Permanent magnet assisted bearingless synchronous reluctance motor multi-frequency vibration compensation control method

Abstract

The application discloses a kind of permanent magnet auxiliary bearingless synchronous reluctance motor multi-frequency vibration compensation control methods, to the suspension force current error and the given x, y direction multi-frequency vibration compensation current given value Fourier decomposition is carried out, and real part coefficient and imaginary part coefficient are obtained, the error between suspension force error current and the multi-frequency vibration compensation current given value is calculated to obtain multi-frequency error current, the multi-frequency current coefficient is obtained by Fourier decomposition to the frequency error current, the real part coefficient amplitude and imaginary part coefficient amplitude are calculated from multi-frequency current coefficient and real part coefficient and imaginary part coefficient, according to real part coefficient amplitude and imaginary part coefficient amplitude Current evaluation function is constructed, and the multi-frequency current coefficient is further optimized using variable angle search algorithm and difference evolution algorithm, and the multi-frequency vibration compensation current is calculated based on the final optimized multi-frequency current coefficient, the multi-frequency vibration compensation of rotor is realized, so that motor realizes high-precision stable suspension operation.

Description

Technical Field

[0001] This invention belongs to the field of electric drive control equipment technology, and relates to a permanent magnet assisted bearingless synchronous reluctance motor suspension system. Specifically, it is a vibration compensation control method for a permanent magnet assisted bearingless synchronous reluctance motor, which is applicable to high-performance control of permanent magnet assisted bearingless synchronous reluctance motor suspension systems. Background Technology

[0002] The permanent magnet assisted bearingless synchronous reluctance motor (PMRM) incorporates a levitation winding into the stator core of a traditional PMRM. Through power electronic devices and a digital control system, it simultaneously possesses rotational and self-levitation support capabilities. In addition to the inherent advantages of synchronous reluctance motors, the PMRM effectively solves the bearing support challenges inherent in conventional high-speed motors operating at high and ultra-high speeds for extended periods. It holds significant application potential in precision CNC machine tools, aerospace, and flywheel energy storage. These specialized fields place higher demands on the levitation operation of the PMRM, and the performance of the levitation system directly impacts the overall system performance.

[0003] Regarding rotor imbalance vibration compensation in permanent magnet assisted bearingless synchronous reluctance motors, most existing technologies compensate for and control the imbalance vibration caused by rotor mass eccentricity. However, in reality, multi-frequency vibrations caused by installation errors of motor sensors and irregular roundness of the rotor surface are also significant, and motor vibration is usually caused by more than one frequency signal. Chinese Patent Publication No. CN112803852A discloses a control method for compensating rotor imbalance vibration using a variable step size and variable angle search algorithm for bearingless motors, compensating for frequency signals caused by rotational speed. Chinese Patent Publication No. CN113037162A compensates for imbalance vibration caused by rotor mass eccentricity and dead zones, compensating for imbalance vibration signals with frequencies of ω and 6ω, thus improving the accuracy of motor imbalance vibration compensation. However, the signals causing motor vibration are often not limited to just a few frequencies, and compensating only a few frequencies makes it difficult to achieve higher vibration compensation accuracy. Furthermore, traditional iterative search algorithms often have excessively large step sizes, leading to low algorithm accuracy, while excessively small step sizes result in excessively long convergence times. Summary of the Invention

[0004] The purpose of this invention is to solve the aforementioned problems in the unbalanced vibration compensation of existing permanent magnet assisted bearingless synchronous reluctance motors, and to propose a multi-frequency vibration compensation control method to achieve high-precision compensation for the rotor.

[0005] The technical solution adopted in the multi-frequency vibration compensation control method for a permanent magnet assisted bearingless synchronous reluctance motor of the present invention includes the following steps:

[0006] Step 1): The difference between the reference displacement and the actual displacement in the x and y directions is used to obtain the levitation force current error. Fourier decomposition is then performed on the levitation force current error and the given multi-frequency vibration compensation current values ​​in the x and y directions to obtain the real part coefficients and the imaginary part coefficients.

[0007] Step 2): Calculate the error between the levitation force error current and the given value of the multi-frequency vibration compensation current to obtain the multi-frequency error current. Perform Fourier decomposition on the multi-frequency error current to obtain the multi-frequency current coefficient.

