A method for obtaining the reduction ratio of a continuous rolling mill unit that can reduce rolling energy consumption

Abstract

A method for obtaining the reduction ratio of a continuous rolling mill unit that can reduce rolling energy consumption includes: inputting process parameters and continuous rolling mill equipment parameters; establishing an optimization objective and constraints for minimum rolling energy consumption; preliminarily setting the reduction ratio of each stand using an optimization algorithm; calculating the entrance thickness and exit thickness of the rolled piece for each stand; calculating the rolling force and energy parameters of each stand, where the rolling force and energy parameters of each stand include the rolling pressure, transmission torque, forward slip value, roll speed, and main motor power of each stand; and determining whether the optimization objective and constraints for minimum rolling energy consumption are simultaneously satisfied. The present invention can be used for online preset calculation of the reduction ratio distribution of a continuous rolling mill unit, avoiding errors caused by the traditional manual experience table setting method, and the calculated reduction ratio distribution can minimize the rolling energy consumption of the continuous rolling mill unit, thereby reducing the total energy consumption of the unit and the production cost.

Description

Technical Field

[0001] This embodiment relates to the field of rolling, and particularly to a method for obtaining the reduction ratio of a continuous rolling mill unit that can reduce rolling energy consumption. Background Art

[0002] In the process of continuous rolling production, the reduction ratio distribution of each stand of the continuous rolling mill is one of the most important continuous rolling process parameters, which directly determines the stability and smoothness of the production process and is also a basic parameter for the control of the rolled piece thickness and the shape control. The existing technology often uses the artificial experience table setting method to distribute the reduction ratio of each stand of the continuous rolling mill. The establishment of this experience table requires a large amount of production practice data and operation experience in the early stage. When the product specifications are changed, it takes a long time to re - establish the new table data, and it cannot perform real - time online preset calculation for the reduction ratio distribution according to the new product specifications. In addition, there is no evaluation standard for the table data for comparison. For example, it is impossible to evaluate whether the rolling energy consumption is optimal, resulting in a large total energy consumption of the unit for the given continuous rolling reduction ratio distribution. Therefore, it is necessary to further develop a method for obtaining the reduction ratio of a continuous rolling mill unit that can optimize the rolling energy consumption.

[0003] Content of the embodiments

[0004] In view of the above problems, this embodiment is proposed to provide a method for obtaining the reduction ratio of a continuous rolling mill unit that can overcome the above problems or at least partially solve the above problems and reduce rolling energy consumption.

[0005] In order to solve the above technical problems, the embodiments of the present application disclose the following technical solutions:

[0006] A method for obtaining the reduction ratio of a continuous rolling mill unit that can reduce rolling energy consumption, comprising:

[0007] S100. Input process parameters and continuous rolling mill equipment parameters;

[0008] S200. Establish an optimization objective of minimum rolling energy consumption and constraint conditions;

[0009] S300. Use an optimization algorithm to preliminarily set the reduction ratios η1, η2... η Q-1 ;

[0010] S400. Calculate the entrance thickness and exit thickness of the rolled piece for each stand;

[0011] S500. Calculate the rolling force and energy parameters for each stand, where the rolling force and energy parameters for each stand include the rolling pressure, transmission torque, forward slip value, roll speed, and main motor power of each stand;

[0012] S600. Determine whether the minimum rolling energy consumption optimization objective and the constraints are satisfied simultaneously: if not, adjust the reduction ratio values of each stand by the optimization algorithm, and then transfer to S400 for recalculation; if satisfied, the calculation ends.

[0013] Further, in S100, the process parameters include the incoming material thickness H (unit: mm), the finished product thickness h (unit: mm), the width B of the rolled piece s (unit: mm), the number of stands Q, and the deformation resistance data of the rolled piece. The continuous rolling mill equipment parameters include the working roll body diameter D of each stand wk (unit: mm), the working roll neck diameter D' wk (unit: mm), the working roll body width B wk (unit: mm), the backup roll body diameter D bk (unit: mm), the backup roll neck diameter D' bk (unit: mm), and the backup roll body width B bk (unit: mm), where the subscript k represents the stand number, 1 ≤ k ≤ Q.

[0014] Further, in S200, the objective function of the minimum rolling energy consumption optimization objective is: In the formula, N k is the main motor power of the k-th stand, unit: KW; ρ is the density of the rolled piece, unit: kg / m 3 ; h k - the rolled piece exit thickness of the k-th stand, unit: mm; v sk is the rolled piece exit speed of the k-th stand, unit: m / min, v sk = v k (1 + f k ), v k is the roll speed of the k-th stand, unit: m / min, f k is the forward slip value of the k-th stand.

[0015] Further, in S400, calculate the rolled piece inlet thickness h 0k and the rolled piece exit thickness h 1k , specifically: when k = 1, h 0k = H, h 1k = (1 - η k )H; when 2 ≤ k ≤ Q - 1, h 0k = h 1(k-1) , h 1k = (1 - η k )h 0k ; when k = Q, h 0k = h 1(k-1) , h 1k = h; the subscript k represents the stand number, 1 ≤ k ≤ Q, η kis the reduction ratio of the k-th stand.

[0016] Further, in S500, the rolling force energy parameters of each stand are calculated. The specific steps are as follows:

[0017] S501. By repeatedly iterating the stress differential equation and the roll gap thickness equation in the roll gap deformation zone, the unit pressure distribution, friction stress distribution, and roll gap thickness distribution in the roll gap deformation zone are calculated;

[0018] S502. Calculate the rolling pressure P k (unit: KN), the driving torque M k (unit: KN×m), and the forward slip value f k ;

[0019] S503. Calculate the roll speed v k (unit: m / min) and the main motor power N k (unit: KW).

[0020] Further, the specific steps of S501 are as follows:

[0021] S5011. Set the initial roll profile curve and determine the entrance position of the rolled piece;

[0022] S5012. Calculate the unit pressure and friction stress of each section in the back slip zone from the entrance to the exit

[0023] S5013. Calculate the unit pressure and friction stress of each section in the forward slip zone from the exit to the entrance;

[0024] S5014. Determine the unit pressure and friction stress of each section in the roll gap deformation zone;

[0025] S5015. Calculate the roll gap thickness distribution from the unit pressure distribution;

[0026] S5016. Judge whether the roll gap thickness distributions obtained from the previous and current calculations converge: If they converge, end the calculation; if not, go to step S5012 for the next round of iterative calculation until the roll gap thickness distribution converges.

