Methods and systems for vibration-based condition monitoring of rotating electrical machines

Abstract

This invention relates to a method and system for vibration-based condition monitoring of a rotating electric machine. The method includes providing historical data, comprising time series of at least two operating parameters and at least one spatial vibration component of the rotating electric machine; identifying stable operating periods in the historical data, wherein the stable operating periods are defined by at least two operating parameters remaining constant for a preset time period; performing cluster analysis on the identified stable operating periods to identify clusters of operating points, wherein different clusters of operating points define different operating states of the rotating electric machine; determining a recommended threshold for at least one spatial vibration component for the defined operating state; and providing the defined operating state and the recommended threshold for the defined operating state.

Description

Technical Field

[0001] The present invention relates to a computer-implemented method for training a model for recommending thresholds for at least one spatial vibration component of a rotating electric machine.

[0002] The present invention also relates to a computer-implemented method for monitoring the status of a rotating electric motor.

[0003] Furthermore, the present invention relates to a computer program product having instructions for performing the aforementioned method, a machine-readable storage medium having such a computer program product, and a data transmission signal carrying the aforementioned instructions.

[0004] Furthermore, the present invention relates to a sensing and computing device for monitoring the state of a rotating electric motor. Background Technology

[0005] In condition monitoring of electric drives, such as electric motors, especially low-voltage motors, and their industrial applications, one of the most important measurement variables is vibration, as it is a good indicator of feasible failures and damage conditions, provided the measurement technique is sufficiently good and the measurement is close enough to the mechanical components. Therefore, many devices used for semi / automatic condition monitoring typically evaluate vibration data and levels to draw conclusions about the drive's "health status." A fundamental problem here is that vibration measurements can be correlated with many external factors, especially when not measured directly at mechanical parts (e.g., bearing housings). In particular, the load conditions of the motor in its intended application have a significant impact on vibration amplitude and speed. Current solutions rely solely on vibration measurements for analysis and attempt (often without knowing the precise operating point) to distinguish different clusters / vibration points based on these measurements.

[0006] However, in order to identify minute changes over a longer period of time and under different load conditions (speed and / or torque variations) in the application, it is necessary to be able to distinguish different load conditions as accurately as possible, and then correlate each vibration measurement with the load condition so that the vibration measurements can be compared at the same operating point over a longer period of time, such as several months, and the deviations can be checked.

[0007] An application is known that processes motor data detected by sensors, calculating and displaying torque, speed, and three-dimensional vibration values. This application can perform threshold checks, where the user can adjust the thresholds. However, the application does not consider operating points when calculating recommended thresholds and checking for exceedances, making it unable to identify small variations, especially at individual operating points. This results in unsatisfactory threshold checks and consequently poor condition monitoring. Summary of the Invention

[0008] To improve state monitoring based on threshold checking, a computer-implemented method for training the model, as mentioned at the beginning, is proposed and improved, wherein, according to the present invention...

[0009] - Provide historical data, which includes time series of at least two operating parameters of the rotating electric motor and at least one spatial vibration component.

[0010] - Identify stable operating periods in historical data, defined by ensuring that at least two operating parameters remain constant during a preset time period (e.g., between 10 and 30 minutes).

[0011] - Perform cluster analysis on the identified stable operating periods to identify operating point clusters, where different operating point clusters define different operating states of the rotating electric machine.

[0012] - Calculate the recommended threshold for at least one spatial vibration component under a defined operating state. (Different recommendations for different operating states).

[0013] - Provides defined operating states and recommended thresholds for those defined operating states.

[0014] One implementation could propose estimating the centroid for each running point cluster (e.g., by calculating the cluster average).

[0015] One implementation could propose: calculating a recommended threshold for each operating state.

[0016] In one implementation, it is possible to propose that historical data comprise time series of two or three spatial vibration components, wherein a recommended threshold for a defined operating state is determined for each of the two or three spatial vibration components. Here, for example, a Bayesian-Gaussian Mixture Model (BGMM) can be applied to the time series of the spatial vibration components.

[0017] One implementation could propose associating different identifiers (e.g., numerical values, numbers, colors, etc.) with different runpoint clusters.

[0018] One implementation could propose associating different identifiers (numbers, codes, colors, etc.) with different periods of stable operation.

