Loading [MathJax]/jax/output/SVG/jax.js
Review Topical Sections

AI-Driven precision in solar forecasting: Breakthroughs in machine learning and deep learning

  • These authors contributed equally to this study.
  • The need for accurate solar energy forecasting is paramount as the global push towards renewable energy intensifies. We aimed to provide a comprehensive analysis of the latest advancements in solar energy forecasting, focusing on Machine Learning (ML) and Deep Learning (DL) techniques. The novelty of this review lies in its detailed examination of ML and DL models, highlighting their ability to handle complex and nonlinear patterns in Solar Irradiance (SI) data. We systematically explored the evolution from traditional empirical, including machine learning (ML), and physical approaches to these advanced models, and delved into their real-world applications, discussing economic and policy implications. Additionally, we covered a variety of forecasting models, including empirical, image-based, statistical, ML, DL, foundation, and hybrid models. Our analysis revealed that ML and DL models significantly enhance forecasting accuracy, operational efficiency, and grid reliability, contributing to economic benefits and supporting sustainable energy policies. By addressing challenges related to data quality and model interpretability, this review underscores the importance of continuous innovation in solar forecasting techniques to fully realize their potential. The findings suggest that integrating these advanced models with traditional approaches offers the most promising path forward for improving solar energy forecasting.

    Citation: Ayesha Nadeem, Muhammad Farhan Hanif, Muhammad Sabir Naveed, Muhammad Tahir Hassan, Mustabshirha Gul, Naveed Husnain, Jianchun Mi. AI-Driven precision in solar forecasting: Breakthroughs in machine learning and deep learning[J]. AIMS Geosciences, 2024, 10(4): 684-734. doi: 10.3934/geosci.2024035

    Related Papers:

    [1] Hanpeng Gao, Yunlong Zhou, Yuanfeng Zhang . Sincere wide τ-tilting modules. Electronic Research Archive, 2025, 33(4): 2275-2284. doi: 10.3934/era.2025099
    [2] Lingzheng Kong, Haibo Chen . Normalized solutions for nonlinear Kirchhoff type equations in high dimensions. Electronic Research Archive, 2022, 30(4): 1282-1295. doi: 10.3934/era.2022067
    [3] Shujie Bai, Yueqiang Song, Dušan D. Repovš . On p-Laplacian Kirchhoff-Schrödinger-Poisson type systems with critical growth on the Heisenberg group. Electronic Research Archive, 2023, 31(9): 5749-5765. doi: 10.3934/era.2023292
    [4] Changmu Chu, Jiaquan Liu, Zhi-Qiang Wang . Sign-changing solutions for Schrödinger system with critical growth. Electronic Research Archive, 2022, 30(1): 242-256. doi: 10.3934/era.2022013
    [5] Bo Chen, Junhui Xie . Harnack inequality for a p-Laplacian equation with a source reaction term involving the product of the function and its gradient. Electronic Research Archive, 2023, 31(2): 1157-1169. doi: 10.3934/era.2023059
    [6] Xing Yi, Shuhou Ye . Existence of solutions for Kirchhoff-type systems with critical Sobolev exponents in R3. Electronic Research Archive, 2023, 31(9): 5286-5312. doi: 10.3934/era.2023269
    [7] Chungen Liu, Huabo Zhang . Ground state and nodal solutions for fractional Kirchhoff equation with pure critical growth nonlinearity. Electronic Research Archive, 2021, 29(5): 3281-3295. doi: 10.3934/era.2021038
    [8] Xinning Niu, Huixin Liu, Dan Li, Yan Yan . Positive periodic solutions for discrete Nicholson system with multiple time-varying delays. Electronic Research Archive, 2023, 31(11): 6982-6999. doi: 10.3934/era.2023354
    [9] Xia Su, Wen Guan, Xia Li . Least energy sign-changing solutions for Kirchhoff-Schrödinger-Poisson system on bounded domains. Electronic Research Archive, 2023, 31(5): 2959-2973. doi: 10.3934/era.2023149
    [10] Shiyong Zhang, Qiongfen Zhang . Normalized solution for a kind of coupled Kirchhoff systems. Electronic Research Archive, 2025, 33(2): 600-612. doi: 10.3934/era.2025028
  • The need for accurate solar energy forecasting is paramount as the global push towards renewable energy intensifies. We aimed to provide a comprehensive analysis of the latest advancements in solar energy forecasting, focusing on Machine Learning (ML) and Deep Learning (DL) techniques. The novelty of this review lies in its detailed examination of ML and DL models, highlighting their ability to handle complex and nonlinear patterns in Solar Irradiance (SI) data. We systematically explored the evolution from traditional empirical, including machine learning (ML), and physical approaches to these advanced models, and delved into their real-world applications, discussing economic and policy implications. Additionally, we covered a variety of forecasting models, including empirical, image-based, statistical, ML, DL, foundation, and hybrid models. Our analysis revealed that ML and DL models significantly enhance forecasting accuracy, operational efficiency, and grid reliability, contributing to economic benefits and supporting sustainable energy policies. By addressing challenges related to data quality and model interpretability, this review underscores the importance of continuous innovation in solar forecasting techniques to fully realize their potential. The findings suggest that integrating these advanced models with traditional approaches offers the most promising path forward for improving solar energy forecasting.



