Control Method and System for Multi-Frequency Intelligent Heat Exchange Unit

Abstract

The present application relates to the technical field of unit control, and particularly to a control method and system for a multi-frequency intelligent heat exchange unit. The method includes: inputting a heat exchange load sequence within a preset time window into a time series prediction model to obtain at least one predicted load; calculating a disturbance coefficient based on the gradient value of the predicted load, and adding a random disturbance to the real-time control parameters according to the disturbance coefficient to obtain a plurality of particles; constructing a fitness function, and using a particle swarm optimization algorithm to iteratively update the plurality of particles to obtain the optimal control parameters of the heat exchange unit. The fitness function is negatively correlated with the energy consumption, as well as the absolute value of the difference between the actual heat exchange amount and the average predicted load. Through the technical solution of the present application, accurate control of the heat exchange unit can be achieved.

Description

Technical Field

[0001] The present application relates to the technical field of unit control, and particularly to a control method and system for a multi-frequency intelligent heat exchange unit. Background Art

[0002] A heat exchange unit is a device for heat transfer and exchange. Its main function is to transfer the heat of one medium (such as steam, hot water or heat-conducting oil) to another medium (such as cold water or air) to achieve the purpose of heating, cooling or temperature regulation, and it is applied in many fields such as industry, building heating, and central air conditioning.

[0003] Currently, the patent application document with the publication number of CN117232317A discloses a remote control method and remote control system for a heat exchange unit. The method includes: establishing a thermal energy monitoring unit inside the heat exchange unit through a sensor network to collect heat distribution data in real time; establishing a communication connection between the heat exchange unit and a remote control center for transmitting the monitored thermal energy data; analyzing the heat distribution data in real time at the remote control center, identifying hot spots, and evaluating the heat exchange efficiency to automatically generate hot spot response control instructions for adjusting the local operating parameters of the heat exchange unit to balance the thermal energy distribution; transmitting the control instructions to the heat exchange unit and ensuring the reliability of the instructions through a redundancy mechanism; the heat exchange unit receiving the control instructions and implementing local parameter adjustment; monitoring the adjustment effect in real time and feeding back the adjusted heat distribution data to the remote control center; the remote control center optimizing the control instructions again based on the feedback data and continuously monitoring the change of hot spots, thus forming a dynamic optimization closed loop for hot spot tracking and management.

[0004] The above method adjusts the control parameters by monitoring the heat distribution data inside the heat exchange unit, thereby reducing energy loss. However, it ignores the influence of the actual working conditions of the heat exchange unit on the control parameters. When the actual working conditions of the heat exchange unit change, it is unable to adjust the control parameters in a timely and accurate manner, resulting in poor control accuracy of the heat exchange unit. Summary of the Invention

[0005] To solve the technical problem of poor control accuracy of the heat exchange unit, the present application provides a control method and system for a multi-frequency intelligent heat exchange unit, which can achieve precise control of the heat exchange unit.

[0006] In the first aspect of the present application, a control method for a multi-frequency intelligent heat exchange unit is provided. The control method includes: inputting a heat exchange load sequence within a preset time window into a time series prediction model to obtain at least one predicted load, where the heat exchange load is the product of the specific heat capacity, mass, and temperature adjustment amount of the target medium, and the temperature adjustment amount is the difference between the target temperature and the current temperature; calculating a disturbance coefficient based on the gradient value of the predicted load, adding a random disturbance to the real-time control parameters according to the disturbance coefficient to obtain a plurality of particles, and the disturbance coefficient is positively correlated with the gradient value; constructing a fitness function, and using the particle swarm optimization algorithm to iteratively update the plurality of particles to obtain the optimal control parameters of the heat exchange unit. The control parameters include the flow rate, inlet temperature, and outlet temperature of the heat exchange medium, and the fitness function is negatively correlated with the energy consumption, as well as the absolute value of the difference between the actual heat exchange amount and the average predicted load.

