A hierarchical modeling method for transient thermal field analysis of high-voltage three-core submarine cables under fluctuating loads

Abstract

This invention discloses a layered modeling method for transient thermal field analysis of high-voltage three-core submarine cables under fluctuating loads. The method simplifies the three-core cable into a coaxial single-core structure based on the principle of equal thermal resistance. Then, based on this simplified equivalent structure, the cable and its external laying environment are layered to determine the heat source intensity, thermal resistance, and heat capacity parameters, establishing a trapezoidal thermal circuit model. The method then simulates the transient temperature rise characteristics of each layer of the cable under fluctuating loads. This method can be applied to any single-core or multi-core submarine cable structure with different insulation types. It can be applied to AC or DC cables, performing transient thermal field analysis under different fluctuating current load waveforms and varying ambient temperatures to accurately determine the steady-state and transient allowable load capacity of the cable.

Description

Technical Field

[0001] This invention relates to the field of transient thermal field analysis technology for high-voltage cables, and specifically to a layered modeling method for transient thermal field analysis of high-voltage three-core submarine cables under fluctuating loads. Background Technology

[0002] Due to the diurnal and seasonal variations in natural wind power, wind turbine output power exhibits significant fluctuations, and the load on wind power transmission cables also fluctuates continuously over a wide range. Currently, the core cross-section of wind power transmission cables is typically determined based on the cable's long-term allowable current-carrying capacity not being less than the maximum transmission current of the wind farm, without considering the cable's short-term overload capacity under fluctuating loads. This results in low equipment utilization, and for offshore wind farm cable lines with investments often exceeding hundreds of millions, this waste of resources is undoubtedly enormous.

[0003] Submarine cables (or submarine cables) and the surrounding soil have a large thermal time constant, typically requiring several days or even months to reach thermal stability. Therefore, high-load operation for a certain period will not cause overheating of the cable core. Studies have shown that under periodic or fluctuating load conditions, the peak current that the cable core can withstand is usually much larger than the 100% load current. This is especially true for submarine cables transmitting power from deeply buried wind farms. Considering the temperature rise characteristics of cables under fluctuating loads, the maximum allowable rated current can be increased by approximately 15%, and the core cross-sectional area can be reduced by 50%.

[0004] The cable temperature rise and current carrying capacity assessment methods widely used in the engineering field today originate from Neher's research. This technique involves stratifying the cable and its installation environment, using a multi-loop trapezoidal network to characterize the corresponding heat sources, thermal resistance, and heat capacity, and establishing thermal circuit equations for solution analysis. Studies have shown that this method is effective in applications involving single-core cables.

[0005] With the development of cable production and laying technology, long-length extruded insulated submarine cables generally adopt a three-core structure to reduce the occupation of submarine channels and save laying costs. For these cables, optimizing the core cross-section selection based on the fluctuation of wind power load is even more important, as it not only significantly affects project costs but may even determine the success or failure of the project. When performing transient thermal analysis on three-core cables using the trapezoidal equivalent thermal circuit method based on the Neher model, the layering technique cannot be directly applied due to the non-concentric structure inside the cable. Therefore, targeted research is urgently needed to propose effective solutions. Summary of the Invention

[0006] To address the limitation of transient thermal field analysis techniques based on trapezoidal equivalent thermal circuits in high-voltage three-core submarine cable structures, this invention aims to provide a layered modeling method for transient thermal field analysis of high-voltage three-core submarine cables under fluctuating loads. This method simplifies the three-core cable to a coaxial single-core structure based on the principle of equal thermal resistance. Then, based on this simplified equivalent structure, the cable and its external laying environment are layered to determine heat source intensity, thermal resistance, and thermal capacity parameters. A trapezoidal thermal circuit model is established, and the transient temperature rise characteristics of each layer of the cable under fluctuating loads are simulated. This method can be applied to any single-core or multi-core submarine cable structure with different insulation types. It can be applied to AC or DC cables, performing transient thermal field analysis under different fluctuating current load waveforms and varying ambient temperatures to accurately determine the cable's steady-state and transient allowable load capacity.

