Governor control method

Abstract

The application discloses a governor control method, which adopts a superposition method of primary frequency modulation opening target value calculation by means of a unit water head, active power and guide vane opening corresponding data table, calculates the primary frequency modulation opening target value, and superposes the primary frequency modulation opening target value on the power PID guide vane opening given value Y pid1 or opening given value y c calculated by a power PID module, aims to unify the primary frequency modulation opening target value, and realizes the unified primary frequency modulation performance effect.

Description

Technical Field

[0001] This invention relates to the field of automatic control technology for hydropower stations, and in particular to a governor control method. Background Technology

[0002] In hydropower generation, the turbine regulation system needs to accurately perform crucial tasks such as speed regulation, opening degree regulation, and power regulation. Governors typically have three control modes: frequency mode, opening degree mode, and power mode. In traditional governor frequency mode, a parallel PID control law is used. The difference between the setpoint frequency and the generator frequency is used as the input to the PID controller. The PID calculates the guide vane opening value and opens the guide vanes to the opening degree corresponding to the setpoint frequency, maintaining the generator frequency within a certain range. In traditional governor power mode, the primary frequency regulation control model also uses a parallel PID control law. It converts the difference between the setpoint frequency and the generator frequency (considering the frequency dead zone) into a primary frequency regulation power target value. This target value is then superimposed with the deviation between the power setpoint and the actual power value, and the result, considering the power dead zone, is used as the input to the PID module. The PID module calculates the PID guide vane opening setpoint Ypid, and the hydraulic servo system adjusts the guide vanes to the opening degree corresponding to the power target value (the sum of the primary frequency regulation power target value and the generator power setpoint). The structure of the traditional governor power mode primary frequency regulation control model is as follows: Figure 1 As shown. The traditional governor opening mode primary frequency regulation control model structure adopts a PI regulation law. It calculates the deviation between the given opening value and the actual guide vane opening value using a permanent slip coefficient Bp. Then, it superimposes the difference between the given frequency value and the unit frequency (considering the frequency dead zone) and inputs it into the opening PID module. The PID module calculates the PID guide vane opening value Ypid, and the hydraulic servo system adjusts the guide vane to the target opening value (the sum of the primary frequency regulation opening target value and the unit opening given value). The structure of the traditional governor opening mode primary frequency regulation control model is as follows: Figure 2 As shown.

[0003] Using the traditional control model, when the unit frequency exceeds the primary frequency regulation dead zone, the target value of primary frequency regulation power in the governor power mode and the target value of primary frequency regulation opening in the opening mode are usually inconsistent, resulting in different performance of primary frequency regulation under the two control modes. In particular, the integral power contributed during the regulation process is significantly different, while the power grid has the same and uniform performance requirements for primary frequency regulation under the two control modes. Summary of the Invention

[0004] To address the problems existing in the prior art, the purpose of this invention is to provide a speed governor control method, which ensures the safe, stable and high-quality operation of the generating unit and the power grid.

[0005] To achieve the above objectives, the technical solution adopted by the present invention is: a speed governor control method, comprising:

[0006] Using a data table that maps unit head, active power, and guide vane opening one-to-one, a unified method for calculating and superimposing the primary frequency regulation opening target value is adopted to calculate the primary frequency regulation opening target value. This target value is then calculated in the power PID module, which generates the power PID guide vane opening setpoint Y. pid1 Or given opening y c The target value of frequency modulation opening is superimposed on top.

[0007] As a further improvement to the present invention, the specific steps include:

[0008] S1, Initialize control parameters E p b p E f And the data in the one-to-one correspondence table of initial head, active power and guide vane opening; where E p This is the coefficient used to convert the frequency difference into the power difference, i.e., the power difference coefficient; b p E is the permanent slip coefficient. f This is a frequency dead zone;

[0009] S2, Speed ​​Controller collects unit power P g Power given P c Given the opening degree y c Unit frequency f g The frequency is given by f c The unit's head is w;

[0010] S3. The speed controller detects whether the speed controller is in frequency mode. If so, it inputs the difference between the frequency setpoint (considering the frequency dead zone) and the unit frequency value into the opening PID module for calculation, and calculates the opening PID guide vane opening setpoint Y. pid2 If the output is positive, the output is sent to the hydraulic servo system and the system returns to S2; otherwise, the system proceeds to S4.

