Submarine cable fault identification and section positioning method based on multi-modulus collaborative analysis

Abstract

The invention belongs to the technical field of cable fault identification, and discloses a submarine cable fault identification and section positioning method based on multi-modulus collaborative analysis, which comprises the following steps of: firstly, acquiring measured values and line parameters of voltage and current at the head end and the tail end of a three-core armored submarine cable; secondly, establishing a distribution parameter model of the three-core armored submarine cable according to line parameters and performing decoupling, performing fault identification on two conditions of cracking and short circuit by utilizing the difference between a measured value and a calculated value and combining a decoupled modulus, and finally calculating the voltage at the position of a cable section joint through the modulus and performing section positioning on a short circuit fault. The method is suitable for fault identification and section positioning of the alternating-current submarine cable in high-voltage long-distance section connection, the calculation precision is high, the speed is high, the method is not affected by power sources at the two ends, fault types and transition resistance, the measured value is convenient to obtain, the implementation cost is low, and the method has high engineering practical significance.

Description

Technical Field

[0001] The invention belongs to the technical field of cable fault identification, and specifically relates to a three-core armored submarine cable fault identification and section positioning method using decoupled multi-modulus collaborative analysis. Background Art

[0002] As an important component of offshore wind power generation systems, submarine cables play a vital role in power transmission and other aspects. However, their operating environment is harsh and they are threatened by biological, tidal, and maritime activities all year round, which can easily lead to damage to the armor layer, insulation layer, and core, causing cable cracking, short circuits, and other faults. These faults will not only affect the normal operation of offshore wind power equipment, but may also cause abnormalities in the power system and even safety accidents, resulting in economic losses. Therefore, accurate and rapid fault identification and ranging are of great significance to ensure the efficient operation of the power system and effectively guarantee the stability and safety of the operating environment.

[0003] At present, the modeling methods for cables can be roughly divided into lumped parameter models and distributed parameter models. The lumped parameter model is only suitable for describing the electrical characteristics of short cables; the distributed parameter model is more suitable for long-distance submarine cables. The fault distance measurement methods for cables can be roughly divided into traveling wave method and impedance method. The traveling wave method uses the traveling wave information generated by the fault to achieve fault distance measurement, but it is easily affected by factors such as wave head recognition and noise; the impedance method is based on the relationship between steady-state measurement phase quantities, line parameters and fault distance to achieve fault location. The results are relatively accurate, but the calculation amount is large for long-distance submarine cables. In addition, most of the above methods are aimed at short-circuit faults, and the identification of cracking and short-circuit faults is rarely involved. Therefore, based on the distributed parameter model, it is of great significance to quickly and accurately identify and locate submarine cable faults in sections. Summary of the invention

[0004] In order to solve the above technical problems and to quickly and accurately identify and locate submarine cable faults, the present invention proposes a method for submarine cable fault identification and section location based on multi-modulus collaborative analysis. The method has high calculation accuracy and fast speed, is not affected by the power supply at both ends, the fault type and the transition resistance, and is easy to obtain measured values, which has engineering practical significance.

[0005] In order to achieve the above object, the present invention is achieved through the following technical solutions:

[0006] The present invention is a method for submarine cable fault identification and section location based on multi-modulus collaborative analysis, comprising the following steps:

[0007] Step 1: Measure the voltage and current at the beginning and end of the three-core armored submarine cable in steady state Subscript i = A, SA, B, SB, C, SC, M, respectively represent the A phase core, A phase shielding layer, B phase core, B phase shielding layer, C phase core, C phase shielding layer and armoring layer, superscript S and R represent the cable head and end quantities, respectively, where the current and three-phase core voltage are obtained by direct measurement, and the voltage of the shielding layer and armoring layer is calculated from the grounding resistance and the current flowing through it:

[0008]

[0009] in R g is the grounding resistance. According to the cable parameters, the impedance matrix Z, admittance matrix Y and cable length L per unit length of the three-core armored submarine cable are obtained.

