Modeling of Selected Lighting Parameters of LED Panel

: Semiconductor light sources are currently the fastest growing and most energy e ﬃ cient group of light sources used in lighting technology. Their lighting parameters, such as luminous ﬂux, correlated color temperature and color rendering index depend on the value of the forward current, as well as the temperature of the junction. LED source manufacturers usually specify, in data sheets, the e ﬀ ect of junction temperature and forward current on the luminous ﬂux for individual light sources. The di ﬃ culty, however, is the correct determination of temperature and then lighting parameters, by simulation methods for multi-source lighting systems. Determining the junction temperature which a ﬀ ects lighting parameters is particulary important in the case of LED panels and luminaires, where thermally coupled LED sources shaping the output lighting parameters are in close proximity to each other. Additionally, other factors inﬂuencing the temperature distribution of sources, such as the design and geometry of the cooling system, the design of the printed circuit and thermal interface material used, should be considered. The article is a continuation of the publication in this journal where the inﬂuence of factors inﬂuencing the temperature distribution of the LED panel is presented. The purpose of the research in this article was to conﬁrm the possibility of using CFD (Computational Fluid Dynamics) software, as well as to determine the accuracy of the results obtained in the temperature analysis of the multi-source LED panel, and in determining the output lighting parameters of the LED panel based on it. In this article, based on previously published research, a LED panel model with a cooling system was made, and then the CFD software determined the junction temperature of all light sources. The determined temperature of the LED sources constituted the basis for determining the output lighting parameters of the panel: luminous ﬂux, color temperature and color rendering index. The simulation results were veriﬁed by real measurements on the constructed LED panel prototype. The LED panel temperature di ﬀ erence between the simulation results and the real results on the prototype did not exceed 5%. Moreover, the error of lighting parameters between the simulation results obtained and the results on the LED panel prototype in the worst case was 4.36%, which proves the validity and accuracy of simulation studies.


Introduction
The dynamic development of semiconductor light sources has made them dominant in lighting technology, displacing the previously used discharge light sources. Increasingly higher values of luminous efficiency mean that LEDs (light-emitting diode) are currently the most energy-efficient group of light sources [1][2][3]. LED light sources are also equipped with advanced lens optics, in order to group of light sources [1][2][3]. LED light sources are also equipped with advanced lens optics, in order to obtain the maximum amount of LED light transmitted by the optical system for the purpose of energy savings, savings in terms of energy consumption, and the cost of electricity bills [4,5]. Electrical light sources are responsible for an energy consumption of around 1/6 to 1/5 of the worldwide electricity production, and the use of semiconductor light sources reduces energy consumption and significantly reduces carbon dioxide emissions into the atmosphere [5,6].
In luminaires where a high luminous flux value is required, and therefore a high luminaire power-from several dozen to several hundred Watts (road, industrial or illumination lighting)-high-power LED sources are most often used. Individual sources are characterized by a power of several Watts, so to obtain the required luminous flux Φ, the sources are grouped close to one another, forming panels containing from several to several hundred LEDs [7][8][9][10][11][12][13].
The main limitations of an even faster development of semiconductor light sources are thermal problems and the high temperature of the semiconductor junction Tj. Only a small part of the power supplied to the LED sources is converted into luminous flux Φ, while the remaining part is lost in the form of heat [14,15]. According to most scientific publications, the optical efficiency of LED sources is 30%, which means that 70% of the power is heat loss [16][17][18][19][20][21]. In a small number of publications, the research results indicate that, for the latest LED sources, the above value of optical efficiency may be higher [22,23]. In addition, the increasingly higher power of LED sources and the small chip area result in high values of thermal density and obstruct effective heat abstraction to the environment. The mentioned thermal problems are more applicable to LED panels, where many sources located in close proximity to one another interact thermally, causing the rise of junction temperature Tj in individual sources.
To limit the junction temperature Tj, cooling systems ensuring effective heat abstraction are used. Among the solutions currently available on the market, passive systems, in which natural convection is the basic heat exchange mechanism, are most commonly used in LED sources [24][25][26]. The decrease of junction temperature Tj prolongs the lifetime of the sources and results in higher values of light efficiency. The basic lighting parameters, such as luminous flux Φ, color temperature CCT (correlated color temperature) and CRI (color rendering index), depend on the junction temperature Tj [14,27]. An example graph, illustrating the correlation between junction temperature, electrical and light parameters, is presented in Figure 1. The current-voltage characteristic of LED sources, which affects the lighting parameters of LED sources, strongly depends on the junction temperature. The change in forward voltage VF as a function of junction temperature Tj is described by the relationship [14]:  The current-voltage characteristic of LED sources, which affects the lighting parameters of LED sources, strongly depends on the junction temperature. The change in forward voltage V F as a function of junction temperature T j is described by the relationship [14]: where: T j -junction temperature, l-is the power factor in the temperature dependence, V G0 -nominal value of the bandgap voltage of the semiconductor material, m-constant (m = 1 for ideal diode), V T -thermal voltage (roughly 26 mV around room temperature 300 K). For single LED sources, the voltage V F change as a function of junction temperature T j is −1 to −3 mV/ • C, and is called the temperature sensitivity of the forward voltage.
