Partial Fraction Research Articles

Efficient Implementation of Multi-Port Frequency Dependent Network Equivalents for Electromagnetic Transient Studies using Norton Equivalent Circuits

Power system electromagnetic transient studies require that a small part of the electrical network be modelled in detail. The rest of the system is represented by a network equivalent taking into account the frequency dependence of the system components (FDNE). With the purpose of getting a FDNE, this paper presents a procedure for including high-order rational models in EMTP-type simulation programs based on Norton equivalent circuits. This approach has the advantage of using few conductance branches compared to those obtained by fitting any particular circuit structure.In relation to the work carried out, this paper presents the following issues: a) a procedure for calculating the admittance matrix of the rest of the system as a function of frequency b) this matrix is synthesized in the form of multi-terminal π-equivalent circuits, whose branches are fitted by rational functions in the pole-residue form c) each partial fraction is converted into a differential equation in time domain. The trapezoidal rule of integration is applied to that differential equation, resulting in a conductance in parallel with a history term current source. Considering all the partial fractions corresponding to a branch, a Norton equivalent circuit is obtained for that branch.This paper also shows how the procedure is implemented in three different ways in an EMTP-type program, along with the validation of such approach through simulations in an electric utility 500 kV transmission system.

An algebraic approach to q-partial fractions and Sylvester denumerants

In 1857, Sylvester established an elegant theory that certain counting functions (which he termed denumerants) are quasi-polynomials by decomposing them into periodic and non-periodic parts. Each component of the decomposition, called a Sylvester wave, corresponds to a root of unity. Recently several researchers, using either combinatorial arguments or complex analytic techniques, obtained explicit formulas for the waves. In this work, we develop an algebraic approach to the Sylvester’s theory. Our methodology essentially relies on deriving q-partial fractions of the generating functions of the denumerants, and thereby obtains new explicit formulas for the waves. The formulas we obtain are expressed in terms of the degenerate Bernoulli numbers and a generalization of the Fourier–Dedekind sum. Further, we also prove certain reciprocity theorems of the generalized Fourier–Dedekind sums and a structure result on the top-order terms of the waves. The proofs rely on our evaluation operator and our far-reaching generalization of the Heaviside’s cover-up method for partial fractions. SageMath code for this work is available in the public domain.

On the experimental vibroacoustic modal analysis of a plate-cavity system

This paper proposes a methodology to the experimental vibroacoustic modal analysis on a coupled plate-cavity system. The method is based on the nonlinear curve fitting to the Frequency Response Functions (FRFs) obtained through a roving hammer test. These FRFs are divided into two groups: one is for the acceleration of the plate; the other is for the sound pressure inside the cavity. The modal parameters are estimated independently in each group using both the single/multi-mode all-curve and the all-mode single-curve fitting techniques. Such a way overcomes the identification difficulties when the natural frequencies of the coupled system slightly change over the roving hammer process. Meanwhile, the mode shapes extraction is eased from the different scales between structural and acoustic responses. The two groups of FRFs are fitted by two different types of FRF models, respectively. The results are in agreement with respect to the natural frequencies and modal damping ratios. Particularly, the FRF models are in alternative forms instead of the conventional ones, and they fit the measured FRFs very well with the estimation outcomes applied. Furthermore, the identified modal parameters are verified by the counterpart estimation based on the conventional FRF models formulated by partial fraction expansions. It is shown that the proposed FRF models contain fewer unknown parameters but do not downgrade the estimation accuracy, making the curve fitting process more practical and efficient. Hence, the two-group strategy with the proposed FRF models provides a practical and efficient way for the experimental modal analysis of vibroacoustic systems.

An evaluation of volatiles and phenolic compounds in conjunction with the antioxidant capacity of endemic endangered species of Erodium hendrikii Alpinar