[0007] Predict the predicted load at future moments based on the heat exchange load sequence within the preset time window. The gradient value of the predicted load can characterize the change of the operating condition of the heat exchange unit at future moments. When the gradient value of the predicted load is large, it means that the change range of the operating condition of the heat exchange unit at future moments is larger. In order to ensure that the heat exchange unit can adapt to the changed operating condition, the real-time control parameters at the current moment need to be adjusted significantly. Therefore, calculate the disturbance coefficient based on the gradient value of the predicted load, add a random disturbance to the real-time control parameters to obtain a plurality of particles of the particle swarm optimization algorithm. The larger the gradient value of the predicted load, the larger the added random disturbance, which can improve the convergence speed of the particle swarm optimization algorithm and improve the real-time performance of the heat exchange unit control; further, construct a fitness function according to the predicted load. The fitness function is negatively correlated with the energy consumption, as well as the absolute value of the difference between the actual heat exchange amount and the average predicted load, so that the particle swarm optimization algorithm can consider the influence of the actual working condition of the heat exchange unit on the control parameters, obtain the optimal control parameters, and realize the precise control of the heat exchange unit.

[0008] Preferably, the time series prediction model is an LSTM model or a GRU model; the training method of the time series prediction model includes: using the historical heat exchange load sequence as a training sample, and using the historical heat exchange loads at multiple moments after the historical heat exchange load sequence as label values; inputting the training sample into the time series prediction model to obtain an output result, and calculating the mean square error loss value between the output result and the label value; using the gradient descent method to update the time series prediction model until the mean square error loss value is less than the preset loss, and then stop to complete the training of the time series prediction model.

[0009] The trained time series prediction model can accurately predict the predicted loads at multiple future moments, so as to accurately judge the operating condition of the heat exchange unit at future moments.

[0010] Preferably, the gradient value of the predicted load is the maximum value of the difference between the predicted load and the heat exchange load at the current moment.

[0011] Preferably, the perturbation coefficient satisfies the relational expression: , being the gradient value.

[0012] Calculate the perturbation coefficient according to the gradient value of the predicted load. The larger the gradient value, the greater the random perturbation added to the real-time control parameter. Taking the real-time control parameter after the random perturbation as a particle can accelerate the optimization speed of the particle swarm optimization algorithm, enabling the heat exchange unit to adapt to the changing operating conditions.

[0013] Preferably, adding random perturbation to the real-time control parameter according to the perturbation coefficient includes: adding any control parameter and the corresponding perturbation value to obtain the perturbed control parameter, and the perturbation value is the product of the perturbation coefficient and a random number generated within a preset range.

[0014] Quantify the change range of the operating conditions according to the gradient value of the predicted load. The larger the change range, the greater the random perturbation added to the real-time control parameter. Taking the real-time control parameter after adding the perturbation as a particle can improve the convergence speed of the particle swarm optimization algorithm.

[0015] Preferably, the fitness function satisfies: , and are the actual heat exchange amount and the average predicted load respectively, being the energy consumption of the heat exchange unit.

[0016] The fitness function is used to accurately evaluate the control effect of each particle; for any particle, the smaller the energy consumption and the closer the actual heat exchange amount is to the predicted load, the better the control effect of the particle.

[0017] Preferably, the actual heat exchange amount is: , and are the flow rate and specific heat capacity of the heat exchange medium respectively, and are the outlet temperature and inlet temperature of the heat exchange medium respectively.

[0018] Preferably, the particle swarm optimization algorithm is used to iteratively update the multiple particles to obtain the optimal control parameters of the heat exchange unit, including: calculating the Euclidean distance between the global optimal position in the current iteration and the previous iteration; in response to the Euclidean distance being less than a preset distance, calculating the position deviation value between the global optimal position and each particle position in the current iteration; taking the ratio of the position deviation value to the fitness function value of each particle position as the selection probability, and selecting a preset number of particles in descending order of the selection probability to add Gaussian noise to obtain the positions of each particle in the next iteration; performing multiple iterations until the fitness function value of the global optimal position is greater than the preset fitness, or when the number of iterations is greater than the preset number, taking the global optimal position as the optimal control parameter of the heat exchange unit.

[0019] To prevent the particle swarm optimization algorithm from falling into a local optimum and failing to obtain accurate optimal control parameters, during the process of iteratively updating particles using the particle swarm optimization algorithm, Gaussian noise is added to some particles to increase the diversity of particles and avoid falling into local optimal points.

