CN103698629A - Real-time on-line prediction method for characteristic parameters of direct current micro grid - Google Patents

Abstract

The invention discloses a real-time on-line prediction method for characteristic parameters of a direct current micro grid. The method comprises the following steps of 1) acquiring a disturbance current value Ik of a disturbed node of a direct current micro grid system in real time at a sampling time interval of Ts, and constructing a disturbance current value ARRAY[I1, I2, I3, I4 and I5] according to the acquired disturbance current value Ik; 2) calculating four order differential derivatives of current values in the disturbance current value ARRAY in real time, i.e. judging that the direct current micro grid system enters a transient process if a first-order differential derivative f(1) is greater than epsilon; or judging that the direct current micro grid system is stable if each order differential derivative is smaller than the preset minute quantity epsilon, continuously acquiring a current value at the next moment, and continuously calculating four order differential derivatives of the current value at the next moment. Compared with the prior art, the method has higher real-time and simplicity, and can be applied to multiple fields such as inhibition of transient impulse current, on-line detection of motor parameters, on-line diagnosis of faults, and on-line detection and inhibition of low-frequency chugging.

Description

A kind of real-time online Forecasting Methodology of DC micro electrical network characteristic parameter

Technical field

The invention belongs to the quality of power supply and signal analysis and processing research field, be specifically related to a kind of real-time online Forecasting Methodology of DC micro electrical network characteristic parameter.

Background technology

Micro power network is the effective measures that realize new forms of energy Distributed Application, is also the main contents of following interactive intelligent electrical network.This system is got up wind-powered electricity generation, photovoltaic, fuel cell distributed energy integration by a large amount of power electronics interface circuits; the energy storage devices such as accumulator, super capacitor and high speed flywheel of take are power adjustments means; under the coordinative role of micro power network central control system, dcs and protective device, by miniature local supply network, for this locality, load clean, lasting, interactively electric energy is provided.

System architecture, networking mode, operational mode, control strategy by analysis micro power network are known, and it has following key property: 1, contain the diversified batch (-type) energy, inevitably exist outside input source disturbance continuation, randomness.The characteristic of 2, loading in micro power network is various, form is changeable, and load variations a large amount of, randomness has caused the characteristic parameter of network constantly to change.3, the type of each assembly varies, and the converters of existing current mode also has the current transformer of voltage-type; The load of existing capacitance-resistance, also has the load of resistance sense; The device of existing silent oscillation, also has the equipment of sports type.4, operational mode is various, both can run on grid-connected pattern, also can be operated in island mode.5, typical multi-scale coupling system, the converters of existing little inertia, also has the rotating machinery of large inertia; Existing lower-powered energy storage device, also has the major network system that capacity is huge; Existing other energy exchange of power frequency level, also has other oscillation of power of switching frequency level.

These characteristics has determined the complicacy of micro power network stability analysis.Existing stability study is nearly all based on certain definite network and adopts certain concrete control strategy.But the analytical approach based on mathematical model seriously relies on the accuracy of systematic parameter and model, so theoretical analysis result may have larger deviation with actual conditions.The more important thing is, micro power network system is always constantly suffering the disturbance of a large amount of randomness, and the characteristic parameter of its control strategy, mode of operation and structure also therefore and constantly changes.Therefore, in the micro power network system under disturbed conditions, the method for analyzing stability based on parameter and model has significant limitation, and existing achievement in research is difficult to meet the needs of micro power network stability on-line analysis.

Summary of the invention

For above-mentioned defect or deficiency, a kind of real-time online disposal route of DC micro electrical network characteristic parameter is provided, can be according to the state of real-time measurement system.

For reaching above object, technical scheme of the present invention is:

Comprise the following steps:

1), Real-time Collection DC micro network system one is subject to the current perturbation value I of disturbance node _k, sampling time interval is T _s; According to gathered current perturbation value I _k, structure current perturbation value array ARRAY[I ₁, I ₂, I ₃, I ₄, I ₅], and in current perturbation value array ARRAY, each element initial value is 0, point of every collection, makes I ₅=I _k; I ₄=I ₅; I ₃=I ₄; I ₂=I ₃; I ₁=I ₂;