[0008] Step 3): Calculate the magnitudes of the real and imaginary coefficients from the multi-frequency current coefficients, real and imaginary coefficients, and construct the current evaluation function based on the magnitudes of the real and imaginary coefficients.

[0009] Step 4): Use a variable angle search algorithm to search for the multi-frequency current coefficients and output the multi-frequency current coefficients after the variable angle search.

[0010] Step 5): The multi-frequency current coefficients obtained by the variable angle search are further optimized using the differential evolution algorithm to output the final optimized multi-frequency current coefficients;

[0011] Step 6): Based on the final optimized multi-frequency current coefficient, calculate the multi-frequency vibration compensation current, and combine the multi-frequency vibration compensation current with the motor's levitation force reference current and levitation force feedback current to achieve multi-frequency vibration compensation of the rotor.

[0012] The advantages of this invention are:

[0013] 1. This invention solves the problem that the vibration of a permanent magnet assisted bearingless synchronous reluctance motor is often multi-frequency vibration in actual operation. The interference force on the rotor can be equivalent to the multi-frequency current input to the suspension winding control. By setting the fundamental frequency signal to the evaluation function of the n-fold frequency signal to be compensated, and using multi-frequency current identification for vibration compensation, the vibration problem caused by rotor eccentricity, sensor position, and rotor surface roundness can be suppressed, thereby improving the suspension control accuracy.

[0014] 2. This invention uses a variable angle iterative algorithm combined with an improved differential algorithm to search for the vibration compensation current signal coefficient of a multi-frequency motor. It takes into account both the calculation speed and accuracy of the algorithm, and can well balance the relationship between the algorithm accuracy and convergence time, so that the algorithm has higher accuracy and the compensation current is more accurate. It suppresses the vibration and offset of the rotor to the greatest extent, enabling the motor to achieve high-precision and stable levitation operation.

[0015] 3. Since the vibration compensation of this invention uses the Maxwell force generated by the current in the suspension winding to balance the centrifugal force, no external equipment is needed to balance the rotor. It can be achieved through software control algorithm, which makes debugging more convenient.

[0016] 4. This invention can suppress rotor vibration to the greatest extent while ensuring the reliability of stable operation of other parts of the system, with fast compensation speed and little interference to the system. Attached Figure Description

[0017] To make the content of this invention clearer and easier to understand, the invention will be described in detail below with reference to the accompanying drawings and specific embodiments:

[0018] Figure 1 This is a block diagram illustrating the implementation structure of the multi-frequency vibration compensation control method for a permanent magnet assisted bearingless synchronous reluctance motor according to the present invention.

[0019] Figure 2 This is a block diagram illustrating the principle of adaptive identification of multi-frequency current in the x-direction;

[0020] Figure 3 This is a block diagram illustrating the principle of adaptive identification of multi-frequency current in the y-direction.

[0021] Figure 4 This is a flowchart illustrating the workflow of multi-frequency current adaptive identification in the x-direction;

[0022] Figure 5 This is a coordinate diagram of the variable angle search and differential evolution algorithm.

[0023] 1. PI controller; 2, 3. PID controller; 4. MTPA module; 5. Torque control module; 6. Clark / Park converter module; 7. Photoelectric encoder; 8. Force / current conversion module; 9. Clark / Park converter module; 10. Suspension force control module; 11. Eddy current sensor; 12. Multi-frequency current adaptive identification module; 13. Multi-frequency vibration compensation module. Detailed Implementation

[0024] See Figure 1 The multi-frequency vibration compensation controller for a permanent magnet assisted bearingless synchronous reluctance motor according to the present invention includes a torque control section, a levitation force control section, and a multi-frequency current adaptive identification module 12. Both the torque control section and the levitation force control section are conventional control sections of a permanent magnet assisted bearingless synchronous reluctance motor control system. The torque control section consists of a PI controller 1, an MTPA (maximum torque / current ratio) module 4, a torque control module 5, a Clark / Park converter module 6, and a photoelectric encoder 7. The levitation force control section consists of PID controllers 2 and 3, a force / current converter module 8, a Clark / Park converter module 9, and a levitation force control module 10.