[0007] To achieve the above objectives, the technical solution adopted by the present invention is as follows:

[0008] A hierarchical modeling method for transient thermal field analysis of a high-voltage three-core submarine cable under fluctuating load includes the following steps:

[0009] Step 1: Establish an equivalent simplified structure:

[0010] Based on the principle of equal thermal resistance, the actual three-core cable structure is equivalent to a simplified single-core cable structure. The specific equivalence process is divided into two steps: (1) The part between the metal sheath and the armor layer is equivalent according to the principle of equal thermal resistance between the metal sheath and the armor layer to obtain the equivalent outer diameter of the metal sheath, calculate the heat capacity of the metal sheath before equivalence, and determine the equivalent inner diameter of the metal sheath according to the heat capacity value of the metal sheath before equivalence; (2) The part between the conductor and the metal sheath is equivalent according to the principle of equal thermal resistance between the conductor and the metal sheath to obtain the equivalent conductor diameter; the part between the metal sheath and the armor layer is defined as the "filling layer", and the "conductor shielding layer" and "insulation shielding layer" are both incorporated into the "insulation layer"; the thermal resistance and heat capacity calculation of each layer structure before equivalence are shown in Table 1, and the structural dimensions and parameters after equivalence are shown in Table 2.

[0011] Table 1. List of Calculation Parameters for Each Layer of High-Voltage Three-Core Submarine Cable Before Equivalent Measurement

[0012]

[0013] Table 2. Calculation list of equivalent structural parameters for each layer of the high-voltage three-core submarine cable.

[0014]

[0015]

[0016] Step 2: Determine the seabed soil domain boundary of the submarine cable

[0017] The boundary of the seabed soil domain is a circle centered on the axis of the submarine cable, and its radius is determined by equation (1):

[0018]

[0019] In the formula, θ lim To determine the temperature threshold of the seabed soil domain boundary; W I The loss per unit length of submarine cable, W / m; ρ T δ is the soil thermal resistivity, K·m / W; δ is the soil thermal diffusivity, m 2 / s; r is the radius corresponding to the boundary circumference of the seabed soil domain, in meters; τ is the duration of the transient load, in seconds;

[0020] In the above formula, E i (x) is an exponential integral function, calculated according to equation (2):

[0021]

[0022] The values ​​of coefficients a0~a5, a′1, a′2, b′1, b′2 in equation (2) can be obtained from the table below:

[0023] <![CDATA[a0]]> -0.57721566 <![CDATA[a′1]]> 2.334733 <![CDATA[a1]]> 0.99999193 <![CDATA[a′2]]> 0.250621 <![CDATA[a2]]> -0.24991055 <![CDATA[b′1]]> 3.330657 <![CDATA[a3]]> 0.05519968 <![CDATA[b′2]]> 1.681534 <![CDATA[a4]]> -0.00976004 <![CDATA[a5]]> 0.00107857

[0024] Step 3: Determine the time step

[0025] The transient process of the conductor temperature of a submarine cable refers to the process in which the conductor temperature changes from t1℃ to t2℃, and the time required for this change is the total duration of the transient process. To calculate the temperature rise at the end of the transient process, it is necessary to calculate the temperature rise at each time step within this process to iteratively calculate the final temperature rise. During the calculation, different time step sizes are selected according to the actual situation.

[0026] (1) When hour

[0027]

[0028] (2) When hour

[0029]

[0030] In the formula, Δτ — time step, s;

[0031] τ—Total duration of the transient process, in seconds;

[0032] RC—the thermal time constant of a submarine cable, defined as the product of the total thermal resistance and total heat capacity of the submarine cable, in seconds;

[0033] Step 4: Layering of Insulation and Soil

[0034] The insulation consists of 5 to 10 layers, and the number of soil layers should meet the following conditions:

[0035]

[0036] In the formula, D t D is the diameter of the boundary of the seabed soil domain. t =2r,m;D e The outer diameter of the submarine cable is in meters (m); n s The number of layers of seabed soil is given by equation (5). As can be seen from equation (5), the larger the area of ​​the seabed soil domain, the more layers should be.

[0037] After determining the total number of layers, the insulation and soil are layered according to two principles: equal thickness or equal thermal resistance. The equal thickness method is to layer each layer with equal thickness; while when layering according to the equal thermal resistance method, the thickness of each layer will be different, with the thickness increasing further away from the center of the submarine cable.

[0038] Step 5: Calculation of Insulation Dielectric Loss

[0039] 1) Equal thickness method

[0040] The thickness of each insulating layer is the same, but the capacitance is different; the voltage of each layer is also different, and it is distributed according to the capacitance; the loss of each layer is calculated accordingly, as shown in equations (6) to (8).