This study considers the chemical composition of an endemic endangered Erodium species (E. hendrikii), which grows naturally in the Turkish flora. The non-polar compounds and fatty acids from air-dried above-ground parts of the plant were analyzed using GC–MS, while phenolic acids and flavonoids were evaluated using UHPLC-MS/MS. Palmitic acid (5.23 mg/g dw) followed by linoleic acid (3.81 mg/g dw) were the major fatty acids, and n-hexadecanal (60.75%, dw) followed by n-tetradecanal (12.99%, dw) were the main non-polar compounds identified in the plant. The major phenolic acids (nmol/g dw) in the plant were protocatechuic acid (90.07) in free, ferulic acid (430.90) in ester, gallic acid (2450.51) in glycoside, and ester-bound (527.05) forms. The crude extract was represented by the highest caffeic acid (3189.56) content. In addition, hesperidin (259.56 nmol/g dw) was determined as the major flavonoid in the crude extract. This crude extract was further fractioned via solid-phase extraction and its three further partial fractions were obtained using water, ethyl acetate, and methanol. The methanol fraction yielded the highest antioxidant capacity among the fractions. A strong correlation and linear regression were determined between the phenolic contents and antioxidant capacity values of the crude extract and the three further partial fractions of the plant. These findings represent the first data concerning the phytochemical composition of E. hendrikii Alpinar and may be useful in the future development of antioxidant and antimicrobial agents for food preservatives against problems of food deterioration.

Characterization of prolamin recycled from the byproduct of the Baijiu brewing industry (Jiuzao) by SDS-PAGE, multispectral analysis, and morphological analysis

Prolamins have attracted increasing attention in recent years because of their potential application in active ingredient encapsulation and food packaging. As a byproduct of the Baijiu brewing industry with huge production (40 million tons every year), Jiuzao is a great source of prolamin because the fermentation process consumes mainly starch instead of protein. However, information on the extraction method and the characteristics of prolamin from Baijiu Jiuzao (PBJ) is still lacking at present. In this study, PBJ was extracted and characterized in terms of composition, secondary structure, morphology, aggregation state and solubility. The results suggested that the PBJ consisted of partial fractions of kafirin without γ-kafirin. PBJ had more α-helix structures, was more hydrophobic, and more easily aggregated together than kafirin. Morphology analysis showed that the surface of the PBJ powder consisted of joint spheres or dumbbell particles. Aggregates were formed at the molecular level with Rg and Dmax of 13.94 nm and 52.61 nm, respectively. These findings updated the new knowledge of prolamin from byproducts of the brewing industry and provided an environmentally friendly way to recycle Jiuzao.

Nonpolarizable Force Fields through the Self-Consistent Modeling Scheme with MD and DFT Methods: From Ionic Liquids to Self-Assembled Ionic Liquid Crystals.

A key to achieve the accuracy of molecular dynamics (MD) simulation is the set of force fields used to express the atomistic interactions. In particular, the electrostatic interaction remains the main issue for the precise simulation of various ionic soft materials from ionic liquids to their supramolecular compounds. In this study, we test the nonpolarizable force fields of ionic liquids (ILs) and self-assembled ionic liquid crystals (ILCs) for which the intermolecular charge transfer and intramolecular polarization are significant. The self-consistent modeling scheme is adopted to refine the atomic charges of ionic species in a condensed state through the use of density functional theory (DFT) under the periodic boundary condition. The atomic charges of the generalized amber force field (GAFF) are effectively updated to express the electrostatic properties of ionic molecules obtained by the DFT calculation in condensed phase, which improves the prediction accuracy of ionic conductivity with the obtained force field (GAFF-DFT). The derived DFT charges then suggest that the substitution of a hydrophobic liquid-crystalline moiety into IL-based cations enhances the charge localization of ionic groups in the amphiphilic molecules, leading to the amplification of the electrostatic interactions among the hydrophilic/ionic groups in the presence of hydrophobic moieties. In addition, we focus on an ion-conductive pathway hidden in the self-assembled nanostructure. The MD results indicate that the ionic groups of cation and anion interact strongly for keeping the bicontinuous nanosegregation of ionic nanochannel. The partial fractions of hydrophilic/ionic and hydrophobic nanodomains are then quantified with the volume difference from referenced IL systems, while the calculated ionic conductivity decreases in the self-assembled ILCs more than the occupied volume of ionic nanodomains. These analyses suggest that the mobility of ions in the self-assembled ILCs remains quite restricted even with small tetrafluoroborate anions because of strong attractive interaction among ionic moieties.