[0025] For the torque control section, an optical encoder 7 is used to detect the actual speed ω of the rotor of the permanent magnet assisted bearingless synchronous reluctance motor, and the actual speed ω is compared with the reference speed ω.* The speed difference is obtained by subtracting the speed difference and inputting it into PI controller 1. Then, the MTPA module 4 calculates the d-axis and q-axis reference currents i. md * i mq * The three-phase current i output from the torque winding ma i mb i mc The actual feedback currents i on the d and q axes are obtained after passing through the Clark / Park transformation module 6. md i mq , will reference command current i md * i mq * The difference between the actual feedback current and the input current is given to torque control module 5. Torque control module 5 then controls the three-phase torque winding current i. ma i mb i mc Torque control is achieved by inputting a permanent magnet assisted bearingless synchronous reluctance motor.

[0026] For the levitation force control section, an eddy current sensor 11 is used to detect the actual displacements in the x and y directions of the permanent magnet assisted bearingless synchronous reluctance motor, and the reference displacement x is used. * y * The difference between the actual displacements x and y is used to obtain the levitation force current error i in the x and y directions. ex i ey Suspension force current error ex i ey Input to PID controllers 2 and 3; output from PID controllers 2 and 3 the levitation force reference value F. x * F y * Suspension force reference value F x * F y * The levitation force reference current i is obtained after force / current conversion module 8. sq * i sd * The three-phase current i output from the levitation winding. sa i sb i sc The actual feedback currents i on the d and q axes are obtained after passing through the Clark / Park transformation module 9. sd i sq .

[0027] In addition, the levitation force current error i ex i eyUsing the rotational speed ω as the input signal to the multi-frequency current adaptive identification module 12, the multi-frequency current adaptive identification module 12 outputs the d-axis and q-axis displacement compensation current i. csd i csq The d-axis and q-axis levitation force reference current i sq * i sd * Compensation current i for d and q axis displacement csd i csq Actual feedback current i on the d and q axes sd i sq The difference between the three values ​​is input into the levitation force control module 10. The levitation force control module 10 then processes the three-phase levitation force winding current i. sa i sb i sc A permanent magnet-assisted bearingless synchronous reluctance motor is used to achieve levitation force control.

[0028] The multi-frequency current adaptive identification module 12, PID controllers 2 and 3, and force / current conversion module 8 together form the multi-frequency vibration compensation control module 13.

[0029] The multi-frequency current adaptive identification module 12 includes adaptive identification in the x and y directions. Its input is the given multi-frequency vibration compensation current setpoint i in the x and y directions. csx i csy , levitation force current error i ex i ey The output is the d-axis and q-axis displacement compensation current i, which is the actual rotational speed ω. csd i csq The specific adaptive recognition control is as follows:

[0030] First, the levitation force current error is obtained by subtracting the reference displacement and actual displacement in the x and y directions. Then, Fourier decomposition is performed on the levitation force current error and the given multi-frequency vibration compensation current values ​​in the x and y directions to obtain real and imaginary coefficients. Next, the error between the levitation force error current and the given multi-frequency vibration compensation current value is calculated to obtain the multi-frequency error current. Fourier decomposition is performed on the multi-frequency error current to obtain the multi-frequency current coefficients. The amplitudes of the real and imaginary coefficients are calculated from the multi-frequency current coefficients and the real and imaginary coefficients. A current evaluation function is constructed based on these amplitudes. Then, a variable-angle search algorithm is used to search for the multi-frequency current coefficients, and the multi-frequency current coefficients after the variable-angle search are output. The multi-frequency current coefficients obtained from the variable-angle search are further optimized using a differential evolution algorithm to output the final optimized multi-frequency current coefficients. Based on the final optimized multi-frequency current coefficients, the multi-frequency vibration compensation current is calculated. The multi-frequency vibration compensation current, together with the motor's levitation force reference current and levitation force feedback current, works to achieve multi-frequency vibration compensation for the rotor. (See also...) Figure 2 Taking the x-direction as an example:

[0031] The error of levitation force current i in the x-direction ex Perform Fourier decomposition, as follows:

[0032]

[0033] In equation (1), α x_uξ β x_uξ The levitation force error current i in the x-direction ex The real part coefficients of the ξ-order Fourier series; β x_vξ α x_vξ Let α be the imaginary coefficient of the ξ-th order Fourier series of the levitation force error current in the x-direction, n be the highest order of the Fourier expansion of the error current, and j be the imaginary unit. From this Fourier decomposition, the real and imaginary coefficients α of the ξ-th order Fourier series are obtained. x_uξ β x_uξ β x_vξ α x_vξ .