[0041]

[0042]

[0043]

[0044] In the formula, W dj For the dielectric loss of the j-th layer of insulation, U0 and U dj These are the overall insulation voltage and the voltage of the j-th layer, respectively, in V and C. d and C dj For the total insulation capacitance and the capacitance of the j-th layer, F / m; ε r tanδ represents the relative permittivity and dielectric loss factor of the insulating material, ω represents the angular frequency, and r represents the dielectric loss factor. j-1 r j Let r be the inner and outer diameters of the j-th layer. j+1 Let J be the outer diameter of the (j+1)th layer, where j = 1…n;

[0045] 2) Equal thermal resistance method

[0046] After the insulation is layered according to the principle of equal thermal resistance, the voltage U of each layer is... j The same applies, as in equation (9), which is distributed equally according to the number of layers; the capacitance and loss of each layer are also equal, and can still be calculated according to equations (6) and (7);

[0047]

[0048] In the formula, n d This refers to the number of insulation layers.

[0049] Step 6: Calculation of thermal resistance and heat capacity

[0050] The submarine cable and its surrounding soil are divided into n thin-walled shells. The thermal resistance T of the j-th layer of the submarine cable and soil domain is given by... j The calculation formula is as follows:

[0051]

[0052] In the formula, ρ T Thermal resistivity;

[0053] Divide the space at the middle of each layer and redistribute the heat capacity to the two adjacent nodes to obtain a π-type equivalent thermal path; the heat capacity Q of each node j The calculation formula is as follows:

[0054]

[0055] In the formula, σ is the volumetric specific heat capacity of the j-th layer material, J / K·m 3 ;

[0056] Step 7: Calculation of conductor, metal sheath, and armor layer losses

[0057] Conductor loss W per unit length of submarine cable c for:

[0058] W c =I 2 R (12) In the formula, I is the current flowing through the submarine cable conductor, in A; R is the AC resistance per unit length of the submarine cable conductor, which is temperature-dependent, in Ω / m.

[0059] Metal sleeve loss W s1 and armor layer loss W s2 Both are based on conductor loss W c It is expressed as a proportion, that is:

[0060] W s1 =K1·W c (13)

[0061] W s2 =K2·W c (14)

[0062] The loss factor K1 of the metal sleeve and the loss factor K2 of the armor layer are both constant values;

[0063] The conductor, metal sheath, and armor layer act as heat sources, and their magnitude varies with the load current and temperature in the submarine cable conductor. When performing transient calculations, these loss values ​​need to be recalculated at each time step.

[0064] Step 8: Constructing the equivalent thermal circuit

[0065] After determining the structure and parameters of each layer of the submarine cable, including thermal resistance, thermal capacity, and loss, the corresponding trapezoidal equivalent thermal path is established.

[0066] Step 9: Calculation of node temperature rise

[0067] After establishing a trapezoidal equivalent thermal path for the submarine cable, the temperature rise at each node can be determined. The basic equations and steps are as follows:

[0068] 1) Calculation of node coefficients

[0069]

[0070] A j,i Let A be the coefficient of the node with node number j and time interval number i, where the value of j ranges from j = 0 to j = n-1, and A -1,i =1; Δτ i For time step; A j,i It will not change throughout the calculation process, unless the time step changes;

[0071] 2) Calculation of initial node values

[0072]

[0073] B j,i Let B be the initial value of the node with node number j and time interval number i. The value of j ranges from j = 0 to j = n-1. -1,i =0; W j,i For the loss corresponding to node j and time interval i; θ j,i-1 For the temperature rise corresponding to node j and time interval number i-1;

[0074] 3) Temperature rise calculation

[0075] θ j,i =A j,i ·θ j+1,i +B j,i (17)

[0076] The value of j ranges from j = 0 to j = n-1, and θ n,i =0;

[0077] 4) Repeat steps 2) and 3) until the preset time is reached, at which point the transient process ends.

[0078] This invention proposes a method to simplify a high-voltage three-core submarine cable into a coaxial single-core structure based on the principle of equal thermal resistance, and to divide the cable and external laying environment into layers based on the equivalent simplified structure, determine the heat source intensity (i.e. loss), thermal resistance and heat capacity parameters, establish a trapezoidal thermal circuit model, and conduct simulation research on the transient temperature rise characteristics of each layer of the cable under fluctuating load. It has the following advantages: 1) This invention can be applied to any single-core or multi-core land cable and submarine cable structure;

[0079] 2) This invention can be applied to cables operating under AC or DC voltage;

[0080] 3) This invention can be applied to transient thermal field analysis of cables under arbitrary fluctuating current load waveforms and varying ambient temperatures, and can also be used for steady-state thermal field analysis under constant load.

