Parameter Extraction for a SPICE-like Delphi4LED Multi-Domain Chip-Level LED Model with an Improved Nelder–Mead Method

Abstract

In this paper, a novel method is presented to estimate the parameters of the SPICE-like multi-domain model of light-emitting diode (LED) chips developed and proposed by the Delphi4LED project. The proposed estimation algorithm employs a modified Nelder-Mead method, as the gradient methods and the original version of Nelder-Mead fail to properly handle this problem. By using the new, modified Nelder-Mead method presented in this paper the parameters are estimated faster, compared to the previously used brute-force algorithm-based parameter extraction process, allowing the same precision of the SPICE-like multi-domain LED model. The modification of the parameter extraction procedure also allows speeding up and simplifying the multi-domain LED characterization method proposed earlier by the Delphi4LED project. The speed and robustness of the new model eliminate the need for time-consuming junction temperature control during measurements by employing a novel extraction strategy that seeks the global minimum, rather than relying on the composition of marginal minima.

1. Introduction

In recent decades, LED devices have taken over other light sources in almost every application, such as automotive headlights, interior lighting, streetlamps, as well as sports lighting. The main reason for this rapid change in the use of light sources is that the efficiency/efficacy of LEDs and LED-based luminaires have surpassed those of highly efficient gas discharge lamps, such as high-pressure sodium lamps. Besides the higher efficiency, LEDs offer higher reliability and longer lifetimes. Additionally, they allow for easy electronic control, including dimming, built-in diagnostics, and health-monitoring options. Unlike the classical, phased-out light sources, the operation of LEDs is strongly temperature dependent with both electrical and light-output characteristics influenced by junction temperature, as is typical in semiconductor technology. The thermal-aware design of LED-based lighting solutions is still a challenge that needs novel solutions [1,2,3]. The design of LED packages and luminaires must consider electrical [4,5], optical [6,7], and thermal [8,9] aspects.

Prior to the LED era, luminaire design did not require a close cooperation between the electrical, optical, and thermal development teams. These different design aspects were considered independently, as cross-domain effects were insignificant. However, when using LEDs as light sources, the temperature significantly influences electrical and optical behavior. As a result, thermal, optical, and electrical design teams must work closely together in the product development workflow [10].

For the thermal management of electronic equipment (also known as electronics cooling), computer model-based simulations became daily practice decades ago. This was facilitated by the availability of CFD (computational fluid dynamics)-based numerical thermal simulation tools and the semiconductor vendors’ ability to create so-called boundary condition, independent, compact thermal models of semiconductor device packages [11]. The achievements of the past European research project DELPHI [10,11,12,13,14,15] became widespread both among EDA (Electronic Design Automation) users and semiconductor vendors. A decade after the completion of the DELPHI project, the compact thermal modeling methodology became an industrial standard [16,17].

DELPHI compact models were primarily aimed at the package design of digital ICs for CFD-based thermal simulations of the steady-state or stationary state of digital electronics. These compact thermal models provide the advantage of accurately predicting the junction temperatures of ICs during system-level numerical simulations without requiring the sharing of proprietary information about package design details from semiconductor vendors. A natural extension of the DELPHI compact thermal models was toward thermal transient simulation capabilities, achieved by the European research project PROFIT [18]. This development became important for power semiconductor device packages, where the Cauer-type thermal RC models of the dominantly one-directional heat-flow path proved to be boundary condition independent as well [19]. Such compact thermal models are well suited for the representation of the thermal impedance of the packages of power LEDs as well and can also be easily combined with the electrical models of LEDs, as suggested in [19].

By reducing the detailed mechanical CAD model of an LED luminaire to a multi-port compact model and combining this with the LED chip + package model, complete LED luminaires can also be simulated. Thus, a natural extension of the compact modeling concept for LED packages was the combination of the chip-level multi-domain model of LEDs and the dynamic compact thermal model of the mechanical structure of an LED package in the form of a netlist suitable for application in an electrical circuit simulation program, such as different versions of SPICE. The main target of the Delphi4LED H2020 ECSEL European research project [20] was to develop appropriate multi-domain LED measurement and modeling methodologies, along with an LED product development workflow in which such compact models are used. A major result of the Delphi4LED project was the refinement and reformulation of a prior multi-domain LED model that was implemented and tested under different environments, with the major purpose of supporting the virtual prototyping of LED-based luminaires [21,22]. This multi-domain model has 27 parameters that should be fitted to a set of measured isothermal IVL characteristics of LEDs.

According to the original recommendations of the Delphi4LED project, the isothermal IVL (current–voltage–luminance) characteristics had to cover the typical operational domain of a power LED determined by 33 pairs of $\left(I_F,T_J\right)$ forward currents and junction temperatures [23]. The huge amount of measurement time needed to obtain the recommended amount of test data and the problems associated with the brute-force fitting techniques used to extract the model parameters have impeded the widespread adoption and use of the proposed multi-domain LED model. Therefore, one of the major targets of the AI-TWILIGHT H2020 ECSEL project [1] (which followed Delphi4LED) was to improve both the measurement processes and the model parameter extraction, both in terms of speed and robustness. Important aspects of the parameter extraction are the number of LED package samples to be measured and a method of finding a single set of model parameters to describe the “LED-type” represented by the measured samples, which was already addressed by the Delphi4LED project [23]. The aim of this paper is to present our recent results that aim to properly address these issues.

As will be shown in Section 2, this model is based on nonlinear formulas with a wide range of possible values in the 27-dimensional parameter space that makes the parameter estimation a real challenge in computing. Our approach, described here, aims to provide a robust, fast, and precise method to extract the model parameters using the least possible sets of test data. Using the extracted parameters and the thermal model described above (the process of thermal model generation is well-established and thus not further detailed here), a digital twin of the LED chip can be developed in a SPICE-like format. This digital twin can be seamlessly integrated with the driver’s model at the electrical port and with the thermal model of the surrounding structure at the thermal port.