[0019] In one implementation, it can be proposed that the two operating plateaus are different if (e.g., if and only if) at least one of the at least two operating parameters is different.

[0020] In one implementation, the operating parameters can be: rotational speed and slip frequency.

[0021] One implementation could propose that historical data be backtracked for a maximum of one month in time series.

[0022] Other parameters related to the operation of rotating electrical machines include: temperature, stator frequency, torque, electrical power and energy, and the effective value of spatial vibration components.

[0023] To improve condition monitoring, according to the present invention, the computer-implemented method for monitoring the condition of a rotating electric motor, as mentioned at the beginning, is improved in the following manner:

[0024] - Provide actual data, which includes time series of at least two operating parameters of the rotating electric motor and at least one spatial vibration component.

[0025] - Identify stable operating periods in actual data, where a stable operating period is defined as at least two operating parameters remaining constant or constant within a preset time period (e.g., between 10 and 30 minutes, especially 15 minutes).

[0026] - Provide a trained model as described above.

[0027] - Associate the identified stationary periods with the operational states defined by the trained model, so as to map the time series / values ​​of at least one spatial vibration component (corresponding to the identified stationary periods) to the operational states defined by the trained model.

[0028] - Check whether the value of at least one spatial vibration component exceeds a threshold recommended by the trained model.

[0029] - If the value of at least one spatial vibration component exceeds the threshold, a warning message will be output according to a predefined standard.

[0030] In one implementation, it can be proposed that each stationary period can be associated with a state, for example by calculating the Euclidean distance between the data point and the centroid of the state.

[0031] In one implementation, it is possible to propose that a warning message be output if at least three successive values ​​of at least one spatial vibration component exceed a threshold.

[0032] In one implementation, it is possible to propose that the actual data includes time series of two or three spatial vibration components, wherein the values ​​of the two or three spatial vibration components are checked to see if they exceed the corresponding thresholds recommended by the trained model.

[0033] In one embodiment, it is possible to: for each value exceeding a threshold of at least one spatial vibration component, calculate an effective value for each spatial vibration component, and calculate a geometric mean from the calculated effective value, wherein a warning message is output when the geometric mean exceeds a preset value, for example, 4 mm / s.

[0034] In one implementation, the warning message can include the number of times exceeded and the associated timestamp.

[0035] One implementation could propose that the time series of actual data be backed up to a maximum of two days.

[0036] In one implementation, it is possible to store the actual data and retrain the provided model according to the training method described above at preset time intervals (e.g., every month).

[0037] Another aspect of the present invention is the sensing and computing device mentioned at the beginning, which is improved according to the present invention to detect data related to the operation of a rotating motor at the rotating motor, wherein the data includes time series of at least two operating parameters of the rotating motor and at least one spatial vibration component, wherein the sensing and computing device includes instructions that, when executed by the sensing and computing device, cause the sensing and computing device to perform according to the training method described above and / or according to the state monitoring method described above.

[0038] In one implementation, it can be proposed that the rotary motor be designed as an electric motor, particularly a low-voltage motor.

[0039] In one embodiment, it can be proposed that the rotating electric motor is under a time-varying load during operation, wherein the time-varying load is preferably characterized by a time-varying torque or a time-varying speed.

[0040] In one implementation, it can be proposed that the sensing and computing device is designed to visualize the calculated data (vibration values, torque, speed).

[0041] This invention is based on the understanding that if different operating points are distinguished and a vibration threshold is calculated for each operating point, threshold-based state monitoring can be improved.

[0042] The calculated vibration threshold should be as close as possible to the expected vibration amplitude and should limit the effective value (RMS value) of the vibration measurement upwards. In electrical engineering, the average of the squares of a physical variable that varies over time is understood as the effective value. However, the limit value should not be chosen too small to reduce the number of false alarms and should only be reacted to when a "relevant" exceedance of the threshold occurs.

[0043] As described above, this can be achieved by filtering by frequent running points in the value domain through clustering based on physical parameters, and simultaneously by filtering by a stationary period identification method in the time domain. Thus, it is feasible to specify vibration-specific thresholds for the most frequent and simultaneously constant running points, enabling the identification of small fluctuations or changes in vibration levels even during moderate time periods (<1 month), while simultaneously filtering out external influences of short-term oscillations in vibration levels, and finally ensuring criticality through comparison with standards.