[0081] 4) This invention can be applied to evaluate the long-term and short-term permissible load capacity of cables;

[0082] 5) The analysis process in this invention can be implemented by technicians using any commercial or specialized software, and has good engineering applicability;

[0083] 6) This invention has a clear theoretical basis, a clear logical concept, a clear implementation process, and is easy to apply and expand. Attached Figure Description

[0084] Figure 1 This is a schematic diagram of the equivalent structure of a high-voltage three-core submarine cable before and after.

[0085] Figure 2 The diagram shows the equivalent thermal circuit of a cable trapezoid. In the diagram, W represents loss, Q represents heat capacity, T represents thermal resistance, θ represents temperature, and n represents a node.

[0086] Figure 3 This is a flowchart for calculating the transient temperature rise of cables.

[0087] Figure 4a , Figure 4b , Figure 4c Examples 1, 2, and 3 illustrate three typical fluctuating current load scenarios, where a 3×1000mm AC 330kV direct-buried cable is installed in the soil under these conditions. 2 Temperature variation curves of different parts of a cross-linked polyethylene insulated submarine cable over time. Detailed Implementation

[0088] The present invention will now be described in more detail with reference to specific embodiments.

[0089] like Figure 3As shown, the present invention provides a hierarchical modeling method for transient thermal field analysis of a high-voltage three-core submarine cable under fluctuating load, comprising the following steps:

[0090] Step 1: Establish an equivalent simplified structure:

[0091] Based on the principle of equal thermal resistance, the actual three-core cable structure is equivalent to a simplified single-core cable structure. The specific equivalence process is divided into two steps: (1) The part between the metal sheath and the armor layer is equivalent according to the principle of equal thermal resistance between the metal sheath and the armor layer to obtain the equivalent outer diameter of the metal sheath, calculate the heat capacity of the metal sheath before equivalence, and determine the equivalent inner diameter of the metal sheath based on the heat capacity value of the metal sheath before equivalence; (2) The part between the conductor and the metal sheath is equivalent according to the principle of equal thermal resistance between the conductor and the metal sheath to obtain the equivalent conductor diameter. The part between the metal sheath and the armor layer is defined as the "filling layer", and the "conductor shielding layer" and "insulation shielding layer" are both incorporated into the "insulation layer". The schematic diagram of the high-voltage three-core submarine cable before and after equivalence is shown in the figure. Figure 1 As shown in Table 1, the thermal resistance and thermal capacity of each layer before equivalence are calculated, and the structural dimensions and parameters after equivalence are calculated, as shown in Table 2.

[0092] Table 1. List of Calculation Parameters for Each Layer of High-Voltage Three-Core Submarine Cable Before Equivalent Measurement

[0093]

[0094]

[0095] Table 2. Calculation list of equivalent structural parameters for each layer of the high-voltage three-core submarine cable.

[0096]

[0097] Step 2: Determine the seabed soil domain boundary of the submarine cable

[0098] The boundary of the seabed soil domain is a circle centered on the axis of the submarine cable, and its radius can be determined by equation (1):

[0099]

[0100] In the formula, θ lim To determine the temperature threshold of the seabed soil domain boundary, it can be set according to requirements; a smaller value, such as 0.1℃, should be used when high thermal field accuracy is required; W I The loss per unit length of submarine cable, W / m; ρ T δ is the soil thermal resistivity, K·m / W; δ is the soil thermal diffusivity, m 2 / s; r is the radius corresponding to the boundary circumference of the seabed soil domain, in meters; τ is the duration of the transient load, in seconds.

[0101] In the above formula, E i(x) is an exponential integral function, which can be calculated according to equation (2):

[0102]

[0103] The values ​​of the coefficients a0~a5, a′1, a′2, b′1, b′2 in equation (2) can be obtained from the table below.