2. LED Modeling: The Governing Equations

Many research groups have dealt with the problem of modeling LED chips and luminaires. Two main approaches exist: empirical modeling, which creates models to fit data, and physics-based modeling, which builds models grounded in physical principles. The problem with the former is justifying its generalization to other structures beyond the one used for fitting. The latter cannot be used without the detailed knowledge of the physical structure of the LED chips that is not likely to be publicly shared by the LED manufacturers.

Theoretical models for the optical behavior are established for GaN-based LEDs in [24] and organic LEDs (OLEDs) in [25]. General behavioral models for both LEDs and OLEDs are presented in [26]. Empirical models for optical behavior are presented in [27] and in [28].

In the Delphi4LED multi-domain model, a hybrid approach is used. This is a quasi-black-box model, where the model is based on physical considerations, where the parameters are not calculated from the actual structure, but are identified from measurement results [29]. This hybridization allows for the generalization of the model, as the inter- and extrapolation are based on physical principles, and no information is provided on the actual structure of the LED. Simulation with this model is computationally effective [30] and so it can be used for design and control purposes as well [24].

2.1. The Delphi4LED Chip-Level Multi-Domain LED Model [31]

The Delphi4LED multi-domain model is aimed at modeling the electrical, optical, and thermal properties of LEDs at the chip level. The model is an improved version of the original model proposed earlier [21]. It relies on Shockley’s model of the IV characteristics of pn-junctions. It is a SPICE-like model, meaning that it can be implemented using standard netlist components of an electrical-only, generic version of the SPICE nonlinear circuit simulation program. As mentioned earlier, the electrical operation depends on the temperature, while the generated heat depends on the electrical operating point and the amount of light emitted. The optical behavior depends primarily on the electrical conditions, but it is also influenced by the LED’s operating temperature. The optical losses, eventually, also result in heat generation within an LED package. As the temperature sensitivity of the electrical and optical behavior is high, the model must reflect the link between these domains and the thermal domain.

In the Delphi4LED model, the electrical, optical, and thermal domains are represented as branches (Figure 1). These branches are represented by electrical circuits that can be simulated by any version of the SPICE program. However, according to the modular modeling approach suggested and used in the Delphi4LED project, the model of the thermal impedance of the junction-to-ambient heat-flow path of the LED package (denoted as $Z_{Th}$ in Figure 1) is not part of the chip-level LED model.

The circuit model representation of the branches basically represents their governing equations that we briefly describe below. Each of the elements can be found in Figure 1.

The thermal branch (Figure 1a) is simple, consisting of a heat source terminated by an external model of the $Z_{Th}$ thermal impedance of the package. From the point of view of the steady-state value of the LED chip’s junction temperature, the steady-state value of this thermal impedance should be considered: $R_{ThJA}=Z_{Th}\left(t=\infty \right)$. The junction temperature can be determined by solving the following energy conservation equation:

$$P_H\left(T_J,I_F,V_F\right)-{\displaystyle \frac{T_J-T_a}{R_{ThJA}}}=0$$

(1)

where $P_H$ is the total heat generated in the LED package (also known as the heating power), which depends on the electrical operating point determined by the applied forward current $\left(I_F\right)$, the total forward voltage drop ($V_F$), and the junction temperature ($T_J$). $T_a$ is the ambient temperature. For modeling the thermal impedance, different approaches have emerged [9], also within the Delphi4LED project [31,32]. The heating power can be calculated from the applied electrical power ($I_F\cdot V_F$, yellow route in Figure 1) subtracting the optical energy leaving the system (${\Phi }_e$, orange route in Figure 1).

The electrical branch (Figure 1b) consists of a current source, a series resistance, an ideal diode, and a voltage generator. This represents the LED as a semiconductor diode, seen barely as an electrical component. Since, in practice, LEDs are typically driven by a current source, our multi-domain LED model follows this and is based on a current-driven approach; thus, as indicated in Figure 1, it is driven by the $I_F$ current source, forcing the forward current into the electrical model. The next netlist item in the model is a built-in SPICE diode model, characterized by Shockley’s ideal diode equation. In the SPICE diode model used, the calculation of the secondary effects is switched off. The most important effect is represented by the inherent electrical series resistance, $R_S$. In the current driven modeling approach, the ideal diode behavior and the effect of the electrical series resistance are jointly described by

$$V_F=m_e\cdot U_T\cdot ln\left({{\displaystyle \frac{I_F}{I_0}}}_{\mathrm{e}}+1\right)+I_F\cdot R_S$$

(2)

where $V_F$ is the forward voltage, $m_e$ is the diode’s ideality factor, $U_T=k_{B}T/q$ is the thermal voltage, $I_F$ is the forward current, $I_{0_e}$ is the saturation current, and $R_S$ is the series resistance, the temperature dependence of which is merged into the overall model of the diode’s temperature sensitivity. It is a known fact that, under forced constant forward-current operating conditions, the forward voltage of the diode diminishes with the increasing junction temperature. This temperature-dependent forward voltage change is represented by the $\Delta V_e$ voltage-controlled voltage generator connected in series with the temperature independent diode and $R_S$ models. The controlling voltage of this voltage generator is the voltage of the junction node, $T_J$, as indicated by the red route in Figure 1. The temperature dependence of the voltage of the $\Delta V_e$ generator is approximated as follows:

$$\begin{array}{c}\Delta V_e=\left(a_e{\cdot I}_F^2+b_e\cdot I_F+c_e\right)\cdot \left({T_J^2-T}_{ref}^2\right)\\ +\left(d_e{\cdot I}_F^2+e_e\cdot I_F+f_e\right)\cdot \left({T_J-T}_{ref}\right)\end{array}$$

(3)

where $T_{ref}$ is the reference temperature for which the parameters of the ideal diode model describing the $I_F-V_F$ relationship and the constant value of the $R_S$ series resistance were identified. Note that this temperature dependence of the change in the diode’s overall forward voltage corresponds to the quadratic relationship of the so-called K-factor function proposed by the most recent thermal testing standard, JESD51-51A, published by JEDEC [33]. Symbols $a_e-f_e$ used in Equation (3) are model parameters to be identified.