[0011] S4. The speed controller calculates the primary frequency regulation power target value P. yctp ;

[0012] S5. The speed controller checks whether the speed controller is in power mode. If so, it sets the primary frequency regulation power target value P. yctp Superimposed on the power given P c and unit power P g If the difference is found, the power PID calculation is entered, and S6 is entered; otherwise, S6 is entered directly.

[0013] S6, the speed controller is based on the unit power P. g The primary frequency modulation power target value P yctp By referring to the data table that corresponds one-to-one between the current unit head w, active power, and guide vane opening, the target value Y of the primary frequency regulation opening is calculated.yctp ;

[0014] S7. The speed controller checks whether the speed controller is in power mode. If so, it sets the primary frequency regulation target value Y. yctp The power PID guide vane opening setpoint Y is superimposed on the power PID module and calculated. pid1 The difference between the above and the unit opening degree y is processed by the permanent slip coefficient b. p After conversion, the value is entered into the opening PID module to calculate the opening PID guide vane setpoint Y. pid2 If the output is sent to the hydraulic servo system, return to S2; otherwise, proceed to S8.

[0015] S8. The speed controller checks whether the speed controller is in open mode. If so, it sets the primary frequency regulation open target value Y. yctp Superimposed on a given opening y c The difference between the above and the unit opening degree y is processed by the permanent slip coefficient b. p After conversion, the value is entered into the opening PID module to calculate the opening PID guide vane setpoint Y. pid2 If the output is sent to the hydraulic servo system, it returns to S2; otherwise, it returns directly to S2.

[0016] As a further improvement of the present invention, in step S4, the primary frequency modulation power target value P is calculated. yctp The method is as follows:

[0017] If f c -f g >E f Then P yctp =(f c -f g -E f ) / E p If f c -f g <-E f Then P yctp =(f c -f g +E f ) / E p ;

[0018] If |f c -f g |≤E f Then P yctp =0.

[0019] As a further improvement of the present invention, in step S1, the one-to-one correspondence between head, active power and guide vane opening is as follows:

[0020]

[0021]

[0022] In the table above, p, q, x, x1, x2, y, y1, and y2 are all positive integers, 1 < x ≤ p, 1 < y ≤ q, and D x,y For W x Water Head G y The guide vane opening corresponding to the active power.

[0023] As a further improvement of the present invention, in step S6, the target value Y of the primary frequency modulation opening is calculated. yctp The method is as follows:

[0024] If W x1-1 ≤w≤W x1 G y1-1 ≤P g ≤G y1 ,but:

[0025] d 表y1-1 =D x1-1，y1-1 +(D x1，y1-1 -D x1-1，y1-1 (wW) x1-1 ) / (W x1 -W x1-1 );

[0026] d 表y1 =D x1-1，y1 +(D x1，y1 -D x1-1，y1 (wW) x1-1 ) / (W x1 -W x1-1 );

[0027] D 表1 =d 表y1-1 +(d 表y1 -d 表y1-1 (P) g -G y1-1 ) / (G y1 -G y1-1 );

[0028] The generator head w and power P can then be calculated. g The corresponding opening Y w,pg =D 表1 =d 表y-1 +(d 表y -d 表y-1 (P) g -G y-1 ) / (G y -G y-1 );

[0029] If W x2-1 ≤w≤W x2 Gy2-1 ≤P g +P yctp ≤G y2 ,but:

[0030] d 表y2-1 =D x2-1，y2-1 +(D x2，y2-1 -D x2-1，y2-1 (wW) x2-1 ) / (W x2 -W x2-1 ).

[0031] d 表y2 =D x2-1，y2 +(D x2，y2 -D x2-1，y2 (wW) x2-1 ) / (W x2 -W x2-1 ).

[0032] D 表2 =d 表y2-1 +(d 表y2 -d 表y2-1 (P) g +P yctp -G y2-1 ) / (G y2 -G y2-1 );

[0033] Similarly, the unit head w and power P can be calculated. g +P yctp The corresponding opening Y w,pg+pyctp =D 表2 =d 表y2-1 +(d 表y2 -d 表y2-1 (P) g -G y2-1 ) / (G y2 -G y2-1 ); then Y yctp =Y w,pg+pyctp -Y w,pg .