A change in the junction temperature also changes the lighting parameters. The change of luminous flux Φ, as a function of junction temperature T j , according to photo-electro-thermal theory, is described by the relationship [28]: where: E 0 -the luminous efficacy at 25 • C, k e -negative coefficient representing the rate of reduction of the luminous efficacy with junction temperature, T j -junction temperature, T 0 -25 • C, P e -electrical power. As with luminous flux, a change in junction temperature affects colorimetric values, such as color temperature CCT and color rendering index CRI. The effect of temperature is more significant in the case of LED sources, in which white color is created using a source of blue and yellow phosphor. A change in the junction temperature value causes a change in the spectrum of the LED source, which results in a change in CCT and CRI. As the temperature of the junction increases, as the electric power increases, the influence of temperature on the energy gap begins to dominate, and the peak wavelength shifts linearly towards longer waves. The above relationship can be described as follows [28]: where: β peak_b -the referenced λ peak b peak wavelength of the blue spectrum at ambient temperature, a and b-positive coefficients dependent on the junction temperature, T j -junction temperature, P e -electrical power.
In the literature, the number of publications related to the effect of temperature on the lighting parameters of LED sources is relatively small, and the focus is mainly laid on studies of individual LED sources. There is a lack of literature in which thermal analysis of luminaires with complex geometry is presented, where the junction temperature of individual sources is not a simple relationship based on superposition. There is also a lack of literature positions in which the determined junction temperature is the basis for determining the lighting parameters of the multi-source LED panel, and not the single light source itself.
The literature [29,30] presents thermal studies on the impact of the heat sink design on the temperature distribution of high-power electronic systems, e.g., high-power LED sources and CPU. In [31,32], the temperature distribution of LED sources was examined depending on the distance and shape of the heat sink fins. In the article [23], in addition to the heat sink design, the effect of the printed circuit used was examined. The effect of forced and free convection on the temperature of LED sources was also studied. In [33], the performance of a conventional heat sink with fins were compared in relation to the heat sink, in which optimized fins with additional holes were used. In the literature [34], the effect of heat sink orientation on convective air flow and temperature distribution was examined. Various heat sink designs and different materials in the LED bulb design have been studied to increase the efficiency of natural convection and improve heat dissipation [35]. A simplified method of thermal analysis of LED sources is the use of compact thermal models (CTM), which usually consist of a very limited number of thermal capacitors and resistors, ensuring satisfactory simulation accuracy. Thermal tests can be carried out in this case using SPICE simulators [36,37]. Unfortunately, ladder models are not independent of boundary conditions, and can only be used at a specific LED operating point and cooling conditions. Currently, research is underway on complex compact multi-domain models that will take into account the interrelationships between multiple LEDs contained in a single luminaire, for example the Delphi4LED European project [38,39]. As of today, however, there is a small number of available research results on the formation of output lighting parameters obtained, on the basis of a complex three-dimensional thermal analysis of multi-source lighting systems.
The modeling method links the heat sink geometrical parameters, as well as the optical, electrical and thermal properties presented in the article [40]. However, the analytical model presented concerned simple heat sink geometry, where a single LED source was included in the analysis and only the luminous flux Φ was considered. The thermal analysis for a single LED source and the effect of temperature on the luminous flux Φ are presented in the article [41], where CFD (Computational Fluid Dynamics) Comsol software was used for the calculations.
The articles listed above mainly concern modeling of cooling systems for effective heat removal from a semiconductor junction. The number of publications in which the resulting temperature distribution with light parameters is associated is relatively small, and the available test results mainly concern individual LED sources.
This article is a continuation of research published in this journal regarding the influence of factors influencing the temperature distribution of the LED panel [23]. This paper deals with the results of the study on modeling the lighting parameters (luminous flux Φ, color temperature and color rendering index) for the LED panel developed on the basis of previous research results [23]. The effect of temperature on the above lighting parameters was determined experimentally for a single LED source on which the LED panel construction was based. The thermal power of the source P H used in the construction of the thermal model of LED panel was also determined. The determined thermal model of the panel and the temperature distributions obtained on its basis for three values of forward current I F provided the basis for determining the final lighting parameters of the tested LED panel. The simulation results were verified by real tests conducted on a prototype.

The Effect of Junction Temperature on the Lighting Parameters of a Single LED Source
The starting point for the studies related to modeling the lighting parameters of LED panel was to determine the effect of junction temperature T j on the lighting parameters of a single light source. One of the most efficient high-power LED sources available on the market was selected for our studies. The LED source was soldered to a printed circuit board with metal core (MCPCB) with dimensions of 25 × 25 × 1.6 mm.