The model of the optical branch (Figure 1c) is based on the idea that the electrons comprising the total forward current ($I_F$) can be divided into two parts, according to the recombination processes to which they are subject to in the active area of the LED’s pn-junction. In this way, we separate the total forward current into a component associated with the radiative recombination processes (direct bandgap transitions of the charge carriers) giving rise to photon emission, and into another component that is associated with non-radiative recombination processes, eventually resulting in heat generation. The current–voltage characteristics of these forward current components can both be described by Shockley’s diode equation [31].

To improve model the optical losses, the ideal light-emitting diode through which the charge carriers pass to perform radiative recombination processes is complemented by a series resistance, denoted as $R_r$ in Figure 1. Thus, similarly to the model of the overall electrical behavior of the LED, the optical branch also comprises a temperature independent electrical series resistance ($R_r$) and a temperature independent ideal diode model, supplemented by a voltage-controlled voltage generator ($\Delta V_r$). The controlling voltage of this generator, again, is the voltage representing the junction temperature of the LED. The model equation is similar to Equation (2), but the characteristic here is presented in a voltage-controlled manner, describing the dependence of this radiative diode on the $V_r$ forward voltage of the internal pn-junction of the LED (excluding the effect of the $R_S$ electrical series resistance):

$$I_r={I_0}_r\cdot \left[exp\left({\displaystyle \frac{V_r-\Delta V_r-I_r{\cdot R}_r}{m_r\cdot U_T}}\right)-1\right]$$

(4)

where $I_r$ is the radiative current component of the overall forward current of the entire LED, ${I_0}_r$ is the saturation current, and $m_r$ is the ideality factor of the radiative diode. The $\Delta V_r$ voltage of the voltage-controlled voltage source represents the temperature-dependent changes of the characteristic of this diode. The junction temperature dependence of the $\Delta V_r$ generator is described with a similar bi-quadratic approximation, as seen earlier for the $\Delta V_e$ generator:

$$\Delta V_r=\left(a_r{\cdot I}_r^2+b_r\cdot I_r+c_r\right)\cdot \left({T_J^2-T}_{ref}^2\right)+\left(d_r{\cdot I}_r^2+e_r\cdot I_r+f_r\right)\cdot \left({T_J-T}_{ref}\right)$$

(5)

The total emitted optical power of the LED, ${\Phi }_e$ (also known as the total emitted radiant flux), is directly proportional to the radiative current:

$${\Phi }_e=I_r\cdot \left[\Delta V_r+m_r\cdot U_T\cdot \ln \left({\displaystyle \frac{I_r}{{I_0}_r}}+1\right)\right]$$

(6)

With the emitted optical power, the value of the $P_H$ heating power can be calculated as:

$$P_H=V_F\cdot I_F-{\Phi }_e$$

(7)

With this last step, the model is closed and the law of the conservation of energy is satisfied.

With additional models, the spectral power distribution or another important light output property, the total emitted luminous flux can be also modeled. The Delphi4LED multi-domain LED model contains a simple, empirical approximation for the total luminous flux, ${\Phi }_v$, with a bi-quadratic formula as follows. The total luminous flux can be determined by:

$${\Phi }_v={\Phi }_e\cdot \mathrm{K}\left(I_F,T_J\right)$$

(8)

where $\mathrm{K}$ is the so-called efficacy of the source of the radiation (for the LED) that can be well approximated by a bi-quadratic formula of the forward current and temperature for most LEDs (in which the luminous flux is a relevant light output property), as follows:

$$\begin{array}{c}{\mathrm{K}\left(I_F,T_J\right)=(a}_L\cdot T_J^2+b_L\cdot T_J+c_L)\cdot I_F^2\\ +{(d}_L\cdot T_J^2+e_L\cdot T_J+f_L)\cdot I_F+{(g}_L\cdot T_J^2+h_L\cdot T_J+l_L))\end{array}$$

(9)

Overall, 27 parameters (excluding the thermal impedance that is directly measured and modeled on the package level rather than being part of this model) should be fitted to the measurement data to establish a particular LED model (circuit schematic/netlist with parameters).

2.2. Measurement

The parameters of the Delphi4LED model were fitted to a sufficiently detailed set of so-called isothermal IVL characteristics. These characteristics were obtained using compliant procedures outlined in JEDEC standards JESD51-51A and JESD51-52A [33,34], which involved combined thermal and radiometric/photometric measurements. During these tests, the LEDs under test (DUT) had their junction temperatures set and kept at a pre-specified value while the LEDs were driven by a pre-specified value of the forward current. Thermal impedance, light output characteristics (total emitted radiant flux and spectral power distribution), and forward voltage were measured. Forward current values covered almost three decades, from the mA range up to the 1000 mA range, and junction temperatures varying from room temperature up to 100–120 °C.

The measurement setup utilized SIEMENS SIMCENTER T3Ster, Budapest, Hungary and SIMCENTER, TeraLED equipment (compliant with JEDEC JESD51-51A and JESD51-52A standards [33,34]), completed with a spectroradiometer (Instrumentsystems CAS-140CT, Munich, Germany).

According to the standards, the measurements were conducted in the hot steady-state of the DUT LED, when the junction temperature stabilized after powering. Waiting for the thermal steady state was time-consuming (minutes for each measurement point), but it was necessary to characterize the diode under stable conditions as required by the CIE’s recommendations for the optical testing of LEDs [35].

The measurement control software of the joint T3Ster-TeraLED setup enabled achieving the LEDs’ hot steady state at a pre-specified value of their $T_J$ junction temperature and capturing the IVL characteristics at such pre-defined, constant-junction temperature values. The problem with such a temperature control was that, once the forward current of the DUT LED changed, its heating power also changed; thus, the temperature of the cold plate holding the DUT LED had to be adjusted in order to maintain the pre-defined junction temperature. The settling time of this temperature control mechanism took several minutes, representing a significant portion of the overall measurement time to characterize a single LED package. One of the goals of this research is to find a way to establish the LED chip-level model without the requirement of the pre-defined junction temperatures during the measurement. A sample of a measurement result is shown in Figure 2. In subsequent sections, the forward voltage ($V_F)$, the radiative flux (${\Phi }_e)$, and the luminous flux (${\Phi }_V)$ will be analyzed at each forward current ($I_F)$ and junction temperature ($T_{junction})$. This measurement was aligned with the aforementioned standards. Therefore, the junction temperature goal was set according to the reference (first column). Not all reference temperatures can be achieved by the measurement apparatus, though. For instance, a junction temperature of 110 °C cannot be achieved with a 100 mA current due to the LED’s limited heating power. In the original extraction method, such data points were excluded, as discussed in Section 3.1. One other goal of this research is to be able to use these points as well for model extraction purposes.