[0034] For a given value i of the multi-frequency vibration compensation current in the x-direction csx Perform Fourier decomposition, as follows:

[0035]

[0036] In equation (2), α c_x_uξ β c_x_uξ β represents the real part coefficients of the ξ-order Fourier series of the multi-frequency vibration compensation current in the x-direction; c_x_vξ α c_x_vξLet ξ be the imaginary coefficient of the Fourier series of the imaginary part of the levitation force error current in the x-direction.

[0037] The real part coefficients α of the ξ-order Fourier series of the multi-frequency vibration compensation current in the x-direction described in this invention are... c_x_uξ β c_x_uξ The imaginary coefficients β of the ξ-order Fourier series with the imaginary part c_x_vξ α c_x_vξ The system is a multi-frequency current coefficient α c_x_uξ β c_x_uξ β c_x_vξ α c_x_vξ Thus, the multi-frequency current coefficient α is obtained. c_x_uξ β c_x_uξ β c_x_vξ α c_x_vξ .

[0038] Calculate the levitation force error current i in the x-direction ex The given value of the multi-frequency vibration compensation current in the x-direction i csx The error between them is used to obtain the multi-frequency error current i. ecx This refers to the error between the compensated multi-frequency current and the multi-frequency current caused by the interference force. The multi-frequency error current i... ecx Perform Fourier decomposition:

[0039]

[0040] In the formula, i ecx The levitation force error current i ex With the multi-frequency vibration compensation current setpoint i csx The difference between them.

[0041] To ensure accurate compensation current, the multi-frequency error current i needs to be minimized as much as possible. ecx Therefore, the error current i of the levitation force in the x-direction ex The given value of the multi-frequency vibration compensation current in the x-direction i csx Each order error after the Fourier expansion is calculated. As can be seen from equation (3), it can be obtained from the multi-frequency current coefficient α. c_x_uξ β c_x_uξ β c_x_vξ α c_x_vξ and the levitation force error current i in the x-direction ex The real and imaginary coefficients α of the ξ-order Fourier series with real part x_uξ β x_uξ β x_vξ α x_vξ The magnitudes of the real and imaginary coefficients of the ξ-th order were calculated as follows:

[0042]

[0043] In equation (4), A x_uξ A x_vξ These represent the levitation force error current i in the x-direction, respectively. ex The given value of the multi-frequency vibration compensation current in the x-direction i csx The magnitudes of the real and imaginary coefficients of the ξth order after Fourier expansion.

[0044] Based on the magnitude A of the real part coefficient of the ξth order x_uξ And the magnitude of the imaginary part coefficient A x_vξ The current evaluation function of order ξ is constructed as follows:

[0045]

[0046] If the multi-frequency current coefficient α c_x_uξ β c_x_uξ β c_x_vξ α c_x_vξ The coefficients α of the real and imaginary parts x_uξ β x_uξ β x_vξ (True value), then the current evaluation function E Aξ It exhibits a divergent trend, i.e., the error current i ecx The ξ-th order current in the equation increases; conversely, as the multi-frequency current coefficients continuously approach the real and imaginary coefficients (true values), the evaluation function E... Aξ The error current i will tend to 0. ecx The ξ-th order current in E will be fully compensated. In other words, E Aξ The magnitude of the multi-frequency current coefficient α was evaluated. c_x_uξ β c_x_uξ β c_x_vξ α c_x_vξ Approaching the real and imaginary part coefficients α x_uξ β x_uξ β x_vξ α x_vξ The degree of compensation and the effectiveness of the algorithm.