[0027] Further, in S502, calculate the rolling pressure P k (unit: KN), the driving torque M k (unit: KN×m), and the forward slip value f k , and the specific calculation formulas are as follows:

[0028]

[0029]

[0030]

[0031] In the formula, n k is the number of discrete segments of the roll gap of the k-th stand, and ΔX k is the length of the discrete segment of the roll gap of the k-th stand, with the unit of mm, and p k (i) is the unit pressure of the i-th segment of the roll gap of the k-th stand, with the unit of MPa, and t k (i) is the frictional stress of the i-th segment of the roll gap of the k-th stand, with the unit of MPa, and Δh k (i) is the thickness difference between the (i + 1)-th segment and the i-th segment of the roll gap of the k-th stand, with the unit of mm, and Δh k (i) = h k (i + 1) - h k (i), and x k (i) is the abscissa of the i-th segment of the roll gap of the k-th stand, with the unit of mm, and m wb is the rolling frictional force arm between the work roll and the backup roll, with the unit of mm, Among them, E wk is the elastic modulus of the work roll, with the unit of MPa, and E bk is the elastic modulus of the backup roll, with the unit of MPa, and L wb is the contact length between the work roll and the backup roll, with the unit of mm. When B wk ≤B bk , L wb = B wk , and when B wk > B bk , L wb = B bk ; ρ bk is the friction circle radius of the backup roll bearing, with the unit of mm, Among them, μ′ k is the rolling friction coefficient of the backup roll bearing; h k (r) is the roll gap thickness of the corresponding segment of the neutral plane of the roll gap of the k-th stand (the r-th segment of the roll gap of the k-th stand), with the unit of mm, and Δh k (r) = h k (r + 1) - h k (r).

[0032] Furthermore, in S503, calculate the roll speed v k (unit: m / min) and the main motor power N k (unit: KW). The specific steps are as follows:

[0033] S5031. Calculate the second flow rate value V k_max using the maximum roll speed v k set for each stand. The calculation formula is: V k = h 1k v k_max (1 + f k );

[0034] S5032. Find the minimum value V of the second flow rate values V1 to V of each stand Q and calculate the roll speed v' of each stand according to the second flow rate theorem from V min , the calculation formula is: min k k , the calculation formula is:

[0035] S5033. Calculate the main motor power N' of each stand k , and the ratio φ of the main motor power N' k to the rated power N of the main motor of this stand k_max , the calculation formula is: k , the calculation formula is: where M k is the transmission torque and D wk is the working roll body diameter;

[0036] S5034. Find the maximum value φ of φ1 to φ Q ; max ;

[0037] S5035. Judge whether φ max is greater than 1: if φ max > 1, then limit the roll speed of each stand by φ max , and the roll speed after limiting is the calculated roll speed of each stand, that is when φ max ≤ 1, then v k = v' k ;

[0038] S5036. Calculate the main motor power N of each stand k , the calculation formula is:

[0039] Furthermore, in S300, the reduction ratio of each stand is initially set by the optimization algorithm and the reduction ratio of each stand is adjusted by the optimization algorithm. The optimization algorithm uses the genetic algorithm or the particle swarm algorithm.

[0040] Furthermore, in S600, judge whether the minimum rolling energy consumption optimization goal and the constraint conditions are simultaneously satisfied. The constraint conditions are specifically that all stands simultaneously satisfy the following inequalities:

[0041] η k_min ≤ η k ≤ η k_max , v k ≤ v k_max , P k ≤ P k_max , M k ≤ M k_max 、Nk ≤ N k_max , where the subscript k represents the stand number, 1 ≤ k ≤ Q,; η k_max is the maximum reduction ratio of the k-th stand, η k_min is the minimum reduction ratio of the k-th stand, v k_max is the maximum roll speed of the k-th stand, P k_max is the maximum rolling pressure of the k-th stand, M k_max is the maximum transmission torque of the k-th stand, N k_max is the rated power of the main motor of the k-th stand.

[0042] The beneficial effects of the above technical solutions provided by the present invention at least include:

[0043] A method for obtaining the reduction ratio of a continuous rolling mill unit that can reduce rolling energy consumption disclosed by the present invention establishes a reduction ratio distribution model of the continuous rolling mill unit with the minimum rolling energy consumption as the optimization goal through theoretical analysis. In the calculation process, an optimization algorithm is used to continuously obtain a better reduction ratio distribution, and the rolling force and energy parameters of each stand under the corresponding reduction ratio distribution are repeatedly calculated according to the given process parameters and equipment parameters, including rolling pressure, transmission torque, forward slip value, roll speed, and main motor power. Finally, the optimal reduction ratio distribution that meets the optimization goal and constraint conditions is obtained.

[0044] The principle of the method disclosed by the present invention is clear and definite, the iterative calculation is stable and rapid, it can be used for the online preset calculation of the reduction ratio distribution of the continuous rolling mill unit, avoiding the errors caused by the traditional manual experience table setting method, and the calculated reduction ratio distribution can make the rolling energy consumption of the continuous rolling mill unit reach the lowest, thereby reducing the total energy consumption of the unit production and reducing the production cost.

[0045] The technical solutions of this embodiment will be further described in detail below through the accompanying drawings and embodiments. Description of the Drawings

[0046] The accompanying drawings are used to provide a further understanding of this embodiment, and constitute a part of the specification. Together with this embodiment, they are used to explain the present invention and do not constitute a limitation to this embodiment. In the accompanying drawings:

[0047] Figure 1 is the calculation flowchart of a method for obtaining the reduction ratio of a continuous rolling mill unit that can reduce rolling energy consumption in Embodiment 1;

[0048] Figure 2 is the calculation flowchart of the rolling force and energy parameters of the k-th stand in Embodiment 1;

[0049] Figure 3 is the calculation flowchart of the repeated iteration of the stress differential equation and the roll gap thickness equation in the roll gap deformation zone in Embodiment 1;

[0050] Figure 4 In Example 1, it is the calculation flow chart of the roll speed and the main motor power;

[0051] Figure 5 In Example 2, it is the curve of the fitness value changing with the number of evolution generations during the iterative calculation using the genetic algorithm;

[0052] Figure 6 In Example 2, it is the curve of the fitness value changing with the number of evolution generations during the iterative calculation using the particle swarm algorithm. Detailed implementation manners

[0053] The exemplary embodiments of the present disclosure will be described in more detail below with reference to the accompanying drawings. Although the exemplary embodiments of the present disclosure are shown in the drawings, it should be understood that the present disclosure can be implemented in various forms and should not be limited by the embodiments set forth herein. On the contrary, these embodiments are provided so that the present disclosure can be more thoroughly understood and the scope of the present disclosure can be fully communicated to those skilled in the art.