As detailed in Section 3.2, the new parameter extraction method allows using isothermal characteristics in terms of the ambient temperature control, ($T_a$), instead of the characteristics measured at a constant junction temperature, ($T_J$). The benefit is that the temperature control loop of the measurement system is much simpler, resulting in shorter overall measurement times, while the level of detail and accuracy of the measurements are maintained, and all the measured data can be used in the extraction phase. Since the chip-level model lacks a thermal component, the junction temperature measurement is still essential.

3. LED Model Parameter Extraction Methods

In the Delphi4LED project, a simple parameter extraction method was implemented in an Excel spreadsheet. This macro was used to find the 27 parameters of the multi-domain model [23]. In this section, the original method is presented first as a baseline for comparing the results obtained with our new, more advanced parameter extraction.

In parallel with the development of the novel extraction algorithm, the extraction strategy was also reformed, enabling us to realize our goals mentioned in the Measurement Section.

3.1. Original Extraction Strategy

The first parameter extraction approach was presented in detail in [22] and briefly described in this section.

As a first step, the reference junction temperature, $T_{ref}$, is defined, at which the parameters of the temperature-independent diodes included in the model of Figure 1b,c are valid. At $T_J=T_{ref}$, the $\mathrm{d}\mathrm{V}\mathrm{F}\_\mathrm{e}\mathrm{l}$ and $\mathrm{d}\mathrm{V}\mathrm{F}\_\mathrm{r}\mathrm{a}\mathrm{d}$ temperature-controlled voltage generators, which describe the temperature dependence of the two electrical-only diode models in Figure 1, according to Equations (3) and (5), present 0 as the output voltage.

The extraction process starts with the electrical branch. At a high forward current, the series resistance dominates in the overall voltage drop across the LED: $I_F\cdot R_S\gg m\cdot U_T\cdot \ln (I_F/I_0+1)$; thus, $I_F\cdot R_S\approx V_F$. Therefore, the slope of the high current range of the measured $I_F-V_F$ characteristic is a good approximation for the value of $R_S$. Once the serial resistance is estimated, the measured $I_F-V_F$ curve can be stripped of its effect, resulting in an almost ideal diode characteristic. The ideality factor and the saturation current can be estimated by curve fitting Equation (2) to the series of $V_F$ values modified this way, yielding the estimated values of $m_e$ and ${I_0}_e$ (values of $I_F$ are pre-defined ones, set during the measurements). This initial estimation of the parameters for Equation (2) is quite rough, so the initially obtained value of $R_S$ has to be fine-tuned in order to achieve a better overall fit of Equation (2) to the measured dataset. This approach requires an exhaustive search in the parameter spaces of $R_S$, ${I_0}_e$, and $m_e$. A good fit at this stage of the parameter extraction procedure is critical since the $V_r$ voltage drop on the entire “radiative diode” denoted as branch (c) in Figure 1 is calculated as $V_r=V_F-R_S\cdot I_F$. The $I_r$ current across this “radiative diode” can be calculated by solving Equation (4), with $\Delta V_r=0$, corresponding to the $T_J=T_{ref}$ assumption.

The optical branch is very similar to the electrical one, but for the $I_r-V_r$ relationship, Equation (4) provides an implicit formula. The relationship between the “experimental values” of $I_r$ and the measured ${\Phi }_e$ total radiant flux values provided by the optical measurements during the physical LED characterization process is provided by Equation (6).

The six parameters of the $\Delta V_e\left(T_J\right)$ relationship are determined by quadratic fitting. However, for any $T_J$ different from the $T_{ref}$ reference temperature, a fitting procedure for the of $R_S$, ${I_0}_e$, and $m_e$ parameters also needs to be performed, requiring substantial computation resources and long execution times, without assuring the best global optimum. The same applies to the parameters of the $\Delta V_r\left(T_J\right)$ relationship and the $R_r$, ${I_0}_r$, and $m_r$ parameters of the “radiative diode” denoted as branch (c), Figure 1. The optimal parameters for the entire multi-domain LED model were obtained using a brute-force method [22]. The parameters of the simple model of the luminous flux—Equations (8) and (9)—are determined by quadratic fitting. This process is straightforward, but takes a long time in its 9-dimensional parameter space.

3.2. A Novel Parameter Extraction Strategy and Its Benefits

The main problems with the original extraction method were:

• $T_J$ had to be set during the measurement, requiring a long settling time to meet the target value of $T_J$ within an error margin less than 0.5 °C by adjusting the temperature of the DUT (device under test) LED holder of the measurement system. Since the measurement system determines $T_J$ indirectly, relying also on an experimentally determined $V_F\left(T_J\right)$ relationship obtained from a prior calibration procedure, the measured value of $T_J$ is loaded by a complex measurement uncertainty budget.
• Including the IVL characteristics measured at $T_J=T_{ref}$ temperature into the global optimization would assure a better overall fit of the model, as the information contained in these characteristics would also contribute to the modeling of temperature dependence.
• The sequential extraction of the parameters leads to a set of marginal optima, but not to the global optimum for all the nine parameters for the electrical and optical branches. An algorithm is required to estimate the global optimum in the 9-dimensional parameter space.

Since we aim at using a fully automatic parameter extraction method, we should avoid relying on an a priori distinction between the low- and high-current regions of the characteristics. To achieve such an algorithm, a new optimization method is needed. First trials with usual gradient methods failed, because the gradient in the direction of the saturation current direction is at least 15 orders of magnitude (the typical value of the saturation current is ${10}^{-32}-{10}^{-15}$) higher than the gradients in any other dimension of the global parameter space. To handle this, semi-heuristic approaches are needed. We selected the Nelder–Mead optimization algorithm as our method [36].