[0104] <![CDATA[a0]]> -0.57721566 <![CDATA[a′1]]> 2.334733 <![CDATA[a1]]> 0.99999193 <![CDATA[a′2]]> 0.250621 <![CDATA[a2]]> -0.24991055 <![CDATA[b′1]]> 3.330657 <![CDATA[a3]]> 0.05519968 <![CDATA[b′2]]> 1.681534 <![CDATA[a4]]> -0.00976004 <![CDATA[a5]]> 0.00107857

[0105] Step 3: Determine the time step

[0106] The transient process of submarine cable conductor temperature refers to the process in which the conductor temperature changes from t1℃ to t2℃, and the time required for this change is the total duration of the transient process. To calculate the temperature rise at the end of the transient process, it is necessary to calculate the temperature rise at each time step within this process, and then iteratively calculate the final temperature rise. During the calculation, different time step sizes can be selected according to the actual situation.

[0107] (1) When hour

[0108]

[0109] (2) When hour

[0110]

[0111] In the formula, Δτ — time step, s;

[0112] τ—Total duration of the transient process, in seconds;

[0113] RC—the thermal time constant of a submarine cable, defined as the product of the cable's total thermal resistance (between the conductor and the outer surface) and total thermal capacity (the entire cable), in seconds.

[0114] Step 4: Layering of Insulation and Soil

[0115] Insulation can generally be divided into 5 to 10 layers. For cables with a thickness of 10mm or more, it is recommended to use 10 layers. For thin insulation, the number of layers can be reduced appropriately.

[0116] The number of soil layers should meet the following conditions:

[0117]

[0118] In the formula, D t D is the diameter of the boundary of the seabed soil domain. t =2r,m;D e The outer diameter of the submarine cable is in meters (m); n s Let be the number of layers of seabed soil. As can be seen from equation (5), the larger the area of ​​the seabed soil domain, the more layers should be.

[0119] Once the total number of layers is determined, the insulation and soil can be layered according to two principles: equal thickness or equal thermal resistance. The equal thickness method involves layering each layer with equal thickness; while the equal thermal resistance method results in layers of varying thickness, with the thickness increasing further away from the center of the submarine cable.

[0120] Step 5: Calculation of Insulation Dielectric Loss

[0121] 1) Equal thickness method

[0122] The thickness of each insulating layer is the same, but the capacitance is different; the voltage of each layer is also different, and it is distributed according to the capacitance; the loss of each layer is calculated accordingly, as shown in equations (6) to (8).

[0123]

[0124]

[0125]

[0126] In the formula, W dj For the dielectric loss of the j-th layer of insulation, U0 and U dj These are the overall insulation voltage and the voltage of the j-th layer, respectively, in V and C. d and C dj For the total insulation capacitance and the capacitance of the j-th layer, F / m; ε r tanδ and ω are the relative permittivity and dielectric loss factor of the insulating material, respectively, and r is the angular frequency. j-1 r j Let r be the inner and outer diameters of the j-th layer. j+1 Let be the outer diameter of the (j+1)th layer (j=1…n), in meters.

[0127] 2) Equal thermal resistance method

[0128] After the insulation is layered according to the principle of equal thermal resistance, the voltage U of each layer is... j The same applies, as in equation (9), which is distributed equally according to the number of layers; the capacitance and loss of each layer are also equal, and can still be calculated according to equations (6) and (7).

[0129]

[0130] In the formula, n d This represents the number of insulation layers.

[0131] Step 6: Calculation of thermal resistance and heat capacity

[0132] The submarine cable and its surrounding soil are divided into n thin-walled shells. The thermal resistance T of the j-th layer of the submarine cable and soil domain is given by... j The calculation formula is as follows:

[0133]

[0134] In the formula, ρ T is the thermal resistivity.

[0135] Dividing the space at the middle of each layer and redistributing the heat capacity to adjacent nodes yields a π-type equivalent thermal path. The heat capacity Q of each node is... j The calculation formula is as follows:

[0136]

[0137] In the formula, σ is the volumetric specific heat capacity of the material layer, J / K·m 3 .

[0138] Step 7: Calculation of conductor, metal sheath, and armor layer losses

[0139] Conductor loss W per unit length of submarine cable c for:

[0140] W c =I 2 R (12) In the formula, I is the current flowing through the submarine cable conductor, in A; R is the AC resistance per unit length of the submarine cable conductor, which is temperature-dependent, in Ω / m.

[0141] Metal sleeve loss W s1 and armor layer loss W s2 Both are based on conductor loss W c It is expressed as a proportion, that is:

[0142] W s1 =K1·W c (13)

[0143] W s2 =K2·W c (14)

[0144] In this method, the metal sheath loss factor K1 and the armor layer loss factor K2 are both constant values, determined by technicians based on the results of relevant steady-state calculations.

