Wind power system measurement and control method based on hardware-in-the-loop simulation

Abstract

The invention relates to the technical field of wind power system measurement and control, and discloses a wind power system measurement and control method based on hardware-in-loop simulation. A wind power system model is established, hardware-in-loop simulation is carried out on the established model, and a result is output and verified. According to the wind power system measurement and control method based on hardware-in-the-loop simulation, development of an HIL platform is described, low-cost hardware is used for developing the HIL platform, a special real-time platform is used, in consideration of hardware cost and the fact that programming is conducted through a high-level graphic language, LabVIEW is used for achieving the platform in a student development suite so that the platform can be copied more easily, and the development efficiency is improved. The wind power system and the controller thereof are embedded in different loops, the control system is allowed to be modified to verify different algorithms without modifying the model of the wind power system or adding new hardware, the nonlinear behavior of the model of the wind power system and the control algorithm thereof are embedded or embedded in different instances of the FPGA using a lookup table, and the method is simple and easy to implement. This allows the model to be solved with high latency using relatively limited hardware.

Description

Technical Field

[0001] The present invention relates to the technical field of wind power system measurement and control, and in particular to a wind power system measurement and control method based on hardware-in-the-loop simulation. Background Art

[0002] In order to mitigate climate change by 2030, renewable energy has been increasing in markets such as power generation, heating and cooling, and transportation. There has been a significant increase in renewable energy production from photovoltaic, wind, and hydroelectric systems. One of the most viable options for producing this energy is large wind systems; however, they also have disadvantages such as spatial availability, impact on utility networks, public acceptance, and transmission and distribution losses. An alternative to reduce some of these disadvantages is the use of small wind turbines. Although research on small wind turbines began in the 1970s, they are now widespread. They are considered a viable option for sustainable energy development in new buildings in urban environments, another reliable, sustainable, and economical technology.

[0003] Microgrid (μG) is an important factor in climate change mitigation. It is defined as a low-voltage distribution network in which renewable energy sources and loads are interconnected through power electronic converters (PECs). There are different types of μGs, depending on the type of current they handle and the mode of operation. However, direct current μG (DCμG) has advantages in terms of efficiency, size, investment and operating costs. On the other hand, although photovoltaic systems are the more common distributed energy resources in μGs, wind power generation has also been widely used. At the same time, photovoltaic, wind power, biomass energy and other ecological and environmentally friendly, simple to deploy, and highly developed are the fastest growing renewable resources in recent years. Wind speed is variable in nature; therefore, wind systems are designed to operate between lower and upper speed limits (called cut-in speed and cut-out speed). A control system is needed to manage these speed limits and extract the maximum amount of energy from the wind, making the wind system more efficient. Between the cut-in and cut-out speeds, the control system performs different tasks. If the system has not yet reached the maximum allowable power (partial load area), the controller will perform the maximum power point tracking (MPPT) task. On the other hand, if the system reaches its maximum power (full load area), the control increases the rotational resistance of the turbine, limiting its angular speed and the extracted power.

[0004] But the existing testbed also includes a HIL platform developed in NImyRIO that simulates a system consisting of a photovoltaic array, a boost converter, and a single-phase inverter. Although these alternatives have the advantage of low cost, the specific languages ​​used in these applications and the high technical level of reporting these implementations can make it difficult for non-expert researchers to replicate them. Summary of the invention

[0005] In view of the deficiencies in the prior art, the present invention provides a wind power system measurement and control method based on hardware-in-the-loop simulation, which has the advantages of cost saving and easier replication, and solves the above-mentioned technical problems.

[0006] To achieve the above object, the present invention provides the following technical solution: a wind power system measurement and control method based on hardware-in-the-loop simulation, comprising the following steps:

[0007] S1. Establish a wind system model;

[0008] S2, performing hardware-in-the-loop simulation on the established model;

[0009] S3. Output the results and verify them.

[0010] As a preferred technical solution of the present invention, the wind power system model established in step S1 includes wind power P ω , Mechanical power generated by wind power generation P m , Power coefficient C p (β,λ), optimal tip speed ratio λ opt , Mechanical torque T m , electromagnetic torque T e , dynamic equations during blade rotation and wind electrical model.

