A Method and System for Establishing a Thermal Model of Bifacial Photovoltaic Modules Based on Dynamic Boundary Enhancement

Abstract

This invention discloses a method and system for establishing a thermal model of bifacial photovoltaic modules based on dynamic boundary enhancement, belonging to the technical field of photovoltaic module thermal state sensing and operation assessment. The method collects data on bifacial irradiance, total horizontal irradiance, ambient temperature, wind speed, wind direction, output power, and module back-side temperature to construct a five-node lumped heat capacity network and an unsteady-state thermal balance equation. It establishes a dynamic asymmetric convection heat transfer model for the front and rear surfaces and a dynamic equivalent ambient radiation temperature memory model for the back side, using these as dynamic boundary quantities in the calculation. Sensitivity analysis and parameter correlation analysis are used to screen parameters for estimation, and based on an augmented state-space model, a dual-timescale recursive filtering method is employed to jointly estimate the rapid temperature change state of the five layers and its strongly correlated parameters. This method can improve the stability and accuracy of bifacial photovoltaic module operating temperature prediction and thermal state sensing under rooftop installation conditions.

Description

Technical Field

[0001] This invention belongs to the field of photovoltaic module thermal state sensing and operation evaluation technology, specifically involving a method and system for establishing a thermal model of a bifacial photovoltaic module based on dynamic boundary enhancement. Background Technology

[0002] During the operation of a photovoltaic (PV) system, module temperature is a crucial factor affecting PV power generation efficiency, the aging rate of encapsulation materials, output power stability, and operational safety. Increased module temperature not only leads to a decrease in PV cell conversion efficiency but may also accelerate the thermal aging of encapsulation materials and the formation of localized hot spots. Therefore, accurate modeling and dynamic estimation of PV module operating temperature are essential foundations for PV system operational status awareness, performance evaluation, and safe operation and maintenance.

[0003] Compared to traditional single-sided photovoltaic (PV) modules, bifacial PV modules are simultaneously irradiated from both the front and back sides, resulting in stronger time-varying and spatially asymmetric heat input. Furthermore, the convective heat transfer conditions and long-wave radiation environments on the front and back surfaces of bifacial modules differ significantly. Especially in rooftop installations, the thermal environment on the back side is further influenced by roof heat storage, installation gaps, rear ventilation channels, and localized airflow conditions, leading to a slow evolution and hysteretic response in the thermal process at the module's back boundary.

[0004] Currently, researchers have proposed various modeling methods for predicting the temperature and analyzing the thermal state of photovoltaic modules. Traditional empirical thermal models typically employ methods such as the Nominal Operating Temperature (NOCT) model, the Sandia empirical model, or linear regression models, directly estimating the module temperature using external meteorological parameters such as ambient temperature, irradiance, and wind speed. These methods are simple in structure and computationally inexpensive, suitable for rapid engineering assessments. However, due to the lack of description of the internal heat transfer process and boundary heat transfer mechanism of the module, they often struggle to guarantee prediction accuracy under complex operating conditions, especially failing to reflect the dynamic temperature changes caused by the difference in thermal environment between the front and back of bifacial modules. While physical mechanism models based on traditional heat balance equations improve the ability to describe thermal processes by introducing heat transfer theory, existing models typically treat the boundary heat transfer coefficient as a constant or a static empirical function of wind speed. This simplification fails to accurately reflect the asymmetric heat transfer characteristics of the front and back boundaries of bifacial photovoltaic modules and cannot effectively characterize the dynamic hysteresis effects caused by fluctuations in ambient wind speed and heat storage in the back space.

[0005] In recent years, data-driven methods have also been used for photovoltaic module temperature prediction. For example, machine learning or deep learning models are used to establish nonlinear mapping relationships between meteorological variables, operating power, and module temperature. These methods can achieve good fitting results under conditions of sufficient training data and stable operating conditions, but their predictive performance is highly dependent on the quality of historical data and the coverage of training samples. When the installation method, roofing material, rear ventilation conditions, or irradiance distribution change, the model is prone to a decline in generalization ability. At the same time, pure data-driven methods usually lack clear thermal mechanism constraints, making it difficult to explain the physical relationship between temperature changes and module structure, boundary heat transfer, and radiation environment. They are also not convenient for further outputting recursive calculation results of dynamic boundary quantities and thermal parameter estimation results with engineering significance.

[0006] To achieve dynamic updates of model parameters to reflect actual operating conditions, some studies have introduced online joint estimation methods based on state observers. However, this approach relies on appropriate parameterized modeling and matching with observational information. Directly incorporating a large number of unknown thermodynamic parameters into the online joint estimation can easily lead to problems such as enhanced parameter correlation, error compensation, and estimation drift under single-point temperature observation conditions. When the parameter dimension is high, it may also result in decreased filtering stability and insufficient model generalization ability. Furthermore, existing research has relatively insufficient consideration of the time-scale characteristics of different thermal parameters, failing to effectively distinguish the dynamic coupling relationship between slowly changing boundary parameters and rapidly changing thermal states. Summary of the Invention

[0007] To address the problem that existing bifacial photovoltaic (PV) module thermal models struggle to simultaneously achieve physical consistency, dynamic characterization of complex boundaries, and stable online estimation under limited observation conditions, this invention provides a method and system for establishing a bifacial PV module thermal model based on dynamic boundary enhancement. This invention establishes a five-node lumped heat capacity network along the module thickness direction. The front dynamic convection heat transfer coefficient, the back dynamic convection heat transfer coefficient, and the back dynamic equivalent ambient radiation temperature are introduced as dynamic boundary quantities into the unsteady-state thermal balance equation, forming a dynamically boundary-enhanced bifacial PV module thermal model. Based on sensitivity analysis and parameter correlation analysis, a set of parameters to be estimated is formed. Using an augmented state-space model and a dual-timescale recursive filtering method, the temperature states of the five layers and the parameters to be estimated are recursively estimated, thereby updating the dynamic boundary quantities and key parameters in the thermal model. Finally, the dynamically boundary-enhanced bifacial PV module thermal model is obtained, and the predicted operating temperature, parameter estimation results, and recursive calculation results of the dynamic boundary quantities are simultaneously obtained.