[0145] The conductor, metal sheath, and armor layer act as heat sources, and their magnitude varies with the load current and temperature in the submarine cable conductor. During transient calculations, these loss values ​​need to be recalculated at each time step.

[0146] Step 8: Constructing the equivalent thermal circuit

[0147] After determining the structure and parameters of each layer of the submarine cable, including thermal resistance, thermal capacity, and loss, through the above steps, the corresponding trapezoidal equivalent thermal path can be established, such as... Figure 2 As shown.

[0148] Step 9: Calculation of node temperature rise

[0149] The transient temperature rise calculation no longer considers dielectric loss. It is assumed that the temperature distribution of the cable body has reached a steady state under the influence of dielectric loss, that is, the initial temperature distribution already includes the effect of dielectric loss.

[0150] After establishing a trapezoidal equivalent thermal path for the submarine cable, the temperature rise at each node can be determined. The basic equations and steps are as follows: 1) Calculation of node coefficients.

[0151]

[0152] A j,i Let A be the coefficient of the node with node number j and time interval number i, where the value of j ranges from j = 0 to j = n-1, and A -1,i =1; Δτ i For time step; A j,i It will not change throughout the calculation process unless the time step changes.

[0153] 2) Calculation of initial node values

[0154]

[0155] B j,i Let B be the initial value of the node with node number j and time interval number i. The value of j ranges from j = 0 to j = n-1. -1,i =0; W j,i For the loss corresponding to node j and time interval i; θ j,i-1 This represents the temperature rise corresponding to node j and time interval number i-1.

[0156] 3) Temperature rise calculation

[0157] θ j,i =A j,i ·θ j+1,i +B j,i (17)

[0158] The value of j ranges from j = 0 to j = n-1, and θ n,i =0.

[0159] 4) Repeat steps 2) and 3) until the preset time is reached, at which point the transient process ends.

[0160] Based on the above analysis of the transient temperature rise of the submarine cable, the corresponding calculation flowchart is drawn, as follows: Figure 3 As shown.

[0161] The proposed AC 330kV 3×1000mm transmission line for a certain wind farm2 Taking a polyethylene insulated submarine cable as an example, the structural parameters and thermal resistance and thermal capacity of each part before and after the equivalent installation are shown in Table 3. The cable is directly buried at a depth of 1.5m, the soil thermal resistivity is 0.8K·m / W, and the ambient temperature is 25℃.

[0162] Table 3 330kV 3×1000mm 2 Parameter list before and after equivalent operation of submarine cables

[0163]

[0164] The actual wind farm output load current was investigated, and three typical fluctuating load conditions with high load rates and long durations were selected for the study of cable transient temperature rise characteristics. Specific settings are as follows:

[0165] Example 1: 750A(8h) + 1000A(8h) + 750A(8h);

[0166] Example 2: 750A(24h) + 1000A(24h) + 750A(24h);

[0167] Example 3: 500A(24h)+1000A(24h)+500A(24h)+1000A(24h)+500A

[0168] (24h)+1000A(24h)+500A(24h).

[0169] Under each load condition, the value before the parentheses is the applied current amplitude, and the value inside the parentheses is the duration of the current. In Example 1, the total application time of the transient current is 24 hours, with each current applied for 8 hours in the order of 750A, 1000A, and 750A. In Example 2, the total application time of the transient current is 72 hours, with each current applied for 24 hours in the order of 750A, 1000A, and 750A. In Example 3, the total application time of the transient current is 168 hours, with 500A and 1000A currents applied alternately, each lasting 24 hours.

[0170] To correspond to the continuous operation of the submarine cable, it is assumed that the cable has reached thermal steady state under rated voltage and 500A load current before the occurrence of high load periods. Starting from this point, three fluctuating load conditions are applied to study the transient temperature changes of the submarine cable.

[0171] According to the invention, a trapezoidal equivalent thermal path for the submarine cable was established, and parameters such as loss, thermal resistance, and heat capacity were determined. A corresponding calculation program was also developed. In the program, three types of time-varying fluctuating loads were used as core current inputs to solve for the transient thermal response of the three-core submarine cable, obtaining the temperature variations over time at the core conductor, lead sheath, and armor. Figure 4a , Figure 4b , Figure 4c As shown. Meanwhile, to verify the calculation results, a 330kV 3×1000mm direct-buried cable was constructed in the soil. 2 The finite element model of the submarine cable, under the same conditions, was used for transient thermal field simulation to obtain the conductor temperature change curve, which is also shown in... Figure 4a , Figure 4b , Figure 4c In the middle, there is the dashed line marked "Conductor (Finite Element)" in the figure.