The tests were carried out on a GL Optic measuring stand, which included an integrating sphere (GL Opti Sphere) with a diameter of 500 mm [42], a GL Spectis 6.0 spectrometer [43], a programmable DC power supply TDK-Lambda GENH300-2.5, and a Peltier module with a 5305 TECSource temperature controller, manufactured by Arroyo Instruments. The accuracy of the temperature measurement of the Peltier module was 0.4 • C, and was the sum of the accuracy of the temperature monitor and thermocouple [44].
The tested LED source, characteristic temperatures and measuring stand are presented in Figure 2. programmable DC power supply TDK-Lambda GENH300-2.5, and a Peltier module with a 5305 TECSource temperature controller, manufactured by Arroyo Instruments. The accuracy of the temperature measurement of the Peltier module was 0.4 °C, and was the sum of the accuracy of the temperature monitor and thermocouple [44]. The tested LED source, characteristic temperatures and measuring stand are presented in Figure 2. The semiconductor light source selected for testing was installed on the surface of a Peltier module (Figure 2b), on which temperature T p was set. The measurements were carried out for four temperatures T p : 25; 45; 65; 85 • C, and for three values of forward currents I F : 350; 700 and 1050 mA.
In order to transform the temperature of the Peltier module T p into the junction temperature T j , the thermal resistance of the MCPCB substrate and the thermal resistance between the junction and case of the tested LED source should be additionally considered, according to the equation: where: T j -junction temperature, T p -Peltier module temperature, Rth MCPCB -MCPCB thermal resistance, Rth j-c -LED source thermal resistance junction to case and P H -heat power. The value of thermal power P H for the LED source was determined experimentally and presented in Table 1. The value of resistance Rth j-c is discussed in Section 2.2 of this article and presented in Table 2, while the thermal resistance of the MCPCB substrate was 0.45 • C/W, and this constituted a negligible part of the total thermal resistance.  Figure 3 shows the changes in luminous flux Φ, CCT and CRI, as a function of Peltier module temperature T p and transformed junction temperature T j .
The increase of temperature T p set on the Peltier module, and the consequent increase of temperature of the junction T j caused the change of lighting parameters for all three tested forward currents I F . The increase of temperature by 60 • C caused a linear decrease of the luminous flux Φ value. The decrease was 9% for I F = 350 mA and 11% for I F = 700 and 1050 mA. The obtained test results (Figure 3b) indicate that the color temperature of CCT depends on the value of the forward current I F and the temperature. The increase of forward current I F caused an increase of color temperature CCT. At I F = 350 mA and T p = 25 • C, the CCT temperature was 4853 K, and at I F = 1050 mA, it increased to 4953 K. For the tested forward currents I F , the increase of temperature T p on the Peltier module from 25 to 85 • C resulted in the rise of color temperature CCT by about 70 K.
In the case of color rendering index CRI, no significant effect of forward current I F on its value was noticed. The temperature increase resulted in a linear increase of the CRI index, where a 60 • C increase caused an increase of CRI from 70.2 to 71.6, and for current I F = 350 mA, this value was slightly higher.
The graphs of the luminous flux Φ change and the color rendering index CRI as a function of junction temperature T j were approximated by the equations of a linear function, while the graph of correlated color temperature CCT was approximated by the equation of a quadratic function ( Figure 3). The determined equations will be used to determine the lighting parameters (Φ, CCT, CRI) of the modeled panel with many LED sources (Section 2.3). The semiconductor light source selected for testing was installed on the surface of a Peltier module (Figure 2b), on which temperature Tp was set. The measurements were carried out for four temperatures Tp: 25; 45; 65; 85 °C, and for three values of forward currents IF: 350; 700 and 1050 mA.
In order to transform the temperature of the Peltier module Tp into the junction temperature Tj, the thermal resistance of the MCPCB substrate and the thermal resistance between the junction and case of the tested LED source should be additionally considered, according to the equation: where: Tj-junction temperature, Tp-Peltier module temperature, RthMCPCB-MCPCB thermal resistance, Rthj-c-LED source thermal resistance junction to case and PH-heat power. The value of thermal power PH for the LED source was determined experimentally and presented in Table 1. The value of resistance Rthj-c is discussed in Section 2.2 of this article and presented in Table 2, while the thermal resistance of the MCPCB substrate was 0.45 °C/W, and this constituted a negligible part of the total thermal resistance. Figure 3 shows the changes in luminous flux Φ, CCT and CRI, as a function of Peltier module temperature Tp and transformed junction temperature Tj.
(a)  In the case of color rendering index CRI, no significant effect of forward current IF on its value was noticed. The temperature increase resulted in a linear increase of the CRI index, where a 60 °C increase caused an increase of CRI from 70.2 to 71.6, and for current IF = 350 mA, this value was slightly higher.