Control-Oriented Modeling of All-Solid-State Batteries Using Physics-Based Equivalent Circuits

Considered as one of the ultimate energy storage technologies for electrified transportation, the emerging all-solid-state batteries (ASSBs) have attracted immense attention due to their superior thermal stability, increased power and energy densities, and prolonged cycle life. To achieve the expected high performance, practical applications of ASSBs require accurate and computationally efficient models for the design and implementation of many onboard management algorithms, so that the ASSB safety, health, and cycling performance can be optimized under a wide range of operating conditions. A control-oriented modeling framework is thus established in this work by systematically simplifying a rigorous partial differential equation (PDE) based model of the ASSBs developed from underlying electrochemical principles. Specifically, partial fraction expansion and moment matching are used to obtain ordinary differential equation based reduced-order models (ROMs). By expressing the models in a canonical circuit form, excellent properties for control design such as structural simplicity and full observability are revealed. Compared to the original PDE model, the developed ROMs have demonstrated high fidelity at significantly improved computational efficiency. Extensive comparisons have also been conducted to verify its superiority to the prevailing models due to the consideration of concentration-dependent diffusion and migration. Such ROMs can thus be used for advanced control design in future intelligent management systems of ASSBs.

Two-loop leading colour helicity amplitudes for W±γ + j production at the LHC

We present the two-loop leading colour QCD helicity amplitudes for the process pp → W (→ lν)γ + j. We implement a complete reduction of the amplitudes, including the leptonic decay of the W-boson, using finite field arithmetic, and extract the analytic finite remainders using a recently identified basis of special functions. Simplified analytic expressions are obtained after considering permutations of a rational kinematic parametrisation and multivariate partial fractioning. We demonstrate efficient numerical evaluation of the two-loop colour and helicity summed finite remainders for physical kinematics, and hence the suitability for phenomenological applications.

Partial Fraction Expansion of Meromorphic Maps

Partial fraction expansion of a meromorphic function or map is a way of expressing it as an infinite series of rational functions and polynomials. This representation is particularly useful in the summation of series in mathematical analysis and has at its heart the Mittag-Leffler’s expansion theorem which holds for arbitrary domains. In this article, the Mittag-Leffler’s expansion theorem was used to obtain partial fraction expansions of the functions tan⁡z and coth⁡z which are meromorphic in the entire complex plane. The expansion of tan⁡z was then used to obtain the Laurent series for it where except for the odd coefficients, all the even coefficients in the series were shown to vanish identically. By comparing these coefficients with those of the well-known Maclaurin series for tan⁡z the exact sum of the odd terms in the important p-series ∑_(n=1)^∞▒1/n^p (p=2,4,6) were obtained in closed form and consequently, those of the series ∑_(n=1)^∞▒1/n^p (p=2,4,6) for all values of n. The partial fraction expansion of coth⁡z was also gainfully employed in finding the exact sum (in closed form) of the p-series ∑_(n=1)^∞▒1/n^2 and ∑_(n=-∞)^∞▒〖1/(n^2+z^2 ) (z≠ni,n=0,1,2,…)〗 to also show its usefulness in application. Finally, since closed form expressions were obtained for those series found from the partial fraction expansions of the functions considered in this article, we suggest that further research in this area should focus on the application of the Mittag-Leffler’s theorem to other meromorphic functions such as sec⁡z, cot⁡z etc to find similar results on the summation of series as done here.

Relating Mandelbrot and Barnes-Jarvis Power Law Noise State Space Models using Partial Fractions

This paper addresses two state space models for generating approximate power law noise and shows that these models are related through partial fractions in frequency domain.

On the Equivalence between Integer- and Fractional Order-Models of Continuous-Time and Discrete-Time ARMA Systems

The equivalence of continuous-/discrete-time autoregressive-moving average (ARMA) systems is considered in this paper. For the integer-order cases, the interrelations between systems defined by continuous-time (CT) differential and discrete-time (DT) difference equations are found, leading to formulae relating partial fractions of the continuous and discrete transfer functions. Simple transformations are presented to allow interconversions between both systems, recovering formulae obtained with the impulse invariant method. These transformations are also used to formulate a covariance equivalence. The spectral correspondence implied by the bilinear (Tustin) transformation is used to study the equivalence between the two types of systems. The general fractional CT/DT ARMA systems are also studied by considering two DT differential fractional autoregressive-moving average (FARMA) systems based on the nabla/delta and bilinear derivatives. The interrelations CT/DT are also considered, paying special attention to the systems defined by the bilinear derivatives.