[0020] Preferably, the selection probability of the particle in the

[0021] -th iteration is: where is the position deviation value of the particle in the -th iteration, is the fitness function value of the particle in the -th iteration, and

[0022]

[0023] In the second aspect of the present application, a control system for a multi-frequency intelligent heat exchange unit is further provided, including a processor and a memory. The memory stores computer program instructions, and when the computer program instructions are executed by the processor, a control method for a multi-frequency intelligent heat exchange unit according to the first aspect of the present application is implemented. The technical solution of the present application has the following beneficial technical effects:

[0024] Predict the predicted load at a future moment based on the heat exchange load sequence within a preset time window. The gradient value of the predicted load can characterize the change in the operating condition of the heat exchange unit at the future moment. When the gradient value of the predicted load is large, it indicates that the change amplitude of the operating condition of the heat exchange unit at the future moment is greater. To ensure that the heat exchange unit can adapt to the changed operating condition, the real-time control parameters at the current moment need to be adjusted significantly. Therefore, calculate the disturbance coefficient based on the gradient value of the predicted load, and add random disturbances to the real-time control parameters to obtain multiple particles of the particle swarm optimization algorithm. The greater the gradient value of the predicted load, the greater the added random disturbance, which can improve the convergence speed of the particle swarm optimization algorithm and improve the real-time performance of the heat exchange unit control. Further, construct a fitness function according to the predicted load. The fitness function is negatively correlated with both the energy consumption and the absolute value of the difference between the actual heat exchange amount and the average predicted load, enabling the particle swarm optimization algorithm to consider the influence of the actual condition of the heat exchange unit on the control parameters, obtain the optimal control parameters, and achieve precise control of the heat exchange unit. Description of the Drawings

[0025] Figure 1 is a flowchart of a control method for a multi-frequency intelligent heat exchange unit according to an embodiment of the present application.

[0026] Figure 2 is a structural block diagram of a control system for a multi-frequency intelligent heat exchange unit according to an embodiment of the present application. Detailed Embodiments

[0027] Next, the technical solutions in the embodiments of the present application will be clearly and completely described in conjunction with the accompanying drawings in the embodiments of the present application. Obviously, the described embodiments are part of the embodiments of the present application. All other embodiments obtained by those of ordinary skill in the art based on the embodiments of the present application without creative efforts shall fall within the protection scope of the present application.

[0028] According to a first aspect of the present application, the present application provides a control method for a multi-frequency intelligent heat exchange unit. Figure 1 is a flowchart of a control method for a multi-frequency intelligent heat exchange unit according to an embodiment of the present application. As Figure 1 shown, the control method for the multi-frequency intelligent heat exchange unit includes steps S101 to S103, which are described in detail below.

[0029] S101, input the heat exchange load sequence within a preset time window into a time series prediction model to obtain at least one predicted load. The heat exchange load is the product of the specific heat capacity, mass, and temperature adjustment amount of the target medium. The temperature adjustment amount is the difference between the target temperature and the current temperature.

[0030] In one embodiment, the heat exchange unit affects the heat exchange process between the exchange medium and the target medium by controlling the temperature of the exchange medium, thereby changing the temperature of the target medium and adjusting the target medium to the target temperature. The energy required to adjust the target medium to the target temperature is defined as the heat exchange load of the heat exchange unit, which can accurately quantify the real-time working condition of the heat exchange unit. Specifically, at time the heat exchange load satisfies: , and are the specific heat capacity and mass of the target medium respectively, is the current temperature at time , is the target temperature.

[0031] Understandably, if the heat exchange load at time is greater than 0, it means that the current temperature of the target medium is lower than the target temperature, and the heat of the exchange medium needs to be exchanged to the target medium to increase the temperature of the target medium; similarly, if the heat exchange load at time is less than 0, it means that the current temperature of the target medium is higher than the target temperature, and the heat of the target medium needs to be exchanged to the exchange medium to reduce the temperature of the target medium; if the heat exchange load at time is equal to 0, it means that the current temperature of the target medium is equal to the target temperature and no heat exchange is required.

[0032] In one embodiment, the heat exchange load sequence within a preset time window is input into a time series prediction model to obtain at least one predicted load, where the preset time window includes the current moment and multiple historical moments before the current moment; the predicted load is the heat exchange load at multiple future moments after the preset time window. In the embodiments of the present application, the number of historical moments within the preset time window is 10.