The graphs of the luminous flux Φ change and the color rendering index CRI as a function of junction temperature Tj were approximated by the equations of a linear function, while the graph of correlated color temperature CCT was approximated by the equation of a quadratic function ( Figure  3). The determined equations will be used to determine the lighting parameters (Φ, CCT, CRI) of the modeled panel with many LED sources (Section 2.3).  To model the LED panel lighting parameters, it is necessary to perform simulation tests on the impact of panel source temperature on lighting parameters. One of the main factors affecting the accuracy of the simulation is correctly determined optical efficiency and directly related to it thermal power of LED sources. Optical efficiency, often called extraction efficiency, is a key factor in influencing LED luminous efficiency. Photons generated in the p-n junction could not escape into free space completely. Extraction efficiency is lower, which is associated with several loss mechanisms such as absorption by the metal electrode, reabsorption in the substrate and and absorption by the bulk material, because of the total internal reflection. Various design solutions for LED sources are currently being developed to increase the efficiency of generating photons in the active region. The main methods that have been tried are as follows: resonant cavity, geometrically shaped chips, distributed Bragg reflector (DBR), patterned substrate and photonic crystal, surface roughening, and so on [45,46].
The extraction efficiency is defined as [45,46]: Finally, the optical efficiency η o , also known as radiant efficiency, is defined as: where: P o -optical power emitted into free space from an LED and P e -electrical power supplied to the LED. Photons that are not emitted into the free space are lost as heat. Heat power can be determined on the basis of the equation: Based on the above equations, the optical efficiency η o and heat power P H of the tested LED source were determined. The results are presented in Table 1. The main factor affecting the optical efficiency η o was the value of the forward current I F . The highest efficiency η o of approx. 48% was obtained for I F = 350 mA, while for I F = 1050 mA, the efficiency η o was 38%. The increase of the temperature by 60 • C resulted in the reduction of η o efficiency by approx. 2%.
The effect of temperature on the spectral distribution of the tested LED source is shown in Figure 4. An increase in the temperature value caused a decrease in the spectral distribution amplitude at a

Thermal Modeling of the LED Panel
An example of an analytical model of a multi-source LED panel is shown in Figure 5. On the common surface of the MCPCB printed circuit and the common heat sink, N LEDs are installed, with the same heat output of each source equal to PH. The total thermal resistance of the LED panel can be determined as the sum of individual thermal resistances [47]: The thermal resistance Rthdie and Rthdie-attach can be determined in a simplified manner, where one-dimensional thermal resistance is assumed for conduction in a flat wall. The equation takes the form [47]: where: g-thickness, k-thermal conductivity, A-surface of the layer. The MCPCB printed circuit is usually made of three material layers (aluminum, dielectric and copper). The thermal resistance RthMCPCB is determined for all layers together, taking into account the phenomenon of heat dissipation. In addition, the thermal resistance of RthTIM thermally conductive material should be considered together with a printed circuit, forming a four-layer structure.
The Palisoc equation for a four-layer system can be used to determine the thermal resistance in a multilayer system. However, this solution is very complex, requiring a large number of calculations. In addition, the problem is analyzed for an infinite plate, and the solution does not take into account the effect of the finite size of the substrate [48].

Thermal Modeling of the LED Panel
An example of an analytical model of a multi-source LED panel is shown in Figure 5.

Thermal Modeling of the LED Panel
An example of an analytical model of a multi-source LED panel is shown in Figure 5. On the common surface of the MCPCB printed circuit and the common heat sink, N LEDs are installed, with the same heat output of each source equal to PH. The total thermal resistance of the LED panel can be determined as the sum of individual thermal resistances [47]: The thermal resistance Rthdie and Rthdie-attach can be determined in a simplified manner, where one-dimensional thermal resistance is assumed for conduction in a flat wall. The equation takes the form [47]: where: g-thickness, k-thermal conductivity, A-surface of the layer. The MCPCB printed circuit is usually made of three material layers (aluminum, dielectric and copper). The thermal resistance RthMCPCB is determined for all layers together, taking into account the phenomenon of heat dissipation. In addition, the thermal resistance of RthTIM thermally conductive material should be considered together with a printed circuit, forming a four-layer structure.
The Palisoc equation for a four-layer system can be used to determine the thermal resistance in a multilayer system. However, this solution is very complex, requiring a large number of calculations. In addition, the problem is analyzed for an infinite plate, and the solution does not take into account the effect of the finite size of the substrate [48]. On the common surface of the MCPCB printed circuit and the common heat sink, N LEDs are installed, with the same heat output of each source equal to P H . The total thermal resistance of the LED panel can be determined as the sum of individual thermal resistances [47]: The thermal resistance Rth die and Rth die-attach can be determined in a simplified manner, where one-dimensional thermal resistance is assumed for conduction in a flat wall. The equation takes the form [47]: where: g-thickness, k-thermal conductivity, A-surface of the layer. The MCPCB printed circuit is usually made of three material layers (aluminum, dielectric and copper). The thermal resistance Rth MCPCB is determined for all layers together, taking into account the phenomenon of heat dissipation. In addition, the thermal resistance of Rth TIM thermally conductive material should be considered together with a printed circuit, forming a four-layer structure. The Palisoc equation for a four-layer system can be used to determine the thermal resistance in a multilayer system. However, this solution is very complex, requiring a large number of calculations. In addition, the problem is analyzed for an infinite plate, and the solution does not take into account the effect of the finite size of the substrate [48].