[0008] The specific technical solution of the present invention is as follows:

[0009] A method for establishing a thermal model of a bifacial photovoltaic module based on dynamic boundary enhancement includes the following steps:

[0010] Step 1: Collect observation data on the front irradiance, back irradiance, total horizontal irradiance, ambient temperature, wind speed, wind direction, output power, and back temperature of bifacial photovoltaic modules under roof installation conditions. Then, perform time alignment, anomaly removal, and sampling period unification on the collected data to obtain the model input dataset and observation dataset.

[0011] Step 2: Equivalently represent the bifacial photovoltaic module along its thickness direction as a five-node lumped heat capacity network consisting of upper glass, upper EVA, silicon cell, lower EVA and lower glass, and establish an unsteady-state thermal balance equation including bifacial net shortwave heat source, interlayer heat conduction, front and rear surface convection heat transfer and front and rear surface longwave radiation heat transfer.

[0012] Step 3: Construct a dynamic boundary enhancement model, which includes a front and rear surface dynamic asymmetric convection heat transfer model and a back dynamic equivalent environmental radiation temperature memory model; wherein, the front dynamic convection heat transfer coefficient, the back dynamic convection heat transfer coefficient, and the back dynamic equivalent environmental radiation temperature are used as dynamic boundary quantities in the calculation of the unsteady-state heat balance equation.

[0013] Step 4: To adapt to the online estimation constraints under the condition of a single back-side temperature measurement point, the back-side temperature prediction error of the bifacial photovoltaic module is first used as the response index to conduct sensitivity analysis on the candidate thermal parameters and obtain the influence intensity of the candidate thermal parameters on the back-side temperature prediction results; then, parameter correlation analysis is performed on the highly sensitive candidate parameters, and parameters whose correlation with the selected parameters exceeds the preset correlation threshold are eliminated to form a set of parameters to be estimated that are related to the five-layer temperature state and whose absolute value of the correlation coefficient between parameters is not higher than the preset correlation threshold.

[0014] Step 5: Construct an augmented state-space model consisting of five temperature states and the parameters to be estimated. The front dynamic convection heat transfer coefficient, the back dynamic convection heat transfer coefficient, and the back dynamic equivalent ambient radiation temperature are used as dynamic boundary quantities in the state transition function for calculation. The back temperature of the component is used as the observation. The five temperature states and the parameters to be estimated are estimated using a dual time-scale recursive filtering method.

[0015] Step 6: Output the predicted operating temperature of the bifacial photovoltaic module, the estimated parameters to be estimated, and the dynamic boundary quantities calculated recursively by the dynamic boundary enhancement model based on the recursive estimation results.

[0016] Preferably, in step 1, the model input dataset includes frontal irradiance, rear irradiance, total horizontal irradiance, ambient temperature, wind speed, wind direction, and output power; the observation dataset includes component rear temperature observation data. Time alignment is used to unify data collected by different sensors into the same time series; anomaly removal is used to remove data points with obvious sensor distortion, negative irradiance, temperatures exceeding reasonable ranges, or abnormal sampling times; unified sampling period processing is used to unify data with different sampling intervals into a preset sampling period.

[0017] Preferably, in step 2, the bifacial photovoltaic module is equivalent to a five-node lumped heat capacity network consisting of an upper glass layer, an upper EVA layer, a silicon cell, a lower EVA layer, and a lower glass layer along the thickness direction of the module. Bifacial irradiation absorption, interlayer heat conduction, front and rear surface convection heat transfer, and long-wave radiation heat transfer are considered respectively, thereby establishing the following unsteady-state energy balance relationship:

[0018]

[0019] The net shortwave heat source term for each node is defined as follows:

[0020]

[0021] In the formula, , , , and These are the temperatures of the upper glass layer, upper EVA layer, silicon solar cell, lower EVA layer, and lower glass layer, respectively. , , , and These are the heat capacities of the corresponding layers; , , and Equivalent thermal resistance between layers; The effective heat exchange area of ​​the component; and These are the convective heat transfer coefficients of the front and rear surfaces, respectively. and These are the effective emissivity of the front and rear surfaces, respectively. It is the Stefan-Boltzmann constant; Ambient air temperature; The equivalent sky radiation temperature of the front surface; The equivalent ambient radiation temperature of the back surface; For the first Net shortwave heat source term for each node; and These are the irradiance values ​​for the front and back sides of the component, respectively. and The first The equivalent net shortwave heat source coefficient of each node for frontal and back irradiation.

[0022] The net shortwave heat source term is used to comprehensively characterize the transmission, absorption, and reflection of bifacial incident irradiation in each layer, as well as the effective proportion of heat ultimately converted into electricity after deducting the output of the solar cells.

[0023] Preferably, in step 3, to characterize the asymmetry and dynamic response characteristics of the heat transfer process on the front and back surfaces of the bifacial photovoltaic module, dynamic asymmetric convection heat transfer models are constructed for the front and back boundaries, respectively. First, the effective wind speeds on the front and back surfaces are determined based on wind speed and direction, satisfying:

[0024]

[0025]

[0026] Then, based on the effective wind speed, the quasi-steady-state target convective heat transfer coefficients for the front and back sides are established respectively, satisfying:

[0027]

[0028]

[0029] Finally, the dynamic convective heat transfer coefficient actually involved in the heat balance calculation is obtained through a first-order hysteresis element, satisfying:

[0030]

[0031]

[0032] In the formula, For ambient wind speed; This is the measured wind direction angle; and These are the feature orientation angles for the front and back sides, respectively. and These are the wind direction correction factors for the front and back sides, respectively; and The effective wind speeds are for the front and back sides, respectively. and These are the reference convective heat transfer terms for the front and back sides, respectively; and These are the wind speed enhancement coefficients for the front and back sides, respectively. and These are the quasi-steady-state target convective heat transfer coefficients for the front and back sides, respectively. and These are the heat transfer hysteresis time constants at the front and back boundaries, respectively; and These are the dynamic convection heat transfer coefficients for the front and back sides, respectively.

[0033] The front dynamic convective heat transfer coefficient and the back dynamic convective heat transfer coefficient are respectively used as dynamic boundary quantities input into the five-node lumped heat capacity network model. The front boundary and the back boundary have independent effective wind speed calculation relationship, quasi-steady-state target heat transfer coefficient calculation relationship and first-order hysteresis time constant, respectively, to characterize the heat transfer establishment process under the action of the front incoming flow and the heat transfer establishment process under the influence of the installation gap, ventilation channel and local flow around the back.