[0010] Step 2: Write the 7th-order matrix equation of voltage and current with respect to distance x according to the distributed parameter model, decouple the U, I, Z, and Y matrices in the equation, and obtain the voltage modulus equation and the current modulus equation;

[0011] Step 3: Based on the modulus equation obtained in step 2, the voltage modulus equation is solved by measuring the voltage and current at the beginning and end of the three-core armored submarine cable as initial conditions to obtain the voltage modulus at any distance x. 7 decoupled voltage modulus expressions

[0012]

[0013] Among them, m represents the modulus after decoupling, and the number corresponds to the serial number of the modulus after decoupling. represents the voltage modulus obtained by taking the decoupled head-end quantity as the initial condition, It represents the voltage modulus obtained by taking the decoupled terminal quantity as the initial condition.

[0014] Step 4: Voltage modulus based on step 3 Calculate the 7 voltage moduli at the end of the three-core armored submarine cable and substitute them into the decoupling formula to obtain the calculated value of the three-phase voltage at the end

[0015] Step 5: Calculate the terminal three-phase voltage based on the value obtained in step 4 The voltage measurement value at the end of the three-core armored submarine cable obtained in step 1 Substitute the calculated value and the measured value of the three-phase voltage at the terminal into the fault judgment criterion to determine whether a fault occurs;

[0016] Step 6: Based on the judgment result of step 5, if it is determined that a fault occurs, the voltage amplitude of the decoupling head end corresponding to the fourth voltage modulus in the steady state after the fault is calculated, and substituted into the cracking criterion to judge whether the fault type is cracking or short circuit, thereby realizing fault identification;

[0017] Step 7: Based on the judgment result of step 6, if a short circuit fault occurs, use the voltage modulus Calculate the voltage modulus at the cable segment joints respectively, and substitute it into the decoupling formula to obtain the calculated value of the voltage at the fault phase joint;

[0018] Step 8: Based on the calculated value of the voltage at the faulty phase joint obtained in step 7, the fault section is determined by sequentially comparing the calculated values ​​of the voltage at different joint positions.

[0019] A further improvement of the present invention is that in step 4, based on the voltage modulus formula obtained in step 3 Let x = L, calculate the 7 voltage moduli at the end of the cable, and substitute them into the decoupling formula to obtain the calculated values ​​of the three-phase voltage at the end.

[0020]

[0021] Among them, V is the calculated value obtained by calculating the voltage modulus and then substituting it into the decoupling formula, α i , β i , γ i They are the i-th element of the 1st, 3rd, and 5th row vectors of the voltage decoupling matrix respectively.

[0022] A further improvement of the present invention is that: based on the calculated value of the terminal three-phase voltage obtained in step 4 The voltage measurement value at the end of the three-core armored submarine cable obtained in step 1 Substitute the calculated and measured amplitudes of the terminal three-phase voltage into the fault criterion respectively:

[0023]

[0024] Where peak is the amplitude, j = A, B, C, nor indicates normal operation. If the judgment condition is met, it is determined that a fault occurs and the faulty phase is determined to be phase j. Otherwise, it is determined that there is no fault.

[0025] A further improvement of the present invention is that in step 6, the method for realizing fault identification is: calculating the decoupling head-end voltage amplitude corresponding to the fourth voltage modulus in the steady state after the fault, and substituting it into the cracking criterion:

[0026]

[0027] in, is the measured value of the decoupling head-end voltage corresponding to the fourth voltage modulus in the steady state after the fault, U S is a column vector composed of the voltage measurement values ​​at the first end, P u4 is the row vector formed by taking the modulus value of the fourth row element of the voltage decoupling matrix, peak represents the amplitude, K is the threshold value, and in order to prevent misjudgment and ensure the reliability of the criterion, the value of the threshold K is obtained by calculating and analyzing the case of slight cracking and large transition resistance short circuit. If the criterion conditions are met, it is determined that a crack fault occurs in the j phase, and the step 5 is used. The size of determines the fault degree, otherwise it is judged as a short circuit fault. At this point, the fault identification is completed.

[0028] A further improvement of the present invention is that: in step 7, the voltage modulus at the cable segment joint position is calculated respectively, and the voltage calculated value is obtained by substituting it into the decoupling formula

[0029]

[0030] Among them, x1, x2, …, x N is the distance between N joints and the head end, η i is the i-th element of the row vector of the fault phase in the voltage decoupling matrix, are the i-th voltage modulus solved with the decoupled head-end quantity and the end-end quantity as initial conditions, the value range of i is 1-7, and L is the cable length.