[0054] In order to solve the problems existing in the prior art, this embodiment provides a method for obtaining the reduction ratio of a continuous rolling mill unit that can reduce rolling energy consumption.

[0055] Example 1

[0056] A method for obtaining the reduction ratio of a continuous rolling mill unit that can reduce rolling energy consumption, such as Figure 1 , includes:

[0057] S100. Input process parameters and continuous rolling mill equipment parameters; specifically, in S100 of this embodiment, the process parameters include the incoming material thickness H (unit: mm), the finished product thickness h (unit: mm), the width B s (unit: mm), the number of stands Q, and the data of the deformation resistance of the rolled piece. The continuous rolling mill equipment parameters include the working roll body diameter D wk (unit: mm) of each stand, the working roll neck diameter D' wk (unit: mm), the working roll body width B wk (unit: mm), the backup roll body diameter D bk (unit: mm), the backup roll neck diameter D' bk (unit: mm), and the backup roll body width B bk (unit: mm), where the subscript k represents the stand number, and 1 ≤ k ≤ Q.

[0058] S200. Establish the minimum rolling energy consumption optimization objective and constraint conditions; in S200 of this embodiment, the objective function of the minimum rolling energy consumption optimization objective is: In the formula, N kis the main motor power of the k-th stand, unit: KW; ρ is the density of the rolled piece, unit: kg / m 3 ; h k - the exit thickness of the rolled piece of the k-th stand, unit: mm; v sk is the exit speed of the rolled piece of the k-th stand, unit: m / min, v sk = v k (1 + f k ), v k is the roll speed of the k-th stand, unit: m / min, f k is the forward slip value of the k-th stand;

[0059] The constraint conditions include the maximum reduction ratio η k_max of each stand, the minimum reduction ratio η k_min , the maximum roll speed v k_max (unit: m / min), the maximum rolling pressure P k_max (unit: KN), the maximum transmission torque M k_max (unit: KN×m) and the rated power N k_max (unit: KW) of the main motor;

[0060] S300. Use the optimization algorithm to preliminarily set the reduction ratios η1, η2... η Q-1 from the 1st stand to the (Q - 1)-th stand; In this embodiment S300, the reduction ratios of each stand are preliminarily set by the optimization algorithm and the reduction ratios of each stand are adjusted by the optimization algorithm. Preferably, the genetic algorithm or the particle swarm algorithm is used; The principles of the genetic algorithm and the particle swarm algorithm belong to the publicly known knowledge in the industry and will not be elaborated here.

[0061] S400. Calculate the entry thickness and exit thickness of the rolled piece of each stand; Specifically, in S400, calculate the entry thickness h 0k and the exit thickness h 1k of the rolled piece of each stand. Specifically: when k = 1, h 0k = H, h 1k = (1 - η k )H; when 2 ≤ k ≤ Q - 1, h 0k = h 1(k-1) , h 1k = (1 - η k )h 0k ; when k = Q, h 0k = h 1(k-1) , h 1k = h; The subscript k represents the stand number, 1 ≤ k ≤ Q, and η k is the reduction ratio of the k-th stand.

[0062] S500. Calculate the rolling force and energy parameters of each rolling mill stand. Among them, the rolling force and energy parameters of each rolling mill stand include the rolling pressure, transmission torque, forward slip value, roll speed, and main motor power of each rolling mill stand;

[0063] Specifically, taking the calculation of the k-th rolling mill stand as an example, as Figure 2 shown, the specific steps are as follows:

[0064] S501. Repeatedly iterate using the stress differential equation and roll gap thickness equation in the roll gap deformation zone to calculate the unit pressure distribution, friction stress distribution, and roll gap thickness distribution in the roll gap deformation zone;

[0065] S502. Calculate the rolling pressure P k (unit: KN), transmission torque M k (unit: KN×m), and forward slip value f k ;

[0066] S503. Calculate the roll speed v k (unit: m / min) and main motor power N k (unit: KW).

[0067] Among them, as Figure 3 shown, S501 is specifically as follows:

[0068] S5011. Set the initial roll profile curve and determine the entrance position of the rolled piece;

[0069] Assume that the roll is undeformed and in the shape of an arc. At this time, the roll flattening amount distribution is δ(x) = 0; Discretize the deformation zone: Divide the deformation zone into n segments along the rolling direction. The roll gap thickness model equation under the arc-shaped roll profile is where x k (i) is the abscissa of the i-th segment, h k (i) is the thickness of the rolled piece in the i-th segment, x k (n) = 0, h k (n) = h 1k , h k (1) = h 0k , so Discrete segment length xk(i) = xk(1) + (i - 1)ΔX, Δh k (i) = h k (i + 1) - h k (i).

[0070] S5012. Calculate the unit pressure and friction stress of each segment in the back slip zone from the entrance to the exit; Specifically:

[0071] Use the back slip zone formula to calculate from the entrance to the exit and judge the partition situation between sliding friction and sticking friction:

[0072] Calculate the unit pressure p of the entrance section (the first section) under sliding friction conditions k (1) b _sli is:

[0073]

[0074] Where K k (1) is the deformation resistance of the rolled piece in the first section of the roll gap, in MPa, and μ k is the roll gap friction coefficient of the kth stand;

[0075] Then use the Aitken iteration method to solve for p k (1) b _sli;

[0076] Judge μ at the entrance section k p k (1) b _sli and size, divided into two cases:

[0077] (i) If Then it means that the entrance section is under sliding friction, and the unit pressure p of the entrance section k (1) b = p k (1) b _sli. Use the stress differential equation in the back slip zone under sliding friction conditions to calculate the unit pressure of the second section, the third section... the nth section in turn, and judge μ at each section k p(i) b _sli(1 ≤ i ≤ n) and size; specifically:

[0078] The unit pressure of the (i + 1)th section in the back slip zone under sliding friction conditions is:

[0079] The frictional stress of the ith section in the back slip zone under sliding friction conditions is: t k (i) b _sli = μ k p k (i) b _sli.