[0034] Specifically, during discrete recursion, the dynamic convective heat transfer coefficient is updated according to the following relationship:

[0035]

[0036]

[0037] In the formula, The sampling period is and All are greater than 0. Through the above discrete update relationship, the dynamic convective heat transfer coefficient can be embedded as a dynamic boundary quantity into the five-node heat balance equation, and participate in the temperature state recursion as a boundary calculation quantity in the state transition function.

[0038] By setting separate parameter sets for the front and back sides, the rapid heat transfer process dominated by direct flow at the front side and the slow-response heat transfer process at the back side affected by the installation structure, ventilation channels, and local flow around the back side are described differently. Unlike the method of uniformly treating the convective heat transfer coefficients of the front and back surfaces as static functions, the method of this invention can describe the dynamic response differences of the heat transfer processes at the front and back boundaries.

[0039] Preferably, in step 3, unlike the approach of equating the back-side radiation environment with ambient temperature or a constant boundary, this method constructs a dynamic equivalent ambient radiation temperature memory model for the back side. This model is specifically designed to characterize the dynamic disturbance of the back-side local heat accumulation process to the radiation boundary under roof installation conditions. The quasi-steady-state target values ​​of the roof surface equivalent temperature and the back-side equivalent ambient radiation temperature satisfy the following:

[0040]

[0041]

[0042]

[0043] Then, the dynamic equivalent environmental radiation temperature on the back side is obtained through a first-order memory circuit, and the long-wave radiation heat transfer term on the back side is determined accordingly:

[0044]

[0045] In the formula, This refers to the total horizontal irradiance. The equivalent temperature of the roof surface; The roof thermal response coefficient; This represents the quasi-steady-state target value of the equivalent ambient radiation temperature on the back side; , and These are the weighting coefficients for ambient air, sky, and roof surface on the back radiation environment, respectively. The thermal environment memory time constant on the back side; The dynamic equivalent environmental radiation temperature on the back side.

[0046] Specifically, during discrete recursion, the dynamic equivalent ambient radiation temperature on the back side is updated according to the following relationship:

[0047]

[0048] In the formula, The time constant for the thermal environment memory on the back side, and The aforementioned dynamic equivalent ambient radiation temperature on the back side does not correspond to a new solid material layer inside the component. Instead, it serves as a low-dimensional dynamic boundary quantity that characterizes the slow evolution of the local thermal environment on the back side under roof installation conditions. This quantity is used to characterize the slow evolution of the local thermal environment on the back side under roof installation conditions.

[0049] Preferably, in step 4, to avoid parameter redundancy and error compensation caused by estimating too many thermal parameters simultaneously under single-point back-side temperature observation conditions, a combination of sensitivity analysis and parameter correlation analysis is used to screen the set of parameters to be estimated. First, the root mean square error of the back-side temperature prediction is used as the response index to calculate the basic effect of the candidate parameters, which characterizes the degree of influence of a single parameter perturbation on the model output:

[0050]

[0051]

[0052] In the formula, This is a candidate parameter vector; The number of sample points; For parameter vectors Next Predicted back surface temperature at each sampling time; For the first Measured back temperature at each sampling time; For the first The candidate parameter in the th... The fundamental effects on a Morris random trajectory.

[0053] Then, based on the mean absolute basic effect and standard deviation, highly sensitive parameters are identified to distinguish the overall influence of the parameters and their nonlinearity and interaction degree.

[0054]

[0055]

[0056] In the formula, For the first The average absolute basic effect of each candidate parameter; The perturbation step size is the parameter. For the first Standard deviation of the basic effects of each candidate parameter; The number of random trajectories.

[0057] Finally, strongly correlated parameters are eliminated using the parameter correlation coefficient to avoid parameter redundancy and error compensation during online joint estimation.

[0058]

[0059] In the formula, For the first The candidate parameter and the first The correlation coefficient between the candidate parameters; Let be the covariance of the two. and These are the standard deviations of the corresponding parameters.

[0060] Preferably, the candidate thermal parameter set includes one or more of the following: front and rear surface reference convective heat transfer term, front and rear surface convective wind speed enhancement coefficient, front and rear boundary heat transfer lag time constant, roof thermal response coefficient, rear thermal environment memory time constant, front and rear surface effective emissivity, and equivalent net shortwave heat source coefficient of each layer. The preset sensitivity threshold can be determined based on the ranking result, quantile, or preset proportion of the average absolute basic effect of the candidate parameters; the preset correlation threshold can be determined based on the upper limit of the absolute value of the parameter correlation coefficient. When a candidate parameter simultaneously satisfies that its average absolute basic effect is higher than the preset sensitivity threshold and its absolute value of the correlation coefficient with the selected parameters is not higher than the preset correlation threshold, it is included in the parameter set to be estimated; the parameter set to be estimated includes the rear reference convective heat transfer term, roof thermal response coefficient, and front convective wind speed enhancement coefficient.

[0061] Specifically, in this embodiment, the samples were selected based on the principle of high sensitivity and weak correlation. , and As the parameter to be estimated, where Used to characterize the back-side reference convective heat transfer term Used to characterize the roof thermal response coefficient Used to characterize the frontal convective wind speed enhancement coefficient.

[0062] Preferably, in step 5, to achieve coordinated updating of rapidly changing temperature states and slowly drifting boundary parameters under single-point back-side temperature observation conditions, an augmented state-space model is constructed, and a dual-time-scale unscented Kalman filter is used for recursive calculation to achieve joint recursive estimation of five-layer temperature states and slowly time-varying parameters. The dynamic boundary quantities are recursively calculated in the state transition function, where the five-layer temperature state vector, the parameter vector to be estimated, and the augmented state vector are respectively represented as:

[0063]

[0064]

[0065]

[0066] In the formula, For the first The rapidly changing state vector at each sampling time contains five layers of temperature states; For the first The vector of parameters to be estimated at each sampling time; This is the augmented state vector.

[0067] The augmented state-space model satisfies the following state transition and observation relationships:

[0068]

[0069]

[0070]

[0071] In the formula, It is a nonlinear state transition function obtained by discretizing the five-node heat balance equation; For fast-state process noise; The parameter to be estimated represents process noise; For backside temperature observation; The observation matrix; To observe noise.

[0072] Based on the augmented state-space model, the recursive relation of the dual-time-scale unscented Kalman filter satisfies:

[0073]

[0074]

[0075]

[0076]

[0077]

[0078]

[0079]

[0080]

[0081] In the formula, These are sampling points for unscented transformation; For augmented state posterior estimation; For the augmented state posterior covariance matrix; The scaling parameter for the unscented transform is given by... Sure, and For the unscented transform adjustment parameters, To augment the state dimension; Kalman gain; This is a priori estimation based on observation.