[0044] The advantages are better identification of error states in rotating electrical machines and, consequently, better identification of error states in applications, because it can better filter out external correlations and / or interfering variables. At the same time, it can reduce the number of error messages, as messages are only generated when there are clearly identifiable criticalities, while still being able to identify and report small changes over time, even during long-term operation with vibrations exceeding standard values.

[0045] This makes it possible to identify deterioration in the "health status" even at operating points with severe damage to the drive or strong vibrations, thereby enabling a more comprehensive on-site inspection or manual data review. Attached Figure Description

[0046] The invention and other advantages are explained in more detail below with reference to exemplary embodiments. As shown in the accompanying drawings:

[0047] Figure 1 A flowchart illustrating a computer-implemented training method is provided.

[0048] Figure 2 The results of stationary period identification are shown.

[0049] Figure 3 The results of the cluster analysis are shown.

[0050] Figure 4 Different threshold recommendations are shown for different operating states.

[0051] Figure 5 Showing the implementation Figure 1 The system of training methods,

[0052] Figure 6 This diagram shows a flowchart of a computer-implemented status monitoring method.

[0053] Figure 7 The results show the correlation between the actual stationary period and the operating state in the trained model, and

[0054] Figure 8 The diagram shows a sensing and computing device for monitoring the status of a rotating electric motor. Detailed Implementation

[0055] Figure 1 A flowchart corresponding to a computer-implemented method according to the invention is shown. In this method, a model is trained to recommend thresholds for at least one spatial vibration component of a rotating electric motor.

[0056] In step S1 of the training method, historical data is provided. The historical data includes time series of at least two operating parameters and at least one spatial vibration component of the rotating electric motor. The rotating electric motor can be, for example, an electric motor, particularly a low-voltage motor.

[0057] The preferred operating parameters are rotational speed and slip frequency.

[0058] Historical time series data, for example, can be traced back to at most one month, thus providing a good overview of the recent behavior of the rotating motor. Here, the one-month time period should not be interpreted as a limitation; on the one hand, if earlier data exists, it can be used; on the other hand, if it makes sense, shorter time intervals can be considered to retrain the model more frequently (see below).

[0059] Other parameters related to the operation of a rotating electric machine may include temperature, stator frequency, torque, electrical power and energy, and the effective value of spatial vibration components.

[0060] All operating parameters can be grouped together. Operating parameters such as temperature and stator frequency are not uncommonly referred to as high-frequency KPIs (Key Performance Identifiers) because they are measured approximately every minute during machine operation. Other operating parameters, such as torque, electrical power, and electrical energy, are called low-frequency KPIs. Low-frequency KPIs are measured at an even lower frequency (approximately every three minutes).

[0061] In another step, S2, a stable operating period is identified in historical data. A stable operating period is defined as follows: at least two operating parameters are constant / remain constant during a preset time period (e.g., between 10 and 30 minutes, particularly a 15-minute period). The values ​​10, 15, and 30 minutes are used as guidelines and are typically motor type-related. For example, a motor may require up to 30 minutes to reach its planned operating state.

[0062] Figure 2 The diagram illustrates an example of stationary period identification. The time from September to the end of December (approximately four months) is plotted on the horizontal axis. Rotational speed and slip frequency are plotted on the vertical axis, respectively.

[0063] from Figure 2The system can identify multiple distinct plateau periods. A plateau period is distinct if at least one of at least two operating parameters (in this case, engine speed and slip frequency) differs from the other.

[0064] from Figure 2 The study also identified that different identifiers could be associated with different plateau periods. In the current example, the plateau periods are numbered consecutively and drawn with different colors (which are difficult to distinguish here due to the black-and-white view).

[0065] In step S3, cluster analysis is performed on the identified stationary periods to identify clusters of running points. For this purpose, machine learning algorithms such as DBSCAN (Density-based spatial clustering of applications with noise) can be used.

[0066] Different operating point clusters define different operating states of the rotating electric machine. In particular, the same threshold for one (or more) vibration components should apply to all points in the operating point cluster.

[0067] Examples of cluster analysis results in Figure 3 The diagram shows the slip frequency plotted on the horizontal axis and the rotational speed plotted on the vertical axis. Three runpoint clusters have been identified. Different identifiers can be associated with different runpoint clusters (numbers, codes, etc.).