[0080] There are also two cases in the calculation process:

[0081] 1) If from the entrance section (the first section) to the exit section (the nth section) all satisfy Then it means that when calculating using the stress differential equation in the back slip zone under sliding friction conditions, it is sliding friction from the entrance to the exit; at this time, the unit pressures of each section calculated using the back slip zone formula are: pk (1) b = p k (1) b _sli, p k (2) b = p k (2) b _sli…p k (n) b = p k (n) b _sli;

[0082] 2) If at the m-th (1 < m ≤ n) section: It indicates that from the m-th section to the outlet is adhesive friction; then, use the stress differential equation in the backward slip zone under the condition of adhesive friction to calculate the unit pressure of the m-th, (m + 1)-th... n-th sections in sequence; specifically:

[0083] The unit pressure of the (i + 1)-th section in the backward slip zone under the condition of adhesive friction is:

[0084]

[0085] The friction stress of the i-th section in the backward slip zone under the condition of adhesive friction is:

[0086] At this time, the unit pressures of each section calculated using the formula in the backward slip zone are respectively: p k (1) b = p k (1) b _sli, p k (2) b = p k (2) b _sli…p k (m - 1) b = p k (m - 1) b _sli, p k (m) b = p k (m) b _sti, p k (m + 1) b = p k (m + 1) b _sti…p k (n) b = p k (n) b _sti;

[0087] (ii) If It indicates that the inlet section is adhesive friction, and from the inlet section to the outlet section is all adhesive friction; the unit pressure of the inlet section Calculate the unit pressure of the second section, the third section... the nth section in turn using the stress differential equation in the back slip zone under the condition of adhesive friction; the unit pressure of each section calculated using the back slip zone formula is p k (1) b = p k (1) b _sti, p k (2) b = p k (2) b _sti... p k (n) b = p k (n) b _sti.

[0088] S5013. Calculate the unit pressure and frictional stress of each section in the forward slip zone from the exit to the entrance; similar to the calculation method of the back slip zone, specifically:

[0089] Calculate from the exit to the entrance using the forward slip zone formula, and judge the zoning situation of sliding friction and adhesive friction.

[0090] 1. Calculate the unit pressure p k (n) f _sli at the exit section under the condition of sliding friction as:

[0091] Solve for p k (n) f _sli using the Aitken iterative method;

[0092] 2. Judge μ k p k (n) f _sli and sizes, divided into two cases:

[0093] (i) If then it means that the exit section is in sliding friction, and the unit pressure p k (n) f = p k (n) f _sli; use the stress differential equation in the forward slip zone under the condition of sliding friction to calculate the unit pressure of the n - 1th section, the n - 2th section... in turn, and judge μ k p k (i) f _sli (1 ≤ i ≤ n) and sizes. Specifically:

[0094] The unit pressure of the ith section in the forward slip zone under the condition of sliding friction is:

[0095]

[0096] Solve for p using the Aitken iterative method k (i) f _sli.

[0097] The friction stress of the i-th section in the forward slip zone under sliding friction conditions is: t k (i) f _sli = -μ k p k (i) f _sli, and the negative sign indicates that the direction of the friction stress in the forward slip zone is towards the entrance side (opposite to the rolling direction).

[0098] Among them, there are two situations in the calculation process:

[0099] 1) If it satisfies from the exit section to the entrance section It indicates that when calculating using the stress differential equation of the forward slip zone under sliding friction conditions, it is sliding friction from the exit to the entrance. At this time, the unit pressures of each section calculated using the forward slip zone formula are: p k (1) f = p k (1) f _sli, p k (2) f = p k (2) f _sli…p k (n) f = p k (n) f _sli;

[0100] 2) If at the s-th section (1 < s ≤ n), there is: It indicates that from the s-th section to the entrance, it is sticking friction; switch to using the stress differential equation of the forward slip zone under sticking friction conditions to calculate the unit pressures of the s-th section, the s - 1-th section... the 1st section in turn; specifically:

[0101] The unit pressure of the i-th section in the backward slip zone under sticking friction conditions is:

[0102]

[0103] The friction stress of the i-th section in the backward slip zone under sticking friction conditions is:

[0104] At this time, the unit pressures of each section calculated using the forward slip zone formula are: p k (1) f = p k (1) f _sti, p k (2)f =p k (2) f _sti…p k (s) f =p k (s) f _sti,p k (s+1) f =p k (s+1) f _sli…p k (n) f =p k (n) f _sli;

[0105] (ii) If This means that the outlet section is adhesive friction, and the friction is adhesive friction from the outlet section to the inlet section; the unit pressure of the outlet section is The stress differential equation of the rear sliding zone under the condition of adhesive friction is used to calculate the unit pressure of the n-1th section, the n-2th section, and the 1st section in sequence;

[0106] At this time, the unit pressure of each section calculated by the forward sliding zone formula is p k (1) f =p k (1) f _sti,p k (2) f =p k (2) f _sti……p k (n) f =p k (n) f _sti.

[0107] S5014. Determine the unit pressure and friction stress of each section of the roll gap deformation zone;

[0108] Compare the two sets of unit pressure p calculated using the front sliding zone formula and the rear sliding zone formula k (1) f 、p k (2) f …… k (n) f and p k (1) b 、p k (2) b ……p k (n) b , find the segment with the smallest difference (assuming it is the rth segment), then this segment is the boundary segment between the front slip zone and the back slip zone (i.e. the segment corresponding to the neutral plane), xk(r)=xk(1)+(r-1)ΔX, retain pk (1) b 、p k (2) b ……、p k (r-1) b 、p k (r) b or p k (r) f 、p k (r+1) f ……p k (n) f ; So far, the unit pressure distribution under the specified roll profile has been calculated, that is, p k (1) = p k (1) b 、p k (2) = p k (2) b ,……,p k (r-1) = p k (r-1) b 、p k (r) = p k (r) b or p k (r) = p k (r) f 、p k (r+1)=p k (r+1) f …… k (n) = p k (n) f , and friction stress distribution t k (1) = t k (1) b ,t k (2) = t k (2) b ……、t k (r-1) = t k (r-1) b ,t k (r) = t k (r) b or k (r) = t k (r) f ,t k (r+1)=t k (r+1) f ……t k (n) = t k (n) f ;

[0109] S5015. Calculate the roll gap thickness distribution from the unit pressure distribution;