[0034] The beneficial effects of this invention are:

[0035] This invention optimizes the traditional control model structure of a turbine governor controller. It uses a data table with a one-to-one correspondence between unit head, active power, and guide vane opening, and employs a unified method for calculating and superimposing the primary frequency regulation opening target value. This method calculates the primary frequency regulation opening target value and integrates it with the power PID module to calculate the power PID guide vane opening setpoint Y. pid1 Or given opening y cThe above-mentioned primary frequency regulation opening target value is superimposed to unify the primary frequency regulation opening target value and achieve a unified primary frequency regulation performance effect. This not only unifies and simplifies the general control model structure of the speed governor under the three control modes, but also unifies the performance of primary frequency regulation under the opening and power control modes. In particular, it unifies the integral power contributed by the regulation process, ensuring the safe, stable and high-quality operation of the unit and the power grid. Attached Figure Description

[0036] Figure 1 A structural diagram of a traditional power mode primary frequency regulation control model for a speed governor;

[0037] Figure 2 A structural diagram of a primary frequency regulation control model for a traditional speed governor opening mode;

[0038] Figure 3 This is a structural diagram of the general control model of the speed governor in the embodiments of the present invention;

[0039] Figure 4 This is a flowchart of the general control method for the speed controller in an embodiment of the present invention. Detailed Implementation

[0040] The embodiments of the present invention will now be described in detail with reference to the accompanying drawings.

[0041] Example

[0042] An electrical control system for a hydro-generator speed governor includes a speed governor relay displacement sensor, a speed governor frequency measuring device, a power transmitter, a water level sensor, a speed governor controller, and a monitoring system.

[0043] The governor relay displacement sensor is used to collect and measure the guide vane opening feedback signal y of the unit; the governor relay displacement sensor adopts the MTS brand model RP-S-1200M-D60-1-A01 relay displacement sensor.

[0044] The speed controller frequency measurement device is used to collect and measure the unit's frequency signal f. g The speed controller frequency measurement device uses a B&R brand PCC2003 controller with a CPU module model of 7CP476.60-1.

[0045] A power transmitter is used to acquire and measure the power signal P of the generator set. g .

[0046] A water level sensor is used to collect and measure the water head w of the unit.

[0047] The speed controller relay displacement sensor, speed controller frequency measuring device, power transmitter, and water level sensor are all connected to the controller, which is connected to the monitoring system.

[0048] The governor controller receives remote control commands from the monitoring system, including commands for setting the opening degree and power; it receives feedback signals of the unit's guide vane opening from the governor relay displacement sensor; it receives unit frequency signals from the governor frequency measurement device; it receives unit power signals from the power transmitter; and it receives unit head signals from the water level sensor.

[0049] The speed governor controller uses a speed governor control method of the present invention to control the hydropower generation process of the hydro-turbine generator set.

[0050] The speed controller uses a B&R brand PCC2005 controller with a CPU module model of 3CP380.60-1.

[0051] The monitoring system sends remote control commands to the speed governor controller, including commands for setting the opening degree and power.

[0052] The power plant monitoring system is generally divided into two levels: the plant level and the local control unit level. The local control unit level is connected to the power plant control network and uses fieldbus technology to complete the local monitoring tasks of designated equipment.

[0053] In this invention, the monitoring system specifically refers to the local control unit layer, distributed according to the controlled object units, and composed of all local control units (LCUs) throughout the plant, including the LCUs of each generating unit, the plant auxiliary power LCU, the public utility LCU, the switchyard LCU, and the dam crest LCU. Each local control unit (LCU) includes PLCs, touch screens, network equipment, cabinets, etc., and is responsible for equipment data acquisition and processing, equipment status monitoring and process monitoring, equipment control and regulation, and equipment information communication. The aforementioned equipment includes the governor electrical control system equipment.

[0054] The speed governor control method of the present invention is applied to the speed governor electrical control system of a power station.

[0055] The general control model structure diagram of the speed governor is as follows: Figure 3 As shown, it optimizes the traditional control model structure of the turbine governor controller. Using a data table that corresponds one-to-one between the unit head, active power, and guide vane opening, it uniformly adopts a method for calculating and superimposing the primary frequency regulation opening target value to calculate the primary frequency regulation opening target value. This target value is then calculated in the power PID module based on the power PID guide vane opening setpoint Y. pid1 Or given opening y c The target value of frequency modulation opening is superimposed on top.