[0011] As a preferred technical solution of the present invention, the wind power P ω , Mechanical power generated by wind power generation P m , Power coefficient C p (β,λ), optimal tip speed ratio λ opt , Mechanical torque T m , electromagnetic torque T e The expressions of the dynamic equations during the rotation of the blades are as follows:

[0012] Wind power

[0013]

[0014] Among them, ρ represents the gas density of the wind, A represents the area generated when the wind turbine rotates, and v represents the wind speed;

[0015] Mechanical power generated by wind power

[0016] P m =Cp(β,λ)P ω

[0017] Among them, C p (β,λ) represents the power coefficient, β represents the angle of attack of the blade, λ represents the tip speed ratio, and the expression of the tip speed ratio λ is as follows:

[0018]

[0019] Among them, ω m represents the angular velocity of the blade during rotation, v represents the wind speed, and r represents the blade radius;

[0020] Power factor

[0021]

[0022] Among them, γ represents the dependency function;

[0023] Mechanical torque

[0024]

[0025] Among them, P m represents the mechanical power generated by wind power generation, ω m It represents the angular velocity of the blade during rotation, and π represents pi;

[0026] Electromagnetic torque

[0027]

[0028] Where, p represents the number of stages of the permanent magnet synchronous generator, φ m represents the constant magnetic flux of the rotor magnet, i q represents the component of the stator current in the q direction;

[0029] Dynamic equations of blade rotation

[0030]

[0031] Where J represents the combined inertia of the blades and rotor of the permanent magnet synchronous generator, Represents the derivative of the angular velocity of the blade rotation process with respect to time t.

[0032] As a preferred technical solution of the present invention, the wind power electrical model expression is as follows:

[0033]

[0034] Among them, ω e Represents the electrical circumferential frequency of the permanent magnet synchronous generator, R s Represents the stator phase resistance, L d and L q represents the orthogonal stator inductance, i d and i q They represent the components of the stator current in the d and q directions, respectively, and v dc Indicates the DC bus level, C dc Represents the DC side capacitance, the electrical circumferential frequency of the permanent magnet synchronous generator ωe The expression is as follows:

[0035]

[0036] Among them, ω m It represents the angular velocity of the blade rotation process, and p represents the number of stages of the permanent magnet synchronous generator.

[0037] As a preferred technical solution of the present invention, the hardware-in-the-loop simulation in step S2 is implemented on the NImyRIO1900 development board based on the HIL platform, and the development version adopts the Xilinx Zynq-7010 system, integrates a real-time processor and an FPGA with a clock frequency of 40 MHz, and discretizes the dynamic equations and wind electrical model during the blade rotation process using the backward Euler approximation, and combines nonlinear functions and nonlinear operations to form a new wind system model, and solves the nonlinear equations in the wind system model through a control algorithm. When performing hardware-in-the-loop simulation, the new wind system model and the control system are divided into two loops, the first loop includes the new wind system model and the function 1D-LUTv in the one-dimensional lookup table in the FPGA 2 And executed every 10μs, the second loop includes D-LUTω m *v, and three PI controllers using corresponding algorithm formulas, running every 25μs.

[0038] As a preferred technical solution of the present invention, the new wind power system model expression is as follows:

[0039]

[0040]

[0041] As a preferred technical solution of the present invention, the algorithm formula in the second loop is as follows:

[0042] u(k)=u(k-1)+Pe(ik)+(TI-P)e(k-1)

[0043] As a preferred technical solution of the present invention, the result output in step S3 is compared with the expected result of simulation in Matlab, and the experimental result is evaluated by the mean absolute error.

[0044] Compared with the prior art, the present invention provides a wind power system measurement and control method based on hardware-in-the-loop simulation, which has the following beneficial effects:

[0045] 1. The invention describes the development of a HIL platform as a research infrastructure in the field of wind system control algorithms. The HIL platform was developed using low-cost hardware and using a dedicated real-time platform compared to commercial HIL platforms. The platform was implemented in a student development kit using LabVIEW making it easier to replicate, taking into account the hardware costs and the fact that programming is done using a high-level graphical language. The wind system and its controller are embedded in different loops, allowing the control system to be modified to verify different algorithms without modifying the model of the wind system or adding new hardware. The nonlinear behavior of the wind system model and its control algorithm are embedded or embedded in different instances of the FPGA using lookup tables, allowing the model to be solved with high latency using relatively limited hardware. BRIEF DESCRIPTION OF THE DRAWINGS