[0110] Calculate the roll gap thickness h using the unit pressure distribution k (i); however, to ensure convergence, a smoothing coefficient e is introduced here to make the unit pressures of each section calculated in two consecutive iterations change smoothly;

[0111] That is, p k m+1 (i) = ep k (i) + (1 - e)p k m (i), 0 < e < 1, p k (i) - the unit pressure of the i-th section of the roll gap under the specified roll profile calculated; p k m (i) - the unit pressure of the i-th section of the roll gap used in the m-th iteration; p k m+1 (i) - the unit pressure of the i-th section of the roll gap used in the (m + 1)-th iteration;

[0112] Calculate the elastic flattening deformation δ(x k m+1 (j)) of the roll using the smoothed unit pressure distribution p k Calculate the elastic flattening of the roll at the abscissa x k (j) by the method of cumulative summation s i is the abscissa corresponding to the unit pressure p k m+1 (i), in mm; E wk is the elastic modulus of the working roll of the k-th stand, in MPa; v wk is the Poisson's ratio of the working roll of the k-th stand;

[0113] Then, the deformed roll profile curve distribution is obtained as:

[0114]

[0115] The roll gap thickness distribution is:

[0116] In the formula, y(x k (j)) min - the ordinate corresponding to the lowest point of the deformed roll profile curve, in mm;

[0117] S5016. Determine whether the roll gap thickness distributions obtained from two consecutive calculations converge: if they converge, end the calculation; if they do not converge, go to step S5012 for the next round of iterative calculation until the roll gap thickness distribution converges;

[0118] Corresponding to the unit pressure distribution, the roll gap thickness is smoothed, i.e., h k m+1 (j) = eh k (j) + (1 - e)h k m (j), where h k (j) - the thickness of the j-th segment of the roll gap calculated; h k m (j) - the thickness of the j-th segment of the roll gap used in the m-th iteration; h k m +1 (j) - the thickness of the j-th segment of the roll gap used in the (m + 1)-th iteration.

[0119] Using the new roll gap thickness distribution (h k m+1 (j)), the unit pressure distribution and the friction stress distribution under the roll profile are re-solved in the same way as above. Such iteration is repeated until convergence; the convergence condition is: the absolute value of the difference in the thickness of the corresponding segments of the roll gap calculated in the previous two times is less than the precision value, i.e., h k (j) - h k m (j) ≤ ε × h k (j), where ε is the convergence precision coefficient.

[0120] After the iteration converges, the unit pressure distribution p k (1), p k (2), p k (3), … p k (n), the friction stress distribution t k (1), t k (2), t k (3), … t k (n) and the roll gap thickness distribution h k (1), h k (2), h k (3), … h k (n) are all calculated.

[0121] In S502 of this embodiment, the rolling pressure P k (unit: KN), the driving torque M k (unit: KN×m) and the forward slip value f k are calculated. The specific calculation formulas are:

[0122]

[0123]

[0124]

[0125] In the formula, n k is the number of discrete segments of the roll gap of the k-th stand, and ΔX k is the length of the discrete segment of the roll gap of the k-th stand, in mm, and p k (i) is the unit pressure of the i-th segment of the roll gap of the k-th stand, in MPa, and t k (i) is the frictional stress of the i-th segment of the roll gap of the k-th stand, in MPa, and Δh k (i) is the thickness difference between the (i + 1)-th segment and the i-th segment of the roll gap of the k-th stand, in mm, and Δh k (i) = h k (i + 1) - h k (i), and x k (i) is the abscissa of the i-th segment of the roll gap of the k-th stand, in mm, and m wb is the rolling frictional force arm between the work roll and the backup roll, in mm, Among them, E wk is the elastic modulus of the work roll, in MPa, and E bk is the elastic modulus of the backup roll, in MPa, and L wb is the contact length between the work roll and the backup roll, in mm. When B wk ≤ B bk , L wb = B wk , when B wk > B bk , L wb = B bk ; ρ bk is the friction circle radius of the backup roll bearing, in mm, Among them, μ′ k is the rolling friction coefficient of the backup roll bearing; h k (r) is the roll gap thickness of the corresponding segment of the neutral plane of the roll gap of the k-th stand (the r-th segment of the roll gap of the k-th stand), in mm, and Δh k (r) = h k (r + 1) - h k (r).

[0126] In S503 of this embodiment, the roll speed v k (unit: m / min) and the main motor power N k (unit: KW) are calculated. As Figure 4 shown, the specific steps are as follows:

[0127] S5031. Calculate the second flow rate value V k_max using the maximum roll speed v k set for each stand. The calculation formula is: V k = h 1k vk_max (1 + f k );

[0128] S5032. Find the minimum value V of the second - flow values V1 to V of each stand Q and calculate the roll speed v' of each stand according to the second - flow theorem from V min . The calculation formula is: min k k S5033. Calculate the main - motor power N' of each stand

[0129] and the ratio φ of the main - motor power N' k to the rated power N of the main - motor of this stand k . The calculation formula is: k_max k Q

[0130] S5034. Find the maximum value φ of φ1 to φ Q ; max

[0131] S5035. Judge whether φ max is greater than 1: If φ max > 1, then limit the roll speed of each stand by φ max . The limited roll speed is the calculated roll speed of each stand, that is max When φ ≤1, then v max = v' k ; k

[0132] S5036. Calculate the main - motor power N of each stand k k . The calculation formula is:

[0133] So far, the rolling - force and energy parameters of the k - th stand have been calculated, including the rolling pressure P k (unit: KN), transmission torque M k (unit: KN×m), forward - slip value f k , roll speed v k (unit: m / min) and main - motor power N k (unit: KW). The calculation methods of the rolling - force and energy parameters of other stands are similar to the above, and will not be elaborated here.

[0134] S600. Judge whether the minimum rolling - energy - consumption optimization goal and constraint conditions are simultaneously met: If not, adjust the reduction ratio values of each stand by the optimization algorithm, and then transfer to S400 for recalculation. If satisfied, the calculation ends.

[0135] Specifically, in S600, it is determined whether the minimum rolling energy consumption optimization goal and the constraint conditions are simultaneously satisfied. The constraint conditions are specifically that all stands simultaneously satisfy the following inequalities:

[0136] η k_min ≤η k ≤η k_max ,v k ≤v k_max 、P k ≤P k_max 、M k ≤M k_max 、N k ≤N k_max where the subscript k represents the stand number, 1 ≤ k ≤ Q; η k_max is the maximum reduction ratio of the k-th stand, η k_min is the minimum reduction ratio of the k-th stand, v k_max is the maximum roll speed of the k-th stand, P k_max is the maximum rolling pressure of the k-th stand, M k_max is the maximum transmission torque of the k-th stand, N k_max is the rated power of the main motor of the k-th stand.