[0068] from Figure 3 The centroid can be identified for each running point cluster. For example, this can be achieved by calculating the cluster mean.

[0069] In step S4, for each defined operating state, a recommended threshold for determining at least one spatial vibration component is preferably calculated. Needless to say, different recommendations can be given for different operating states.

[0070] For example, to derive recommendations, a Bayesian Gaussian mixture model can be applied to the time series of spatial vibration components.

[0071] The determined threshold is preferably made as close as possible to the expected value of the corresponding vibration amplitude, preferably within, for example, three standard deviations, and should limit the RMS value (root mean square RMS) of the vibration measurement upwards.

[0072] What is effective is to calculate a recommended threshold for each operating state.

[0073] If the historical data includes time series of two or three spatial vibration components, it is also possible to determine the corresponding threshold suggestions for the defined operating states for the vibration components in other spaces.

[0074] Figure 4 This section describes three recommended thresholds for the X-vibration component under three operating conditions. Figure 3 The system can identify the three operating states. Time is plotted on the horizontal axis. Axial vibration (X) is plotted on the vertical axis.

[0075] In step S5, a defined operating state and a recommended threshold for that defined operating state are provided. Therefore, the model is trained and can be used.

[0076] Figure 5 A system 1 suitable for implementing the above training method is schematically shown. System 1 includes a storage medium 2 for storing, for example, temporarily storing machine-executable components, and a processor unit 3 (e.g., one or more CPUs) operatively coupled to the storage medium 2 to implement machine-executable components.

[0077] Storage medium 2 includes machine-executable instructions 4 that can be processed by the processor unit, and the learning method described above is implemented based on historical data 5 when processing the instructions. The historical data can be provided to or stored on storage medium 2. Processor unit 3 can also be configured to download historical data 5 from a database.

[0078] Figure 6 A flowchart is shown of a computer-implemented method corresponding to a method for monitoring the state of a rotating electric motor according to the present invention.

[0079] In step S01, actual data is provided, which includes time series of at least two operating parameters of the rotating motor and at least one spatial vibration component.

[0080] As in the training methods already discussed, the preferred operating parameters are speed and slip frequency. The rotating electrical machine can be an electric motor, especially a low-voltage electric motor.

[0081] Actual data can also include time series of two or three spatial components (X, Y, and Z). Time series of other high-frequency and / or low-frequency KPIs can also be included in the actual data.

[0082] The actual time series data typically goes back a maximum of two days, but no longer. This time series also only includes data that was examined during the last 24 hours of machine uptime.

[0083] In step S02, a stable operating period is identified in the actual data, wherein the stable operating period is defined as follows: at least two operating parameters are constant / remain constant during a preset time period (e.g., between 10 and 30 minutes, particularly 15 minutes). The detection of the stable operating period in the actual data is performed in a similar manner and method to the identification of the stable operating period in historical data.

[0084] In step S03, a trained model as described above is provided. This model can also be trained first on-site, for example at the machine, so that it can be applied in the same way to the machine's real-time conditions or continuous operation.

[0085] In step S04, the identified stationary phase is associated with the operating state defined by the trained model. The purpose of this association is to map the time series or values ​​of at least one spatial vibration component corresponding to the identified stationary phase to the operating state defined by the trained model.

[0086] Here, each actual stationary period can be associated with the operating state derived from the model. This can be achieved, for example, by calculating the Euclidean distance between the centroids of the actual stationary period and the operating state derived from the model.

[0087] The result of this association is Figure 7 The example is shown below. Figure 7 The following operational states can be identified, which are defined during the cluster analysis of historical data (see [link]). Figure 3 Historical data is identified using square dots. Stationary periods in the actual data can be identified as circular (orange) dots. For example, stationary periods can be associated with clusters of operating points or operating state "0" based on a small Euclidean distance from the corresponding centroid.

[0088] After association, the threshold suggestions obtained by the model for each running state during training can now be used to search for thresholds that have been exceeded.

[0089] Therefore, in step S05, it is checked whether the value (from the time series) of at least one spatial vibration component exceeds the threshold recommended by the trained model.

[0090] In step S06, if the value of at least one spatial vibration component exceeds a threshold, a warning message is output according to a predetermined standard.