[0031] A further improvement of the present invention is that in step 8, based on the voltage calculation value at the fault phase joint obtained in step 7, the fault section is determined by sequentially comparing the voltage amplitudes at different joint positions, specifically:

[0032] Step 8.1: Substitute the positions of N joints from the head end into the criterion formula Where n = 1, 2, ..., N, k2 is a threshold value selected according to the calculation error. This value takes the maximum calculation error at different transition resistances and different joint positions and leaves a certain margin. If the criterion is met, it is determined that the fault occurs near the nth joint. Let x∈(x n -Δx,x n +Δx), where Δx is the maximum ranging error under different conditions and a certain margin is left. If all x n If the criterion formula is not satisfied, it is considered that the fault occurs at a non-joint location and the subsequent steps are continued;

[0033] Step 8.2: If the criteria are met Then it is determined that the fault occurs between the cable head end and the first connector, that is, x∈(0,x1);

[0034] Step 8.3: If the criteria are met Then the fault is determined to occur between the cable end and the Nth connector, that is, x∈(x N ,L);

[0035] Step 8.4: If the two criteria of step 8.2 and step 8.3 are not met, calculate the two adjacent joint positions in turn. and If the value of

[0036]

[0037] Where n=1,2,…,N-1, then let n=n+1 and repeat the above calculation until

[0038]

[0039] It is determined that the fault occurs between the nth and n+1th connectors of the cable, that is, x∈(x n ,x n+1 ), so far, the fault section location is completed.

[0040] The beneficial effects of the present invention are as follows: the present invention is based on the distributed parameter model of the three-core armored submarine cable, and improves the calculation accuracy by decoupling the three-phase core, the shielding layer and the armoring layer;

[0041] The present invention utilizes the decoupled modulus to realize the identification of the fault type; the section where the fault is located is located in advance before accurate distance measurement, which greatly improves the speed and accuracy of calculation.

[0042] The present invention is applicable to high-voltage long-distance segmentally connected AC submarine cables, is not affected by power supplies at both ends, fault types and transition resistance, is easy to obtain measured values, has low implementation costs, and has high engineering practice significance. BRIEF DESCRIPTION OF THE DRAWINGS

[0043] Figure 1 It is a flow chart of the present invention.

[0044] Figure 2 Schematic diagram of the distributed parameter model of three-core armored submarine cable.

[0045] Figure 3 Schematic diagram of the segmented structure of a three-core armored submarine cable with two connectors. DETAILED DESCRIPTION

[0046] The following will disclose the embodiments of the present invention with drawings. For the purpose of clear description, many practical details will be described together in the following description. However, it should be understood that these practical details should not be used to limit the present invention. That is to say, in some embodiments of the present invention, these practical details are not necessary.

[0047] The schematic diagram of the distributed parameter model of the three-core armored submarine cable is as follows Figure 2 As shown in the figure, the electromagnetic coupling between the conductors is represented by impedance and admittance. The impedance is composed of the self-impedance of a certain phase core, the mutual impedance between different phase cores, and the mutual impedance between the corresponding phase core and the shielding layer and armor layer. Similarly, the admittance is composed of the self-admittance of a certain phase core, the mutual admittance between different phase cores, and the mutual admittance between the corresponding phase core and the shielding layer and armor layer. The schematic diagram of the segmented structure of a three-core armored submarine cable with two joints is shown in Figure 3 As shown, the measuring points are set at the beginning and end of the cable, the total length L of the cable is 30 km, and there is a joint every 10 km.

[0048] like Figure 1 As shown, a method for submarine cable fault identification and section location based on multi-modulus collaborative analysis of the present invention comprises the following steps:

[0049] Step 1: Measure the voltage and current at the beginning and end of the three-core armored submarine cable in steady state Subscript i = A, SA, B, SB, C, SC, M, respectively represent the A phase core, A phase shielding layer, B phase core, B phase shielding layer, C phase core, C phase shielding layer and armoring layer, superscript S and R represent the cable head and end quantities, respectively, where the current and three-phase core voltage are obtained by direct measurement, and the voltage of the shielding layer and armoring layer is calculated from the grounding resistance and the current flowing through it:

[0050]

[0051] in R g is the grounding resistance, which is 1Ω. The actual measured value is close to 0 during normal operation. To facilitate the implementation of subsequent steps, the value is taken as 0 during normal operation. According to the HYJQF41-F 127 / 220kV cable parameters, the impedance matrix Z, admittance matrix Y and cable length L per unit length of the three-core armored submarine cable are obtained.

