Integrating Green Renewable Energy Sources (GRES) as substitutes for fossil fuel-based energy sources is essential for reducing harmful emissions. The GRES are intermittent and their integration into the conventional IEEE 30 bus configuration increases the complexity and nonlinearity of the system. The Grey Wolf optimizer (GWO) has excellent exploration capability but needs exploitation capability to enhance its convergence speed. Adding particle swarm optimization (PSO) with excellent convergence capability to GWO leads to the development of a novel algorithm, namely a Grey Wolf particle swarm optimization (GWPSO) algorithm with excellent exploration and exploitation capabilities. This study utilizes the advantages of the GWPSO algorithm to solve the optimal power flow (OPF) problem for adaptive IEEE 30 bus systems, including thermal, solar photovoltaic (SP), wind turbine (WT), and small hydropower (SHP) sources. Weibull, Lognormal, and Gumbel probability density functions (PDFs) are employed to forecast the output power of WT, SP, and SHP power sources after evaluating 8000 Monte Carlo possibilities, respectively. The multi-objective green economic optimal solution consisted of 11 control variables to reduce the cost, power losses, and harmful emissions. The proposed method to address the OPF problem is validated using an adaptive IEEE bus system. The proposed GWPSO algorithm is evaluated by comparing it with PSO and GWO optimization algorithms in terms of achieving an optimal green economic solution for the adaptive IEEE 30 bus system. This evaluation is conducted within the confines of the same test system using identical system constraints and control variables. The integration of a small SHP with WT and SP sources, along with the proposed GWPSO algorithm, led to a yearly cost reduction ranging from $19,368 to $30,081. Simulation findings endorsed that the proposed GWPSO algorithm executes fruitfully compared to alternative algorithms regarding a consistent convergence curve and robustness, proving its potential as a viable choice for achieving cost-effective solutions in power systems incorporating GRES.
Citation: Hisham Alghamdi, Lyu-Guang Hua, Muhammad Riaz, Ghulam Hafeez, Safeer Ullah, Monji Mohamed Zaidi, Mohammed Jalalah. An optimal power flow solution for a power system integrated with renewable generation[J]. AIMS Mathematics, 2024, 9(3): 6603-6627. doi: 10.3934/math.2024322
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Integrating Green Renewable Energy Sources (GRES) as substitutes for fossil fuel-based energy sources is essential for reducing harmful emissions. The GRES are intermittent and their integration into the conventional IEEE 30 bus configuration increases the complexity and nonlinearity of the system. The Grey Wolf optimizer (GWO) has excellent exploration capability but needs exploitation capability to enhance its convergence speed. Adding particle swarm optimization (PSO) with excellent convergence capability to GWO leads to the development of a novel algorithm, namely a Grey Wolf particle swarm optimization (GWPSO) algorithm with excellent exploration and exploitation capabilities. This study utilizes the advantages of the GWPSO algorithm to solve the optimal power flow (OPF) problem for adaptive IEEE 30 bus systems, including thermal, solar photovoltaic (SP), wind turbine (WT), and small hydropower (SHP) sources. Weibull, Lognormal, and Gumbel probability density functions (PDFs) are employed to forecast the output power of WT, SP, and SHP power sources after evaluating 8000 Monte Carlo possibilities, respectively. The multi-objective green economic optimal solution consisted of 11 control variables to reduce the cost, power losses, and harmful emissions. The proposed method to address the OPF problem is validated using an adaptive IEEE bus system. The proposed GWPSO algorithm is evaluated by comparing it with PSO and GWO optimization algorithms in terms of achieving an optimal green economic solution for the adaptive IEEE 30 bus system. This evaluation is conducted within the confines of the same test system using identical system constraints and control variables. The integration of a small SHP with WT and SP sources, along with the proposed GWPSO algorithm, led to a yearly cost reduction ranging from $19,368 to $30,081. Simulation findings endorsed that the proposed GWPSO algorithm executes fruitfully compared to alternative algorithms regarding a consistent convergence curve and robustness, proving its potential as a viable choice for achieving cost-effective solutions in power systems incorporating GRES.