New insights into the mantle source of a large igneous province from highly siderophile element and Sr-Nd-Os isotope compositions of carbonate-rich ultramafic lamprophyres

Despite being volumetrically minor components, carbonate-rich ultramafic magmas like aillikites represent good candidates to investigate the compositional variations in plume and/or lithospheric mantle sources because they represent low-degree melts which preferentially sample highly fusible components including recycled crustal material. To gain new insights into the composition of the plume-related magmas and, more broadly, the petrogenesis of ultramafic lamprophyres, we have undertaken the first comprehensive study of bulk rock and mineral (olivine and Ti-magnetite) highly siderophile element (HSE) abundances and Re-Os isotopes combined with in situ major-, trace-element and Sr-Nd isotope analyses of apatite and perovskite from the Permian Wajilitag aillikites of the Tarim large igneous province, China. The Wajilitag aillikites have high PPGE (Pt and Pd) contents relative to IPGE (Os, Ir and Ru), which can be ascribed to low-degree partial melting and/or fractionation of olivine and laurite. Measured 187Os/188Os ratios are moderately to highly radiogenic (0.186–0.313) with age-corrected γOs values up to +113. In situ Sr and Nd isotope analyses of apatite phenocrysts (87Sr/86Sr(i) = 0.70349–0.70384; εNd(i) = +1.3 to +4.9) and fresh perovskite grains (87Sr/86Sr(i) = 0.70340–0.70390; εNd(i) = +1.3 to +3.8) exhibit limited variability both within and across samples from different aillikite dykes and the only volcanic pipe in the area. These Nd isotopic values resemble those from bulk-rock samples (εNd(i) = +1.9 to +5.2), whereas Sr in apatite and perovskite extends to marginally less radiogenic values than the bulk-rock compositions (87Sr/86Sr(i) = 0.70362–0.70432). The moderately depleted Sr-Nd isotope compositions of magmatic apatite and perovskite, and the previously reported mantle-like C isotope values of these samples suggest that the aillikites and their carbon probably derived from a sub-lithospheric (plume) source with minimal contribution of deeply subducted material. Conversely, the radiogenic Os isotope compositions of the Tarim aillikites and separated minerals require some contribution from recycled crustal material in the plume source. Mass balance calculations suggest that the radiogenic Os isotopes and moderately depleted Sr-Nd isotopes can be reproduced by less than one third of eclogite component addition to a moderately depleted mantle source. We conclude that the combination of complementary isotopic systems can enlighten contributions from different components to mantle-derived magmas and, in this case, clarifies the occurrence of carbon-free subducted oceanic crust in the Tarim plume.

Controls on the metallogenesis of the Lhasa–Mozugongka district, Gangdese Belt, Tibetan Plateau: Constraints on melt distribution and viscosity from the 3-D electrical structure of the lithosphere

Some of the largest and most significant Miocene porphyry copper systems in China are within the Gangdese metallogenic belt on the southern Tibetan Plateau. It has been recognized that the crustal architecture and rheology, derived from regional tectonic events, has direct implications for the evolution and transport of fluids and magmas, and thus for the metallogenesis and prospectivity. Using data from a magnetetolluric array, which intersects the Lhasa–Mozugongka district of the Gangdese metal belt, a 3-D electrical resistivity model was generated, with the goal of investigating the tectonic and rheological controls on the magmatic mineral system. The lithospheric temperature distribution was estimated by applying the Arrhenius equation to conductivity profiles generated from 1-D Monte-Carlo models of long-period magnetotelluric data. The conductivity of partial melts in the lower and middle crust (30–60 km depth) was estimated for local conditions by applying the experimentally-derived equation of X. Guo et al. (2018). Subsequently, we estimated the melt fraction required to explain the observed bulk resistivity in each part of the study area. Variations in the effective viscosity of the lower and middle crust were constrained by the electrical resistivity model by applying the empirical relation of Liu and Hasterock (2016). Beneath the Miocene Cu–Mo deposits in the Lhasa terrane, conductive features in the lower and middle crust are attributed to partial melt fractions of more than 5% and viscosity reductions of 1–2 orders of magnitude. These conductive features may represent the signatures of (ore-controlling) melt/fluid migration channels and deeper melt/fluid source regions in the form of extensive crustal reservoirs of partial melt. Based on the interpretations of the geophysical model, and other available geological and geochemical evidence, a model of the metallogenic dynamics of the Miocene porphyry Cu–Mo deposits is proposed. Overall, the study highlights the applicability of electromagnetic geophysical methods to reliably link resistivity structures to melt/fluid transport channels and sources within a mineral system and supports the hypothesis that crustal rheology exerts a major control on the distribution of ore deposits.