[0052] Step 2: Write the 7th-order matrix equation of voltage and current with respect to distance x according to the distributed parameter model, decouple the U, I, Z, and Y matrices in the equation, and obtain the modulus equation of voltage and current;

[0053] Step 3: Decouple the measured head-end quantity and end-end quantity by the method in step 2, and use the decoupled head-end quantity and end-end quantity as initial conditions to solve the voltage modulus equation in step 2, and obtain 7 decoupled voltage modulus expressions respectively:

[0054]

[0055] The subscript m represents the modulus after decoupling, and the number corresponds to the serial number of the modulus after decoupling. It represents the calculation expression of voltage modulus obtained by taking the decoupled head-end quantity as the initial condition, It represents the voltage modulus calculation expression obtained by taking the decoupled terminal quantity as the initial condition. The specific values ​​of the row vector of the 4th row of the voltage decoupling matrix of the HYJQF41-F 127 / 220kV three-core armored submarine cable are as follows:

[0056] σ=[0.0465-0.0441i,-0.3137+0.0465i,0.0465-0.0441i,-0.3137+0.0465i,

[0057] 0.0465-0.0441i,-0.3137+0.0465i,0.8277]

[0058] Step 4: Based on the voltage modulus expression obtained in step 3 Let x = L, calculate the 7 voltage moduli at the end of the cable, and substitute them into the decoupling formula to obtain the calculated values ​​of the three-phase voltage at the end.

[0059]

[0060] Where V represents the calculated value obtained by calculating the voltage modulus and then substituting it into the decoupling formula, α i , β i , γ i They are the i-th element of the 1st, 3rd, and 5th row vectors of the voltage decoupling matrix respectively.

[0061] Step 5: Based on the calculated value of the terminal three-phase voltage in step 4 The voltage measurement value at the end of the three-core armored submarine cable obtained in step 1 Substitute the calculated and measured amplitudes of the terminal three-phase voltage into the fault criterion respectively:

[0062]

[0063] Where peak represents the amplitude, j = A, B, C, nor means normal operation. After multiple calculations, it is known that the threshold value k1 of HYJQF41-F 127 / 220kV three-core armored submarine cable is 0.00275kV. If the judgment condition is met, it is determined that a fault has occurred and the fault phase is determined to be phase j, otherwise it is determined that there is no fault.

[0064] Step 6: Based on the judgment result of step 5, if it is determined that a fault occurs, calculate the decoupling head-end voltage amplitude corresponding to the fourth voltage modulus in the steady state after the fault, and substitute it into the cracking criterion:

[0065]

[0066] in is the measured value of the decoupling head-end voltage corresponding to modulus 4 in the steady state after the fault, U S is a column vector composed of the voltage measurement values ​​at the first end, P u4 is the row vector formed by taking the modulus value of the fourth row element of the voltage decoupling matrix. The subscript peak represents the amplitude, and K is the threshold. To prevent misjudgment and ensure the reliability of the criterion, calculation and analysis are performed on the cases of 0.1Ω cracking and 1000Ω transition resistance short circuit. The threshold K is 0.15kV. If the criterion conditions are met, it is determined that a crack fault occurs in the j phase, and the step 5 is performed. The size of determines the fault degree, otherwise it is judged as a short circuit fault. At this point, the fault identification is completed.

[0067] Step 7: Based on the judgment result of step 6, if it is determined that a short circuit fault occurs, the voltage modulus at the two cable segment joints x1 = 10km and x2 = 20km is calculated respectively, and the voltage calculation value is obtained by substituting it into the decoupling formula.

[0068]

[0069] where η i is the i-th element of the row vector of the fault phase in the voltage decoupling matrix, are the i-th voltage modulus solved with the decoupled head-end quantity and the end-end quantity as initial conditions, the value range of i is 1-7, and L is the cable length.