Credit rating migration risk is an important kind of credit risk, referring to the possibility of potential losses due to changes in credit rating. The phenomenon of rating movements has long attracted the attention of academia and industry and there have been many studies on the performance and patterns of credit rating transitions [1,2,3,4,5,6]. As for the valuation of credit migration risk, the used models are mainly categorized into two types: the reduced-form model and the structural model. The former employs a transition density matrix to characterize the process of credit rating migrations, such as [7,8,9] while the latter utilizes the company's own financial status as the determinant of credit rating changes, such as [10,11,12,13]. Each kind of model has its advantages, but the structural model elucidates the mechanism of credit migration by explicitly linking credit ratings to the company's assets and liabilities.
Liang and Zeng ([10], 2015) constructed the first structural model for assessing credit migration risk by pricing a corporate bond with this risk. By predefining an asset threshold as the boundary for credit migration, the company's asset values are divided into high and low rating regions. In different regions, the asset values follow geometric Brownian motions with different volatilities. The model was derived as an initial value problem of parabolic differential equations which are coupled on a fixed inner boundary. Further, Hu et al. ([11], 2015) took the asset-liability ratio as a threshold for credit migration and deduced the migration boundary as a free boundary. On these bases, more theoretical analyses are provided, such as the asymptotic traveling wave solution [14], convergence rate of the difference scheme [15], multi-credit rating case [16], steady-state solution [17], etc (see also [18,19,20,21,22]). There are also some empirical results for the single threshold model, for instance, the credit migration boundary was identified by pricing long-term bonds in the U.S. corporate bond market [23]. In those models with the fixed boundaries or free boundaries, the thresholds for upgrades and downgrades are the same. However, this single threshold may lead to infinite frequent changes in credit ratings within a short period, due to the assumption that asset values follow Brownian motions which can cross any level infinitely many times within any time interval. Chen and Liang ([12], 2021) made an improvement based on the free boundary model by introducing different asset-liability ratio thresholds for upgrades and downgrades, resulting in a pair of migration boundaries. As a buffer zone is formed between these two boundaries, the frequency of credit rating migrations per unit time becomes finite. Liang and Lin ([13], 2023) made a similar modification, applying a pair of asymmetric asset thresholds to the fixed boundary model, and obtaining a system of partial differential equations coupled with each other on a pair of fixed boundaries. Liang and Lin ([24], 2023) further explored an asymptotic traveling wave solution with a buffer zone for this kind of model.
These structural credit migration models with thresholds are somewhat similar in form to the credit barrier model considered in Albanese and Chen ([3], 2006). A stochastic process with state-dependent volatilities was used to model the credit quality process, which directly indicated the dynamic of credit rating, with barrier crossings corresponding to credit migrations and default events, in Albanese and Chen ([3], 2006). The variable of credit quality might capture the firm's fundamental information but did not articulate specific implications in finance. In the structural models of Liang et al., the credit migrations were delineated by the crossings of the thresholds as well, with the driving factors explicitly identified as the asset value or the asset-liability ratio. Since the asset value was assumed to follow the Brownian motion with different volatilities in different ratings, the distribution of future asset value depends on only the current rating and value of asset, and not on the history of previous states, even in the structural models with buffer zones. Therefore, these models we analyze in this paper cannot capture non-Markov effects in rating transition probabilities, such as those identified in Lando and Skødeberg ([4], 2002), Nickell et al. ([6], 2000), etc.
According to whether or not the upgrade threshold and downgrade threshold are the same, the structural models are classified into the single threshold model and the model with different upgrade and downgrade thresholds. We already know that substituting a pair of asymmetric thresholds for a single threshold avoids high-frequency fluctuations in credit ratings. However a natural question arises: What is the relationship between those two kinds of models? Intuitively, when upgrade threshold and downgrade threshold are very close, the asymmetric threshold model's behavior should be similar to that of a single threshold model. This article attempts to answer this question rigorously within the scope of the credit migration problem with fixed boundaries. We consider two fixed boundary models: one model from [10] with X(X>0) as a critical threshold for credit migration, and the other model from [13] with X as a threshold for downgrades, and Xε:=X+ε (depending on a small parameter ε>0) as a threshold for upgrades. By techniques of partial differential equations, we prove that the solution of the asymmetric threshold problem converges to that of the single threshold problem when the upgrade threshold Xε approaches the downgrade threshold X as ε→0. To our knowledge, it is the first time that the asymptotic relationship between the single threshold and a pair of asymmetric thresholds has been shown in credit migration problems. Therefore, when ε is small enough, the different upgrade and downgrade threshold model can be approximated by a single threshold model, which already has more theoretical and empirical results.