New families of mock theta functions and partial fraction decomposition

In 2000, Andrews challenged mathematicians in the 21st century to elucidate the overlap between classes of q-series and modular forms. Motivated by Andrews' challenge, seeking q-hypergeometric series which are mock theta functions in the modern sense and finding their alternative representations have attracted the attention of many mathematicians in recent years. In this paper, we use partial fraction decomposition to give rise to several families of two-parameter mock theta functions and express them in the form of Appell-Lerch sums. In particular, we establish some identities involving several families of mock theta functions and Appell-Lerch sums which imply three identities proved by Cui and Gu.

The neutrosophic integrals by partial fraction

The neutrosophic integrals by partial fraction

On the rational approximation of Markov functions, with applications to the computation of Markov functions of Toeplitz matrices

We investigate the problem of approximating the matrix function f(A) by r(A), with f a Markov function, r a rational interpolant of f, and A a symmetric Toeplitz matrix. In a first step, we obtain a new upper bound for the relative interpolation error 1 − r/f on the spectral interval of A. By minimizing this upper bound over all interpolation points, we obtain a new, simple, and sharp a priori bound for the relative interpolation error. We then consider three different approaches of representing and computing the rational interpolant r. Theoretical and numerical evidence is given that any of these methods for a scalar argument allows to achieve high precision, even in the presence of finite precision arithmetic. We finally investigate the problem of efficiently evaluating r(A), where it turns out that the relative error for a matrix argument is only small if we use a partial fraction decomposition for r following Antoulas and Mayo. An important role is played by a new stopping criterion which ensures to automatically find the degree of r leading to a small error, even in presence of finite precision arithmetic.

Imaging the heat source of the Kangding high-temperature geothermal system on the Xianshuihe fault by magnetotelluric survey

The Kangding geothermal field along the Xianshuihe fault is a typical high-temperature geothermal system with an enormous capacity for geothermal power generation. The Xianshuihe fault is a major active sinistral strike-slip fault in eastern Tibet Plateau, where the fault splits into three parallel branches near Kangding. We collected data from 17 magnetotelluric stations in the Kangding geothermal system. After data processing, the dimensionality of the subsurface structure was qualitatively estimated using the phase tensor. The electrical resistivity model inverted with ModEM shows the Kangding geothermal system from the surface to a depth of 20 km, which correlates closely with the near-surface geology. The synthetic tests confirmed that the main resistivity anomalies identified in the electrical resistivity model were robust. There are several significant low resistivity anomalies of less than 10 Ωm beneath the Kangding geothermal system. The cap rock and liquid-dominated reservoir can be interpreted as shallow conductive zones. The conductors beneath the Kangding geothermal field at a depth of 10 km are interpreted as an underlying zone of partial melt, which provides fluids and heat for the geothermal system. The partial melt fractions are in the range of 4–19%, confirming that there is a magmatic heat source beneath the Kangding geothermal system.

Immunotherapy of COVID-19: Inside and Beyond IL-6 Signalling.

Acting on the cytokine cascade is key to preventing disease progression and death in hospitalised patients with COVID-19. Among anti-cytokine therapies, interleukin (IL)-6 inhibitors have been the most used and studied since the beginning of the pandemic. Going through previous observational studies, subsequent randomised controlled trials, and meta-analyses, we focused on the baseline characteristics of the patients recruited, identifying the most favourable features in the light of positive or negative study outcomes; taking into account the biological significance and predictivity of IL-6 and other biomarkers according to specific thresholds, we ultimately attempted to delineate precise windows for therapeutic intervention. By stimulating scavenger macrophages and T-cell responsivity, IL-6 seems protective against viral replication during asymptomatic infection; still protective on early tissue damage by modulating the release of granzymes and lymphokines in mild-moderate disease; importantly pathogenic in severe disease by inducing the proinflammatory activation of immune and endothelial cells (through trans-signalling and trans-presentation); and again protective in critical disease by exerting homeostatic roles for tissue repair (through cis-signalling), while IL-1 still drives hyperinflammation. IL-6 inhibitors, particularly anti-IL-6R monoclonal antibodies (e.g., tocilizumab, sarilumab), are effective in severe disease, characterised by baseline IL-6 concentrations ranging from 35 to 90 ng/mL (reached in the circulation within 6 days of hospital admission), a ratio of partial pressure arterial oxygen (PaO2) and fraction of inspired oxygen (FiO2) between 100 and 200 mmHg, requirement of high-flow oxygen or non-invasive ventilation, C-reactive protein levels between 120 and 160 mg/L, ferritin levels between 800 and 1600 ng/mL, D-dimer levels between 750 and 3000 ng/mL, and lactate dehydrogenase levels between 350 and 500 U/L. Granulocyte-macrophage colony-stimulating factor inhibitors might have similar windows of opportunity but different age preferences compared to IL-6 inhibitors (over or under 70 years old, respectively). Janus kinase inhibitors (e.g., baricitinib) may also be effective in moderate disease, whereas IL-1 inhibitors (e.g., anakinra) may also be effective in critical disease. Correct use of biologics based on therapeutic windows is essential for successful outcomes and could inform future new trials with more appropriate recruiting criteria.