[0091] To reduce the number of feasible warning messages, it is essential to first distinguish between "oscillating" and "non-oscillating" (actual) stable operating periods. Here, if the slip frequency does not change very drastically during a stable operating period, for example, if its standard deviation is less than 0.5, then the stable operating period is considered non-oscillating. Therefore, checks are only performed during non-oscillating stable operating periods.

[0092] Furthermore, a warning message can only be output when at least three consecutive values ​​(three consecutive data points in the time series) of at least one spatial vibration component exceed the threshold. If a measurement is performed every three minutes, this corresponds to exceeding the threshold for nine minutes.

[0093] As already discussed, real-world data can include time series of two or three spatial vibrational components (X, Y, and Z components). Here, it is possible to check whether the values ​​of the two or three spatial vibrational components exceed the corresponding thresholds recommended by the trained model.

[0094] In this case, if, for example, one of the three spatial vibration components exceeds a threshold, an effective value (RMS value) can be calculated to determine the geometric mean. If the geometric mean exceeds a preset value (e.g., 4 mm / s), a warning message is output. The value of 4 mm / s corresponds to the value for medium-sized machines in DIN ISO 10816-3.

[0095] Alert messages can include the number of times the alert was exceeded and the associated timestamp.

[0096] If the rotating motor operates for a long period of time (e.g., several months or years), what is effective is to retrain or familiarize the model. Here, it can be proposed to store the actual data and retrain the provided model with “new” historical data at preset time intervals (e.g., every month) as described above.

[0097] Figure 8 A sensing and computing device 10 for monitoring the status of a rotary motor 11 is shown. The machine is designed as an electric motor. The sensing and computing device includes a sensor device 12 disposed on the electric motor 11. For example, the sensor device 12 can be fixed to a heat sink on the motor housing.

[0098] The electric motor 11 can be implemented as a low-voltage motor.

[0099] During operation, the electric motor 11 is subjected to a time-varying load, wherein the time-varying load is preferably characterized by a time-varying torque or a time-varying speed.

[0100] The sensor device 12, which can be implemented as a battery-powered smart box, is designed to detect data related to the operation of the electric motor. In the example shown, the sensor device 12 detects multiple data points indirectly. For example, it receives vibrations of the bearings not directly, but via the housing. This also relates to the temperature of the rotor, etc.

[0101] The detected data can exist in the form of a time series. The data includes time series of at least two operating parameters of the electric motor (e.g., speed and slip frequency) and at least one spatial vibration component.

[0102] The sensor device 12 can be configured in the field, for example via a short-range radio connection (e.g., via Bluetooth).

[0103] The sensor device 12 has a data interface (e.g., WiFi) that enables data transmission to the computing device 13.

[0104] For example, computing device 13 can be based on a cloud platform that provides different services App#1, App#2, ... to monitor and / or manage rotating motors, such as electric motors, through users.

[0105] The sensing and computing device 10, preferably the computing device 13, includes instructions 14, which, when executed by the sensing and computing device, cause the sensing and computing device to implement the above-described training method and / or the above-described state monitoring method.

[0106] Instruction 14 can be designed as part of an application.

[0107] In other words, the sensing and computing device 10 is configured to: detect drive data (motor data) and calculate vibration values ​​based on the drive data, and perform state monitoring of the electric drive based on threshold checks according to the calculated vibration values. To this end, the sensing and computing device 10 has a trained and learning algorithm 14 that can be executed on the sensing and computing device, wherein the algorithm 14, when executed on the sensing and computing device:

[0108] - Determine at least two distinct operating states of the electric drive based on the drive data.

[0109] and

[0110] - Calculate the vibration threshold based on the calculated vibration value for each operating point.

[0111] - Perform threshold-based condition monitoring on the electric actuator, taking into account the calculated vibration threshold.

[0112] Figure 8An embodiment of the sensing and computing device 10 is shown, which is designed as a system including workshop components (e.g., sensor devices 12 designed as smart boxes) and cloud components (e.g., computing devices 13).

[0113] However, the sensing and computing device 10 can also be implemented as a structural unit that includes sensing devices for detecting data and computing resources for executing algorithm 14 and processing data.

[0114] Figures 2 to 4 , Figure 7 and Figure 8 It is clearly indicated that the sensing and computing device 10 is configured to: prepare or visualize the calculated data (vibration value, torque, speed) for visualization purposes.