[0047] This invention employs a variable angle search algorithm to search for the multi-frequency current coefficient α. c_x_uξ β c_x_uξ β c_x_vξ α c_x_vξ With the real part coefficient α x_uξ β x_uξ For example, see Figure 5 , Figure 5 The origin of the coordinate system is O, and the horizontal axis α represents the real part coefficient α. x_uξ The vertical axis represents the real part coefficient β. x_uξ , with the set point M(α) x_uξ ,β x_uξ Let R be the step size of the search algorithm and θ be the angle. aThe objective evaluation function value is E obj1 The first search step has a step size of R and an angle θ1 of 0. The first search ends at point A. According to the formula... Calculate the current evaluation function E at point A. Aξ The current evaluation function E Aξ With the target evaluation function value E obj1 For comparison, if the current evaluation function E Aξ Greater than the target price function value E obj1 Then, the variable-angle iterative search algorithm is used to continue the search. In the second iteration, the evaluation function is first calculated with point A as the starting point and the angle still θ1, and the step size R is increased. Then, the evaluation function is calculated with point A as the starting point and the angle θ2 = θ1 + θ a The evaluation function for the result with step size R is used to compare the evaluation functions of the calculation results in different directions. The iteration direction with the smaller evaluation function is selected, and the endpoint of the second iteration calculation is finally determined to be point B. Subsequent iteration results can be obtained using a similar method. If the current evaluation function E... Aξ Less than or equal to the target evaluation function value E obj1 Then the variable angle iterative search ends, and n multi-frequency current coefficients (α) are output. c_x_u1 β c_x_u1 ), (α) c_x_u1 β c_x_u1 ...(α) c_x_un β c_x_un ).

[0048] The n multi-frequency current coefficients (α) obtained by the variable angle search c_x_u1 β c_x_u1 ), (α) c_x_u1 β c_x_u1 ...(α) c_x_un β c_x_un The differential evolution algorithm was used for further optimization. The real part coefficients α were used as the basis for the optimization. x_uξ β x_uξ For example, see Figure 5 Based on the variable angle iterative search results, with its coordinate origin O, in the set R... L Further optimization is performed within the square region bounded by the differential evolution algorithm. First, the objective evaluation function E of the differential evolution algorithm is defined. obj2 , with the current evaluation function E Aξ For comparison, if the current evaluation function E Aξ Less than the target evaluation function E obj2 The differential evolution algorithm optimization is complete, outputting n final optimized multi-frequency current coefficients (α). oc_x_u1 β oc_x_u1 ), (α) oc_x_u1 β oc_x_u1 ...(α) oc_x_unβ oc_x_un Conversely, if the current evaluation function E Aξ Greater than or equal to the target evaluation function E obj2 If the current evaluation function E is not found, continue optimizing until the current evaluation function E is found. Aξ Less than the target evaluation function E obj2 until.

[0049] The search for the imaginary part coefficients is similar to the search for the real part coefficients. Similarly, the differential evolution algorithm optimization for the imaginary part coefficients is similar to the differential evolution algorithm optimization for the real part coefficients. The result is n final optimized multi-frequency current coefficients [(α...]. oc_x_v1 β oc_x_v1 ),(β oc_x_v1 α oc_x_v1 )]、[(α oc_x_u2 β oc_x_u2 ),(β oc_x_v2 α oc_x_v2 )]、...、[(α oc_x_νn β oc_x_un ),(β oc_x_vn α oc_x_vn )).

[0050] The compensation current is calculated based on the final optimized multi-frequency current coefficients obtained from the real and imaginary parts using the differential evolution algorithm, and the multi-frequency vibration compensation current i is generated by the inverter. csd :

[0051]

[0052] This invention uses a method similar to that used in the x-direction to obtain the multi-frequency vibration compensation current i in the y-direction. csq ,like Figure 3 As shown, therefore, no further details will be provided.

[0053] See Figure 1 The multi-frequency vibration compensation current i is obtained from the multi-frequency current adaptive identification module 12. csd i csq , and the levitation force reference current i sd * i sq * and levitation force feedback current i sd i sq The combined effect results in the levitation force input current i c_sd i c_sq The suspension force is controlled by the suspension force control module 10 to eliminate unbalanced forces, thereby achieving multi-frequency vibration compensation of the rotor.