A Masana analytical solution can also be used in this case, which is not limited by the number of layers. However, it is associated with simplifications regarding the omission of the heat transfer coefficient h, and thermal couplings between the inner layers [49].
Determining the heat resistance of the heat sink N·Rth heat_sink is a significant difficulty, where the thermal resistance associated with the arrangement of LED sources should be taken into account. Muzychka's solution is used to determine it ( Figure 6): Energies 2020, 13, x FOR PEER REVIEW 10 of 22 A Masana analytical solution can also be used in this case, which is not limited by the number of layers. However, it is associated with simplifications regarding the omission of the heat transfer coefficient h, and thermal couplings between the inner layers [49].
Determining the heat resistance of the heat sink • ℎ _ is a significant difficulty, where the thermal resistance associated with the arrangement of LED sources should be taken into account. Muzychka's solution is used to determine it ( Figure 6): The temperature distribution on the surface of the heat sink is obtained using the superposition method. For N installed heat sources, the surface temperature distribution is represented by [50]: where: and is replaced by λ, δ and β, respectively. The analytical solutions presented are a generalized description of thermal processes and can only be used for the non-complex temperature analysis of LED sources. The above solutions contain significant simplifications, limiting the possibility of their use in complex structures, including the ability to define the simple geometry of the radiation system or heat sources (usually a rectangle or circle).
For a detailed temperature analysis of LED panels with a complex structure, software based on FEM (finite element method) or CFD (computational fluid mechanics) can be used. The above software enables the analysis of complex geometric constructions, taking into account all types of heat transfer, as well as a three-dimensional analysis of the temperature distribution.
In the next stage of research related to the modeling of LED panel lighting parameters, thermal simulation tests were carried out to determine the junction temperature of individual LED sources, which affects the lighting parameters, as presented in Section 2.1. The temperature distribution on the surface of the heat sink is obtained using the superposition method. For N installed heat sources, the surface temperature distribution is represented by [50]: A i mn = 16Q i cos(λ m x i ) sin 1 2 λ n c i cos(δ n y i ) sin 1 2 δ n d i abc i d i kβ mn λ m δ n ϕ(β mn ) (14) where: and ω is replaced by λ, δ and β, respectively. The analytical solutions presented are a generalized description of thermal processes and can only be used for the non-complex temperature analysis of LED sources. The above solutions contain significant simplifications, limiting the possibility of their use in complex structures, including the ability to define the simple geometry of the radiation system or heat sources (usually a rectangle or circle).
For a detailed temperature analysis of LED panels with a complex structure, software based on FEM (finite element method) or CFD (computational fluid mechanics) can be used. The above software enables the analysis of complex geometric constructions, taking into account all types of heat transfer, as well as a three-dimensional analysis of the temperature distribution.
In the next stage of research related to the modeling of LED panel lighting parameters, thermal simulation tests were carried out to determine the junction temperature of individual LED sources, which affects the lighting parameters, as presented in Section 2.1.
For detailed thermal tests, a panel with 12 LED sources was used. Its parameters were determined in Table 1. The sources were installed on MCPCB in two rows, each of which consisted of 6 sources. The size of the MCPCB printed circuit board was 146 × 44 mm. The heat sink of LED panel was made of an Al5052 aluminum alloy. Its base had dimensions of 183 × 80 mm with a thickness H BH = 4 mm. On the upper surface of the base, 13 fins of the height H f = 29 mm were formed. In addition, all fins, except the outer ones, have deformations in the form of cylinders (Figure 7). The geometry and shape of the heat sink were developed based on conducted and published research in this journal [23]. For detailed thermal tests, a panel with 12 LED sources was used. Its parameters were determined in Table 1. The sources were installed on MCPCB in two rows, each of which consisted of 6 sources. The size of the MCPCB printed circuit board was 146 × 44 mm. The heat sink of LED panel was made of an Al5052 aluminum alloy. Its base had dimensions of 183 × 80 mm with a thickness HBH = 4 mm. On the upper surface of the base, 13 fins of the height Hf = 29 mm were formed. In addition, all fins, except the outer ones, have deformations in the form of cylinders (Figure 7). The geometry and shape of the heat sink were developed based on conducted and published research in this journal [23]. The tests were implemented in Mentor Graphics FloEFD 16.2 software [51]. The selected tool has, among others, an embedded CAD module creating the analyzed 3D geometry, as well as a flow simulation module based on computational fluid mechanics, which performs advanced thermal calculations for all types of heat transport, i.e., conduction, convection, and radiation [52,53].