Wave excitation and dynamics in non-Hermitian disordered systems

Dynamic and steady-state aspects of wave propagation are deeply connected in lossless open systems in which the scattering matrix is unitary. There is then an equivalence among the energy excited within the medium through all channels, the Wigner time delay, which is the sum of dwell times in all channels coupled to the medium, and the density of states. But these equivalences fall away in the presence of material loss or gain. In this paper, we use microwave measurements, numerical simulations, and theoretical analysis to discover the changing relationships among fundamental wave properties with loss and gain, and their dependence upon dimensionality and spectral overlap. We begin with the demonstrations that the transmission time in random 1D media is equal to the density of states even in the presence of ultrastrong absorption and that its ensemble average is independent of the strengths of scattering and absorption. In contrast, the Wigner time becomes imaginary in the presence of loss, with real and imaginary parts that fall with absorption. In multichannel media, the transmission time remains equal to the density of states and is independent of the scattering strength in unitary systems but falls with absorption to a degree that increases with the strengths of absorption and scattering, and the number of channels coupled to the medium. We show that the relationships between key propagation variables in non-Hermitian systems can be understood in terms of the singularities of the phase of the determinant of the transmission matrix. The poles of the transmission matrix are the same as those of the scattering matrix, but the transmission zeros are fundamentally different. Whereas the zeros of the scattering matrix are the complex conjugates of the poles, the transmission zeros are topological: in unitary systems they occur only singly on the real axis or as conjugate pairs. We follow the evolution and statistics of zeros in the complex plane as random samples are deformed. The sensitivity of the spacing of zeros in the complex plane with deformation of the sample has a square-root singularity at a zero point at which two single zeros and a complex pair interconvert. The transmission time is a sum of Lorentzian functions associated with poles and zeros. The sum over poles is the density of states with an average that is independent of scattering and dissipation. But the sum over zeros changes with loss, gain, scattering strength and the number of channels in ways that make it possible to control ultranarrow spectral features in transmission and transmission time. We show that the field, including the contribution of the still coherent incident wave, is a sum over modal partial fractions with amplitudes that are independent of loss and gain. The energy excited may be expressed in terms of the resonances of the medium and is equal to the dwell time even in the presence of loss or gain.

LIPASE-CATALYZED TRANSESTERIFICATION OF MEDIUM-LONG-MEDIUM STRUCTURED LIPID (MLM-SL) USING PALM OLEIN AND PALM KERNEL OIL IN BATCH AND CONTINUOUS SYSTEMS

Lipase-catalyzed transesterification between refined bleached deodorized palm olein (RBDO) and palm kernel oil (RBDPKO) has been investigated to produce medium-long-medium structured lipid (MLM-SL). The synthesis was catalyzed by a specific sn-1,3 commercial lipase (Lipozyme TL IM) in batch and continuous systems. Progress of the transesterification of this study was monitored as triacylglycerol (TAG) with equivalent carbon number (ECN) 40, presumably that of 1,3-dilauryl-2-oleoyl-sn-glycerol (LaOLa). The results showed that lipase-catalyzed transesterification using RBDO and RBDPKO could potentially be used as the main substrate for MLM-SL synthesis, both for batch and continuous systems. In batch system, transesterification of RBDO and RBDPKO at the ratio of 1:2 at 50oC yielded in the highest concentration of ECN 40 (LaLaO/LaOLa, 7.34%) but also a higher total concentration of partial acylglycerol fractions (Di- and Monoacylglycerols; DAGs and MAGs). Thus, this condition also obtained transesterified lipid rich in EC N40 with a lower slip melting point as compared to other substrate ratios. In a continuous system, transesterification at RDBO:RBDPKO of 1:1 at 50oC and 15 min of residence time were selected as the optimum conditions, resulting in 5.39% EC N40 with a minimum concentration of DAGs and MAGs.