The remainder of the paper is organized as follows. In Section 2, the single threshold and the different downgrade and upgrade threshold model for credit migration problems with fixed boundaries are reviewed. In Section 3, we prove a series of lemmas to establish ε-independent estimates for the solution of the asymmetric threshold problem. In Section 4, a key step is shown that the first-order derivatives of the asymmetric threshold problem's solution on both sides of x=X tend to be equal, as ε approaches zero. It follows that the solution of this model converges to that of a single threshold model by compactness. Section 5 is a summary of this paper.
The structural models with fixed migration boundaries assess credit rating migration risk by pricing a zero-coupon corporate bond.
Let (Ω,F,P) be a complete probability space. Assuming that the company only issues one bond with a face value of 1 and a maturity of T, the bond is considered as a contingent claim of the company's asset value on the space (Ω,F,P). At maturity, the bond will default and pay out the remaining assets if the company's asset value is less than the face value of the bond. Let St denote the company's asset value in the risk-neutral world. It satisfies
dSt={rStdt+σHStdWt, in high rating region, rStdt+σLStdWt, in low rating region, |
where r is the risk-free interest rate, and
σH<σL | (2.1) |
represent volatilities (positive constants) of the company under the high and low credit grades respectively. Wt is the standard Brownian motion which generates the filtration {Ft}. High and low rating regions are determined by the company's asset value. Inequality (2.1) captures the characteristics of the high and low credit ratings, with the volatility of asset return in the lower rating region being greater than the volatility in the higher rating region.
Liang and Zeng ([10], 2015) (referred to as LZ's model hereafter) gave a single predetermined threshold to divide asset value into high and low rating regions. Through a standard variable transformation x=logS and renaming T−t as t, we set the migration boundary as X(X>0) and represented the values of low and high-rated bonds as vL(x,t) and vH(x,t), respectively. The model is derived as the following partial differential equation problem:
{∂vH∂t−12σ2H∂2vH∂x2−(r−12σ2H)∂vH∂x+rvH=0,for x>X,t>0,∂vL∂t−12σ2L∂2vL∂x2−(r−12σ2L)∂vL∂x+rvL=0,for x<X,t>0,vH(x,0)=min{ex,1},for x>X,vL(x,0)=min{ex,1},for x<X,vH(X,t)=vL(X,t),for t>0,∂vH∂x(X,t)=∂vL∂x(X,t),for t>0. | (2.2) |
We define a solution v over the entire region (−∞,∞)×[0,∞) as follows:
v={vH(x,t), forx≥X,t≥0,vL(x,t), forx<X,t≥0. | (2.3) |
Liang and Lin ([13], 2023) (referred to as LL's model hereafter) proposed a pair of asymmetric thresholds for credit migrations: one threshold for downgrades and the other slightly higher threshold for upgrades. After the same change of variables as in Section 2.1, we have set the downgrade threshold as X(X>0) and the upgrade threshold as Xε:=X+ε,ε>0. Denote by uLε(x,t) and uHε(x,t) the bond values in low rating and high rating, respectively. They are captured by the following PDE problem:
{∂uHε∂t−12σ2H∂2uHε∂x2−(r−12σ2H)∂uHε∂x+ruHε=0,for x>X,t>0,∂uLε∂t−12σ2L∂2uLε∂x2−(r−12σ2L)∂uLε∂x+ruLε=0,for x<Xε,t>0,uHε(x,0)=min{ex,1},for x>X,uLε(x,0)=min{ex,1},for x<Xε,uHε(X,t)=uLε(X,t),for t>0,uLε(Xε,t)=uHε(Xε,t),for t>0. | (2.4) |
Note that vLε and vHε overlap in [X,Xε]×[0,∞). To be comparable with v, we rearrange uε over the entire region as the following formula:
uε={uHε(x,t), forx≥X,t≥0,uLε(x,t), forx<X,t≥0. | (2.5) |
Letting the upgrade threshold Xε approach the downgrade threshold X, we find an asymptotic behavior of LL's model, i.e., as ε→0, uε converges to v. This actuallly constructs an asymptotic relationship between LL's model and LZ's model. To prove this, we first establish some ε-independent estimates for the solution (uLε,uHε). Based on this, |uLεx(X−,t)−uHεx(X+,t)| approaching 0 as ε→0 is verified as a key step. Further, we suggest that any convergent subsequence of uε tends to the solution v of LZ's model as ε→0.