[0172] from Figure 4a , Figure 4b , Figure 4c It can be seen that as the applied current amplitude increases, the temperature of each layer of the submarine cable gradually rises; the higher the load and the longer the duration, the more significant the temperature rise of the submarine cable; however, the temperature change lags significantly behind the current change, indicating that there is a large thermal time constant in the submarine cable line. In Example 3, it was observed that when the load current repeats in a cycle of 48 hours (24 hours each for 500A and 1000A), on the one hand, the temperature of each layer of the submarine cable shows a corresponding periodic change with the load change; on the other hand, the temperature gradually increases with the increase of the number of cycles, showing the heat accumulation effect generated by the periodic high load acting on the submarine cable. As the number of cycles continues to increase, this effect will gradually tend to saturate.

[0173] Figure 4a , Figure 4b , Figure 4c In the simulation, the conductor temperature change curve obtained by modeling and programming the equivalent structure of the three-core submarine cable basically coincides with the output curve of the finite element simulation. The maximum conductor temperature values ​​are close, with a maximum deviation of 3%, as shown in Table 4. This indicates that the method proposed in this invention, which uses the equal thermal resistance method to perform single-core structural equivalence analysis of the three-core submarine cable and establishes a layered trapezoidal thermal circuit model for transient temperature response analysis of the submarine cable under fluctuating loads, can provide sufficient accuracy to meet the needs of engineering applications.

[0174] Table 4. Maximum core temperature and deviation calculated by thermal circuit method and finite element method

[0175]