[0046] Figure 1 A schematic diagram of a wind power system of the present invention;

[0047] Figure 2 It is a schematic diagram of the power coefficient curve of the present invention;

[0048] Figure 3 Schematic diagram of equivalent circuit of the present invention;

[0049] Figure 4 This is a schematic diagram of the wind power system model of the present invention;

[0050] Figure 5 is a schematic diagram of a power curve of a wind power system of the present invention;

[0051] Figure 6 It is a schematic diagram of the control algorithm curve of the present invention;

[0052] Figure 7 It is a schematic diagram of the HIL implementation of the wind power system of the present invention;

[0053] Figure 8 It is a schematic diagram of wind system parameters of the present invention;

[0054] Fig. 9 This is a schematic diagram of the wind power system output of the present invention;

[0055] Fig.10 It is a schematic diagram of the power and control algorithm curve of the present invention;

[0056] Fig.11 This is a schematic diagram of the power trajectory of the control algorithm of the present invention;

[0057] Fig.12 The present invention uses the ARMA algorithm to realize the gust schematic diagram;

[0058] Fig.13 It is a schematic diagram of the gust trajectory of the present invention;

[0059] Fig.14Schematic diagram of angular velocity and speed ratio of the present invention;

[0060] Fig.15 It is a flow chart of the present invention; DETAILED DESCRIPTION

[0061] The following will be combined with the drawings in the embodiments of the present invention to clearly and completely describe the technical solutions in the embodiments of the present invention. Obviously, the described embodiments are only part of the embodiments of the present invention, not all of the embodiments. Based on the embodiments of the present invention, all other embodiments obtained by ordinary technicians in this field without creative work are within the scope of protection of the present invention.

[0062] See also Figure 1-15 ,A wind system measurement and control method based on hardware-in-the-loop simulation includes the following steps:

[0063] S1. Establish a wind system model;

[0064] like Figure 1 The wind power system model established in step S1 includes the wind power P ω , Mechanical power generated by wind power generation P m , Power coefficient C p (β,λ), optimal tip speed ratio λ opt , Mechanical torque T m , electromagnetic torque T e , dynamic equations during blade rotation and wind electrical model.

[0065] As a preferred technical solution of the present invention, the wind power P ω , Mechanical power generated by wind power generation P m , Power coefficient C p (β,λ), optimal tip speed ratio λ opt , Mechanical torque T m , electromagnetic torque T e The expressions of the dynamic equations during the rotation of the blades are as follows:

[0066] Wind power

[0067]

[0068] Among them, ρ represents the gas density of the wind, A represents the area generated when the wind turbine rotates, and v represents the wind speed;

[0069] Mechanical power generated by wind power

[0070] P m =C p (β,λ)P ω

[0071] Among them, C p (β,λ) represents the power coefficient, β represents the angle of attack of the blade, λ represents the tip speed ratio, and the expression of the tip speed ratio λ is as follows:

[0072]

[0073] Among them, ω m represents the angular velocity of the blade during rotation, v represents the wind speed, and r represents the blade radius;

[0074] Power factor

[0075]

[0076] Among them, γ represents the dependency function;

[0077] The power coefficients for different angles of attack and speed ratios described by the above formula are as follows: Figure 2 As shown, when C p (β,λ) reaches the given maximum λ, then it becomes the optimal λ, that is, λ opt , and λ opt When the value is 8.1, C p The maximum value of (β,λ) is 0.48, β=0, so λ opt The speed ratio represents the maximum degree of conversion of wind power into mechanical power. This means that for a given wind speed, the angular velocity ω of the turbine opt In order to extract the maximum possible energy, this angular velocity can be determined by substituting

[0078]

[0079] Mechanical torque

[0080]

[0081] Among them, P m represents the mechanical power generated by wind power generation, ω m It represents the angular velocity of the blade during rotation, and π represents pi;

[0082] Electromagnetic torque

[0083]

[0084] Where, p represents the number of stages of the permanent magnet synchronous generator, φ m represents the constant magnetic flux of the rotor magnet, i q represents the component of the stator current in the q direction;

[0085] Dynamic equations of blade rotation

[0086]

[0087] Where J represents the combined inertia of the blades and rotor of the permanent magnet synchronous generator, Represents the derivative of the angular velocity of the blade rotation process with respect to time t;