[0056] like Figure 4 As shown, a general control method for a speed governor is as follows:

[0057] S1, Control parameter K f K pg Ky E p b p E f E pg The head, active power and guide vane opening correspondence table data is initialized, and then the process enters S2.

[0058] Control parameter K f Convert frequency sample values ​​to code value conversion coefficients; control parameter K pg The power sample value is converted into a code value conversion coefficient; control parameter K y The conversion coefficients for converting the openness sampled values ​​into calculated code values; control parameter E p This is the coefficient used to convert the frequency difference into the power difference, i.e., the power difference coefficient; control parameter b p The permanent slip coefficient; control parameter E f E pg These are the frequency dead zone and the power dead zone, respectively.

[0059] S2, Speed ​​Controller collects unit power P g Power given P c Given the opening degree y c Unit frequency f g The frequency is given by f c The generator head w enters S3.

[0060] S3. The speed controller checks whether the speed controller is in frequency mode. If so, it inputs the difference between the frequency setpoint and the unit frequency (considering the frequency dead zone) into the opening PID module for calculation, and calculates the opening PID guide vane opening setpoint Y. pid2 If the output is sent to the hydraulic servo system, return to S2; otherwise, proceed to S4.

[0061] S4. The speed controller calculates the primary frequency regulation power target value P. yctp Enter S5.

[0062] S5. The speed controller checks whether the speed controller is in power mode. If so, it sets the primary frequency regulation power target value P. yctp Superimposed on the power given P c and unit power P g If the difference is found, the power PID calculation is initiated, and step S6 is entered; otherwise, step S6 is entered directly.

[0063] S6, the speed controller is based on the unit power P. g The primary frequency modulation power target value P yctp By referring to the data table that corresponds one-to-one between the current unit head w, active power, and guide vane opening, the target value Y of the primary frequency regulation opening is calculated. yctp Enter S7.

[0064] S7. The speed controller checks whether the speed controller is in power mode. If so, it sets the primary frequency regulation target value Y. yctp The power PID guide vane opening setpoint Y is superimposed on the power PID module and calculated. pid1 The difference between the above and the unit opening degree y is processed by the permanent slip coefficient b. p After conversion, the value is entered into the opening PID module to calculate the opening PID guide vane setpoint Y. pid2 If the output is sent to the hydraulic servo system, return to S2; otherwise, proceed to S8.

[0065] S8. The speed controller checks whether the speed controller is in open mode. If so, it sets the primary frequency regulation open target value Y. yctp Superimposed on a given opening y c The difference between the above and the unit opening degree y is processed by the permanent slip coefficient b. p After conversion, the value is entered into the opening PID module to calculate the opening PID guide vane setpoint Y. pid2 If the output is sent to the hydraulic servo system, it returns to S2; otherwise, it returns directly to S2.

[0066] In S4, the primary frequency modulation power target value P is calculated. yctp The method is as follows:

[0067] If f c -f g >E f Then P yctp =(f c -f g -E f ) / E p If f c -f g <-E f Then P yctp =(f c -f g +E f ) / E p ;

[0068] If |f c -f g |≤E f Then P yctp =0.

[0069] In S6, the target value Y of the primary frequency modulation opening is calculated. yctp The method is as follows:

[0070] Table 1 shows the one-to-one correspondence between head, active power, and guide vane opening. In Table 1, p, q, x, x1, x2, y, y1, and y2 are all positive integers, 1 < x ≤ p, 1 < y ≤ q, and D. x,y For Wx Water Head G y The guide vane opening corresponding to active power;

[0071] Table 1. Correspondence between water head, active power, and guide vane opening.