[0082] The recursive method of the dual-timescale unscented Kalman filter improves the stability and identifiability of the joint state-parameter estimation under single-point back-side temperature observation conditions by configuring process noise covariances of different magnitudes for the temperature state and the parameters to be estimated. This allows the temperature state to be updated on a fast timescale and the parameters to be estimated to be updated by drifting on a slow timescale. Specifically, the five-layer temperature state is updated using a larger process noise covariance, while the parameters to be estimated are updated by drifting slowly using a smaller process noise covariance. The front dynamic convective heat transfer coefficient, the back dynamic convective heat transfer coefficient, and the back dynamic equivalent environmental radiation temperature are recursively calculated by the dynamic boundary enhancement model and participate in the temperature state recursion as dynamic boundary quantities in the state transition function. The process noise covariance matrix corresponding to the five-layer temperature state... The value is greater than the process noise covariance matrix corresponding to the parameter to be estimated. This allows the temperature state to follow minute-level thermal dynamic changes, while the parameters to be estimated drift slowly within a small range in the form of random walks, thereby achieving coordinated recursion across two time scales.

[0083] A system for establishing a thermal model of a bifacial photovoltaic module based on dynamic boundary enhancement includes: a data processing module, a thermal model construction module, a joint estimation module, and a result output module;

[0084] The data processing module is used to collect observation data of the front irradiance, back irradiance, total horizontal irradiance, ambient temperature, wind speed, wind direction, output power and back temperature of the bifacial photovoltaic module, and to perform time alignment, anomaly removal and sampling period unification processing on the observation data.

[0085] The aforementioned thermal model construction module is used to establish a five-node lumped heat capacity network model along the thickness direction of the component, consisting of the upper glass, upper EVA, silicon solar cell, lower EVA, and lower glass. It establishes an unsteady-state thermal balance equation that takes into account the net short-wave heat source on both sides, interlayer heat conduction, convective heat transfer on the front and rear surfaces, and long-wave radiation heat transfer on the front and rear surfaces. It also constructs a dynamic asymmetric heat transfer model of the front and rear surfaces based on wind speed and direction, and constructs a dynamic equivalent environmental radiation temperature memory model for the back side for recursively calculating the dynamic equivalent environmental radiation temperature on the back side.

[0086] The joint estimation module is used to screen the set of parameters to be estimated using sensitivity analysis and parameter correlation analysis; based on the augmented state-space model consisting of five temperature states and the parameters to be estimated, the dual-time-scale unscented Kalman filter method is used to recursively estimate the temperature states of each layer and the parameters to be estimated, and the front dynamic convection heat transfer coefficient, the back dynamic convection heat transfer coefficient and the back dynamic equivalent environmental radiation temperature are calculated according to the dynamic boundary enhancement model.

[0087] The result output module is used to output the predicted operating temperature of the bifacial photovoltaic module, the estimated parameters to be estimated, and the front dynamic convection heat transfer coefficient, the back dynamic convection heat transfer coefficient, and the back dynamic equivalent ambient radiation temperature calculated by the dynamic boundary enhancement model.

[0088] Compared with the prior art, the present invention has the following advantages:

[0089] (1) The present invention converts the bifacial photovoltaic module into a five-node lumped heat capacity network along the thickness direction, and introduces a net short-wave heat source term that takes into account the electrical output deduction in the silicon cell node, which can describe the energy relationship between bifacial light incident, interlayer heat conduction and electrothermal conversion while keeping the model structure simple.

[0090] (2) The present invention constructs the front dynamic convection heat transfer coefficient and the back dynamic convection heat transfer coefficient respectively, and uses the two as dynamic boundary quantities to participate in the calculation of the five-node heat balance equation, which can describe the asymmetric heat transfer dynamics caused by the direct flow on the front and the back affected by the installation gap, ventilation channel and local flow around.

[0091] (3) The present invention constructs a dynamic equivalent environmental radiation temperature memory model on the back side, and dynamically corrects the long-wave radiation heat transfer term on the back side according to the total horizontal irradiance, roof thermal response and the radiation weights of the sky, air and roof, which can characterize the influence of roof heat storage and the slow evolution of the local thermal environment on the thermal boundary of the back side of the component.

[0092] (4) Under the condition of a single back temperature measurement point, the present invention forms a set of parameters to be estimated through sensitivity analysis and parameter correlation analysis, and constructs an augmented state space model so that the five-layer temperature state and the parameters to be estimated can be updated in a coordinated manner. At the same time, the boundary quantity recursive calculation results are obtained through the dynamic boundary enhancement model, thereby forming a bifacial photovoltaic module thermal model with updatable parameters and recursive boundaries, which improves the stability of bifacial photovoltaic module thermal state perception and operating temperature prediction under roof installation conditions. Attached Figure Description

[0093] To more clearly illustrate the technical solutions in the embodiments of this application, the accompanying drawings used in the description of the embodiments will be briefly introduced below. Obviously, the accompanying drawings described below are only some embodiments of this application. For those skilled in the art, other drawings can be obtained based on these drawings without creative effort.

[0094] Figure 1 This is a flowchart illustrating the overall process of an embodiment of the present invention.

[0095] Figure 2 This is a schematic diagram of a five-node thermal network coupled with a dynamic boundary according to an embodiment of the present invention.

[0096] Figure 3 This is a schematic diagram of dynamic boundary modeling in an embodiment of the present invention;

[0097] Figure 4 This is a flowchart of parameter selection and dual-timescale joint estimation in an embodiment of the present invention;

[0098] Figure 5 This is a schematic diagram of the system structure according to an embodiment of the present invention. Detailed Implementation

[0099] To gain a deeper understanding of this invention, we will provide a comprehensive and detailed description. However, this invention has various implementations and is not limited to the specific examples listed herein. These examples are presented to enhance a full understanding of the disclosure of this invention.

[0100] The specific embodiments of the present invention will be further described below with reference to the accompanying drawings.