[0033] Among them, the time series prediction model is an LSTM model or a GRU model; the training method of the time series prediction model includes: using the historical heat exchange load sequence as a training sample, and using the historical heat exchange load at multiple moments after the historical heat exchange load sequence as a label value; inputting the training sample into the time series prediction model to obtain an output result, and calculating the mean square error loss value between the output result and the label value; using the gradient descent method to update the time series prediction model until the mean square error loss value is less than the preset loss and then stop, completing the training of the time series prediction model. The trained time series prediction model can accurately predict the predicted load at multiple future moments.

[0034] Among them, the preset loss is 0.01; the mean square error loss value is:

[0035] , is the number of moments after the historical heat exchange load sequence in the tag value, is the historical heat exchange load at the moment after the historical heat exchange load sequence in the tag value . is the predicted load at the moment in the output result .

[0036] In this way, the trained time series prediction model is used to accurately obtain the predicted load, and the operating conditions of the heat exchange unit at future moments are predicted.

[0037] S102. Calculate the perturbation coefficient according to the gradient value of the predicted load, and add random perturbation to the real-time control parameter according to the perturbation coefficient to obtain a plurality of particles. The perturbation coefficient is positively correlated with the gradient value.

[0038] In one embodiment, the gradient value of the predicted load is the maximum value of the difference between the predicted load and the heat exchange load at the current moment. The larger the gradient value, the greater the change range of the operating conditions of the heat exchange unit at future moments. In order to ensure that the heat exchange unit can adapt to the changed operating conditions, the real-time control parameter at the current moment needs to be adjusted greatly. When the gradient value is 0, it means that the operating conditions of the heat exchange unit remain unchanged, and the real-time control parameter at the current moment does not need to be adjusted to achieve precise control of the heat exchange unit.

[0039] Therefore, calculate the perturbation coefficient according to the gradient value of the predicted load. The larger the gradient value, the greater the random perturbation added to the real-time control parameter. Taking the real-time control parameter after random perturbation as a particle can accelerate the optimization speed of the particle swarm optimization algorithm, improve the real-time performance of the heat exchange unit control, and enable the heat exchange unit to adapt to the continuously changing operating conditions. Specifically, the perturbation coefficient satisfies the relational expression: , is the gradient value.

[0040] In one embodiment, after determining the perturbation coefficient, add random perturbation to the real-time control parameter. When the gradient value is large, theoretically, the difference between the optimal control parameter at the next moment and the real-time control parameter is greater. Therefore, a larger random perturbation is added to the real-time control parameter.

[0041] Specifically, adding random perturbation to the real-time control parameter according to the perturbation coefficient includes: adding any control parameter and the corresponding perturbation value to obtain the perturbed control parameter. The perturbation value is the product of the perturbation coefficient and a random number generated within a preset range.

[0042] The preset range is 0-1. Each perturbation will obtain a set of perturbed control parameters. Taking the perturbed control parameters as a particle and performing multiple perturbations on the real-time control parameters can obtain multiple particles.

[0043] It can be understood that the larger the preset range, the greater the random perturbation added to the real-time control parameters. Therefore, in other embodiments, the random perturbation of the real-time control parameters can also be increased by adjusting the preset range.

[0044] In this way, according to the gradient value of the predicted load, the change amplitude of the operating condition is quantified. The larger the change amplitude, the greater the random perturbation added to the real-time control parameters. Taking the real-time control parameters with added perturbation as particles can improve the convergence speed of the particle swarm optimization algorithm and enhance the real-time performance of the heat exchange unit control.

[0045] S103. Construct a fitness function and use the particle swarm optimization algorithm to iteratively update the multiple particles to obtain the optimal control parameters of the heat exchange unit. The control parameters include the flow rate, inlet temperature, and outlet temperature of the heat exchange medium. The fitness function is negatively correlated with the energy consumption, as well as the absolute value of the difference between the actual heat exchange amount and the average predicted load.

[0046] In one embodiment, a fitness function in the particle swarm optimization algorithm is constructed. The fitness function is used to evaluate the control effect of each particle. For any particle, the smaller the energy consumption and the closer the actual heat exchange amount is to the predicted load, the better the control effect of the particle. Therefore, the fitness function is negatively correlated with the energy consumption, as well as the absolute value of the difference between the actual heat exchange amount and the average predicted load.