[0137] A method for obtaining the reduction ratio of a continuous rolling mill unit that can reduce rolling energy consumption disclosed in this embodiment establishes a reduction ratio distribution model of a continuous rolling mill unit with the minimum rolling energy consumption as the optimization goal through theoretical analysis. In the calculation process, an optimization algorithm is used to continuously obtain a better reduction ratio distribution, and the rolling force and energy parameters of each stand under the corresponding reduction ratio distribution are repeatedly calculated according to the given process parameters and equipment parameters, including rolling pressure, transmission torque, forward slip value, roll speed, and main motor power. Finally, the optimal reduction ratio distribution that meets the optimization goal and constraint conditions is obtained.

[0138] The principle of the method disclosed in this embodiment is clear and definite, the iterative calculation is stable and rapid, it can be used for the online preset calculation of the reduction ratio distribution of a continuous rolling mill unit, avoiding the errors brought by the traditional manual experience table setting method, and the calculated reduction ratio distribution can make the rolling energy consumption of the continuous rolling mill unit reach the lowest, thereby reducing the total energy consumption of the unit production and reducing the production cost.

[0139] Embodiment 2

[0140] For ease of understanding, the following further illustrates with Embodiment 2. In this embodiment, the process parameters include: the incoming material thickness H = 17 mm, the finished product thickness H = 2 mm, the width of the rolled piece B s = 1600 mm, and the number of stands Q = 5. The equipment parameters are shown in Table 1. The constraint condition parameters are shown in Table 2.

[0141] Table 1 shows the equipment parameters of Embodiment 2

[0142]

[0143] Table 2 shows the constraint condition parameters of Example 2

[0144]

[0145] Other parameters include: the number of segments n = 500 divided along the rolling direction in the roll gap deformation zone, the convergence precision coefficient ε of the roll gap thickness distribution = 0.001, the smoothing coefficient e = 0.3, and the density ρ of the rolled piece = 7800 kg / m 3 , the roll gap friction coefficients of each stand μ1 = μ2 = μ3 = μ4 = μ5 = 0.3, and the elastic modulus E of the work rolls of each stand w1 = E w2 = E w3 = E w4 = E w5 = 206000 MPa, and the elastic modulus E of the backup rolls of each stand b1 = E b2 = E b3 = E b4 = E b5 = 206000 MPa, and the Poisson's ratio v of the work rolls of each stand w1 = v w2 = v w3 = v w4 = v w5 = 0.3, and the rolling friction coefficients μ′1 = μ′2 = μ′3 = μ′4 = μ′5 = 0.002 of the backup roll bearings of each stand.

[0146] As shown in Table 3, the data of the deformation resistance of the rolled piece are given in the form of a data table of the deformation resistance of the rolled piece varying with the reduction ratio.

[0147] Table 3 shows the data of the deformation resistance of the rolled piece

[0148]

[0149] The deformation resistance of the rolled piece at each position of the roll gap of each stand is calculated by the linear interpolation method:[[]]

[0150] Taking the 3rd stand as an example, assuming that the calculated inlet thickness of the rolled piece of the 3rd stand is 5 mm and the outlet thickness is 3.2 mm , then the total reduction ratio of the inlet rolled piece of this stand is The total reduction ratio of the outlet rolled piece is Then, the deformation resistance of the inlet rolled piece of this stand can be calculated by the linear interpolation method as The deformation resistance of the outlet rolled piece of this stand is

[0151] Further, taking the i-th section (1 ≤ i ≤ 500) of the roll gap of the 3rd stand as an example, the reduction ratio per pass of this stand is Assume that the thickness h3(i) of the rolled piece in the i-th section of the roll gap of the 3rd stand is 4 mm, then the reduction ratio of the rolled piece in the i-th section is Then, further using the linear interpolation method, the deformation resistance of the rolled piece in the i-th section of the roll gap of the 3rd stand can be calculated as:

[0152]

[0153] The deformation resistance of the rolled piece at each position of the roll gap of each other stand can be calculated according to the above method, which will not be elaborated here.

[0154] For comparison, Table 4 shows the reduction ratio distribution of each stand and the calculated force and energy parameters given by a traditional manual experience table setting method in a certain steel plant. The reduction ratio of each stand and the calculated rolling energy consumption per ton of steel of each stand are shown in the 4th column and the 11th column of Table 4 respectively. It can be further calculated that the total rolling energy consumption per ton of steel under this reduction ratio distribution is 42.107 KWh.

[0155] Table 4 Reduction ratio distribution of each stand and calculated force and energy parameters given by the traditional manual experience table setting method

[0156]

[0157] Specifically, in this embodiment, the optimization algorithms respectively adopt the genetic algorithm and the particle swarm algorithm. When the genetic algorithm is used as the optimization algorithm, the principle of the genetic algorithm belongs to the publicly known knowledge in the industry and will not be elaborated here. The relevant calculation parameters of the genetic algorithm adopted in this embodiment are: the population size is 80, the crossover probability is 0.95, the mutation probability is 0.1, the selection method is roulette wheel selection, the crossover method is single-point crossover, the maximum number of evolution generations is 100, and the curve of the fitness value changing with the number of evolution generations during the iterative calculation process is as Figure 5 shown.

[0158] The reduction ratio of each stand and the corresponding rolling energy consumption calculated in this embodiment are shown in the 4th column and the 11th column of Table 5 respectively. It is further calculated that the total rolling energy consumption per ton of steel is 36.284 KWh. Comparing with the data (Table 4) given by the traditional manual experience table setting method, it shows that the method of this embodiment can reduce the rolling energy consumption by about 13.83%.

[0159] Table 5 Reduction ratio distribution of each stand and force and energy parameters calculated in Embodiment 2

[0160]

[0161] When the optimization algorithm adopts the particle swarm optimization algorithm, the principle of the particle swarm optimization algorithm belongs to the publicly known knowledge in the industry and will not be elaborated here. The relevant calculation parameters of the particle swarm optimization algorithm adopted in this embodiment are as follows: the population size is 30, the maximum number of evolutionary generations is 50, the inertia weight is 0.5, the acceleration coefficient of the particle's optimal position is 2, the acceleration coefficient of the global optimal position is 2, and the curve of the fitness value changing with the number of evolutionary generations during the iterative calculation process is as Figure 6 shown.