[0101] like Figure 1 As shown, the method for establishing a thermal model of a bifacial photovoltaic module based on dynamic boundary enhancement according to the present invention includes the following steps:

[0102] Step 1: Collect and preprocess operational data of bifacial photovoltaic (PV) modules under rooftop installation conditions. The collected data includes front irradiance, back irradiance, total horizontal irradiance, ambient temperature, wind speed, wind direction, output power, and back-side temperature observation data of the bifacial PV modules. The collected data undergoes time alignment, anomaly removal, and sampling period unification. Time alignment unifies data from different sensors to the same time series; anomaly removal removes data points with significant sensor distortion, negative irradiance values, temperatures exceeding reasonable ranges, or abnormal sampling times; and sampling period unification unifies data from different sampling intervals to a preset sampling period.

[0103] Specifically, in this embodiment, the sampling period can be uniformly set to 1 minute. The model input dataset includes front irradiance, back irradiance, total horizontal irradiance, ambient temperature, wind speed, wind direction, and output power. The observation dataset includes the observed temperature of the back of the component.

[0104] Step 2: Construct a five-node lumped heat capacity network model.

[0105] like Figure 2 The diagram illustrates a five-node thermal network and dynamic boundary coupling structure according to an embodiment of the present invention. This embodiment focuses on a bifacial double-glass photovoltaic module installed on a roof. Considering its typical symmetrical double-glass encapsulation structure, the module is equivalently represented along its thickness as a five-node physical topology consisting of an upper glass layer, an upper EVA layer, silicon solar cells, a lower EVA layer, and a lower glass layer. This topology describes the interlayer heat transfer process under bifacial heating conditions. Adjacent layers are coupled through equivalent thermal conductivity and thermal resistance. The front and back sides exchange heat with the ambient air and corresponding radiation environment, respectively. Its unsteady-state thermal balance equation is as follows:

[0106]

[0107]

[0108] In the formula, , , , and These are the temperatures of the upper glass layer, upper EVA layer, silicon solar cell, lower EVA layer, and lower glass layer, respectively. , , , and These are the heat capacities of the corresponding layers; , , and Equivalent thermal resistance between layers; The effective heat exchange area of ​​the component; and These are the convective heat transfer coefficients of the front and rear surfaces, respectively. and These are the effective emissivity of the front and rear surfaces, respectively. It is the Stefan-Boltzmann constant; Ambient air temperature; The equivalent sky radiation temperature of the front surface; The equivalent ambient radiation temperature of the back surface; For the first Net shortwave heat source term for each node; and These are the irradiance values ​​for the front and back sides of the component, respectively. and The first The equivalent net shortwave heat source coefficient of each node for frontal and back irradiation.

[0109] Specifically, in this embodiment, the net shortwave heat source term is used to comprehensively characterize the transmission, absorption, and reflection of bifacial incident irradiation in each layer, as well as the effective proportion of heat ultimately converted after deducting the power generation output of the solar cells; the energy deduction caused by the power generation output of the solar cells can be included in the equivalent net shortwave heat source coefficient corresponding to the silicon solar cell node. Furthermore, the module output power can also be converted into an electrical energy output term per unit area and deducted from the absorbed irradiation heat source of the silicon solar cell node to maintain the energy balance between light absorption, electrical output, and thermal conversion.

[0110] Specifically, in this embodiment, the unsteady-state thermal balance equation is discretized according to the sampling period Δt to obtain the discrete update relationship of the temperature of each thermal node. Explicit Euler method can be used for discretization, i.e.:

[0111]

[0112] In the formula, Indicates the first The sampling time of the first sampling moment The temperature of each hot node. To improve computational stability, multiple internal integration steps can also be set within a single sampling period.

[0113] Step 3: Construct a dynamic asymmetric heat transfer model for the front and rear surfaces and a dynamic equivalent environmental radiation temperature memory model for the back surface.

[0114] like Figure 3 As shown, considering that the front boundary is directly affected by the external flow, while the rear boundary is more significantly affected by the ventilation channel, installation structure, and local thermal environment of the roof, this invention establishes the effective wind speed, quasi-steady-state target convective heat transfer coefficient, and their hysteresis response relationships for the front and rear boundaries respectively:

[0115]

[0116]

[0117]

[0118]

[0119]

[0120]

[0121] In the formula, and The dynamic convective heat transfer coefficient that actually participates in the heat balance calculation. and The heat transfer build-up rates at the front and back boundaries are determined separately. By setting different sets of front and back parameters, the distinctly different heat transfer response characteristics at the front and back boundaries can be characterized.

[0122] Specifically, in this embodiment, the dynamic convection heat transfer coefficient can be discretized and recursively derived according to the following relationship:

[0123]

[0124]

[0125] In the formula, The sampling period is and All are greater than 0.

[0126] To address the characteristics of memory and gradual change in the local thermal environment on the back side in rooftop installation scenarios, this invention constructs a dynamic equivalent environmental radiation temperature memory model on the back side:

[0127]

[0128]

[0129]

[0130]

[0131]

[0132] In the formula, Used to characterize the equivalent thermal state of a roof under irradiation excitation; This represents the dynamic equivalent ambient radiation temperature experienced at the rear boundary. This model allows the long-wave radiation boundary at the rear to be dynamically adjusted as the roof thermal environment changes slowly, rather than being solely determined by the instantaneous ambient temperature.

[0133] Specifically, in this embodiment, the dynamic equivalent ambient radiation temperature on the back side can be discretized and recursively calculated according to the following relationship:

[0134]

[0135] The back-side dynamic equivalent environmental radiation temperature does not correspond to a new solid material layer inside the component, but rather serves as a low-dimensional dynamic boundary quantity characterizing the slow evolution of the local thermal environment on the back side under roof installation conditions, participating in the calculation of the five-node thermal balance equation.

[0136] Step 4: Screen candidate thermal parameters.

[0137] like Figure 4 As shown, to avoid parameter redundancy and error compensation caused by simultaneously estimating too many parameters under single-point back-side temperature observation conditions, this invention first performs sensitivity analysis on the candidate thermal parameters. Firstly, the root mean square error of the back-side temperature prediction is used as the response index, and the basic effects of the candidate parameters are calculated:

[0138]

[0139]

[0140] Then, based on the mean absolute basic effect and standard deviation, highly sensitive parameters are identified to distinguish the overall influence of the parameters and their nonlinearity and interaction degree.

[0141]

[0142]

[0143] Finally, strongly correlated parameters are eliminated using the parameter correlation coefficient to avoid parameter redundancy and error compensation during online joint estimation.