[0072] <![CDATA[G1]]> <![CDATA[D 1,1 ]]> <![CDATA[D 2,1 ]]> … <![CDATA[D x-1，1 ]]> <![CDATA[D x，1 ]]> … <![CDATA[D p-1,1 ]]> <![CDATA[D p，1 ]]> <![CDATA[G2]]> <![CDATA[D 1,2 ]]> <![CDATA[D 2,2 ]]> … <![CDATA[D x-1，2 ]]> <![CDATA[D x，2 ]]> … <![CDATA[D p-1,2 ]]> <![CDATA[D p，2 ]]> … … … … … … … … … <![CDATA[G y-1 ]]> <![CDATA[D 1，y-1 ]]> <![CDATA[D 2，y-1 ]]> … <![CDATA[D x-1，y-1 ]]> <![CDATA[D x，y-1 ]]> … <![CDATA[D p-1,y-1 ]]> <![CDATA[D p，y-1 ]]> <![CDATA[G y ]]> <![CDATA[D 1，y ]]> <![CDATA[D 2，y ]]> … <![CDATA[D x-1，y ]]> <![CDATA[D x，y ]]> … <![CDATA[D p-1,y ]]> <![CDATA[D p，y ]]> … … … … … … … … … <![CDATA[G q-1 ]]> <![CDATA[D 1，q-1 ]]> <![CDATA[D 2，q-1 ]]> … <![CDATA[D x-1，q-1 ]]> <![CDATA[D x，q-1 ]]> … <![CDATA[D p-1,q-1 ]]> <![CDATA[D p，q-1 ]]> <![CDATA[G q ]]> <![CDATA[D 1，q ]]> <![CDATA[D 2，q ]]> … <![CDATA[D x-1，q ]]> <![CDATA[D x，q ]]> … <![CDATA[D p-1,q ]]> <![CDATA[D p，q ]]>

[0073] If W x1-1 ≤w≤W x1 G y1-1 ≤P g ≤G y1 ,but

[0074] d 表y1-1 =D x1-1，y1-1 +(D x1，y1-1 -D x1-1，y1-1 (wW) x1-1 ) / (W x1 -W x1-1 ).

[0075] d 表y1 =D x1-1，y1 +(D x1，y1 -D x1-1，y1 (wW) x1-1 ) / (W x1 -W x1-1 ).

[0076] D 表1 =d 表y1-1 +(d 表y1 -d 表y1-1 (P) g -G y1-1 ) / (G y1 -G y1-1 );

[0077] The generator head w and power P can then be calculated. g The corresponding opening Y w,pg =D 表1 =d 表y-1 +(d 表y -d 表y-1 (P) g -G y-1 ) / (G y -G y-1 )

[0078] If W x2-1 ≤w≤W x2 G y2-1 ≤P g +P yctp ≤G y2 ,but

[0079] d表y2-1 =D x2-1，y2-1 +(D x2，y2-1 -D x2-1，y2-1 (wW) x2-1 ) / (W x2 -W x2-1 ).

[0080] d 表y2 =D x2-1，y2 +(D x2，y2 -D x2-1，y2 (wW) x2-1 ) / (W x2 -W x2-1 ).

[0081] D 表2 =d 表y2-1 +(d 表y2 -d 表y2-1 (P) g +P yctp -G y2-1 ) / (G y2 -G y2-1 );

[0082] The same method can be used to calculate the unit's head w and power P. g +P yctp The corresponding opening Y w,pg+pyctp =D 表2 =d 表y2-1 +(d 表y2 -d 表y2-1 (P) g -G y2-1 ) / (G y2 -G y2-1 ).

[0083] Then Y yctp =Y w,pg+pyctp -Y w,pg .

[0084] The embodiments described above are merely illustrative of specific implementations of the present invention, and while the descriptions are detailed, they should not be construed as limiting the scope of the present invention. It should be noted that those skilled in the art can make various modifications and improvements without departing from the concept of the present invention, and these modifications and improvements all fall within the scope of protection of the present invention.