3.3. The Nelder–Mead Method

The Nelder–Mead method is a gradient-free, simplex-based direct search method. Let us use the following notations:

• Let $\mathsf{{\rm N}}$ denote the number of parameters to be extracted;
• let $f(\mathit{x}$) denote the cost function to be minimized during the search for the best fitting parameters in the $\mathsf{{\rm N}}$-dimensional parameter space, and;
• let $\mathit{x}$ denote a point in the $\mathsf{{\rm N}}$-dimensional parameter space.

During the search for the global minimum of the cost function $f(\mathit{x}$), an initial guess of $\mathsf{{\rm N}}+1$ point (vectors) was made, each being a candidate for the final set of parameters, where $f(\mathit{x}$) reaches its minimum. This collection of $\mathsf{{\rm N}}+1$ parameter sets is referred to as a simplex: $S={\mathit{x}}_0,{\mathit{x}}_1,{\mathit{x}}_2\dots {\mathit{x}}_{\mathsf{{\rm N}}}\in {\mathbb{R}}^{\mathsf{{\rm N}}}$. The evaluation and modification of $S$—as the optimization process involves the following steps:

• Identify the data points for which the $f(\mathit{x}$) cost function provides the largest, the second largest, and the smallest values. Denote these points as follows: (${\mathit{x}}_{\mathit{max}}|\underset{i}{\max }f\left({\mathit{x}}_{\mathit{i}}\right))$, (${\mathit{x}}_{\mathit{sec}}|\underset{i\ne \mathit{max}}{\max }f\left({\mathit{x}}_{\mathit{i}}\right))$, and (${\mathit{x}}_{\mathit{min}}|\underset{i\ne \mathit{max},\mathit{sec}}{\min }f\left({\mathit{x}}_{\mathit{i}}\right))$.
• Calculate the coordinates of the centroid of the N best vectors: ${\mathit{x}}_{\mathit{c}}={\displaystyle \frac{1}{\mathsf{{\rm N}}\sum _{i\ne max}{\mathit{x}}_{\mathit{i}}}}$.
• Replace ${\mathit{x}}_{\mathit{m}\mathit{a}\mathit{x}}$ with a new point in the vertex by creating a line between ${\mathit{x}}_{\mathit{c}}$ and ${\mathit{x}}_{\mathit{m}\mathit{a}\mathit{x}}$, and try out different points on this line. The points are called the reflection, the expansion, and the contraction points of ${\mathit{x}}_{\mathit{m}\mathit{a}\mathit{x}}$ with respect to ${\mathit{x}}_{\mathit{c}}$, calculated as follows:
  • Reflection: ${\mathit{x}}_r={\mathit{x}}_c+\alpha \cdot \left({\mathit{x}}_c-{\mathit{x}}_{max}\right)$. The reflection point replaces the maximum point if $f\left({\mathit{x}}_{min}\right)\le f\left({\mathit{x}}_r\right)<f\left({\mathit{x}}_{sec}\right).$
  • Expansion: If $f\left({\mathit{x}}_r\right)<f\left({\mathit{x}}_{min}\right)$, there is a chance to find a better point if we move elsewhere on the line, so ${\mathit{x}}_e={\mathit{x}}_c+\gamma \left({\mathit{x}}_{max}-{\mathit{x}}_c\right)$. If ${\mathit{x}}_e$ is better than ${\mathit{x}}_r$, the maximum point is replaced with that value; otherwise, we keep the reflection point.
  • Contraction: If $f\left({\mathit{x}}_{max}\right)>f\left({\mathit{x}}_r\right)>f\left({\mathit{x}}_{sec}\right)$, the contraction point will be ${\mathit{x}}_{co}={\mathit{x}}_c+\beta \left({\mathit{x}}_r-{\mathrm{x}}_c\right)$; otherwise, if $f\left({\mathit{x}}_r\right)>f\left({\mathit{x}}_{max}\right)$, then ${\mathit{x}}_{co}={\mathit{x}}_c+\beta \left({\mathit{x}}_{max}-{\mathit{x}}_c\right)$. If ${f(\mathit{x}}_{co})<f({\mathit{x}}_{max})$, the condition holds.
  • If no replacement point is found, shuffle all the vectors except the best, ${\mathit{x}}_i={\mathit{x}}_{min}+\sigma \left({\mathit{x}}_i-{\mathit{x}}_{min}\right)$.

The Nelder–Mead method can be terminated under one of these conditions:

• A lack of improvement after numerous iterations.
• If the minimum of $f(\mathit{x}$) is close enough to 0.
• The number of function evaluations or iteration steps reaches its preset maximum number.

The function to be minimized is the following for the electrical part:

$$E_e=\sum _i{\displaystyle \frac{{\left(V_{{calc}_i}-V_{{meas}_i}\right)}^2}{V_{{meas}_i}^2}}$$

(10)

where the summation is applied to every data point of the measured I–V characteristics, with $V_{{meas}_i}$ denoting the $i$-th measured forward voltage and $V_{{calc}_i}$ denoting the $i$-th forward voltage value calculated for the same operating point, where $V_{{meas}_i}$ is measured. The division by the measured voltage square is not necessary, but it is needed to avoid low voltage points, especially for the optical part of the model, for which the cost function to be minimized is as follows (with notations similar to the ones applied in Equation (10), with ${\Phi }_e$ representing radiant flux values):

$$E_r=\sum _i{\displaystyle \frac{{\left({\Phi }_{{e,calc}_i}-{\Phi }_{{e,meas}_i}\right)}^2}{{\Phi }_{{e,meas}_i}}}$$

(11)

Unfortunately, the original Nelder–Mead algorithm fails for the cost functions described in Equations (10) and (11), i.e., it is stuck at a local minimum. With a few modifications, however, we succeeded to overcome this problem.