Claims

1. A hierarchical modeling method for transient thermal field analysis of a high-voltage three-core submarine cable under fluctuating load, characterized in that: Includes the following steps: Step 1: Establish an equivalent simplified structure: Based on the principle of equal thermal resistance, the actual three-core cable structure is equivalent to a simplified single-core cable structure. The specific equivalence process is divided into two steps: (1) The part between the metal sheath and the armor layer is equivalent according to the principle of equal thermal resistance between the metal sheath and the armor layer to obtain the equivalent outer diameter of the metal sheath, calculate the heat capacity of the metal sheath before equivalence, and determine the equivalent inner diameter of the metal sheath according to the heat capacity value of the metal sheath before equivalence; (2) The part between the conductor and the metal sheath is equivalent according to the principle of equal thermal resistance between the conductor and the metal sheath to obtain the equivalent conductor diameter; the part between the metal sheath and the armor layer is defined as the "filling layer", and the "conductor shielding layer" and "insulation shielding layer" are both incorporated into the "insulation layer"; the thermal resistance and heat capacity calculation of each layer structure before equivalence are shown in Table 1, and the structural dimensions and parameters after equivalence are shown in Table 2. Table 1. List of Calculation Parameters for Each Layer of High-Voltage Three-Core Submarine Cable Before Equivalent Measurement Table 2. Calculation list of equivalent structural parameters for each layer of the high-voltage three-core submarine cable. Step 2: Determine the seabed soil domain boundary of the submarine cable The boundary of the seabed soil domain is a circle centered on the axis of the submarine cable, and its radius is determined by equation (1): where θ lim is the temperature threshold for determining the boundary of the seabed soil domain; W I is the loss per unit length of the submarine cable, W / m; p T is the soil thermal resistance, K-m / W; δ is the soil thermal diffusivity, m 2 / s; r is the radius corresponding to the circumference of the boundary of the seabed soil domain, m; and τ is the time of action of the transient load, s. In the above formula, E i (x) is an exponential integral function, calculated according to equation (2): The values ​​of coefficients a0~a5, a'1, a'2, a'1, b'2 in equation (2) can be obtained from the table below: Step 3: Determine the time step The transient process of conductor temperature in a submarine cable refers to the process in which the conductor temperature changes from t1℃ to t2℃, and the time required for this change is the total duration of the transient process. To calculate the temperature rise at the end of the transient process, it is necessary to calculate the temperature rise at each time step within the process and iteratively calculate the final temperature rise. During the calculation, different time steps are selected according to the actual situation; (1) When hour (2) When hour In the formula, Δτ — time step, s; τ—Total duration of the transient process, in seconds; RC—the thermal time constant of a submarine cable, defined as the product of the total thermal resistance and total heat capacity of the submarine cable, in seconds; Step 4: Layering of Insulation and Soil The insulation consists of 5 to 10 layers, and the number of soil layers should meet the following conditions: In the formula, D t D is the diameter of the boundary of the seabed soil domain. t =2r,m;D e The outer diameter of the submarine cable is in meters (m); n s The number of layers of seabed soil is given by equation (5). As can be seen from equation (5), the larger the area of ​​the seabed soil domain, the more layers should be. After determining the total number of layers, the insulation and soil are layered according to two principles: equal thickness or equal thermal resistance. The equal thickness method is to layer each layer with equal thickness; while when layering according to the equal thermal resistance method, the thickness of each layer will be different, with the thickness increasing further away from the center of the submarine cable. Step 5: Calculation of Insulation Dielectric Loss 1) Equal thickness method The thickness of each insulating layer is the same, but the capacitance is different; the voltage of each layer is also different, and it is distributed according to the capacitance; the loss of each layer is calculated accordingly, as shown in equations (6) to (8). In the formula, W dj For the dielectric loss of the j-th layer of insulation, U0 and U dj These are the overall insulation voltage and the voltage of the j-th layer, respectively, in V and C. d and C dj For the total insulation capacitance and the capacitance of the j-th layer, F / m; ε r tanδ represents the relative permittivity and dielectric loss factor of the insulating material, ω represents the angular frequency, and r represents the dielectric loss factor. j-1 r j Let r be the inner and outer diameters of the j-th layer. j+1 Let J be the outer diameter of the (j+1)th layer. 2) Equal thermal resistance method After the insulation is layered according to the principle of equal thermal resistance, the voltage U of each layer is... j The same applies, as in equation (9), which is distributed equally according to the number of layers; the capacitance and loss of each layer are also equal, and can still be calculated according to equations (6) and (7); In the formula, n d This refers to the number of insulation layers. Step 6: Calculation of thermal resistance and heat capacity The submarine cable and its surrounding soil are divided into n thin-walled shells. The thermal resistance T of the j-th layer of the submarine cable and soil domain is given by... j The calculation formula is as follows: In the formula, ρ T Thermal resistivity; Divide the space at the middle of each layer and redistribute the heat capacity to the two adjacent nodes to obtain a π-type equivalent thermal path; the heat capacity Q of each node j The calculation formula is as follows: In the formula, σ is the volumetric specific heat capacity of the j-th layer material, J / K·m 3 ; Step 7: Calculation of conductor, metal sheath, and armor layer losses Conductor loss W per unit length of submarine cable c for: W c =I 2 R (12) In the formula, I is the current flowing through the submarine cable conductor, in A; R is the AC resistance per unit length of the submarine cable conductor, which is temperature-dependent, in Ω / m. Metal sleeve loss W s1 and armor layer loss W s2 Both are based on conductor loss W c It is expressed as a proportion, that is: IN s1 =K1·W c (13) IN s2 =K2·W c (14) The loss factor K1 of the metal sleeve and the loss factor K2 of the armor layer are both constant values; The conductor, metal sheath, and armor layer act as heat sources, and their magnitude varies with the load current and temperature in the submarine cable conductor. When performing transient calculations, these loss values ​​need to be recalculated at each time step. Step 8: Constructing the equivalent thermal circuit After determining the structure and parameters of each layer of the submarine cable, including thermal resistance, thermal capacity, and loss, the corresponding trapezoidal equivalent thermal path is established. Step 9: Calculation of node temperature rise After establishing a trapezoidal equivalent thermal path for the submarine cable, the temperature rise at each node can be determined. The basic equations and steps are as follows: 1) Calculation of node coefficients A j,i Let A be the coefficient of the node with node number j and time interval number i, where the value of j ranges from j = 0 to j = n-1, and A -1,i =1; Δτ i For time step; A j,i It will not change throughout the calculation process, unless the time step changes; 2) Calculation of initial node values B j,i Let B be the initial value of the node with node number j and time interval number i. The value of j ranges from j = 0 to j = n-1. -1,i =0; W j,i For the loss corresponding to node j and time interval i; θ j,i-1 For the temperature rise corresponding to node j and time interval number i-1; 3) Temperature rise calculation i j,i =A j,i ·i j+1,i +B j,i (17) The value of j ranges from j = 0 to j = n-1, and θ n,i =0; 4) Repeat steps 2) and 3) until the preset time is reached, at which point the transient process ends.

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