[0088] See also Figure 3 , in order to define the component of the stator current in the q direction, the equivalent circuit of the electrical part of the wind power system is considered, including the generator and the three-phase rectifier R with resistive load connected on the DC side, connected in dq coordinates, and Kirchhoff's law is applied to Figure 3 The circuit in the wind power model is expressed as follows:

[0089]

[0090] Among them, ω e Represents the electrical circumferential frequency of the permanent magnet synchronous generator, R s Represents the stator phase resistance, L d and L q represents the orthogonal stator inductance, i d and i q They represent the components of the stator current in the d and q directions, respectively, and v dc Indicates the DC bus level, C dc Represents the DC side capacitance, the electrical circumferential frequency of the permanent magnet synchronous generator ω e The expression is as follows:

[0091]

[0092] Among them, ω m It represents the angular velocity of the blade rotation process, and p represents the number of stages of the permanent magnet synchronous generator.

[0093] Combination Figure 1-4 ,ω m * is the target angular velocity determined by the MPPT algorithm. One of the typical algorithms for controlling a wind system starts with measuring the wind speed. This variable determines the angular velocity, the angle of attack of the blades and / or the transmission ratio between the turbine and the generator, thus maximizing the power factor C p (β,λ), which ensures the extraction of the maximum possible mechanical power. However, in systems such as the one implemented in this study, it is not possible to control a fixed angle of attack or a transfer ratio. Only the angular velocity of the blades is controlled by regulating the electrical torque of the generator. Moreover, since the system is designed for low power generation (low wind), it operates in a limited range of mechanical power, blade angular velocity and wind speed. Once these limits are reached, the angular velocity acquires a value that regulates them, without necessarily extracting the maximum available power. Figure 5The operating curve of the wind system implemented in this study is shown. It has two cut-off wind speeds: a lower wind speed of 3.5 m / s, below which the generator cannot extract enough energy to transfer it to the DC side of the rectifier, and a higher wind speed of 17.5 m / s, above which the current required by the generator to keep the mechanical power at the limit exceeds its capacity. It is also observed that the curve within the power generation range is divided into two regions: one below the limit power, where the maximum available power is extracted and / or the maximum angular velocity of the blades is adjusted, and another region where the system adjusts the mechanical power to keep it at the limit value. For the system studied in this study, the maximum angular velocity of the blades is 48.69 rad / s and the maximum mechanical power allowed is 3500 W. In order to compensate for the limitations of the system and determine the angular velocity of the blades as a function of the wind, the above equations were solved for different operating conditions described by the generated curves and obtained Figure 6 The relationship is shown. This relationship is divided into five cases, identified as A to E. In cases A and E, the system is out of production, so it is disconnected. In case B, the angular speed of the blade maximizes the power factor and extracts the maximum possible mechanical power (ω opt =ω m * ), in case C the angular speed limit has been reached, so it remains unchanged. Finally, in case D, the mechanical power extracted at constant angular speed reaches the permissible limit, so the speed is reduced to maintain constant power output, ω m * For each value of v defined in the operating range, the control algorithm simplifies the measurement and reference value of the wind speed. This article focuses on how to implement the HIL platform using relatively low-cost, student-oriented hardware that is described in a high-level description language and has relatively limited hardware resources. Implementing well-known control strategies lays a solid foundation for HIL platform verification

[0094] S2, performing hardware-in-the-loop simulation on the established model;

[0095] The hardware-in-the-loop simulation in step S2 is implemented on the NImyRIO1900 development board based on the HIL platform, and the development version adopts the Xilinx Zynq-7010 system, which integrates a real-time processor and an FPGA with a clock frequency of 40 MHz. The dynamic equations and wind electrical model in the blade rotation process are discretized using the backward Euler approximation, and a new wind system model is formed by combining nonlinear functions and nonlinear operations. The nonlinear equations in the wind system model are solved through the control algorithm. When performing hardware-in-the-loop simulation, the new wind system model and the control system are divided into two loops. The first loop includes the new wind system model and the function 1D-LUTv in the one-dimensional lookup table in the FPGA. 2And executed every 10μs, the second loop includes D-LUTω m *v, and three PI controllers using corresponding algorithm formulas, running every 25μs

[0096] The new wind system model expression is as follows:

[0097]

[0098]

[0099] As a preferred technical solution of the present invention, the algorithm formula in the second loop is as follows:

[0100] u(k)=u(k-1)+Pe(k)+(TI-P)e(k-1)

[0101] S3. Output the results and verify them.