Define Li=∂∂t−12σi2∂2∂x2−(r−12σi2)∂∂x+r⋅,i=H,L; QLε=(−∞,Xε)×(0,∞), and QH=(X,∞)×(0,∞).
For any given ε, we iteratively define a sequence {uLεk,uHεk}∞k=0, which is proved to decrease with k and converge to (uLε,uHε) as k→0 in Liang and Lin ([13], 2023). In detail, the sequence satisfies equations
LL[uLεk]=0 in QLε, LH[uHεk]=0 in QH, |
with the initial value min{ex,1}. Starting from uLε0(Xε,t)≡1 and by the induction assumption uHεk(X,t)=uLεk(X,t) and uLε(k+1)(Xε,t)=uHεk(Xε,t), we have completed the definition of the sequence.
The estimations of the maximum norm of (uLε,uHε) and its first-order derivative with respect to time t are carried out by induction on the sequence {uLεk,uHεk}∞k=0. As for the boundary estimate of (uLεx,uHεx), we introduce barrier functions.
Lemma 3.1.
0≤uLε≤min{ex,1} in QLε, 0≤uHε≤min{ex,1} in QH. | (3.1) |
Proof. By the induction and maximum principle, for each k≥0, 0≤uHεk≤min{ex,1} in QH, and 0≤uLεk≤min{ex,1} in QLε.
Lemma 3.2.
−C1≤∂uLε∂t≤0 in QLε∖¯Qρ, −C1≤∂uHε∂t≤0 in QH, | (3.2) |
where Qρ=(−ρ/2,ρ/2)×(0,ρ2/4) and 0<ρ<X.
Proof. We claim that −C1≤uLεkt,uHεkt≤0 for any k∈N and C1 is independent of ε and k.
Differentiating uLεk and uHεk with respect to t, they satisfy
LL[uLεkt]=0 in QLε, LH[uLεkt]=0 in QH. |
On the migration boundaries, uLεkt(Xε,t)=uHε(k−1)t(Xε,t) and uHεkt(X,t)=uLεkt(X,t).
It is also clear that initially
uLεkt(x,0)=0 forx<0,uLεkt(x,0)=−r for0<x<Xε, | (3.3) |
uHεkt(x,0)=−r forx>X. | (3.4) |
At x=0,uLεkxx(x,0) produces a Dirac measure of density −1. Thus, in the distribution sense,
uLεkt(x,0)≤0 forx<Xε. | (3.5) |
From the standard parabolic estimates (see e.g. [25]), there exists constants c1,c2 independent of ε and k, such that
uLεkt≥−c2−c2√texp(−c1x2t) for|x|<ρ2,0<t≤ρ24, |
where 0<ρ<X. Take C1≥r such that
C1≥c2+c2√texp(−c1x2t) on {|x|=ρ2,0<t≤ρ24}∪{|x|<ρ2,t=ρ24}. |
It follows that on {x≤−ρ2,t=0}∪{|x|=ρ2,0<t≤ρ24}∪{|x|<ρ2,t=ρ24}∪{ρ2≤x≤X,t=0},
uLεkt≥−C1. | (3.6) |
When k=0, uε0(Xε,t)=1 and we have uε0t(Xε,t)=0 on x=Xε for t>0.
By further approximating the initial data with smooth functions if necessary, we conclude by (3.5) and maximum principle that
uLε0t≤0 in QLε. |
We conclude by (3.6) and minimum principle that
uLε0t≥−C1 in QLε∖¯Qρ, |
where Qρ=(−ρ/2,ρ/2)×(0,ρ2/4).