[0115] The purpose of this specification is solely to provide illustrative examples and to demonstrate other advantages and particularities of the invention, and therefore it should not be construed as a limitation on the scope of application of the invention or on the patent rights claimed herein. In particular, the features disclosed in conjunction with the methods described herein can be used to improve the systems described herein, and vice versa.

[0116] In the embodiments and drawings, elements that are the same or serve the same function may be provided with the same reference numerals. The reference numerals are only provided to simplify the location of elements that are also referenced, and do not have a limiting effect on the protected subject matter.

Claims

1. A computer-implemented method for training a model for recommending thresholds for at least one spatial vibration component of a rotating electrical machine, wherein, Provide historical data, including The historical data includes time series of at least two operating parameters of the rotating electric motor and at least one spatial vibration component; A stable operating period is identified in the historical data, wherein the stable operating period is defined by the fact that at least two operating parameters are constant during a preset time period; Cluster analysis is performed on the identified stable operating periods to identify operating point clusters, wherein different operating point clusters define different operating states of the rotating motor; For at least one of the spatial vibration components, a recommended threshold for the defined operating state is determined, and the determined threshold is as close as possible to the expected value of the corresponding vibration amplitude. Provides the defined operating state and the recommended threshold for the defined operating state.

2. The method according to claim 1, wherein, Estimate the centroid of the cluster for each running point.

3. The method according to claim 1 or 2, wherein, The historical data includes time series of two or three spatial vibration components, wherein a recommended threshold is determined for the defined operating state for the two or three spatial vibration components.

4. The method according to claim 1 or 2, wherein, The operating parameters are rotational speed and slip frequency.

5. A computer-implemented method for monitoring the state of a rotating electric machine, wherein, Provide actual data, among which, The actual data includes time series of at least two operating parameters of the rotating motor and at least one spatial vibration component; A stable operating period is identified in the actual data, wherein the stable operating period is defined by the fact that at least two operating parameters are constant during a preset time period; Provide a model trained by the method of any one of claims 1 to 4; The identified stable operating period is associated with the operating state defined by the trained model, so that the time series of the at least one spatial vibration component can be mapped to the operating state defined by the trained model. Check whether the value of the at least one spatial vibration component exceeds the threshold recommended by the trained model; If the value of at least one spatial vibration component exceeds the threshold, a warning message is output according to a predetermined standard.

6. The method according to claim 5, wherein, If at least three consecutive values ​​of the at least one spatial vibration component exceed the threshold, a warning message is output.

7. The method according to claim 5, wherein, The actual data includes time series of two or three spatial vibration components, wherein the values ​​of the two or three spatial vibration components are checked to see if they exceed the corresponding thresholds recommended by the trained model.

8. The method according to claim 7, wherein, For each value of the at least one spatial vibration component that exceeds the threshold, an effective value is calculated for each spatial vibration component, and a geometric mean is calculated from the calculated effective value, wherein a warning message is output when the geometric mean exceeds a preset value.

9. The method according to claim 8, wherein, A warning message is output when the geometric mean exceeds 4 mm / s.

10. The method according to any one of claims 5 to 9, wherein, The warning message includes the number of times the limit has been exceeded and the associated timestamp.

11. The method according to any one of claims 5 to 9, wherein, The time series of the actual data can be traced back up to two days at most.

12. The method according to any one of claims 5 to 9, wherein, The actual data is stored, and the provided model is retrained at preset time intervals according to the method described in any one of claims 1 to 4.

13. The method according to claim 12, wherein, The provided model is retrained monthly according to the method described in any one of claims 1 to 4.

14. A computer program product comprising instructions (14) that, when executed by a computer, cause the computer to perform the method according to any one of claims 1 to 4 or the method according to any one of claims 5 to 13.

15. A machine-readable storage medium (2) comprising a computer program product according to claim 14.

16. A sensing and computing device (10) for monitoring the state of a rotating electric motor (11), said sensing and computing device being designed to: detect data at the rotating electric motor (11) related to the operation of the rotating electric motor (11), wherein, The data includes a time series of at least two operating parameters of the rotating motor (11) and at least one spatial vibration component, wherein the sensing and computing device includes an instruction (14) that, when executed by the sensing and computing device, causes the sensing and computing device to perform the method according to any one of claims 1 to 4 or the method according to any one of claims 5 to 13.

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