2), the Four order difference derivative of current value in real-time calculation perturbation current value array ARRAY: if first order difference derivative f ⁽¹⁾be greater than ε, judge that DC micro network system has entered transient state process; If all-order derivative is all less than default small quantity ε, judge that DC micro network system is steady, continue to gather next current value constantly, and continue to calculate next Four order difference derivative of current value constantly, wherein, in described current perturbation value array ARRAY, the Four order difference derivative of current value is respectively f ⁽¹⁾, f ⁽²⁾, f ⁽³⁾, f ⁽⁴⁾:

$f^{\left(1\right)}=\frac{I_5-I_4}{T_s}$

$f^{\left(2\right)}=\frac{I_5-{2I}_4+I_3}{{\left(T_s\right)}^2}$

$f^{\left(3\right)}=\frac{I_5-{3I}_4+{3I}_3-I_2}{{\left(T_s\right)}^3}$

$f^{\left(4\right)}=\frac{I_5-{4I}_4+{6I}_3-{4\mathrm{<figure-callout\; id="2"\; label="I"\; filenames="HDA0000436297290000011.png"\; state="\{\{state\}\}">I</figure-callout>}}_2+I_1}{{\left(T_s\right)}^4};$

3), as first order difference derivative f ⁽¹⁾be greater than ε, continue to gather next current value constantly, calculate one of next moment current value and arrive Four order difference derivative, calculate intermediate variable a, b, c:

$\left\{\begin{array}{c}a=f^{\left(1\right)}{f}^{\left(2\right)}-f^{\left(1\right)}{f}^{\left(3\right)}\\ b=f^{\left(2\right)}{f}^{\left(3\right)}-f^{\left(1\right)}-f^{\left(4\right)}\\ c=f^{\left(3\right)}{f}^{\left(3\right)}-f^{\left(2\right)}{f}^{\left(4\right)}\end{array}\right.$

(1) if result of calculation a=0, the transient state process of judging DC micro network system is steady dead-beat transient state process, and transient state process is only containing a main mode, the time domain specification of described main mode is that exponential law changes;

(2) if the discriminant Δ=b of transient state process type is calculated in result of calculation a ≠ 0 ²-4ac:

(2.1) if Δ >0, the transient state process of judging DC micro network system is steady dead-beat transient state process, and steady dead-beat transient state process contains two main mode, the time domain specification of two master modes is exponential law to be changed;

(2.2) if Δ <0, the transient state process of judging DC micro network system is change in oscillation transient state process, and described change in oscillation transient state process contains a main Oscillatory mode shape, the time domain specification of Oscillatory mode shape is to change in the cycle, and oscillation amplitude is pressed exponential law and changed.

The characteristic parameter that contains the steady dead-beat transient state process of a main mode in (1) in step 3) characterizes with the attenuation coefficient σ of index mode, and its computing formula is as follows:

$\mathrm{\&sigma;}=-\frac{f^{\left(2\right)}}{f^{\left(1\right)}}.$

Steady dead-beat transient state process containing a main mode finishes stable electrical flow valuve i afterwards _a:

$i_A=I_5-\frac{{\left(f^{\left(1\right)}\right)}^2}{f^{\left(2\right)}}.$

(2.1) attenuation coefficient ρ, the μ for characteristic parameter that in, contain the steady dead-beat transient state process of two main mode represent, attenuation coefficient ρ, μ computing formula are as follows:

$\left\{\begin{array}{c}\mathrm{\&rho;}=\frac{\sqrt{b^2-4\mathrm{ac}}-b}{2a}\\ \mathrm{\&mu;}=\frac{\sqrt[-]{b^2-4\mathrm{ac}}-b}{2a}\end{array}\right.$

If ρ >0 and μ >0, judge the convergence of DC micro network system transient state process;

Otherwise the transient state process of decision-making system is dispersed, and causes the most at last system unstability.

If ρ >0 and μ >0, judge DC micro network system transient state process convergence, and the steady current of transient state process after finishing is i _b, i _bcomputing formula as follows:

$i_B=\frac{{\mathrm{\&rho;\&mu;I}}_5+(\mathrm{\&mu;}+\mathrm{\&rho;})f^{\left(1\right)}+f^{\left(2\right)}}{\mathrm{\&rho;\&mu;}}.$

(2.2) in the characteristic parameter of change in oscillation transient state process be oscillation period ω and its computing formula of attenuation coefficient λ as follows:

$\left\{\begin{array}{c}\mathrm{\&omega;}=\frac{\sqrt{4\mathrm{ac}-{\mathrm{<figure-callout\; id="2"\; label="b"\; filenames="HDA0000436297290000011.png"\; state="\{\{state\}\}">b</figure-callout>}}^2}}{2a}\\ \mathrm{\&lambda;}=\frac{b}{2a}.\end{array}\right.$

When Δ <0, and λ >0, transient state process convergence that can decision-making system, the steady current after the transient state process of convergence finishes is i _c:

$i_C=I_5+\frac{f^{\left(1\right)}{f}^{\left(1\right)}{f}^{\left(4\right)}-{2f}^{\left(1\right)}{f}^{\left(2\right)}{f}^{\left(3\right)}+f^{\left(2\right)}{f}^{\left(2\right)}{f}^{\left(2\right)}}{f^{\left(3\right)}{f}^{\left(3\right)}-f^{\left(2\right)}{f}^{\left(4\right)}}.$

Compared with the prior art, beneficial effect of the present invention is:

The present invention adopts the transient current that detects a certain disturbed node in DC micro electrical network, the electric current detecting is in real time configured to current perturbation, then by real-time current perturbation value array is calculated, the real-time online prediction of realization to DC micro electrical network, obtain the real-time status of system, and then realized the separated of transient state composition and stable state composition; Compared with prior art, the inventive method has better real-time and simplicity, can be applicable to the numerous areas such as inhibition, parameter of electric machine on-line testing, the on-line fault diagnosis of transient state dash current, the online detection of low frequency power oscillation and inhibition.

Further, the present invention can calculate the situation of change of the local feature parameter of micro power network system quickly and accurately according to the transient current of real-time measurement, thereby the system that dopes stands characteristic value and the stability of disturbance transient response process afterwards.And then, the control system of micro power network can be according to the disturbing influence rule doping, decide strategy and the steering order of disturbance suppression, within the shortest time of vibration harm early period of origination, decision signal composition rapidly, the line number of going forward side by side separation, for follow-up attachment device such as processes in real time, online at the work, information support is provided, and then reduces loss and the harm to electrical network of power.

Accompanying drawing explanation

Fig. 1 is the real-time online Forecasting Methodology of DC micro electrical network characteristic parameter of the present invention;

Fig. 2 is DC micro network system structural representation of the present invention;

Fig. 3 is the realistic model figure of DC micro electrical network of the present invention;

Fig. 4 is the prediction oscillogram of load current of the present invention.

Embodiment:

Below in conjunction with accompanying drawing, the present invention is described in detail.

As shown in Figure 1, the invention provides a kind of real-time online Forecasting Methodology of DC micro electrical network characteristic parameter, comprise the following steps:

1), Real-time Collection DC micro network system one is subject to the current perturbation value I of disturbance node _k, sampling time interval is T _s=0.01s; According to gathered current perturbation value I _k, structure current perturbation value array ARRAY[I ₁, I ₂, I ₃, I ₄, I ₅], and in current perturbation value array ARRAY, each element initial value is 0, point of every collection, makes I ₅=I _k; I ₄=I ₅; I ₃=I ₄; I ₂=I ₃; I ₁=I ₂;

2), the Four order difference derivative of current value in real-time calculation perturbation current value array ARRAY: if first order difference derivative f ⁽¹⁾be greater than ε, judge that DC micro network system has entered transient state process; If all-order derivative is all less than default small quantity ε, judge that DC micro network system is steady, continue to gather next current value constantly, and continue to calculate next Four order difference derivative of current value constantly, wherein, in described current perturbation value array ARRAY, the Four order difference derivative of current value is respectively f ⁽¹⁾, f ⁽²⁾, f ⁽³⁾, f ⁽⁴⁾:

$f^{\left(1\right)}=\frac{I_5-I_4}{T_s}$

$f^{\left(2\right)}=\frac{I_5-{2I}_4+I_3}{{\left(T_s\right)}^2}$

$f^{\left(3\right)}=\frac{I_5-{3I}_4+{3I}_3-I_2}{{\left(T_s\right)}^3}$

$f^{\left(4\right)}=\frac{I_5-{4I}_4+{6I}_3-{4\mathrm{<figure-callout\; id="2"\; label="I"\; filenames="HDA0000436297290000011.png"\; state="\{\{state\}\}">I</figure-callout>}}_2+I_1}{{\left(T_s\right)}^4};$

Wherein, small quantity ε considers to get ε=0.001 after the combined factors such as computational accuracy requirement and measuring error.