[0047] Specifically, the control parameters include the flow rate, inlet temperature, and outlet temperature of the heat exchange medium, and the actual heat exchange amount is: , and are the flow rate and specific heat capacity of the heat exchange medium respectively, and are the outlet temperature and inlet temperature of the heat exchange medium respectively.

[0048] Among them, the control of the flow rate can be achieved by adjusting the frequency of the variable frequency pump. The control of the inlet temperature can be achieved by controlling the heating process of the exchange medium. The control of the outlet temperature can be achieved by adjusting the heat exchange area. The actual heat exchange amount is the heat actually exchanged between the heat exchange medium and the target medium under the control parameters.

[0049] The fitness function satisfies: , and are the actual heat exchange amount and the average predicted load respectively, is the energy consumption of the heat exchange unit.

[0050] In one embodiment, the particle swarm optimization algorithm iteratively updates each particle multiple times. After completing one iteration update, the individual optimal position of each particle is obtained. The individual optimal position is the position corresponding to the maximum value of the fitness function during all the iteration updates of the particle. And the individual optimal position corresponding to the maximum value of the fitness function among all the individual optimal positions is taken as the global optimal position.

[0051] In the th iteration, the moving speeds and positions of all particles are updated. The update process of the particle is as follows:

[0052] ;

[0053] ;

[0054] where is the inertia weight, and are the moving speeds of the particle in the th iteration and the th iteration respectively. and are the learning factors of individual experience and social experience respectively. and are random numbers between [0, 1]. and are the positions of the particle in the th iteration and the th iteration respectively. is the individual optimal position of the particle in the th iteration. is the global optimal position of the th iteration. Among them, the learning factor of individual experience and the learning factor of social experience take the value of 2. The random numbers and are used to increase the search randomness of the particle swarm algorithm, and the value of the inertia weight is 0.5.

[0055] All particles are updated according to the above update process. Each time they are updated, each particle will reach a new position, and at the same time, the global optimal position and the individual optimal position of each particle are updated. To prevent the particle swarm optimization algorithm from falling into a local optimum and failing to obtain accurate optimal control parameters, during the process of iteratively updating particles using the particle swarm optimization algorithm, Gaussian noise is added to some particles to increase the diversity of the particles and avoid falling into local optimal points.

[0056] Specifically, the step of iteratively updating the multiple particles using the particle swarm optimization algorithm to obtain the optimal control parameters of the heat exchange unit includes: calculating the Euclidean distance between the global optimal positions in the current iteration and the previous iteration; in response to the Euclidean distance being less than a preset distance, calculating the position deviation value between the global optimal position and each particle position in the current iteration; using the ratio of the position deviation value to the fitness function value of each particle position as the selection probability, and selecting a preset number of particles to add Gaussian noise in descending order of the selection probability to obtain the positions of each particle in the next iteration; performing multiple iterations until the fitness function value of the global optimal position is greater than a preset fitness, or when the number of iterations is greater than a preset number, taking the global optimal position as the optimal control parameter of the heat exchange unit.

[0057] Among them, the larger the position deviation value, the greater the difference between the current particle's position and the global optimal position, the smaller the probability of it falling into a local optimal solution, and the greater the possibility of exploring a new optimal solution by adding Gaussian noise to the current particle; the smaller the fitness function value, the worse the control effect of the current particle. Adding Gaussian noise to the current particle can explore new optimal solutions while maintaining the current optimal solution, thereby ensuring that the particle swarm optimization algorithm can converge quickly and determine the optimal control parameters of the heat exchange unit; therefore, if a particle has a larger position deviation value and a smaller fitness function value, then the selection probability of this particle is greater.

[0058] Specifically, the selection probability of particle in the -th iteration is: , where is the position deviation value of particle in the -th iteration, is the fitness function value of particle in the -th iteration, and is a minimum value. Exemplarily, the value of the minimum value

[0059] In one embodiment, the optimal control parameters include the optimal flow rate, the optimal inlet temperature, and the optimal outlet temperature of the heat exchange medium. By adjusting the operating state of the heat exchange unit to the optimal flow rate, the optimal inlet temperature, and the optimal outlet temperature, the intelligent control of the heat exchange unit can be achieved.

[0060] In this way, the predicted load at a future moment is predicted based on the heat exchange load sequence within a preset time window, and a fitness function is constructed based on the predicted load, enabling the particle swarm optimization algorithm to take into account the influence of the actual working conditions of the heat exchange unit on the control parameters and obtain accurate control parameters.