[0102] To verify the operation of the wind system, the results obtained from the hardware implementation were compared with those from the continuous model simulation in Matlab. The first experiment evaluated the response of the system to a stepped wind speed profile, starting from a value of 6 m / s, then transitioning to 17.5 m / s and finally returning to 8 m / s. The variables compared were the angular velocity of the blades, the currents in the q and d directions, and the DC voltage. These comparisons were Fig. 9 The accuracy of the platform in reproducing the behavior of these variables is measured by calculating the mean absolute error using the simulated values ​​as a reference. m 、i q 、i d and v dc The error corresponds to 0.7rad / s, 1.9A, 0.04A and 9.5V

[0103] Once the correct reproduction of the wind system behavior in hardware has been verified, testing is performed over the entire power generation range to verify the operation of the control algorithm. Fig.10 and 11 shown. Fig.10 a Comparison of the angular velocity of the proposed control algorithm and the angular velocity of the implemented system. The angular velocity satisfies Figure 6 , extracting maximum power when possible and keeping the system within its operating limits as power is increased. Fig.10 b shows the operating curves of the implemented system compared with the theoretical system. This figure shows that the controller can extract the maximum power in the entire operating range. Fig.11shows the trajectory followed by the system mechanical power. In this figure, the red dashed line marked MPPT selection represents the theoretical optimal trajectory, which reaches the maximum value of the extracted mechanical power function at any wind speed. The extracted mechanical function is represented by the gray dashed lines at different speeds. The shape of these curves is similar to Figure 3 C shown in p (β,λ) functions are similar. Fig.11 It can be seen that for each wind speed value, there is an angular velocity that reaches the maximum value of the extraction, and ideally, the wind system should extract the maximum power for each wind speed. However, the mechanical power that the turbine can support has reached a specific value. The red dashed line Pmax represents the limit power that can be extracted by the wind system under study. The green line MPPTt represents the theoretical trajectory of the system under study. The trajectory follows the MPPT selection line until the limiting angular velocity is reached, and then rises vertically while keeping wm constant and increasing Pm until the trajectory intersects the Pmax line. From this point on, wm begins to decrease, maintaining the trajectory on the Pmax line until the upper cut-off wind speed of the operating range is reached. Finally, the blue line MPPt represents the trajectory followed by the wind system implemented using the control algorithm. As shown Fig.11 As shown, the MPPTh trajectory matches well with the MPPTt trajectory; this means that the controller behaves as expected. Therefore, this means that the HIL platform is very useful for testing wind systems over the entire operating range. It is important to emphasize here that this controller is used to verify the usefulness of the platform, but not only this controller can be tested. Finally, in order to test the wind behavior of the system, an autoregressive moving average model was implemented. This model generates wind profiles with a normal distribution according to the specified mean and variance. The algorithm models the wind speed v(t) as the sum of two components according to the equation: the mean speed μw and the instantaneous turbulence vt(t), which is modeled as a first-order Gaussian noise excitation filter as shown in the formula. After filtering out the noise, the instantaneous turbulence is as shown in the formula, where st is the standard deviation of the wind speed and Jt is the ARMA time series model, as shown in the formula. The implementation block diagram of the algorithm is shown in Fig.12 shown.

[0104] The result output in step S3 is compared with the expected result of the simulation in Matlab, and the experimental result is evaluated by the mean absolute error;

[0105] The control algorithm was tested using the ARMA model with a wind profile of 8 m / s mean wind speed and 0.5 m / s standard deviation. The experiment lasted 5 minutes, during which 300 wind speed values ​​were generated, one per second. The purpose of this experiment was to verify the controller’s ability to maintain the turbine at ω m * The variables analyzed are the angular velocity of the blades and the speed ratio. Fig.13 is the wind speed used in the experiment, Fig.14 a Fig.14 b shows the angular velocity response, Fig.14 b shows the comparison with the optimal speed ratio.

[0106] Although embodiments of the present invention have been shown and described, it will be appreciated by those skilled in the art that various changes, modifications, substitutions and variations may be made to the embodiments without departing from the principles and spirit of the present invention, and that the scope of the present invention is defined by the appended claims and their equivalents.