Claims

1. A speed governor control method, characterized in that, include: Using a data table that maps unit head, active power, and guide vane opening one-to-one, a unified method for calculating and superimposing the primary frequency regulation opening target value is adopted to calculate the primary frequency regulation opening target value. This target value is then calculated in the power PID module, which generates the power PID guide vane opening setpoint Y. pid1 Or given opening y c The above is superimposed with the target value of the frequency modulation opening; specifically, it includes the following steps: S1, Initialize control parameters E p b p E f And the data in the one-to-one correspondence table of initial head, active power and guide vane opening; among which, E p This is the coefficient used to convert the frequency difference into the power difference, i.e., the power difference coefficient; b p E is the permanent slip coefficient. f This is a frequency dead zone; S2, Speed ​​Controller collects unit power P g Power given P c Given the opening degree y c Unit frequency f g The frequency is given by f c The unit's head is w; S3. The speed controller detects whether the speed controller is in frequency mode. If so, it inputs the difference between the frequency setpoint (considering the frequency dead zone) and the unit frequency value into the opening PID module for calculation, and calculates the opening PID guide vane opening setpoint Y. pid2 If the output is positive, the output is sent to the hydraulic servo system and the system returns to S2; otherwise, the system proceeds to S4. S4. The speed controller calculates the primary frequency regulation power target value P. yctp ; S5. The speed controller checks whether the speed controller is in power mode. If so, it sets the primary frequency regulation power target value P. yctp Superimposed on the power given P c and unit power P g If the difference is found, the power PID calculation is entered, and S6 is entered; otherwise, S6 is entered directly. S6, the speed controller is based on the unit power P. g The primary frequency modulation power target value P yctp By referring to the data table that corresponds one-to-one between the current unit head w, active power, and guide vane opening, the target value Y of the primary frequency regulation opening is calculated. yctp ; S7. The speed controller checks whether the speed controller is in power mode. If so, it sets the primary frequency regulation target value Y. yctp The power PID guide vane opening setpoint Y is superimposed on the power PID module and calculated. pid1 The difference between the above and the unit opening degree y is processed by the permanent slip coefficient b. p After conversion, the value is entered into the opening PID module to calculate the opening PID guide vane setpoint Y. pid2 If the output is sent to the hydraulic servo system, return to S2; otherwise, proceed to S8. S8. The speed controller checks whether the speed controller is in open mode. If so, it sets the primary frequency regulation open target value Y. yctp Superimposed on a given opening y c The difference between the above and the unit opening degree y is processed by the permanent slip coefficient b. p After conversion, the value is entered into the opening PID module to calculate the opening PID guide vane setpoint Y. pid2 If the output is sent to the hydraulic servo system, it returns to S2; otherwise, it returns directly to S2.

2. The speed governor control method according to claim 1, characterized in that, In step S4, the primary frequency modulation power target value P is calculated. yctp The method is as follows: If f c -f g >E f Then P yctp =(f c -f g -E f ) / E p If f c -f g <-E f Then P yctp =(f c -f g +E f ) / E p ; If |f c -f g |≤E f Then P yctp =0.

3. The speed governor control method according to claim 1, characterized in that, In step S1, the one-to-one correspondence between head, active power, and guide vane opening is shown in the table below: ； In the table above, p, q, x, and y are all positive integers, 1 < x ≤ p, 1 < y ≤ q, and D x,y For W x Water Head G y The guide vane opening corresponding to active power; In step S6, the target value Y of the primary frequency modulation opening is calculated. yctp The method is as follows: If W x1-1 ≤w≤W x1 G y1-1 ≤P g ≤G y1 ,but: d 表y1-1 =D x1-1，y1-1 +(D x1，y1-1 -D x1-1，y1-1 )(w-W x1-1 ) / (W x1 -W x1-1 ); d 表y1 =D x1-1，y1 +(D x1，y1 -D x1-1，y1 )(w-W x1-1 ) / (W x1 -W x1-1 ); D 表1 =d 表y1-1 +(d 表y1 -d 表y1-1 )(P g - G y1-1 ) / (G y1 -G y1-1 ); The generator head w and power P can then be calculated. g The corresponding opening Y w , pg =D 表1 =d 表y1-1 +(d 表y1 -d 表y1-1 (P) g -G y1-1 ) / (G y1 -G y1-1 ); If W x2-1 ≤w≤W x2 G y2-1 ≤P g +P yctp ≤G y2 ,but: d 表y2-1 =D x2-1，y2-1 +(D x2，y2-1 -D x2-1，y2-1 )(w-W x2-1 ) / (W x2 -W x2-1 ); d 表y2 =D x2-1，y2 +(D x2，y2 -D x2-1，y2 )(w-W x2-1 ) / (W x2 -W x2-1 ); D 表2 =d 表y2-1 +(d 表y2 - d 表y2-1 )(P g +P yctp -G y2-1 ) / (G y2 -G y2-1 ); Similarly, the unit head w and power P can be calculated. g +P yctp The corresponding opening Y w , pg+pyctp =D 表2 =d 表y2-1 +(d 表y2 -d 表y2-1 (P) g +P yctp -G y2-1 ) / (G y2 -G y2-1 ); then Y yctp =Y w , pg+pyctp -Y w , pg ; Where x1, x2, y1, and y2 are all positive integers.

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