The parameters of the materials used in the simulation model and their physical characteristics are presented in Table 2.  The tests were implemented in Mentor Graphics FloEFD 16.2 software [51]. The selected tool has, among others, an embedded CAD module creating the analyzed 3D geometry, as well as a flow simulation module based on computational fluid mechanics, which performs advanced thermal calculations for all types of heat transport, i.e., conduction, convection, and radiation [52,53].
The parameters of the materials used in the simulation model and their physical characteristics are presented in Table 2.  (Table 1). To determine the junction temperature T j , the real value of the thermal resistance Rth j-c [55,56], described in detail and determined in [23,57], was used.
The size of the computational domain was selected based on the heat transfer characteristics and computation time. The domain length and width were set 2L p and 2W p , and the height of the domain was set as 7H p , where L p , W p and H p denote the length, width and height of the LED panel, respectively. Natural convection was used as the boundary conditions for simulation. The ambient temperature Ta was 24 • C and the air pressure surrounding the panel p = 101.325 kPa. The simulation calculations were conducted with regard to thermal radiation and the emissivity factor ε for the aluminum plate was 0.2. To define the calculation grid, we used an advanced module for adaptive meshing, which is built into the software FloEFD 16.2 software. Figure 8 shows the computational domain, while Figure 9 shows the temperature distribution of the LED panel at I F = 700 mA.  (Table 1). To determine the junction temperature Tj, the real value of the thermal resistance Rthj-c [55,56], described in detail and determined in [23,57], was used.
The size of the computational domain was selected based on the heat transfer characteristics and computation time. The domain length and width were set 2Lp and 2Wp, and the height of the domain was set as 7Hp, where Lp, Wp and Hp denote the length, width and height of the LED panel, respectively. Natural convection was used as the boundary conditions for simulation. The ambient temperature Ta was 24 °C and the air pressure surrounding the panel p = 101.325 kPa. The simulation calculations were conducted with regard to thermal radiation and the emissivity factor ε for the aluminum plate was 0.2. To define the calculation grid, we used an advanced module for adaptive meshing, which is built into the software FloEFD 16.2 software. Figure 8 shows the computational domain, while Figure 9 shows the temperature distribution of the LED panel at IF = 700 mA.  A very even temperature distribution on the heat sink surface was obtained. The temperature difference between the extreme and central fin of the heat sink did not exceed 1 °C. The difference between the temperature of the junction LED panel source and the temperature on the fins of the heat sink (Figure 9) is small and is about 13 °C, which proves effective heat dissipation.
The results of the temperature distribution of the LED panel were verified by measurements conducted on the prototype of the panel. The real measurement was made using a L200-TC Labfacility 8-channel laboratory temperature monitor [58]. Seven measuring channels were installed on the heat sink and MCPCB, with a temperature sensor in the form of a K-type thermocouple. The LED panel measuring points were located as follows: on the upper base in the central part of the heat sink (1), on the central fin of the heat sink (2), on the fin protrusion above the central LED source (3), on the fin protrusion above the outer LED source (4), on the lower base of heat sink (5), on MCPCB at a distance of 1 mm from the central LED source (6) and on MCPCB at a distance of 1 mm from the outer LED source (7) (Figure 9).
Temperature measurements at the designated measuring points were carried out for three values of forward current IF: 350; 700 and 1050 mA. During the measurements, the LED panel was suspended in a horizontal position, and heat exchange with the ambient environment took place by free convection. The ambient temperature during the measurements was the same as the simulation temperature and was Ta = 24 °C. Temperature values were automatically saved every 10 s. The measurement time was one hour, so that the temperature for all measuring channels could reach the steady value. Figure 10 shows the view of the LED panel prototype and the temperature curves of the measuring channels, for forward current IF = 700 mA. A very even temperature distribution on the heat sink surface was obtained. The temperature difference between the extreme and central fin of the heat sink did not exceed 1 • C. The difference between the temperature of the junction LED panel source and the temperature on the fins of the heat sink ( Figure 9) is small and is about 13 • C, which proves effective heat dissipation.
The results of the temperature distribution of the LED panel were verified by measurements conducted on the prototype of the panel. The real measurement was made using a L200-TC Labfacility 8-channel laboratory temperature monitor [58]. Seven measuring channels were installed on the heat sink and MCPCB, with a temperature sensor in the form of a K-type thermocouple. The LED panel measuring points were located as follows: on the upper base in the central part of the heat sink (1), on the central fin of the heat sink (2), on the fin protrusion above the central LED source (3), on the fin protrusion above the outer LED source (4), on the lower base of heat sink (5), on MCPCB at a distance of 1 mm from the central LED source (6) and on MCPCB at a distance of 1 mm from the outer LED source (7) (Figure 9).