Claims

1. A wind power system measurement and control method based on hardware-in-the-loop simulation, characterized in that: The following steps are involved: S1. Establish a wind system model; S2, performing hardware-in-the-loop simulation on the established model; S3. Output the results and verify them.

2. A wind power system measurement and control method based on hardware-in-the-loop simulation according to claim 1, characterized in that: The wind power system model established in step S1 includes wind power P ω , Mechanical power generated by wind power generation P m , Power coefficient C p (β, λ), optimal tip speed ratio λ opt , Mechanical torque T m , electromagnetic torque T e , dynamic equations during blade rotation and wind electrical model.

3. The wind power system measurement and control method based on hardware-in-the-loop simulation according to claim 2 is characterized in that: The wind power P ω , Mechanical power generated by wind power generation P m , Power coefficient C p (β, λ), optimal tip speed ratio λ opt , Mechanical torque T m , electromagnetic torque T e The expressions of the dynamic equations during the rotation of the blades are as follows: Wind power Among them, ρ represents the gas density of the wind, A represents the area generated when the wind turbine rotates, and v represents the wind speed; Mechanical power generated by wind power P m =C p (β,λ)P ω Among them, C p (β, λ) represents the power coefficient, β represents the angle of attack of the blade, λ represents the tip speed ratio, and the expression of the tip speed ratio λ is as follows: Among them, ω m represents the angular velocity of the blade during rotation, v represents the wind speed, and r represents the blade radius; Power factor Among them, γ represents the dependency function; Mechanical torque Among them, P m represents the mechanical power generated by wind power generation, ω m It represents the angular velocity of the blade during rotation, and π represents pi; Electromagnetic torque Where, p represents the number of stages of the permanent magnet synchronous generator, φ m represents the constant magnetic flux of the rotor magnet, i q represents the component of the stator current in the q direction; Dynamic equations of blade rotation Where J represents the combined inertia of the blades and rotor of the permanent magnet synchronous generator, Represents the derivative of the angular velocity of the blade rotation process with respect to time t.

4. The wind power system measurement and control method based on hardware-in-the-loop simulation according to claim 3 is characterized in that: The wind power electrical model expression is as follows: Among them, ω e Represents the electrical circumferential frequency of the permanent magnet synchronous generator, R s Represents the stator phase resistance, L d and L q represents the orthogonal stator inductance, i d and i q They represent the components of the stator current in the d and q directions, respectively, and v dc Indicates the DC bus level, C dc Represents the DC side capacitance, the electrical circumferential frequency of the permanent magnet synchronous generator ω e The expression is as follows: Among them, ω m It represents the angular velocity of the blade rotation process, and p represents the number of stages of the permanent magnet synchronous generator.

5. A wind power system measurement and control method based on hardware-in-the-loop simulation according to claim 4, characterized in that: The hardware-in-the-loop simulation in step S2 is implemented on the NImyRIO1900 development board based on the HIL platform, and the development board adopts the Xilinx Zynq-7010 system, integrates a real-time processor and an FPGA with a clock frequency of 40 MHz, and discretizes the dynamic equations and wind electrical model during the blade rotation process using the backward Euler approximation, and combines nonlinear functions and nonlinear operations to form a new wind system model, and solves the nonlinear equations in the wind system model through the control algorithm. When performing hardware-in-the-loop simulation, the new wind system model and the control system are divided into two loops, the first loop includes the new wind system model and the function 1D-LUTv in the one-dimensional lookup table in the FPGA 2 And executed every 10μs, the second loop includes D-LUTω m *v, and three PI controllers using corresponding algorithm formulas, running every 25μs.

6. A wind power system measurement and control method based on hardware-in-the-loop simulation according to claim 5, characterized in that: The new wind system model expression is as follows:

7. The wind power system measurement and control method based on hardware-in-the-loop simulation according to claim 5 is characterized in that: The algorithm formula in the second loop is as follows: u(k)=u(k-1)+Pe(k)+(TI-P)e(k-1).

8. The wind power system measurement and control method based on hardware-in-the-loop simulation according to claim 1 is characterized in that: The result output in step S3 is compared with the expected result of simulation in Matlab, and the experimental result is evaluated by the mean absolute error.

Images