[0162] The reduction ratios of each rolling mill stand and the corresponding rolling energy consumption calculated in this embodiment are shown in the 4th column and the 11th column of Table 6 respectively. Further calculated, the total rolling energy consumption per ton of steel is 36.281 KWh. Comparing with the data (Table 4) given by the traditional method of setting with artificial experience tables, it shows that the method of this embodiment can reduce the rolling energy consumption by about 13.84%.

[0163] Table 6 Reduction ratio distribution and energy and power parameters of each rolling mill stand calculated in Embodiment 2

[0164]

[0165] It should be understood that the specific order or hierarchy of steps in the disclosed process is an example of an exemplary method. Based on design preferences, it should be understood that the specific order or hierarchy of steps in the process can be rearranged without departing from the scope of the present disclosure. The appended method claims present the elements of the various steps in an exemplary order and are not intended to be limited to the specific order or hierarchy recited.

[0166] In the above detailed description, various features are combined in a single embodiment to simplify the present disclosure. This method of disclosure should not be construed as reflecting an intention that the embodiments of the claimed subject matter require more features than are expressly stated in each claim. On the contrary, as reflected by the appended claims, this embodiment is in a state with fewer features than all the features of the disclosed single embodiment. Therefore, the appended claims are hereby expressly incorporated into the detailed description, where each claim stands alone as a separate preferred embodiment of this embodiment.

[0167] Those skilled in the art should also understand that all the illustrative logical blocks, modules, circuits, and algorithmic steps described in conjunction with the embodiments herein can be implemented as electronic hardware, computer software, or a combination thereof. To clearly illustrate the interchangeability between hardware and software, the various illustrative components, blocks, modules, circuits, and steps have been generally described in terms of their functions above. Whether such a function is implemented as hardware or software depends on the specific application and the design constraints imposed on the overall system. Skilled technicians can implement the described functions in a flexible manner for each specific application. However, such implementation decisions should not be construed as departing from the scope of protection of this disclosure.

[0168] The steps of the methods or algorithms described in conjunction with the embodiments herein can be directly embodied as hardware, software modules executed by a processor, or a combination thereof. The software modules can be located in a RAM memory, a flash memory, a ROM memory, an EPROM memory, an EEPROM memory, a register, a hard disk, a removable disk, a CD-ROM, or any other form of storage medium well-known in the art. An exemplary storage medium is connected to the processor, enabling the processor to read information from and write information to the storage medium. Of course, the storage medium can also be a component of the processor. The processor and the storage medium can be located in an ASIC. The ASIC can be located in a user terminal. Of course, the processor and the storage medium can also exist as discrete components in the user terminal.

[0169] For software implementation, the techniques described in this application can be implemented using modules (e.g., procedures, functions, etc.) that perform the functions described in this application. These software codes can be stored in a memory unit and executed by a processor. The memory unit can be implemented inside the processor or outside the processor. In the latter case, it is communicatively coupled to the processor by various means, which are well-known in the art.

[0170] The above description includes examples of one or more embodiments. Of course, it is impossible to describe all possible combinations of components or methods for the purpose of describing the above embodiments. However, those of ordinary skill in the art should recognize that the various embodiments can be further combined and arranged. Therefore, the embodiments described herein are intended to cover all such changes, modifications, and variations that fall within the scope of protection of the appended claims. In addition, with respect to the term "comprising" used in the specification or claims, the manner in which this term is encompassed is similar to the term "including," as is explained when "including" is used as a transitional word in the claims. In addition, any term "or" used in the claims or the specification is intended to mean "non-exclusive or."

Claims

1. A method for obtaining the reduction ratio of a continuous rolling mill unit capable of reducing rolling energy consumption, characterized in that Including: S100. Input process parameters and continuous rolling mill equipment parameters; S200. Establish the minimum rolling energy consumption optimization objective and constraints; S300. Initially set the reduction ratios η1, η2…η of the first stand to the (Q - 1)-th stand by using an optimization algorithm Q-1 ; S400. Calculate the entrance thickness and exit thickness of the rolled piece for each stand; S500. Calculate the rolling force and energy parameters for each stand, where the rolling force and energy parameters for each stand include the rolling pressure, transmission torque, forward slip value, roll speed, and main motor power for each stand; In S500, the specific steps for calculating the rolling force and energy parameters for each stand are as follows: S501. Through repeated iteration of the stress differential equation and roll gap thickness equation in the roll gap deformation zone, calculate the unit pressure distribution, friction stress distribution, and roll gap thickness distribution in the roll gap deformation zone; S502. Calculate the rolling pressure P k , in units of KN, the driving torque M k , in units of KN×m, and the forward slip value f k , where the subscript k represents the stand number, 1 ≤ k ≤ Q, and Q is the number of stands; S503. Calculate the roll speed v k , in m / min, and the main motor power N k , in KW; In S503, calculate the roll speed v k , in the unit of m / min and the main motor power N k , in the unit of KW. The specific steps are as follows: S5031. Calculate the second flow rate value V with the maximum roll speed v set for each stand k_max The calculation formula is: V k =h k v 1k (1 + f k_max ), where h k is the thickness of the rolled piece at the exit, in mm; 1k ​ S5032. Find the minimum value V of the second flow rate values V1 to V of each stand Q and, according to the second flow rate theorem, calculate the roll speed v' of each stand from V min . The calculation formula is as follows: min where h k is the thickness of the rolled piece at the outlet, in mm; h 1k is the thickness of the rolled piece at the outlet, in mm; S5033. Calculate the main motor power N of each stand k ′, and the ratio φ of the main motor power N k ′ to the rated power N of the main motor of this stand k_max is calculated by the formula: k The calculation formula is as follows: where M k is the transmission torque, and D wk is the diameter of the working roll body; S5034. Find the maximum value φ of φ1 to φ Q ; max ; S5035. Determine φ max Whether it is greater than 1: If φ max > 1, then limit the roll speed of each stand with φ max The limited roll speed is the calculated roll speed of each stand, that is When φ max ≤ 1, then v k = v' k ; S5036. Calculate the main motor power N of each rack k , and the calculation formula is as follows: S600. Determine whether the minimum rolling energy consumption optimization objective and constraints are simultaneously satisfied: If not, adjust the reduction ratio value of each stand by the optimization algorithm, and then transfer to S400 for recalculation. If satisfied, the calculation ends.