[0144]

[0145] In the formula, For parameter vectors The root mean square error of the back-side temperature prediction; For the first The candidate parameter in the th... The fundamental effects on the Morris random trajectory; For the first The average absolute basic effect of each candidate parameter; For the first Standard deviation of the basic effects of each candidate parameter; For the first The candidate parameter and the first The correlation coefficients between the candidate parameters.

[0146] In this embodiment, the candidate thermal parameter set may include one or more of the following: front and rear surface reference convective heat transfer terms, front and rear surface convective wind speed enhancement coefficients, front and rear boundary heat transfer hysteresis time constants, roof thermal response coefficients, rear thermal environment memory time constants, front and rear surface effective emissivity, and equivalent net shortwave heat source coefficients for each layer. The preset sensitivity threshold may be determined based on the ranking results, quantiles, or preset proportions of the average absolute basic effects of the candidate parameters, and the preset correlation threshold may be determined based on the upper limit of the absolute value of the parameter correlation coefficient.

[0147] Specifically, in this embodiment, based on the principle of high sensitivity and weak correlation, the following were selected: , and As the parameter to be estimated, where, Used to characterize the back-side reference convective heat transfer term Used to characterize the roof thermal response coefficient Used to characterize the frontal convective wind speed enhancement coefficient.

[0148] Step 5: Construct an augmented state-space model and perform joint estimation using unscented Kalman filtering at two time scales.

[0149] After parameter selection, an augmented state-space model is constructed, and recursive estimation is performed using a dual-time-scale unscented Kalman filter.

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[0151]

[0152]

[0153]

[0154]

[0155]

[0156]

[0157] In the formula, Update the temperature status of the fifth layer; Update the parameters to be estimated. For the model input vector, For fast-state process noise, For the process noise of the parameter to be estimated, To observe noise.

[0158] Specifically, in this embodiment, the temperature observation on the back of the component corresponds to the temperature of the underlying glass node. Observation matrix Used to select from the augmented state vector Corresponding components, namely:

[0159]

[0160] The core recursive process of dual-timescale unscented Kalman filtering is as follows:

[0161]

[0162]

[0163]

[0164]

[0165]

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[0167]

[0168]

[0169]

[0170]

[0171]

[0172] In the formula, The scale parameter for the unscented transform can be obtained from... Determine α and For the unscented transform adjustment parameters, To augment the state dimension; and These are the mean weight and the covariance weight, respectively.

[0173] Specifically, in this embodiment, the process noise covariance matrix corresponding to the five temperature states is... The value is greater than the process noise covariance matrix corresponding to the parameter to be estimated. This allows the temperature state of the five layers to follow the minute-level thermal dynamic changes, while the parameters to be estimated drift slowly within a small range in the form of random walks, thereby achieving coordinated recursion across two time scales.

[0174] Step 6: Output the predicted operating temperature of the bifacial photovoltaic module, the estimated parameters to be estimated, and the recursive calculation results of the dynamic boundary quantities.

[0175] Specifically, based on the recursive estimation results, the system outputs the five-layer temperature status, the predicted temperature of the back side of the module, the estimated results of the parameters to be estimated, and the dynamic convective heat transfer coefficients of the front and back sides, and the dynamic equivalent ambient radiation temperature of the back side, calculated recursively by the dynamic boundary enhancement model. These results can be used for thermal state sensing, operating temperature prediction, and operating status assessment of bifacial photovoltaic modules.

[0176] To verify the temperature prediction effectiveness of the method of this invention, tests were conducted on a rooftop bifacial photovoltaic experimental platform. The test subject was a bifacial double-glass photovoltaic module, installed at a fixed tilt angle at a height of approximately 0.4m above the ground. The data acquisition system simultaneously collected operational data such as front irradiance, back irradiance, total horizontal irradiance, ambient temperature, wind speed, wind direction, module back temperature, and output power at 60-second sampling intervals. During the test, a fixed-parameter thermal model was used as a comparison method, and the measured value of the module back temperature was used as the evaluation benchmark. The root mean square error (RMSE), mean absolute error (MAE), and coefficient of determination (R²) were used to evaluate the prediction effectiveness.

[0177]

[0178] Test results show that the RMSE, MAE, and R² of the fixed-parameter model are 2.3540℃, 1.8742℃, and 0.9531, respectively. After adopting the dynamic boundary enhancement-based thermal model establishment method for bifacial photovoltaic modules described in this invention, the RMSE decreases to 1.0041℃, the MAE decreases to 0.7058℃, and the R² increases to 0.9934. This demonstrates that the method of this invention can effectively reduce the temperature prediction error caused by the fixed-parameter model and improve the accuracy of temperature prediction on the back side of the module and the stability of thermal state sensing under the condition of bifacial photovoltaic modules installed on rooftops.

[0179] like Figure 5 As shown, a system for establishing a thermal model of a bifacial photovoltaic module based on dynamic boundary enhancement is presented, including a data processing module, a thermal model construction module, a joint estimation module, and a result output module.

[0180] The data processing module is used to collect observation data on the front irradiance, back irradiance, total horizontal irradiance, ambient temperature, wind speed, wind direction, output power, and back temperature of the bifacial photovoltaic module, and to perform time alignment, anomaly removal, and sampling period unification processing on the observation data.

[0181] The thermal model construction module is used to establish a five-node lumped heat capacity network model along the thickness direction of the component, consisting of the upper glass, upper EVA, silicon solar cell, lower EVA and lower glass. It establishes an unsteady-state thermal balance equation that takes into account double-sided irradiation absorption, interlayer heat conduction, front and rear surface convection heat transfer and front and rear surface long-wave radiation heat transfer. It also constructs a dynamic asymmetric heat transfer model of the front and rear surfaces based on wind speed and direction, constructs a dynamic equivalent environmental radiation temperature memory model of the back side, and recursively calculates the dynamic equivalent environmental radiation temperature of the back side.

[0182] The joint estimation module is used to screen the set of parameters to be estimated using sensitivity analysis and parameter correlation analysis; based on the augmented state-space model consisting of five temperature states and the parameters to be estimated, the dual-time-scale unscented Kalman filter method is used to recursively estimate the temperature states of each layer and the parameters to be estimated, and the front dynamic convection heat transfer coefficient, the back dynamic convection heat transfer coefficient and the back dynamic equivalent environmental radiation temperature are calculated according to the dynamic boundary enhancement model.