3.4. Modified Nelder–Mead Method

In a 9-dimensional space, 10 points can easily approach a false minimum (i.e., a local minimum), or the vertex can contract too much into a set of points that are too close to each other, resulting in the premature termination of the optimization process. A possible approach to overcome this problem is to increase the number of data points in the parameter space from Ν + 1 to a higher value. As found in the literature [36,37,38], a potential solution is to combine the Nelder–Mead method with particle swarm optimization (PSO) or with other methods, as described in other papers [39,40]. These approaches certainly provide better results, but the computational effort for this problem is very high, as the number of iterations, which involves solving implicit equations, increases linearly with the number of trial data points used within the Nelder–Mead method.

In our present approach, the Nelder–Mead algorithm is modified to use more points in the simplex. We found that this increased the success rate of the optimization, i.e., the termination of the algorithm with convergence to a true optimum. The initialization (step 1 described in Section 3.3) is modified as follows: The vectors of measured data included in the simplex are chosen randomly from a growing hyper-volume around the initial vector ${\mathit{x}}_0$:

$${\mathit{x}}_i\left[k\right]=\left(1.001+{\displaystyle \frac{i}{10\cdot \mathrm{N}}}rand\left(-{\displaystyle \frac{1}{2}},{\displaystyle \frac{1}{2}}\right)\right)\cdot {\mathit{x}}_0[k]$$

(12)

where $rand\left(\right)$ is a random value from a uniform distribution. The initialization ensures that the points are not too close to each other. The resulting initial simplex is shown in Figure 3.

Increasing the number of vectors in the simplex will not greatly increase the number of cost function evaluations, as only one vector changes over an iteration, except for contraction (see step 3b in Section 3.3), where all vectors change, and the cost function must be evaluated for all of them. Therefore, this modification does not significantly slow down the original Nelder–Mead method. Although the number of cost function evaluations per iteration does not change significantly, the overall optimization’s convergence becomes slower since the central point cannot move as quickly as it could with only $\mathsf{{\rm N}}+1$ elements in the simplex. The number of iterations is assumed to increase linearly with the number of vectors in the simplex.

The convergence probability of the method is defined as the ratio of successful evaluations to all evaluations. The extraction was performed 100 times with different random simplexes on 10 different LED samples of various kinds for testing purposes. The success rate is defined as the ratio of successful extractions (finding the global minimum, obtained by an exhaustive search) to the total number of attempts. The best heuristic modifications and initial parameter settings of the Nelder–Mead algorithm were identified through experiments on more than 100 sets of measured LED characteristics obtained from the archived data of the Delphi4LED project.

First, we investigated how the Nelder–Mead algorithm behaved with the randomly selected initial $\mathsf{{\rm N}}+1$ points (for the parameter extraction of the electrical branch $\mathsf{{\rm N}}=9$). The success rate of the algorithm was approximately 50%, even if we followed the original extraction strategy. Increasing the number of points in the simplex resulted in an increased success rate, reaching 96% when the number of points was $4\cdot \left(\mathsf{{\rm N}}+1\right)=40$. Further increasing the number of points provided even better success rates; we found that $5\cdot \left(\mathsf{{\rm N}}+1\right)=50$ yielded a success rate of 98%. The results are summarized in

Table 1. Dependence of the success rate of the modified Nelder–Mead algorithm on the size of the simplex.

Number of Vectors	Success Rate [%]
N = 10 (original method)	51.4
N = 20	86.8
N = 30	92.4
N = 40	95.8
N = 50	98.5

and shown in Figure 4, where we can see that the error with the parameters from an unsuccessful extraction is also reduced as the number of vectors increases.

As an illustration, the excess error with respect to the best case of $5\cdot \left(\mathsf{{\rm N}}+1\right)=50$ points is shown for the parameters of the optical branch of the multi-domain model. For the extraction of the parameters of the optical branch, the number of objective function evaluations for the original Nelder–Mead simplex with $\mathsf{{\rm N}}+1=10$ data points in the parameter space was cca. 14,000 and for the 5-fold increased simplex size of $5\cdot \left(\mathsf{{\rm N}}+1\right)=50$, the number of evaluations of the objective function increased to cca. 52,000, slightly less than the expected linear increase.

The success rate of the Nelder–Mead algorithm depends on the actual initial simplex. Therefore, it is essential to investigate how the initial simplex $\left({\mathit{x}}_0\right)$ influences the success rate. This dependence was investigated by increasing and decreasing the values of the initial parameters by an order of magnitude, except the values of the series resistances and the ideality factors in order to avoid non-physical solutions or no solution for Equation (4). The initial values of the series resistances can be reduced to 0.1 and 0.05, respectively, for electric and optic branches. Ideality factors must be greater than 1.

Decreasing the initial simplex with the above limitations has no adverse effect on the extraction process, other than a slight increase in computational resources, due to 20–30% more iterations being needed to achieve a global optimum. Increasing the parameters, however, greatly increases the probability of the successful execution of the new parameter extraction procedure. These statements are based on our initial numerical experiments using 100+ sets of archived LED measurement data detailed in

Table 2. Dependence of the success rate of the modified Nelder–Mead algorithm on the choice of the initial central points.

Simplex Size	Success Rate with an Increased Central Point (10×) [%]	Success Rate with a Decreased Central Point (10×) [%]	Success Rate with a Decreased Central Point (5×) [%]
N = 10	58.1	0	15.7
N = 20	89.3	15.2	37.5
N = 30	94.5	41.4	59.9
N = 40	95.2	55.8	72.3
N = 50	98.8	78.2	95.1

.

Reducing the values of the parameter values forming the initial central vector in the parameter space had an adverse effect on the success rate if the parameter extraction process had a 10-fold decrease moderated to a 5-fold reduction. These results are also summarized in Table 2. These experimental results suggest that there are many local minima of the objective function at parameter sets with reduced values. Decreasing the parameter values defining the central vector also narrows the possible range within which additional vectors in the parameter space can be randomly selected. Additionally, overly aggressive upscaling the parameter values results in a decreased success rate. This indicates that using typical parameter sets of the chip-level multi-domain LED model as initial values ensures a high likelihood of success in the extraction process. However, the tested heuristics are not sufficiently robust against higher errors in the measurement data or against unusual LED characteristics. Further steps need to be taken to make our solver robust enough for unsupervised working conditions.