Temperature measurements at the designated measuring points were carried out for three values of forward current I F : 350; 700 and 1050 mA. During the measurements, the LED panel was suspended in a horizontal position, and heat exchange with the ambient environment took place by free convection. The ambient temperature during the measurements was the same as the simulation temperature and was Ta = 24 • C. Temperature values were automatically saved every 10 s. The measurement time was one hour, so that the temperature for all measuring channels could reach the steady value. Figure 10 shows the view of the LED panel prototype and the temperature curves of the measuring channels, for forward current I F = 700 mA. The fastest temperature increase was observed in the first ten minutes. After a time of approx. 45 min., the temperature reached a steady state. Even temperature distribution was recorded on the heat sink surface. The temperature values recorded at the measuring points did not show large differences, only at measuring point No. 4, located on the outer fin of the heat sink, was the temperature slightly lower. The highest temperature value was obtained for measuring points No. 6 and No. 7 located on the MCPCB, in close proximity to semiconductor light sources.
The results of simulations and real measurements of temperature for all three values of IF forward current are presented in Table 3. The real results refer to temperature measurement after 60 min when the LED panel has reached a steady state. The error of simulation results was determined as represented below: A high compliance of simulations and real results was obtained. The largest error was recorded at the sixth and seventh measuring point, located directly at the semiconductor light sources. Its value did not exceed 5% for all three tested values of forward current IF. A bigger difference between simulation and real results obtained for the measuring points can be attributed to a relatively larger temperature gradient in close proximity of LED sources. The fastest temperature increase was observed in the first ten minutes. After a time of approx. 45 min., the temperature reached a steady state. Even temperature distribution was recorded on the heat sink surface. The temperature values recorded at the measuring points did not show large differences, only at measuring point No. 4, located on the outer fin of the heat sink, was the temperature slightly lower. The highest temperature value was obtained for measuring points No. 6 and No. 7 located on the MCPCB, in close proximity to semiconductor light sources.
The results of simulations and real measurements of temperature for all three values of I F forward current are presented in Table 3. The real results refer to temperature measurement after 60 min when the LED panel has reached a steady state. The error of simulation results was determined as represented below: A high compliance of simulations and real results was obtained. The largest error was recorded at the sixth and seventh measuring point, located directly at the semiconductor light sources. Its value did not exceed 5% for all three tested values of forward current I F . A bigger difference between simulation and real results obtained for the measuring points can be attributed to a relatively larger temperature gradient in close proximity of LED sources. The presented thermal model of the LED panel and high compliance of the simulation and real results is the basis for modeling the panel's luminous parameters, which depend on the temperature of individual sources.

Modeling of LED Panel Luminous Parameters
The influence of temperature on the lighting parameters of a single LED source and a reliable thermal analysis of the panel presented in the preceding part of the paper provide the basis for determining the lighting parameters of the LED panel.
For the LED panel studied, the junction temperature T j of individual sources was simulated for three values of forward current I F : 350, 700 and 1050 mA. An example of the panel temperature distribution and the junction temperature T j for I F = 700 mA is shown in Figure 11.
Energies 2020, 13, x FOR PEER REVIEW 15 of 22 The presented thermal model of the LED panel and high compliance of the simulation and real results is the basis for modeling the panel's luminous parameters, which depend on the temperature of individual sources.

Modeling of LED Panel Luminous Parameters
The influence of temperature on the lighting parameters of a single LED source and a reliable thermal analysis of the panel presented in the preceding part of the paper provide the basis for determining the lighting parameters of the LED panel.
For the LED panel studied, the junction temperature Tj of individual sources was simulated for three values of forward current IF: 350, 700 and 1050 mA. An example of the panel temperature distribution and the junction temperature Tj for IF = 700 mA is shown in Figure 11. Luminous flux Φ, color temperature CCT and color rendering index CRI of individual LED panel sources were determined, based on the simulation of the source junction temperature and the approximation equations specified in Section 2.1 ( Figure 3).
As an example, the following is the calculation for the LED1 source for IF = 700 mA: The output lighting parameters of the LED panel were determined based on:  Luminous flux Φ, color temperature CCT and color rendering index CRI of individual LED panel sources were determined, based on the simulation of the source junction temperature and the approximation equations specified in Section 2.1 ( Figure 3).
As an example, the following is the calculation for the LED1 source for I F = 700 mA: The output lighting parameters of the LED panel were determined based on: CCT panel_LED = CCT LED1 + CCT LED2 + . . . + CCT LED12 number o f LED panel sources (20)  where: Φ panel_LED -luminous flux of panel LED, Φ LED -luminous flux of LED source, CCT panel_LED -correlated color temperature of panel LED, CCT LED -correlated color temperature of LED source, CRI panel_LED -color rendering index of panel LED, CRI LED -color rendering index of LED source. Table 4 presents the calculated lighting parameters of the analyzed LED panel for all three values of I F forward current. For all three considered values of the forward current, the uniform temperature distribution and the approximate value of the junction temperature affecting the lighting parameters of the LED panel were obtained. For 350 mA, the temperature was about 41 • C, for 700 mA about 58 • C, and for 1050 mA about 79 • C. An even temperature distribution proves the correct heat sink design and effective heat dissipation. A similar junction temperature of all panel sources influenced similar values of lighting parameters of individual light sources. In this case, it is possible to estimate the output light parameters of the panel based on the information contained in the data sheet of light sources. Manufacturers usually only declare the effect of forward current and junction temperature on the luminous flux, however, there is no information regarding the color temperature and color rendering index. For panels containing a larger number of light sources, the temperature difference between the extreme and central light sources may be much larger, also resulting in an uneven distribution of light parameters. The uneven temperature distribution of LED sources can also be affected by incorrect design of the cooling system. In this case, determining the output light parameters based on catalog data may be subject to a significant error. The determination of the correct lighting parameters with an acceptable error should be supported by a complex thermal analysis using, e.g., CFD software.