[0183] The results output module is used to output the predicted operating temperature of the bifacial photovoltaic module, the estimated parameters to be estimated, and the front dynamic convection heat transfer coefficient, the back dynamic convection heat transfer coefficient, and the back dynamic equivalent ambient radiation temperature obtained by recursion calculation from the dynamic boundary enhancement model.

[0184] Contents not described in detail in this specification are prior art known to those skilled in the art. Although illustrative specific embodiments of the invention have been described above to facilitate understanding by those skilled in the art, it should be understood that the invention is not limited to the scope of the specific embodiments. Various modifications are readily apparent to those skilled in the art as long as they fall within the spirit and scope of the invention as defined and determined by the appended claims, and all inventions utilizing the concept of this invention are protected.

Claims

1. A method for establishing a thermal model of a bifacial photovoltaic module based on dynamic boundary enhancement, characterized in that: Includes the following steps: Step 1: Collect observation data on the front irradiance, back irradiance, total horizontal irradiance, ambient temperature, wind speed, wind direction, output power, and back temperature of bifacial photovoltaic modules under roof installation conditions. Then, perform time alignment, anomaly removal, and sampling period unification on the collected data to obtain the input dataset and observation dataset of the model. Step 2: To characterize the asymmetric heat exchange between the front and back of the bifacial photovoltaic module and the hysteresis effect of the thermal environment on the back side, the bifacial photovoltaic module is equivalent to a five-node lumped heat capacity network model along the thickness direction, consisting of the upper glass, upper EVA, silicon cell, lower EVA and lower glass. An unsteady-state thermal balance equation is established, including the net short-wave heat source on both sides, interlayer heat conduction, convective heat transfer between the front and back surfaces and long-wave radiation heat transfer between the front and back surfaces. Step 3: Construct a dynamic boundary enhancement model, which includes a dynamic asymmetric convection heat transfer model for the front and rear surfaces and a dynamic equivalent environmental radiation temperature memory model for the back surface. The dynamic convection heat transfer coefficient for the front surface, the dynamic convection heat transfer coefficient for the back surface, and the dynamic equivalent environmental radiation temperature for the back surface are used as dynamic boundary quantities in the calculation of the unsteady-state heat balance equation. Step 4: To adapt to the online estimation constraints under the condition of a single back-side temperature measurement point, the back-side temperature prediction error of the bifacial photovoltaic module is first used as the response index to conduct sensitivity analysis on the candidate thermal parameters and obtain the influence intensity of the candidate thermal parameters on the back-side temperature prediction results; then, parameter correlation analysis is performed on the highly sensitive candidate parameters, and parameters whose correlation with the selected parameters exceeds the preset correlation threshold are eliminated to form a set of parameters to be estimated that are strongly correlated with the rapid temperature change state of the five layers and whose absolute values ​​of the correlation coefficients between parameters are not higher than the preset correlation threshold. Step 5: Construct an augmented state-space model consisting of five layers of rapidly changing temperature states and their strongly correlated parameters. The front dynamic convective heat transfer coefficient, the back dynamic convective heat transfer coefficient, and the back dynamic equivalent ambient radiation temperature are used as dynamic boundary quantities in the state transition function for calculation. The back temperature of the component is used as the observation. The five layers of rapidly changing temperature states and the parameters to be estimated are estimated using a dual-time-scale recursive filtering method. Step 6: Output the predicted operating temperature of the bifacial photovoltaic module, the parameter estimation results, and the front dynamic convection heat transfer coefficient, the back dynamic convection heat transfer coefficient, and the back dynamic equivalent ambient radiation temperature obtained by recursion calculation from the dynamic boundary enhancement model, based on the recursive estimation results.

2. The method for establishing a thermal model of a bifacial photovoltaic module based on dynamic boundary enhancement according to claim 1, characterized in that: The unsteady thermal balance equation in step 2 is specifically: ； ； ； ； ； The net shortwave heat source term for each node is defined as follows: ； In the formula, , , , and These are the temperatures of the upper glass layer, upper EVA layer, silicon solar cell, lower EVA layer, and lower glass layer, respectively. , , , and These are the heat capacities of the corresponding layers; , , and Equivalent thermal resistance between layers; The effective heat exchange area of ​​the component; and These are the convective heat transfer coefficients of the front and rear surfaces, respectively. and These are the effective emissivity of the front and rear surfaces, respectively. It is the Stefan-Boltzmann constant; Ambient air temperature; The equivalent sky radiation temperature of the front surface; The equivalent ambient radiation temperature of the back surface; For the first Net shortwave heat source term for each node; and These are the irradiance values ​​for the front and back sides of the component, respectively. and The first The equivalent net shortwave heat source coefficient of each node for frontal and back irradiation; The net shortwave heat source term for each node is used to comprehensively characterize the transmission, absorption, and reflection of bifacial incident irradiance in each layer, as well as the effective proportion of heat ultimately converted after deducting the power output of the solar cell. The net shortwave heat source term of the silicon solar cell node includes the absorbed heat caused by the front and back irradiance, and deducts the electrical output term obtained by converting the module output power, so that the heat source input of the solar cell node satisfies the energy balance relationship between light absorption, electrical output and thermal conversion.

3. The method for establishing a thermal model of a bifacial photovoltaic module based on dynamic boundary enhancement according to claim 1, characterized in that: In step 3, the construction of the dynamic asymmetric convection heat transfer model of the front and rear surfaces is specifically as follows: First, determine the effective wind speeds on the front and rear surfaces based on wind speed and direction, satisfying the following: ； ； Then, based on the effective wind speeds of the front and rear surfaces, quasi-steady-state target convective heat transfer coefficients for the front and rear surfaces are established, respectively, satisfying: ； ； Finally, the dynamic convective heat transfer coefficient actually involved in the heat balance calculation is obtained through a first-order hysteresis element, satisfying: ； ； In the formula, and The effective wind speeds are for the front and back sides, respectively. For ambient wind speed; and These are the wind direction correction factors for the front and back sides, respectively; This is the measured wind direction angle; and These are the feature orientation angles for the front and back sides, respectively. and These are the quasi-steady-state target convective heat transfer coefficients for the front and back sides, respectively. and These are the reference convective heat transfer terms for the front and back sides, respectively; and These are the wind speed enhancement coefficients for the front and back sides, respectively. and These are the heat transfer hysteresis time constants at the front and back boundaries, respectively; and These are the dynamic convection heat transfer coefficients for the front and back sides, respectively.