The conclusions drawn from our numerical experiments regarding the heuristic modifications of the original Nelder–Mead method are as follows:

• Including more random vectors in the simplex increases the success rate of the parameter extraction process, but involves an increased computational cost.
• Using points from the parameter space with values much less than those forming a valid parameter set for a typical LED chip decreases the success rate of the parameter extraction process.
• An initial parameter set with parameter values not too far above the values from the expected solution of the parameter extraction process increases the success rate, involving only a slight increase in the computational cost.
• The above heuristic modifications result in high success rates for the parameter extraction, but do not assure high robustness.

We found that the success rate of the parameter extraction method using these modifications strongly depends on the initial central vector. However, even without increasing the values in the initial set, a high success rate may still be achieved, as presented in Table 2. A thorough investigation of the failed extraction processes revealed that the process often becomes stuck in a local minimum, resulting in much higher cost function values than the global minimum.

To ensure robustness, an additional heuristic approach was added to the initialization step of the parameter extraction process: the number of the initial vectors was increased to 100, covering a significantly extended range, as follows:

$${\mathit{x}}_i\left[k\right]=\left[1.001+\mathrm{r}and\left(rand\left(-{\displaystyle \frac{i}{100}},-{\displaystyle \frac{i-1}{100}}\right),rand\left({\displaystyle \frac{i-1}{100}},{\displaystyle \frac{i}{100}}\right)\right)\right]{\cdot \mathit{x}}_0\left[k\right],$$

(13)

where $i$ is the index between 1 and includes 100. The cost function for these points is evaluated, and the 30 points with the smallest cost function values are kept, for which we used the Nelder–Mead algorithm modified with the above-described heuristics. With this addition, the success rate of the parameter extraction process reached 100% in the case of 5200 extractions performed with measurement data from 52 different LED samples of various types, including color LEDs and phosphor-converted white LEDs. The choice of the central point has a much smaller effect than before. The success rate for increased parameter values remains 2600/2600 (50 extraction per LED sample); with 10-fold reduced values, the success rate was 2595/2600, nearly 100% This suggests that the Nelder–Mead algorithm, modified with these heuristics, is now robust enough and likely to work well for other LED types beyond those for which we had archived measurement data from the Delphi4LED project.

4. Results Obtained with the Modified Nelder–Mead Method

We investigated the behavior of our modified Nelder–Mead parameter extraction method with two kinds of benchmarks.

The first benchmark compares the overall error of the parameter fitting performed with the new method with the overall error of the prior brute-force approach used in the Delphi4LED project. This comparison was chosen because the brute-force-based method—though very slow and computationally intensive—provides very good fits (less than a 1% error for $V_F$ and less than a 5% error for ${\Phi }_e$) for all the LED samples characterized in the Delphi4LED project [41]. Therefore, this comparison offers a valuable insight into the fit errors of the new parameter extraction procedure. However, caution is needed for this comparison, as there are several significant differences between the two methods.

• The old method requires predefined junction temperatures, so the measurement points that do not meet this requirement are not taken into account.
• Slight differences in junction temperatures (±0.5 °C) are neglected in the old method.
• The old and the new parameter extraction methods also differ in the objective functions that they minimize during the fitting process. The old method uses the sum of square errors as the objective function to be minimized, while in the new method, the sum of the relative square error is used. Therefore, for the used figure of merit for the comparison of the two methods, the maximal relative error is much higher at low forward currents when applied to evaluate the fit error of the old method.

The new model uses the sum of the relative square error, because many end-users of the SPICE-like multi-domain LED model may require accurate simulations across a wide range of forward currents, spanning two decades (for our sample LEDs: from 20 mA up to 1000 mA). As a result, the LED model should be equipped with a parameter set that ensures accurate results, even below the greatest decade in the forward current ($I_F<100 \mathrm{m}\mathrm{A})$. To address this need, we compared the performance of the two parameter extraction methods, both with and without including low-forward-current operating points. In the comparisons for the high current domain, we completely discarded the low-current operating points of the LEDs during the fitting procedures.

The parameter extraction procedures were applied to 52 high-power LED samples measured in the Delphi4LED project: 11 phosphor-converted white LED samples of the same type, 5 red LED samples of the same type, 5 amber LED samples of the same type, and 31 blue LED samples with 11 samples from the same LED type and 20 other samples of other types. The maximum relative error values obtained with the old and new parameter extraction procedures are shown in Figure 5 for the electrical branch, Figure 6 for the optical branch, and Figure 7 for luminous flux, applied to the high-forward-current operating points only (I_F > 100 mA) and for the full range of operating points (involving the low-forward-current ones as well).

In the high-forward-current region (above 100 mA), where the “linearization effect” of the series resistance starts to appear in the LED characteristics, the strong dependence of the accuracy of the parameters of the pn-junction-related parts of the multi-domain LED model is increasingly suppressed as the forward current increases.

Consequently, the simulation results obtained using the parameter sets identified by the previous method and those identified by the new, modified Nelder–Mead-based parameter extraction process agree very well in this current range.

Considering the entire set of LED operating points, including forward currents below 100 mA, the simulation results obtained from the two different kinds of parameter sets show greater differences. Besides the stronger dependence of the accuracy of the model calculations on the accuracy of the parameter sets at lower forward currents, it is worth mentioning that the previous and the new parameter extraction processes use different objective functions.

The new method uses the relative square difference as the cost function, ensuring acceptable accuracy across the entire range of operating forward currents. The statistics for the maximum relative errors obtained by both parameter extraction processes are summarized in

Table 3. Relative errors (REs) statistics for 52 different sets of LED samples.