3) judgement DC micro network system has entered the concrete state of transient state process:

If first order difference derivative f ⁽¹⁾be greater than ε, judge that DC micro network system has entered after transient state process, also comprises:

Continue to gather next current value constantly, calculate one of next moment current value and arrive Four order difference derivative, calculate intermediate variable a, b, c:

$\left\{\begin{array}{c}a=f^{\left(1\right)}{f}^{\left(2\right)}-f^{\left(1\right)}{f}^{\left(3\right)}\\ b=f^{\left(2\right)}{f}^{\left(3\right)}-f^{\left(1\right)}-f^{\left(4\right)}\\ c=f^{\left(3\right)}{f}^{\left(3\right)}-f^{\left(2\right)}{f}^{\left(4\right)}\end{array}\right.$

(1) if result of calculation a=0, the transient state process of judging DC micro network system is steady dead-beat transient state process, and transient state process is only containing a main mode, the time domain specification of described main mode is that exponential law changes; The characteristic parameter that contains the steady dead-beat transient state process of a main mode characterizes with the attenuation coefficient λ of index mode, and its computing formula is as follows:

$\mathrm{\&sigma;}=-\frac{f^{\left(2\right)}}{f^{\left(1\right)}}.$

Steady dead-beat transient state process containing a main mode finishes stable electrical flow valuve i afterwards _a:

$i_A=I_5-\frac{{\left(f^{\left(1\right)}\right)}^2}{f^{\left(2\right)}};$

(2) if result of calculation a ≠ 0, the discriminant of calculating transient state process type, that is: Δ=b ²-4ac:

(2.1) if Δ >0, the transient state process of judging DC micro network system is steady dead-beat transient state process, and nonoscillating transient state process contains two main mode, the time domain specification of two master modes is exponential law to be changed; Attenuation coefficient ρ, the μ for characteristic parameter of the steady dead-beat transient state process that contains two main mode represent, attenuation coefficient ρ, μ computing formula are as follows:

$\left\{\begin{array}{c}\mathrm{\&rho;}=\frac{\sqrt{b^2-4\mathrm{ac}}-b}{2a}\\ \mathrm{\&mu;}=\frac{\sqrt[-]{b^2-4\mathrm{ac}}-b}{2a}\end{array}\right..$

When Δ >0, if ρ >0 and μ >0 judge DC micro network system transient state process convergence, and the steady current of transient state process after finishing is i _b, i _bcomputing formula as follows:

$i_B=\frac{{\mathrm{\&rho;\&mu;I}}_5+(\mathrm{\&mu;}+\mathrm{\&rho;})f^{\left(1\right)}+f^{\left(2\right)}}{\mathrm{\&rho;\&mu;}}.$

Otherwise transient state process that can decision-making system is dispersed, and causes the most at last system unstability, now needs to take urgent human intervention measure, to guarantee system-wide safety and stability.

(2.2) if Δ <0, the transient state process of judging DC micro network system is change in oscillation transient state process, and described change in oscillation transient state process contains a main Oscillatory mode shape, the time domain specification of Oscillatory mode shape is to change in the cycle, and oscillation amplitude is pressed exponential law and changed; The characteristic parameter of change in oscillation transient state process be oscillation period ω and its computing formula of attenuation coefficient λ as follows:

$\left\{\begin{array}{c}\mathrm{\&omega;}=\frac{\sqrt{4\mathrm{ac}-{\mathrm{<figure-callout\; id="2"\; label="b"\; filenames="HDA0000436297290000011.png"\; state="\{\{state\}\}">b</figure-callout>}}^2}}{2a}\\ \mathrm{\&lambda;}=\frac{b}{2a}.\end{array}\right.$

When Δ <0, and λ >0 judges the convergence of DC micro network system transient state process; Steady current after the transient state process of this convergence finishes is i _c:

$i_C=I_5+\frac{f^{\left(1\right)}{f}^{\left(1\right)}{f}^{\left(4\right)}-{2f}^{\left(1\right)}{f}^{\left(2\right)}{f}^{\left(3\right)}+f^{\left(2\right)}{f}^{\left(2\right)}{f}^{\left(2\right)}}{f^{\left(3\right)}{f}^{\left(3\right)}-f^{\left(2\right)}{f}^{\left(4\right)}}.$

Otherwise the transient state process of decision-making system is dispersed, and causes the most at last system unstability.