We assume that −C1≤uLεkt≤0 in QLε∖¯Qρ holds for k≥1. Since uHεkt(X,t)=uLεkt(X,t) for t>0 and (3.4), we conclude by extremum principle
−C1≤uHεkt≤0 in QH. |
From uLε(k+1)t(Xε,t)=uHεkt(Xε,t) and (3.5), (3.6), applying the extremum principle gives
−C1≤uLε(k+1)t≤0 in QLε∖¯Qρ. |
Thus, by induction, we derive that −C1≤uLεkt≤0 in QLε∖¯Qρ and −C1≤uHεkt≤0 in QH for any k≥0 and C1 is independent of ε and k. The lemma's results can be obtained by taking limits with k.
Lemma 3.3.
|∂uLε∂x|≤C2 in QLε, |∂uHε∂x|≤C3 in QH. | (3.7) |
Proof. We estimate uLεx as an example and the same process works for uHεx.
We fixed ε and first deal with the uLεx(Xε,t). Let K1(ρ<K1<X) be a constant and η=X−K1. Define wLε=uLε−Xε−xXε−K1uLε(K1,t)−x−K1Xε−K1uLε(Xε,t). Thus, wLε(K1,t)=wLε(Xε,t)=0 for t>0 and wLε(x,t)=0 for K1≤x≤Xε. In QLεK1={K1<x<Xε,t>0},
LL0wLε=−ruLε−Xε−xXε−K1uLεt(K1,t)−x−K1Xε−K1uLεt(Xε,t)+(r−12σ2L)uLε(Xε,t)−uLε(K1,t)Xε−K1, |
where Li0=∂∂t−12σi2∂2∂x2−(r−12σi2)∂∂x. Actually, LL0wLε can be bounded by a number G independent of ε,
supQLεK1|LL0wLε|≤rsupQLεK1|uLε|+supQLεK1|uLεt|+2r+σ2LηsupQLεK1|uLε|≤r+C1+2r+σ2Lη≜G. |
In the case σ2L≠2r, we introduce the function
zLε(x,t)=2Cσ2L−2r(1−exp((1−2rσ2L)(x−Xε)))+2Gσ2L−2r(x−Xε), |
where C is a constant to be determined later. Obviously, LL0zLε=G≥LL0wLε in QLεK1 and zLε(Xε,t)=0 for t>0. To make
zLεx=−2Cσ2Lexp((1−2rσ2L)(x−Xε))+2Gσ2L−2r≤0 |
hold for K1<x<Xε,t>0, we choose C=σ2LGσ2L−2rexp((1−2rσ2L)(η+ε)). Thus, on the parabolic boundary of QLεK1 we clearly have zLε≥±wLε. It follows by comparison principle that
zLε≥±wLε in QLεK1. |
Since zLε(Xε,t)=±wLε(Xε,t)=0,
±(wLε(x,t)−wLε(Xε,t))Xε−x≤zLε(x,t)−zLε(Xε,t)Xε−x |
for x<Xε. Letting x→Xε, we have
|wLεx(Xε,t)|≤−zLεx(Xε,t)=−2Gσ2L−2r(1−exp((1−2rσ2L)(η+ε))). |
Then, by uLεx(Xε,t)=wLεx(Xε,t)+[uLε(Xε,t)−uLε(K1,t)]/(Xε−K1) there exists a constant C2>1 independent of ε such that
|uLεx(Xε,t)|≤−2Gσ2L−2r(1−exp((1−2rσ2L)(η+1)))+2/η≤C2, |
when ε<1.
In the case σ2L=2r, we can introduce the function
zLε(x,t)=−Gσ2L(x−Xε)2−2Gσ2L(η+ε)(x−Xε), |
then by a similar analysis as above, we can conclude that there exists a number C2>1 independent of ε such that
|uLεx(Xε,t)|≤2Gσ2L(η+1)+2/η≤C2, |
when ε<1.
It is known that uLεx(x,0)=ex for x<0 and uLεx(x,0)=0 for 0<x<Xε. Thus it follows by maximum principle that |uLεx(x,t)|≤C2 in QLε when ε<1.
Lemma 3.4.
|∂2uLε∂x2|≤C4 in QLε∖¯Qρ, |∂2uHε∂x2|≤C5 in QH. | (3.8) |
Proof. The corollary of Lemmas 3.1–3.3.
In this section, we establish the asymptotic relationship between LL's model and LZ's model, when Xε approaches X. A key lemma is proved to show |uLεx−uHεx|→0 on x=X along ε→0. Then, by compact embedding theorem, we deduce that there exists a subsequence uεj converging a function u which is examined as a solution of the single threshold problem (2.2). By the uniqueness of the solution, we obtain v≡u and uε→v uniformly as ε→0.