[0070] Step 8: Based on the voltage calculation value at the fault phase joint obtained in step 7, the fault section is determined by comparing the voltage amplitudes at different joint positions in sequence. The specific steps are as follows:

[0071] (a) Substitute the distance between the two joints and the head end into the criterion formula in turn. Where n = 1, 2, k2 is a threshold value selected according to the calculation error. After multiple calculations, k2 is taken as 0.05 kV. If the criterion is met, it is determined that the fault occurs near the nth joint. Let x∈(xn -Δx,x n +Δx), where Δx = 0.3 km according to the maximum ranging error in different situations. If all x n If the criterion formula is not satisfied, it is considered that the fault occurs at a non-joint location and the subsequent steps are continued;

[0072] (b) If the criteria are met Then it is determined that the fault occurs between the cable head end and the first connector, that is, x∈(0,10), in km;

[0073] (c) If the criteria are met Then it is determined that the fault occurs between the cable end and the second connector, that is, x∈(20,30), in km;

[0074] (d) If the above two criteria are not met, calculate the and If the value of

[0075]

[0076] It is determined that the fault occurs between the first and second joints of the cable, that is, x∈(10,20), in km. At this point, the fault section location is completed.

[0077] Simulation Verification

[0078] In order to verify the effectiveness and reliability of the present invention, a transmission model of a three-core armored submarine cable was built on PSCAD / EMTDC. The core crack fault and the core-shield short circuit fault were set respectively. Fault simulation was performed under the influence of different fault types, fault distances, and fault resistances. According to the method of the present invention, MATLAB was used to calculate the relevant parameters and fault sections. The fault identification results are shown in Table 1, and the fault section location results are shown in Table 2, where x r is the actual fault distance, R f represents the fault resistance,

[0079]

[0080] Table 1 Fault identification results under different fault conditions

[0081]

[0082] Table 2 Fault section location results under different fault conditions

[0083]

[0084] According to the data in Table 1 and Table 2, the present invention has high calculation accuracy and high speed, is not affected by the power supply at both ends, the fault type and the transition resistance, and is easy to obtain the measured value.

[0085] The above description is only an embodiment of the present invention and is not intended to limit the present invention. For those skilled in the art, the present invention may have various modifications and variations. Any modification, equivalent substitution, improvement, etc. made within the spirit and principle of the present invention should be included in the scope of the claims of the present invention.

Claims

1. A method for submarine cable fault identification and section location based on multi-modulus collaborative analysis, characterized in that: The submarine cable fault identification and section location method comprises the following steps: Step 1, measuring the voltage and current at the beginning and the end of the three-core armored submarine cable, and obtaining the impedance matrix Z, the admittance matrix Y and the cable length L per unit length of the three-core armored submarine cable according to the cable parameters; Step 2: Write the 7th-order matrix equation of voltage and current with respect to distance x according to the distributed parameter model, decouple the U, I, Z, and Y matrices in the equation, and obtain the voltage modulus equation and the current modulus equation; Step 3: Based on the modulus equation obtained in step 2, the voltage modulus equation is solved by measuring the voltage and current at the beginning and end of the three-core armored submarine cable as initial conditions to obtain the voltage modulus at any distance x. Where S and R represent the cable head and end quantities respectively, and m represents the modulus after decoupling; Step 4: Voltage modulus obtained based on step 3 Calculate the 7 voltage moduli at the end of the three-core armored submarine cable and substitute them into the decoupling formula to obtain the calculated value of the three-phase voltage at the end Step 5: Calculate the terminal three-phase voltage based on the value obtained in step 4 The voltage measurement value at the end of the three-core armored submarine cable obtained in step 1 Substitute the calculated value and the measured value of the three-phase voltage at the terminal into the fault judgment criterion to determine whether a fault occurs; Step 6: Based on the judgment result of step 5, if it is determined that a fault occurs, the voltage amplitude of the decoupling head end corresponding to the fourth voltage modulus in the steady state after the fault is calculated, and substituted into the cracking criterion to judge whether the fault type is cracking or short circuit, thereby realizing fault identification; Step 7: Based on the judgment result of step 6, if a short circuit fault occurs, use the voltage modulus Calculate the voltage modulus at the cable segment joints respectively, and substitute it into the decoupling formula to obtain the calculated value of the voltage at the fault phase joint; Step 8: Based on the calculated value of the voltage at the faulty phase joint obtained in step 7, the fault section is determined by sequentially comparing the calculated values ​​of the voltage at different joint positions.