Claims

1. A method for multi-frequency vibration compensation control of a permanent magnet assisted bearingless synchronous reluctance motor, characterized in that: Includes the following steps: Step 1): The difference between the reference displacement and the actual displacement in the x and y directions is used to obtain the levitation force current error. Fourier decomposition is then performed on the levitation force current error and the given multi-frequency vibration compensation current values ​​in the x and y directions to obtain the real part coefficients and the imaginary part coefficients. Step 2): Calculate the error between the levitation force error current and the given value of the multi-frequency vibration compensation current to obtain the multi-frequency error current. Perform Fourier decomposition on the multi-frequency error current to obtain the multi-frequency current coefficient. Step 3): Calculate the magnitudes of the real and imaginary coefficients from the multi-frequency current coefficients, real and imaginary coefficients, and construct the current evaluation function based on the magnitudes of the real and imaginary coefficients. Step 4): Use a variable angle search algorithm to search for the multi-frequency current coefficients and output the multi-frequency current coefficients after the variable angle search. Step 5): The multi-frequency current coefficients obtained by the variable angle search are further optimized using the differential evolution algorithm to output the final optimized multi-frequency current coefficients; Step 6): Based on the final optimized multi-frequency current coefficient, calculate the multi-frequency vibration compensation current, and combine the multi-frequency vibration compensation current with the motor's levitation force reference current and levitation force feedback current to achieve multi-frequency vibration compensation of the rotor.

2. The multi-frequency vibration compensation control method for a permanent magnet assisted bearingless synchronous reluctance motor according to claim 1, characterized in that: In step 1), taking the x-direction as an example, the levitation force current error i in the x-direction is... ex Fourier decomposition is: α x_uξ β x_uξ The levitation force error current i in the x-direction ex The real coefficients of the ξ-order Fourier series with real part, β x_vξ α x_vξ Let ξ be the imaginary coefficient of the Fourier series of the levitation force error current in the x-direction, n be the highest order of the Fourier expansion of the error current, and j be the imaginary unit.

3. The multi-frequency vibration compensation control method for a permanent magnet assisted bearingless synchronous reluctance motor according to claim 2, characterized in that: For a given value i of the multi-frequency vibration compensation current in the x-direction csx Fourier decomposition is: α c_x_uξ β c_x_uξ β c_x_vξ α c_x_vξ This is the multi-frequency current coefficient.

4. The multi-frequency vibration compensation control method for a permanent magnet assisted bearingless synchronous reluctance motor according to claim 3, characterized in that: For multi-frequency error current i ecx Fourier decomposition is:

5. The multi-frequency vibration compensation control method for a permanent magnet assisted bearingless synchronous reluctance motor according to claim 4, characterized in that: Real part coefficient magnitude Imaginary part coefficient amplitude 6. The multi-frequency vibration compensation control method for a permanent magnet assisted bearingless synchronous reluctance motor according to claim 5, characterized in that: Current evaluation function 7. The multi-frequency vibration compensation control method for a permanent magnet assisted bearingless synchronous reluctance motor according to claim 6, characterized in that: Variable angle search with real part coefficient α x_uξ β x_uξ For example, let the horizontal axis represent the real part coefficient α. x_uξ The vertical axis represents the real part coefficient β. x_uξ The current evaluation function E is calculated by searching for the endpoint based on the first step size and angle to obtain the endpoint. Aξ The current evaluation function E Aξ Compare the current evaluation function value with the target evaluation function value; if the current evaluation function E... Aξ Greater than the target evaluation function value E obj1 If the result is positive, continue the search; otherwise, end the search and output the n multi-frequency current coefficients (α) of the real part. c_x_u1 β c_x_u1 ), (α) c_x_u1 β c_x_u1 ...(α) c_x_un β c_x_un ).

8. The multi-frequency vibration compensation control method for a permanent magnet assisted bearingless synchronous reluctance motor according to claim 7, characterized in that: Define the objective evaluation function E of the differential evolution algorithm. obj2 , with the current evaluation function E Aξ For comparison, if the current evaluation function E Aξ Less than the target evaluation function E obj2 The optimization ends, and the n final optimized multi-frequency current coefficients (α) with real parts are output. oc_x_u1 β oc_x_u1 ), (α) oc_x_u1 β oc_x_u1 ...(α) oc_x_un β oc_x_un Similarly, the n final optimized multi-frequency current coefficients of the imaginary part are obtained [(α oc_x_v1 β oc_x_v1 ),(β oc_x_v1 α oc_x_v1 )]、[(α oc_x_u2 β oc_x_u2 ),(β oc_x_v2 α oc_x_v2 )]、...、[(α oc_x_νn β oc_x_un ),(β oc_x_vn α oc_x_vn )).

9. The multi-frequency vibration compensation control method for a permanent magnet assisted bearingless synchronous reluctance motor according to claim 8, characterized in that: According to the formula Calculate the multi-frequency vibration compensation current i csd .

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