injection power of the node is 0, and if the generator is broken down, the injection power of a corresponding generator terminal bus is negative generator breaking power; but unlike the load node, the node has no power frequency regulation capability;
equation (1) is rewritten and expressed as an incremental equation:
in the formula (I), the compound is shown in the specification,
is generator node G i And load node L j Forming an extended load node set, namely the original system nodeThe number of nodes n = m + l; Δ represents the increment of the corresponding variable;
the expression for the generator electromagnetic power increment from equation (2) is:
in the formula, B S Is an inter-machine oscillation matrix and is provided with a plurality of oscillation matrixes,
distributing a matrix for the disturbance power, which respectively comprises:
s2: the network equation is based on a formula (3) in S1, the units all adopt SFR models, a multi-machine system is simplified into a two-machine equivalent system, and the incremental equation is expressed as follows:
in the formula, the parameter with the "'" at the upper left represents a two-machine equivalent system, and the parameter without the "'" represents a multi-machine system; h ' and D ' are generator inertia time constants H ' k Equivalent generator damping D' k Forming a diagonal matrix with the unit of s; b' S Is an inter-machine oscillation matrix with the unit of pu; delta P' d ，ΔP′ M Respectively distributing power delta P 'for disturbance' dk Prime mover-governor augmented power Δ P' Mk The unit of the formed column vector is pu; delta delta 'is the generator rotor angle delta' k The unit of the formed column vector is rad; delta omega 'is generator rotor angular speed omega' k The unit of the formed column vector is rad/s; omega 0 Is the rated generator angular frequency; s is the complex frequency; k is the equivalent unit number;
delta P 'of' d ，ΔP′ M The expression of (a) is as follows:
ΔP′ M ＝-K′ m R′ -1 (1+T′ R s) -1 (1+F′ H T′ R s)Δω′ (8)
of formula (II) F' H ，K′ m R ' is respectively a work doing proportion F ' of a high-pressure cylinder of the generator ' Hk Mechanical power gain coefficient K' mk Reheat time constant R k The diagonal matrix is formed and has no dimension; t' R Is a difference-adjusting coefficient T' Rk Forming a diagonal matrix with the unit of s;
distributing a matrix for disturbance power, wherein the unit is pu;
for disturbing power
The unit of the formed column vector is pu;
s3: let the power system multiple machine frequency response model parameters be X = { H, D, R -1 }，Y＝{F H ,T R The equivalent parameters of the two-machine equivalent frequency response model are X '= { H', D ', R' -1 }、Y′＝{F′ H ,T′ R The computational expressions of are:
in the formula, P Ni Rated active power for the ith generator; v k When the set is a set contained in the kth equivalent set and only one set is contained in the set, the actual system is a dual-machine model; wherein λ is i The influence factors of the unit parameters on the equivalent system parameters are as follows:
inter-machine oscillation matrix B 'of two-machine equivalent model' S Perturbation distribution matrix
The expression of (a) is:
s4: equation (6) is:
2H′Δδ′s 2 +B′ S Δδ′＝ΔP′ d -ΔP′ D +ΔP′ T (14)
in formula, proportional feedback Power increment Δ P' D And a first-order inertia link feedback power increment delta P' T The expression of (a) is as follows:
ΔP′ D ＝D′ eq Δω′ (15)
ΔP′ T ＝-K′ m R′ -1 (1+T′ R s) -1 (1-F′ H )Δω′ (16)
wherein, D' eq Equivalent damping for the analytical model; f' H Is generator high pressure cylinder working ratio F' Hk Forming a diagonal matrix; t' R Is a difference modulation coefficient T' Rk Forming a diagonal matrix with the unit of s;
D′ eq ＝R′ -1 F′ H K′ m +D′ (17)
s5: making SFR model parameters X = { H, D, R corresponding to multi-machine frequency response model of power system -1 }，Y＝{F H ,T R And (5) substituting the equation into the left side of the equation (9) to obtain a set V = { V = 1 ∪V 2 The solution of the corresponding parameter;
obtaining a frequency response analytic solution of the SFR model by adopting an ASFR polymerization method; performing loop release at the frequency difference delta omega 'feedback position of the ASF model, and equivalently replacing by adopting an SFR analytic frequency expression, so that the proportional feedback power increment delta P' in the ASF model D Feedback power increment delta P' with first-order inertia link T The approximate analytical expression of (a) is as follows:
ΔP″ D ＝(R′ -1 F′ H K′ m +D′)Δω (18)
ΔP″ T ＝-K′ m R′ -1 (1+T′ R s) -1 (1-F′ H )Δω (19)
according to the property of the SFR, the delta omega is the system inertia center frequency, the frequency passes through the center of a delta omega' curve, and the system frequency oscillation cannot be described to generate errors; proportional feedback can directly bring the error into calculation, while first-order inertial link feedback can reduce frequency oscillation accumulation, and the influence of oscillation error is reduced along with the increase of frequency offset accumulation; thus, the Δ P ″' is retained D And use of Δ P ″) T Substitute delta P' T ；
Let Delta P d ＝ΔP′ d1 +ΔP′ d2 (ii) a Rewriting the formula (14) as:
2H′Δδ′s 2 +D′ eq Δδ′s+B′ S Δδ′＝ΔP′ d +ΔP″ T (20)
s6: according to the modal superposition method, the undamped free vibration equation of equation (20) is:
2H′Δδ′s 2 +B′ S Δδ′＝0 (21)
introducing a regular transformation:
Δδ′＝Φy (25)
substituting (25) into (20) and multiplying by phi on both sides of the equation T (ii) a Assuming that the system is in a steady state before disturbance, performing inverse Laplace transformation, and obtaining a time domain expression of a kinetic equation described by a modal coordinate as follows:
in the formula, M is a main mass matrix, K is a main rigidity matrix, C is a modal damping matrix, and F (t) is excitation after transformation;
s7: when 2H' 2 D′ eq1 ＝2H′ 1 D′ eq2 When C is classical damping, the formula (26) is a decoupling equation, otherwise, the velocity term is coupled, a forced decoupling method is adopted, and the non-diagonal elements of C are ignored, so that the formula (26) is expressed as follows:
wherein, the formula (31) is a vibration equation of the system inertia center, and the formula (32) is an oscillation equation of the system;
solving equation (31) yields:
in the formula:
because, in formula (32),. DELTA.P ″ T1 (t) and. DELTA.P ″) T2 (t) the resulting response is much less than Δ P' d1 And Δ P' d2 Generated response, ignore, etcDelta P' in the formula T1 (t) and. DELTA.P ″) T2 (t), solving the equation to obtain:
in the formula:
s8: the frequency difference approximate analysis expression of the two equivalent model nodes can be obtained by the formulas (6), (25), (33) and (35):
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