2. The method for submarine cable fault identification and section location based on multi-modulus collaborative analysis according to claim 1 is characterized in that: Based on the calculated value of the terminal three-phase voltage obtained in step 4 The voltage measurement value at the end of the three-core armored submarine cable obtained in step 1 Substitute the calculated and measured amplitudes of the terminal three-phase voltage into the fault criterion respectively: Where peak is the amplitude, j = A, B, C, nor indicates normal operation. If the judgment condition is met, it is determined that a fault occurs and the faulty phase is determined to be phase j. Otherwise, it is determined that there is no fault.

3. The method for submarine cable fault identification and section location based on multi-modulus collaborative analysis according to claim 2 is characterized in that: In step 6, the method for realizing fault identification is: calculating the voltage amplitude of the decoupling head end corresponding to the fourth voltage modulus in the steady state after the fault, and substituting it into the cracking criterion: in, is the measured value of the decoupling head-end voltage corresponding to the fourth voltage modulus in the steady state after the fault, U S is a column vector composed of the voltage measurement values ​​at the first end, P u4 is the row vector formed by taking the modulus value of the fourth row element of the voltage decoupling matrix, peak represents the amplitude, K is the threshold, and the value of the threshold K is obtained by calculating and analyzing the cracking and short-circuit conditions. If the judgment condition is met, it is determined that a cracking fault occurs in the j phase, and the step 5 is performed. The size of determines the fault degree, otherwise it is judged as a short circuit fault. At this point, the fault identification is completed.

4. The method for submarine cable fault identification and section location based on multi-modulus collaborative analysis according to claim 1 is characterized in that: In step 7, the voltage modulus at the cable segment joints is calculated and substituted into the decoupling formula to obtain the voltage calculation value. Among them, x1, x2, …, x N is the distance between N joints and the head end, η i is the i-th element of the row vector of the fault phase in the voltage decoupling matrix, are the i-th voltage modulus solved with the decoupled head-end quantity and the end-end quantity as initial conditions, the value range of i is 1-7, and L is the cable length.

5. The method for submarine cable fault identification and section location based on multi-modulus collaborative analysis according to claim 4 is characterized in that: In step 8, based on the voltage calculation value at the fault phase joint obtained in step 7, the fault section is determined by comparing the voltage amplitudes at different joint positions in sequence, specifically: Step 8.1: Substitute the positions of N joints from the head end into the criterion formula Where n = 1, 2, ..., N, k2 is a threshold value selected according to the calculation error. This value takes the maximum calculation error at different transition resistances and different joint positions and leaves a margin. If the criterion is met, it is determined that the fault occurs near the nth joint. Let x∈(x n -Δx,x n +Δx), where Δx is the maximum ranging error under different conditions and a margin is left. If all x n If the criterion formula is not satisfied, it is considered that the fault occurs at a non-joint location and the subsequent steps are continued; Step 8.2: If the criteria are met Then it is determined that the fault occurs between the cable head end and the first connector, that is, x∈(0,x1); Step 8.3: If the criteria are met Then the fault is determined to occur between the cable end and the Nth connector, that is, x∈(x N ,L); Step 8.4: If the two criteria of step 8.2 and step 8.3 are not met, calculate the two adjacent joint positions in turn. and If the value of Where n=1,2,…,N-1, then let n=n+1 and repeat the above calculation until It is determined that the fault occurs between the nth and n+1th connectors of the cable, that is, x∈(x n ,x n+1 ), so far, the fault section location is completed.

6. A method for submarine cable fault identification and section location based on multi-modulus collaborative analysis according to any one of claims 1 to 5, characterized in that: In step 4, based on the voltage modulus formula obtained in step 3 Let x = L, calculate the 7 voltage moduli at the end of the cable, and substitute them into the decoupling formula to obtain the calculated values ​​of the three-phase voltage at the end. Among them, V is the calculated value obtained by calculating the voltage modulus and then substituting it into the decoupling formula, α i , β i , γ i They are the i-th element of the 1st, 3rd, and 5th row vectors of the voltage decoupling matrix respectively.

7. The method for submarine cable fault identification and section location based on multi-modulus collaborative analysis according to claim 6 is characterized by: 7 decoupled voltage modulus expressions Among them, m represents the modulus after decoupling, and the number corresponds to the serial number of the modulus after decoupling. represents the voltage modulus obtained by taking the decoupled head-end quantity as the initial condition, It represents the voltage modulus obtained by taking the decoupled terminal quantity as the initial condition.

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