Lemma 4.1. As ε→0,
|∂uLε∂x(X−,t)−∂uHε∂x(X+,t)|→0 | (4.1) |
uniformly for 0≤t≤T.
Proof. Let
hε(t)=uLε(X+ε,t)−uLε(X,t)ε=uHε(X+ε,t)−uHε(X,t)ε, |
for 0≤t≤T. By the mean value theorem, there is μ1(t)∈[0,ε] such that hε(t)=uLεx(X+μ1(t),t), then,
|uLεx(X+,t)−hε(t)|=|uLεx(X,t)−uLεx(X+μ1(t),t)|=|uLεxx(X+μ2(t),t)|μ1(t)≤εsupQLε∖¯Qρ|uLεxx|, |
where 0≤μ2(t)≤μ1(t). The second equal sign holds by the mean value theorem. Similarly, we can derive
|uHεx(X+,t)−hε(t)|≤εsupQH|uHεxx|. |
Considering the continuity of uLεx across x=X, we have for 0≤t≤T,
|uLεx(X−,t)−uHεx(X+,t)|=|uLεx(X+,t)−uHεx(X+,t)|=|uLεx(X+,t)−hε(t)+hε(t)−uHεx(X+,t)|≤ε(C4+C5). |
As C4,C5 are independent of the small parameter ε, the result is seen by letting ε→0.
Theorem 4.1. As ε→0, uε(x,t) tends to v(x,t) uniformly in (−∞,∞)×[0,T].
Proof. Treat uLε|x≤X as uLε restricted on (−∞,X]×[0,T]. From Lemmas 3.1–3.4, uLε|x≤X is bounded in W2,1∞((−∞,X)×[0,T]∖¯Qρ). By the compact embedding theorem, there exists uL and a subsequence εj of ε such that, as εj→0,
uLεj|x≤X→uL in C1+α,1+α2(([−A,X]×[0,T])∖Qρ), 0<α<1, | (4.2) |
for any A>1 and ρ<X. Similarly, uHε is bounded in W2,1∞((X,∞)×[0,T]) and there exists uH such that
uHεj→uH in C1+α,1+α2([X,A]×[0,T]), | (4.3) |
along a subsequence of εj if necessary.
Thus, it is clear that
uLεjx(X−,t)→uLx(X−,t), uHεjx(X+,t)→uHx(X+,t), | (4.4) |
uniformly for 0≤t≤T. By Lemma 4.1, we conclude that
uLx(X−,t)=uHx(X+,t). | (4.5) |
It then can be verified that u=(uL,uH) is a solution of the problem (2.2). By the uniqueness of the solution to this problem, we have v≡u. This implies that any convergent subsequence of uLε|x≤X or uHε has the same limit and uLε|x≤X or uHε converges as ε→0.
It follows that as ε→0,
uLε(X−,t)→vL(X−,t), uHε(X+,t)→vH(X+,t), | (4.6) |
uniformly for 0≤t≤T. Since uLε(x,0)|x≤X=vL(x,0) and uHε(x,0)=vH(x,0), implying the maximum norm estimation in their regions, respectively, gives that
uLε|x≤X→vL uniformly in (−∞,X]×[0,T], | (4.7) |
uHε→vH uniformly in [X,∞)×[0,T], | (4.8) |
as ε→0.
By PDE techniques, we showed an asymptotic relationship between two kinds of structural models for credit migration problems with fixed boundaries. When the downgrade threshold was locked and the upgrade threshold approached to it, the solution of the model with a pair of asymmetric thresholds converged to that of the single threshold model in which the downgrade threshold serves as a unique migration threshold. Symmetrically, by fixing the upgrade threshold and moving the downgrade threshold, a similar conclusion will be obtained. As far as we know, it was the first time that the relationship between the single threshold and a pair of asymmetric thresholds was studied in credit migration problems. This may contribute to generalizing existing research results by using a single threshold model to approximate models with different upgrade and downgrade thresholds.
The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.
This work was supported by the National Natural Science Foundation of China (Grant No.12071349).
All authors declare no conflicts of interest in this paper.
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