As shown in Figure 2, on a certain load branch of real-time current detection device micro power network.Fig. 3 has provided the realistic model of a DC micro electrical network, for sake of convenience, DC micro electrical network is simplified herein.DC bus passes through PWM(Pulse Width Modulation, pulse-length modulation by public electric wire net) rectifier power supply, by voltage close loop, guarantee that DC bus-bar voltage is stable.Some loads access in parallel DC bus.At a time, large inertia load access micro power network, its capacity is unknown, the Forecasting Methodology that the present invention proposes can real-time online calculation perturbation after the steady-state operation value of system, the stability of prediction DC micro network system.

Fig. 4 has provided the load current waveform of actual measurement and by the method for the invention, has calculated the waveform of the load current steady-state value predicting.As seen from the figure, the on-line prediction value being calculated by the method for the invention is predict steady-state value well, and can and then calculate the perturbation load power putting into operation.

Hence one can see that, and the method for the invention can be applicable to the inhibition of transient state dash current.The moment just having occurred in oscillatory process is taked control strategy in advance, suitable according to the characteristic of transient state process, realize the correction of transient state process, to reach good static and dynamic performance, especially can avoid the impact of transient current, avoid micro power network to occur voltage fluctuation by a relatively large margin, thereby improve the stability margin of micro power network.

In the present invention, provided the Forecasting Methodology of real-time online computing system characteristic parameter in a kind of DC micro electrical network.And utilize MATLAB/Simulink to carry out simulating, verifying to the method.From simulation result, can see, in transient state process, just just start, calculation of characteristic parameters method given in this article has just online calculated the important parameter in this transient state process, and has realized the separated of transient state composition and stable state composition.Than additive method, there is better real-time and simplicity.The method proposing can be applied to the numerous areas such as inhibition, parameter of electric machine on-line testing, the on-line fault diagnosis of transient state dash current, the online detection of low frequency power oscillation and inhibition; The method that the present invention proposes can be within the shortest time of vibration harm early period of origination, decision signal composition rapidly, the line number of going forward side by side separation, for follow-up attachment device such as processes in real time, online at the work, information support is provided, and then reduces loss and the harm to electrical network of power.

Claims (7)

1. a real-time online Forecasting Methodology for DC micro electrical network characteristic parameter, is characterized in that, comprises the following steps:

1), Real-time Collection DC micro network system one is subject to the current perturbation value I of disturbance node _k, sampling time interval is T _s; According to gathered current perturbation value I _k, structure current perturbation value array ARRAY[I ₁, I ₂, I ₃, I ₄, I ₅], and in current perturbation value array ARRAY, each element initial value is 0, point of every collection, makes I ₅=I _k; I ₄=I ₅; I ₃=I ₄; I ₂=I ₃; I ₁=I ₂; K is positive integer;

2), the Four order difference derivative of current value in real-time calculation perturbation current value array ARRAY: if first order difference derivative f ⁽¹⁾be greater than ε, judge that DC micro network system has entered transient state process; If all-order derivative is all less than default small quantity ε, judge that DC micro network system is steady, continue to gather next current value constantly, and continue to calculate next Four order difference derivative of current value constantly, wherein, in described current perturbation value array ARRAY, the Four order difference derivative of current value is respectively f ⁽¹⁾, f ⁽²⁾, f ⁽³⁾, f ⁽⁴⁾:

$f^{\left(1\right)}=\frac{I_5-I_4}{T_s}$

$f^{\left(2\right)}=\frac{I_5-{2I}_4+I_3}{{\left(T_s\right)}^2}$

$f^{\left(3\right)}=\frac{I_5-{3I}_4+{3I}_3-I_2}{{\left(T_s\right)}^3}$

$f^{\left(4\right)}=\frac{I_5-{4I}_4+{6I}_3-{4I}_2+I_1}{{\left(T_s\right)}^4};$

3), as first order difference derivative f ⁽¹⁾be greater than ε, continue to gather next current value constantly, calculate one of next moment current value and arrive Four order difference derivative, calculate intermediate variable a, b, c:

$\left\{\begin{array}{c}a=f^{\left(1\right)}{f}^{\left(2\right)}-f^{\left(1\right)}{f}^{\left(3\right)}\\ b=f^{\left(2\right)}{f}^{\left(3\right)}-f^{\left(1\right)}-f^{\left(4\right)}\\ c=f^{\left(3\right)}{f}^{\left(3\right)}-f^{\left(2\right)}{f}^{\left(4\right)}\end{array}\right.$

(1) if result of calculation a=0, the transient state process of judging DC micro network system is steady dead-beat transient state process, and transient state process is only containing a main mode, the time domain specification of described main mode is that exponential law changes;

(2) if the discriminant Δ=b of transient state process type is calculated in result of calculation a ≠ 0 ²-4ac:

(2.1) if Δ >0, the transient state process of judging DC micro network system is steady dead-beat transient state process, and steady dead-beat transient state process contains two main mode, the time domain specification of two master modes is exponential law to be changed;

(2.2) if Δ <0, the transient state process of judging DC micro network system is change in oscillation transient state process, and described change in oscillation transient state process contains a main Oscillatory mode shape, the time domain specification of Oscillatory mode shape is to change in the cycle, and oscillation amplitude is pressed exponential law and changed.

2. the real-time online Forecasting Methodology of DC micro electrical network characteristic parameter according to claim 1, it is characterized in that, the characteristic parameter that contains the steady dead-beat transient state process of a main mode in (1) in step 3) characterizes with the attenuation coefficient σ of index mode, and its computing formula is as follows:

$\mathrm{\&sigma;}=-\frac{f^{\left(2\right)}}{f^{\left(1\right)}}.$

3. the real-time online Forecasting Methodology of DC micro electrical network characteristic parameter according to claim 1 and 2, is characterized in that, the steady dead-beat transient state process that contains a main mode finishes stable electrical flow valuve i afterwards _a:

$i_A=I_5-\frac{{\left(f^{\left(1\right)}\right)}^2}{f^{\left(2\right)}}.$

4. the real-time online Forecasting Methodology of DC micro electrical network characteristic parameter according to claim 1, it is characterized in that, (2.1) attenuation coefficient ρ, the μ for characteristic parameter that in, contain the steady dead-beat transient state process of two main mode represent, attenuation coefficient ρ, μ computing formula are as follows:

$\left\{\begin{array}{c}\mathrm{\&rho;}=\frac{\sqrt{b^2-4\mathrm{ac}}-b}{2a}\\ \mathrm{\&mu;}=\frac{\sqrt[-]{b^2-4\mathrm{ac}}-b}{2a}\end{array}\right.$

If ρ >0 and μ >0, judge the convergence of DC micro network system transient state process;

Otherwise the transient state process of decision-making system is dispersed, cause the most at last DC micro network system unstability.

5. the real-time online Forecasting Methodology of DC micro electrical network characteristic parameter according to claim 4, it is characterized in that, if ρ >0 and μ >0, judge DC micro network system transient state process convergence, and the steady current of transient state process after finishing is i _b, i _bcomputing formula as follows:

$i_B=\frac{{\mathrm{\&rho;\&mu;I}}_5+(\mathrm{\&mu;}+\mathrm{\&rho;})f^{\left(1\right)}+f^{\left(2\right)}}{\mathrm{\&rho;\&mu;}}.$

6. the real-time online Forecasting Methodology of DC micro electrical network characteristic parameter according to claim 1, is characterized in that, in (2.2) characteristic parameter of change in oscillation transient state process be oscillation period ω and its computing formula of attenuation coefficient λ as follows:

$\left\{\begin{array}{c}\mathrm{\&omega;}=\frac{\sqrt{4\mathrm{ac}-b^2}}{2a}\\ \mathrm{\&lambda;}=\frac{b}{2a}.\end{array}\right.$

7. the real-time online Forecasting Methodology of DC micro electrical network characteristic parameter according to claim 6, it is characterized in that, when Δ <0, and λ >0, transient state process convergence that can decision-making system, the steady current after the transient state process of convergence finishes is i _c:

$i_C=I_5+\frac{f^{\left(1\right)}{f}^{\left(1\right)}{f}^{\left(4\right)}-{2f}^{\left(1\right)}{f}^{\left(2\right)}{f}^{\left(3\right)}+f^{\left(2\right)}{f}^{\left(2\right)}{f}^{\left(2\right)}}{f^{\left(3\right)}{f}^{\left(3\right)}-f^{\left(2\right)}{f}^{\left(4\right)}}.$

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