4. The method for establishing a thermal model of a bifacial photovoltaic module based on dynamic boundary enhancement according to claim 1, characterized in that: In step 3, the back dynamic equivalent environmental radiation temperature memory model is used to characterize the dynamic influence of the back local heat accumulation process on the back long-wave radiation boundary under roof installation conditions. Among them, the quasi-steady-state target values ​​of the equivalent temperature of the roof surface and the equivalent ambient radiation temperature on the back side satisfy: ； ； ； Then, the dynamic equivalent ambient radiation temperature on the back side is obtained through a first-order memory circuit, and the long-wave radiation heat transfer term on the back side is determined accordingly: ； In the formula, The equivalent temperature of the roof surface; The roof thermal response coefficient; This refers to the total horizontal irradiance. This represents the quasi-steady-state target value of the equivalent ambient radiation temperature on the back side; , and These are the weighting coefficients for ambient air, sky, and roof surface on the back radiation environment, respectively. The thermal environment memory time constant on the back side; The dynamic equivalent ambient radiation temperature on the back side; The dynamic equivalent ambient radiation temperature on the back side does not correspond to the internal solid material layer of the component, but is introduced as a low-dimensional dynamic boundary quantity into the five-node lumped heat capacity network model to correct the long-wave radiation heat transfer term on the back side caused by the coupling of double-sided light incident and roof heat storage.

5. The method for establishing a thermal model of a bifacial photovoltaic module based on dynamic boundary enhancement according to claim 1, characterized in that: The specific steps in step 4 involve forming a set of parameters to be estimated that are strongly correlated with the rapid temperature change state of the five layers and whose absolute values ​​of correlation coefficients do not exceed a preset correlation threshold. First, using the root mean square error of the back-side temperature prediction as the response index, the basic effect of the candidate parameters is calculated to characterize the degree of influence of a single parameter perturbation on the model output: ； ； In the formula, This is a candidate parameter vector; The number of sample points; For parameter vectors Next Predicted back surface temperature at each sampling time; For the first Measured back temperature at each sampling time; Then, based on the mean absolute basic effect and standard deviation, highly sensitive parameters are identified, and the overall influence strength of the parameters and their nonlinearity and interaction degree are distinguished: ； ； In the formula, For the first The candidate parameter in the th... The fundamental effects on the Morris random trajectory; For the first The average absolute basic effect of each candidate parameter; For the first Standard deviation of the basic effects of each candidate parameter; The number of random trajectories; Finally, parameters with strong correlations are eliminated using the parameter correlation coefficient to avoid parameter redundancy and error compensation during online joint estimation. ； In the formula, For the first The candidate parameter and the first The correlation coefficient between the candidate parameters; Let be the covariance of the two. and These are the standard deviations of the corresponding parameters; When a candidate parameter simultaneously satisfies the following conditions: its average absolute basic effect is higher than a preset sensitivity threshold, and the absolute value of its correlation coefficient with the selected parameters is not higher than a preset correlation threshold, it is included in the set of parameters to be estimated.

6. The method for establishing a thermal model of a bifacial photovoltaic module based on dynamic boundary enhancement according to claim 1, characterized in that: In step 5, the augmented state-space model satisfies the following state transition and observation relationships: ； ； ； in, For the first The five-layer temperature rapid change state vector at each sampling time, containing the five temperature states, is represented as follows: ; For the first The parameter vector at each sampling time point contains the parameters to be estimated obtained through sensitivity analysis and parameter correlation analysis, denoted as: ; The augmented state vector is represented as: ; The nonlinear state transition function is obtained by discretizing the five-node heat balance equation. Noise in rapidly changing states; For parameter process noise; For backside temperature observation; The observation matrix; To observe noise; The front and rear surface dynamic asymmetric convection heat transfer model and the back dynamic equivalent environmental radiation temperature memory model are used as the nonlinear state transition function. The dynamic boundary calculation relationship is involved in the calculation; Based on the augmented state-space model described above, the recursive relation for the dual-time-scale unscented Kalman filter satisfies: ； ； ； ； In the formula, These are sampling points for unscented transformation; For augmented state posterior estimation; For the augmented state posterior covariance matrix; The scaling parameter for the unscented transformation; Kalman gain; For prior estimation of observations; The recursive method of the dual-timescale unscented Kalman filter improves the stability and identifiability of the joint state-parameter estimation under single-point back-side temperature observation conditions by configuring process noise covariance of different magnitudes for the temperature state and the parameter to be estimated, so that the temperature state is updated on a fast time scale and the parameter to be estimated is updated by drift on a slow time scale.

7. A bifacial photovoltaic module operating temperature prediction system based on dynamic boundary enhancement, characterized in that, include: The module includes a data processing module, a thermal model construction module, a joint estimation module, and a results output module. The data processing module is used to collect observation data of the front irradiance, back irradiance, total horizontal irradiance, ambient temperature, wind speed, wind direction, output power and back temperature of the bifacial photovoltaic module, and to perform time alignment, anomaly removal and sampling period unification processing on the observation data. The aforementioned thermal model construction module is used to establish a five-node lumped heat capacity network model along the thickness direction of the component, consisting of the upper glass, upper EVA, silicon solar cell, lower EVA, and lower glass. It establishes an unsteady-state thermal balance equation that takes into account the net short-wave heat source on both sides, interlayer heat conduction, convective heat transfer on the front and rear surfaces, and long-wave radiation heat transfer on the front and rear surfaces. It also constructs a dynamic asymmetric heat transfer model of the front and rear surfaces based on wind speed and direction, and constructs a dynamic equivalent environmental radiation temperature memory model for the back side for recursively calculating the dynamic equivalent environmental radiation temperature on the back side. The joint estimation module is used to screen the set of parameters to be estimated using sensitivity analysis and parameter correlation analysis; Based on the augmented state-space model consisting of five layers of rapidly changing temperature states and parameters to be estimated, a dual-time-scale unscented Kalman filter method is used for recursive estimation. The dynamic convective heat transfer coefficients at the front and back are calculated based on the dynamic boundary enhancement model, as well as the dynamic equivalent ambient radiation temperature at the back. The result output module is used to output the predicted operating temperature of the bifacial photovoltaic module, the parameter estimation results, and the front dynamic convection heat transfer coefficient, the back dynamic convection heat transfer coefficient, and the back dynamic equivalent ambient radiation temperature calculated by the dynamic boundary enhancement model.

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