To verify the simulation values of lighting parameters, real measurements were carried out on the prototype ( Figure 10). The tests were conducted on a measuring stand, which included an integrating sphere with a diameter of 2 m (Figure 12a) and a Konica Minolta CS-2000 spectroradiometer [59] ( Figure 12b). The relative uncertainty of measurement did not exceed 4%. Measurements of lighting parameters were made after 60 min, when the LED panel reached a thermal steady state. Table 5 summarizes the results obtained in real measurements and the results obtained from simulation tests. parameters based on catalog data may be subject to a significant error. The determination of the correct lighting parameters with an acceptable error should be supported by a complex thermal analysis using, e.g., CFD software. To verify the simulation values of lighting parameters, real measurements were carried out on the prototype (Figure 10). The tests were conducted on a measuring stand, which included an integrating sphere with a diameter of 2 m (Figure 12a) and a Konica Minolta CS-2000 spectroradiometer [59] (Figure 12b). The relative uncertainty of measurement did not exceed 4%. Measurements of lighting parameters were made after 60 min, when the LED panel reached a thermal steady state. Table 5 summarizes the results obtained in real measurements and the results obtained from simulation tests.    The obtained results (luminous flux Φ, color temperature CCT, color rendering index CRI) show the high compliance of simulation results with experimental tests-the error value did not exceed 4.4%. The smallest error was determined for I F = 350 mA, with the largest for I F = 1050 mA.

Conclusions
The junction temperature T j of semiconductor light sources significantly affects the lifetime of the sources, as well as their electrical and lighting parameters. Real tests carried out for a single LED source showed the influence of T j on the luminous flux Φ, color temperature CCT and color rendering index CRI. The increase of temperature by 60 • C resulted in the reduction of luminous flux Φ by approx. 10%, and the shift of color temperature CCT towards the cooler direction by 70 K. The temperature increase also improved the value of color rendering index CRI by about 1.5. A high value of optical efficiency η o ranging from 37% to 50%, depending on the value of forward current I F , was obtained for the tested LED source, which proves that the optical efficiency η o of latest commercial LED light sources can be higher than 30%-often reported in the literature.
It is difficult to determine the output light parameters of the LED panel, as their value depends on the junction temperature T j of many sources thermally coupled on a common heat sink. The influence of junction temperature and forward current on the lighting parameters of individual sources declared by manufacturers can be used to determine approximate panel values in simple systems and refer mainly to the luminous flux. In the case of complex panels with many LED sources, as well as to determine additional lighting parameters of the panel, it is necessary to carry out a complex thermal analysis using, e.g., CFD software.
The simulation results of the LED panel temperature distribution presented in the paper showed high compliance with the real results, and the error did not exceed 5%. The compliance was achieved by determining the real thermal power P H for the tested LED source, and taking into account the real thermal resistance of the Rth j-c determined in the calculations. Thermal working conditions of LED panel sources are shaped by many factors, including the number of LEDs, their arrangement, the design of the printed circuit, the design of the heat sink, the thermal interface material used or the type of heat exchange with the ambient environment. The presented thermal model allows for their analysis and can be used to optimize the design of the panel or lighting fixture with LED sources. As a result, the lowest T j temperature can be obtained. The presented thermal model, together with the results of the temperature distribution of the LED panel, provided the basis for determining the initial lighting parameters of the panel. The determined T j temperature of individual sources and the results of tests for a single LED source enabled the determination of the luminous flux Φ, color temperature CCT and color rendering index CRI for the LED panel. The simulation results were confirmed by real tests on the prototype, and the error did not exceed 4.4%. The results presented in the paper confirm the validity of computational tools used to design and evaluate the thermal operating conditions of semiconductor light sources. The use of correct model parameters, including, among others, heat power P H or thermal resistance Rth j-c , makes it possible to obtain reliable simulation results related to shaping the distribution of both temperature and output parameters of the LED panel.
Author Contributions: The measurements of LED source parameters were made by H.W. The results of simulation studies presented in this manuscript and their assessment was carried out by K.B. Finally, A.R. and S.R. supervised the research and prepared the manuscript. All authors have read and agreed to the published version of the manuscript.
Funding: This research received no external funding.