[0061] According to the second aspect of the present application, the present application also provides a control system for a multi-frequency intelligent heat exchange unit. Figure 2 FIG. is a structural block diagram of a control system for a multi-frequency intelligent heat exchange unit according to an embodiment of the present application. As Figure 2 shown, the system 50 includes a processor and a memory. The memory stores computer program instructions, and when the computer program instructions are executed by the processor, the control method for a multi-frequency intelligent heat exchange unit according to the first aspect of the present application is implemented. The system also includes a communication bus and a communication interface, as well as other components well-known to those skilled in the art. Their settings and functions are known in the art, and thus will not be elaborated herein.

[0062] It should be noted that for those of ordinary skill in the art, without departing from the concept of the present application, several modifications and improvements can still be made, and these all fall within the protection scope of the present application.

Claims

1. A control method for a multi-frequency intelligent heat exchange unit, characterized in that The control method includes: Inputting a heat exchange load sequence within a preset time window into a time series prediction model to obtain at least one predicted load, where the heat exchange load is the product of the specific heat capacity, mass, and temperature adjustment amount of a target medium, and the temperature adjustment amount is the difference between the target temperature and the current temperature; Calculate the perturbation coefficient according to the gradient value of the predicted load, add random perturbations to the real-time control parameters according to the perturbation coefficient to obtain multiple particles, and the perturbation coefficient is positively correlated with the gradient value; the gradient value of the predicted load is the maximum value of the difference between the predicted load and the heat exchange load at the current moment; adding random perturbations to the real-time control parameters according to the perturbation coefficient includes: adding any control parameter and the corresponding perturbation value to obtain the perturbed control parameter, and the perturbation value is the product of the perturbation coefficient and a random number generated within a preset range; the perturbation coefficient satisfies the relational expression: , is the gradient value; Constructing a fitness function and calculating the Euclidean distance between the globally optimal positions in the current iteration and the previous iteration; in response to the Euclidean distance being less than a preset distance, calculating the position deviation values between the globally optimal position in the current iteration and each particle position; using the ratio of the position deviation value to the fitness function value of each particle position as the selection probability, and selecting a preset number of particles in descending order of the selection probability to add Gaussian noise to obtain the positions of each particle in the next iteration; performing multiple iterations until the fitness function value of the globally optimal position is greater than a preset fitness or the number of iterations is greater than a preset number, taking the globally optimal position as the optimal control parameters of the heat exchange unit; the control parameters include the flow rate, inlet temperature, and outlet temperature of the heat exchange medium, and the fitness function is negatively correlated with the energy consumption and the absolute value of the difference between the actual heat exchange amount and the average predicted load.

2. The control method of a multi-frequency intelligent heat exchange unit according to claim 1, characterized in that, The time series prediction model is an LSTM model or a GRU model; The training method of the time series prediction model includes: using a historical heat exchange load sequence as a training sample and using the historical heat exchange loads at multiple moments after the historical heat exchange load sequence as label values; inputting the training sample into the time series prediction model to obtain an output result, and calculating the mean square error loss value between the output result and the label values; using the gradient descent method to update the time series prediction model until the mean square error loss value is less than a preset loss and then stopping to complete the training of the time series prediction model.

3. The control method of a multi-frequency intelligent heat exchange unit according to claim 1, characterized in that, The fitness function Satisfies: , and are the actual heat exchange amount and the average predicted load respectively, is the energy consumption of the heat exchange unit.

4. The control method of a multi-frequency intelligent heat exchange unit according to claim 3, characterized in that The actual heat exchange amount is as follows: , and are the flow rate and specific heat capacity of the heat exchange medium respectively, and are the outlet temperature and inlet temperature of the heat exchange medium respectively.

5. The control method of a multi-frequency intelligent heat exchange unit according to claim 1, characterized in that, The selection probability of the particle in the th iteration is as follows: , is the position deviation value of the particle in the th iteration, is the fitness function value of the particle in the th iteration, and is the minimum value.

6. A control system for a multi-frequency intelligent heat exchange unit, characterized in that, It includes a processor and a memory, and the memory stores computer program instructions, which, when executed by the processor, implement a control method for a multi-frequency intelligent heat exchange unit according to any one of claims 1 to 5.

Images