2. The method for obtaining the reduction ratio of a continuous rolling mill unit capable of reducing rolling energy consumption according to claim 1, characterized in that, In S100, the process parameters include the incoming material thickness H, unit: mm, the finished product thickness h, unit: mm, the width B of the rolled piece s , unit: mm, the number Q of stands, and the data of the deformation resistance of the rolled piece. The equipment parameters of the tandem mill include the working roll body diameter D wk of each stand, unit: mm, the working roll neck diameter D′ wk , unit: mm, the working roll body width B wk , unit: mm, the backup roll body diameter D bk , unit: mm, the backup roll neck diameter D′ bk , unit: mm, and the backup roll body width B bk , unit: mm, where the subscript k represents the stand number, 1 ≤ k ≤ Q.

3. The method for obtaining the reduction ratio of a continuous rolling mill unit capable of reducing rolling energy consumption according to claim 1, wherein In S200, the objective function of the minimum rolling energy consumption optimization target is as follows: In the formula, N k is the main motor power of the k-th stand, in kW; ρ is the density of the rolled piece, in kg / m 3 ; h k is the exit thickness of the rolled piece of the k-th stand, in mm; v sk is the exit speed of the rolled piece of the k-th stand, in m / min, v sk = v k (1 + f k ), v k is the roll speed of the k-th stand, in m / min, f k is the forward slip value of the k-th stand.

4. A method for obtaining the reduction ratio of a continuous rolling mill unit capable of reducing rolling energy consumption according to claim 2, characterized in that In S400, calculate the entry thickness h of the rolled piece for each stand 0k and the exit thickness h 1k of the rolled piece. Specifically: when k = 1, h 0k = H, h 1k = (1 - η k )H; when 2 ≤ k ≤ Q - 1, h 0k = h 1(k-1) , h 1k = (1 - η k )h 0k ; when k = Q, h 0k = h 1(k-1) , h 1k = h; the subscript k represents the stand number, 1 ≤ k ≤ Q, and η k is the reduction ratio of the k-th stand.

5. The method for obtaining the reduction ratio of a continuous rolling mill unit capable of reducing rolling energy consumption according to claim 1, characterized in that, The specific steps of S501 are as follows: S5011. Set the initial roll profile curve and determine the entrance position of the rolled piece; S5012. Calculate the unit pressure and friction stress for each section in the back slip zone from the entrance to the exit S5013. Calculate the unit pressure and friction stress for each section in the forward slip zone from the exit to the entrance; S5014. Determine the unit pressure and friction stress for each section in the roll gap deformation zone; S5015. Calculate the roll gap thickness distribution from the unit pressure distribution; S5016. Determine whether the roll gap thickness distributions obtained from the previous and current calculations converge: If they converge, end the calculation; if not, transfer to step S5012 for the next round of iterative calculation until the roll gap thickness distribution converges.

6. The method for obtaining the reduction ratio of a continuous rolling mill unit capable of reducing rolling energy consumption according to claim 1, characterized in that, In S502, calculate the rolling pressure P k , unit KN, transmission torque M k , unit KN×m and forward slip value f k , the specific calculation formula is: Wherein, n k is the number of discrete segments of the roll gap of the k-th stand, ΔX k is the length of the discrete segment of the roll gap of the k-th stand, in mm, p k (i) is the unit pressure of the i-th segment of the roll gap of the k-th stand, 1 ≤ i ≤ n, and n is the number of segments divided along the rolling direction in the roll gap deformation zone, in MPa, t k (i) is the frictional stress of the i-th segment of the roll gap of the k-th stand, in MPa, Δh k (i) is the thickness difference between the (i + 1)-th segment and the i-th segment of the roll gap of the k-th stand, in mm, Δh k (i) = h k (i + 1) - h k (i), h k (i + 1) and h k (i) are the thicknesses of the (i + 1)-th segment and the i-th segment of the roll gap of the k-th stand respectively, in mm, x k (i) is the abscissa of the i-th segment of the roll gap of the k-th stand, in mm, m wb is the rolling frictional force arm between the work roll and the backup roll, in mm, Wherein, E wk is the elastic modulus of the work roll, in MPa, E bk is the elastic modulus of the backup roll, in MPa, L wb is the contact length between the work roll and the backup roll, in mm. When B wk ≤ B bk , L wb = B wk . When B wk > B bk , L wb = B bk ; ρ bk is the friction circle radius of the backup roll bearing, in mm, Wherein, μ′ k is the rolling friction coefficient of the backup roll bearing, D′ bk is the diameter of the backup roll neck, in mm; Δh k (r) is the thickness difference between the (r + 1)-th segment and the r-th segment of the roll gap of the k-th stand, in mm, Δh k (r) = h k (r + 1) - h k (r), h k (r) is the roll gap thickness of the corresponding segment of the neutral plane of the roll gap of the k-th stand, and the corresponding segment of the neutral plane of the roll gap of the k-th stand is the r-th segment of the roll gap of the k-th stand, in mm, h k (r + 1) is the (r + 1)-th segment of the roll gap of the k-th stand, in mm.

7. A method for obtaining the reduction ratio of a continuous rolling mill unit capable of reducing rolling energy consumption according to claim 1, characterized in that In S300, the reduction ratio of each stand is initially set by the optimization algorithm and the reduction ratio of each stand is adjusted by the optimization algorithm. The optimization algorithm uses a genetic algorithm or a particle swarm algorithm.

8. The method for obtaining the reduction ratio of a continuous rolling mill unit capable of reducing rolling energy consumption according to claim 1, wherein In S600, determine whether the minimum rolling energy consumption optimization objective and constraints are simultaneously satisfied. The constraints are specifically that all stands simultaneously satisfy the following inequalities: η k_min ≤ η k ≤ η k_max 、v k ≤ v k_max 、P k ≤ P k_max 、M k ≤ M k_max 、N k ≤ N k_max where the subscript k represents the stand number, 1 ≤ k ≤ Q, η k_max is the maximum reduction ratio of the k-th stand, η k_min is the minimum reduction ratio of the k-th stand, v k_max is the maximum roll speed of the k-th stand, P k_max is the maximum rolling pressure of the k-th stand, M k_max is the maximum transmission torque of the k-th stand, N k_max is the rated power of the main motor of the k-th stand.

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