Method	Avg. of REs, Electric [%]	Variance of REs, Electric [10−4]	Avg. of REs, Optic [%]	Variance of REs, Optic [10−4]	Avg. of REs, Luminous [%]	Variance of REs Luminous [10−4]
$\mathrm{Brute} \mathrm{force} {\mathit{I}}_{\mathit{F}}\mathbf{\ge }\mathbf{0.1}$ A	0.11	0.002	0.88	0.427	5.19	12.48
$\mathrm{Nelder}–\mathrm{Mead} {\mathit{I}}_{\mathit{F}}\mathbf{\ge }\mathbf{0.1}$ A	0.11	0.001	0.43	0.029	0.62	0.053
Brute force, all currents	1.55	0.289	12.48	18.84	10.68	14.60
Nelder–Mead, all currents	0.25	0.005	2.69	1.40	3.89	1.93

, where the statistics includes the relative errors of the model and the measurements for all operating points that are included in the extraction.

Consequently, the simulation results obtained using parameter sets identified by the previous method and those identified by the new, modified Nelder–Mead-based parameter extraction process agree very well in this current range.

The statistics show that the multi-domain LED model with parameter sets obtained through the new extraction methodology accurately describes the operation of various high-power LEDs, including color and phosphor-converted white types, with minimal errors. Specifically:

• For the forward voltage, errors are below 0.5%.
• For simulated radiant fluxes, errors are below 5%.
• For the luminous flux, errors are below 8%.

These error levels are comparable to the industrial average uncertainties associated with measuring these LED properties.

The second benchmark evaluates how closely the measured LED characteristics are approximated by the multi-domain LED model equipped with the extracted parameter set. In these evaluations, we have to account for the effects of measurement uncertainties and noise, as their impact is absorbed into the overall error of the parameter fit. Since these measurement-related issues are consistent across any extraction process, and the theoretical accuracy of the multi-domain model is independent of the extraction method, the calculated error of the fit can be considered indicative solely of the quality of the parameter extraction process itself.

Figure 8 and Figure 9 illustrate the accuracy of parameter extraction by comparing the measurement and simulation results for two phosphor-converted white power LEDs—one with the best fit (Figure 8) and one with the worst fit (Figure 9). The parameter extraction for these LEDs was performed for the full sets of forward currents applied during measurements. However, for visibility, only the measurement values are plotted that have a junction temperature equal to the target. This allows us to display the isothermal lines and better present model errors. The extraction process, however, utilizes all measured values, which is a significant advantage of the new extraction methodology enabled by the new method. The difference of the results obtained by the model compared with the measurements in Figure 8 and Figure 9 shows only slight differences; however, the first has a maximum relative error of 1.5% and the second has a 4.7% radiant flux, but these differences appear in the low-current area. This confirms the hypothesis that the model adequately describes both the low- and high-current regions. There is only one original goal left that must be investigated: The new parameter extraction method should also effectively address sample-to-sample variations when the goal is to provide a single model parameter set representative of a given LED type or binning class. Because the extraction process is unknown for the specific LED samples, it can incorporate characteristics from all samples into the extended vertex used in the modified Nelder–Mead algorithm-based parameter extraction process. Figure 10 shows the extraction for one bin (white Nos. 1, 3, 4, 5, and 6). The resulting model provides a good fit, even with significant variance between the chips.

Note that the parameter sets obtained from the aggregate data of a larger number of LED samples of the same type differ from the parameter sets obtained by averaging individual samples. This highlights another advantage of our new approach: a significant reduction in required computational resources, particularly when extracting model parameter sets representative of an LED-type or binning class.

It is conspicuous but not surprising that the parameter sets obtained for multiple LED samples from the same type (and same binning class) provide a fit with more or less the same accuracy, as one LED can replace another, i.e., the model-to-measurement error is in the same range as the measurement-to-measurement error. This holds both for the prior brute-force method-based and for the new modified Nelder–Mead algorithm-based parameter extraction methods. Not only are the accuracies of the simulations similar for the LED samples of the same type, but the actual parameter values are also close to one another, e.g., the different $I_0$ values are within $\pm 10\%$ in a logarithmic scale. In the case of a larger population of phosphor-converted white LEDs of the same type (and presumed binning class), much larger sample-to-sample variations in parameter values were observed. This finding aligns with the previous results of the Delphi4LED project, which suggest that this population is a mix of two LED types exhibiting similar light output properties at the applied binning forward current and junction temperature [23].

The new parameter extractor was coded in Python 3.9. Extracting parameters for an LED sample measured at 33 operating points takes 10–18 s on a standard desktop computer (i5-12400, 3.2 GHz, 32 GB RAM) using a single core.

5. Conclusions

In this paper, a novel parameter extraction procedure was presented that is based on a modified Nelder–Mead optimization algorithm. The modifications greatly improved the robustness of the optimization process, as, due to the applied heuristics, the success rate increased from 50% to 100%.

The reproducibility of the new method was tested with different initializations. Over 10,000 extraction runs were performed on the measured IVL characteristics of 52 high-power LEDs from various types.

Compared to the previous parameter extraction process used in the previous Delphi4LED project, the new parameter extraction procedure provides multiple advantages, where, at least by an order of magnitude faster (reduced computational resource need), the original method lasts for 2–6 h (and needs manual operations). The new method finds the required data within a minute (typically 10–18 s), while the accuracy does not decrease or fall into the range of the supposed model and measurement accuracy for both optical and electrical domains, that is, 1% and 5% for maximum relative errors for the forward voltage and emitted flux, respectively.

The new method demonstrates a high level of robustness, achieving a 100% success rate in a benchmark test. This allows for an autonomous, unsupervised operation, making it suitable for automated web-based model parameter extraction services, as it can be found on https://extractor.ai-twilight.org/ (accessed on 10 August 2024), which has been publicly available and in use since March 2023 without any robustness issues.

Additionally, the new parameter extraction process simplifies the use of measurement data from less complex and faster LED IVL tests. It eliminates the need for standard isothermal characterization processes that require precise junction temperature control. Instead, it is sufficient to wait for any equilibrium state, rather than ensuring the junction temperature aligns with a reference point. This approach allows for the use of all measured data points, even if the reference junction temperature is not achieved, thereby reducing the number of measurements needed to create the model. Moreover, parameter extraction for a specific LED type can be completed in a single session using the test data from multiple samples of that type.