    Neuronal activities generate the electrical current in the brain, and further result in the potential changes over the scalp. Electroencephalography (EEG) is a technique used to record the potential changes on the scalp. Even though fMRI, PET, MEG and other brain-imaging tools are widely used in brain research, they are limited by low spatial/temporal resolution, cost, mobility and suitability for long-term monitoring. For example, fMRI has the advantage of providing spatially-resolved data, but suffers from an ill-posed temporal inverse problem, i.e., a map with regional activations does not contain information about when and in which order these activations have occurred [1]. In contrast, EEG signals have been successfully used to obtain useful diagnostic information (neural oscillations and response times) in clinical contexts. Further, they present the advantage to be highly portable, inexpensive, and can be acquired at the bedside or in real-life environments with a high temporal resolution. Because of the lack of significant patient risks, EEG is additionally suited for long-term monitoring.

    EEG offers the possibility of measuring the electrical activity of neuronal cell assemblies on the sub-millisecond time scale [2,3,4]. EEG source imaging further identifies the positions or distributions of electric fields based on EEG signals collected on the scalp [5]. This new tool is widely used in cognitive neuroscience research, and has also found important applications in clinical neuroscience such as neurology, psychiatry and psychopharmacology [6,7]. In cognitive neuroscience, the majority of the studies investigate the temporal aspects of information processing by analyzing event related potentials (ERP). In neurology, the study of sensory or motor evoked potentials is of increasing interest, but the main clinical application concerns with the localization of epileptic foci. In psychiatry and psychopharmacology, a major focus of interest is the localization of sources of certain EEG frequency bands. Localizing the activity sources of a given scalp EEG measurement is achieved by solving the so-called inverse problem [8]. These kinds of inverse problems are usually ill-posed and their solutions are non-unique [9,10].

    Leahy et al. [11] investigated the accuracy of forward and inverse techniques for EEG and MEG dipole localization using a human skull phantom. El Badia and Ha-Duong [12] established an algebraic method to identify the number, locations and moments of electrostatic dipoles in 2D or 3D domain from the Cauchy data on the boundary. Chafik et al. [13] further provided an error estimate without proof. Nara and Ando [14] provided a new projective method for 3D source reconstruction by projecting the sources onto a Riemann sphere. Kang and Lee [15] proposed an algorithm for solving the inverse source problem of a meromorphic function and apply their method to an electrical impedance tomography (EIT) problem. El Badia [16] established a uniqueness result and a local Lipschitz stability estimate for an anisotropic elliptic equation, assuming that the sources are a linear combination of a finite number of monopoles and dipoles. The author also proposed a global Lipschitz stability estimate for dipolar sources. Baratchart et al. [17] solved the inverse source problem by locating the singularities of a meromorphic function from the 2D boundary measurements using best rational or meromorphic approximations.

    Chung and Chung [18] proposed an algorithm for detecting the combination of monopolar and multipolar point sources for elliptic equations in the 2D domain from the Neumann and Dirichlet boundary data. Kandasmamy et al. [19] proposed a novel technique, called "analytic sensing", to estimate the positions and intensities of point sources in 2D for a Poisson's equation. Analytic sensing also used the reciprocity gap principle, but with a novel design of an analytic function which behaved like a sensor. The authors evaluated their estimation accuracy by Cramér-Rao lower bound. Nara and Ando [20] proposed an algebraic method to localize the positions of multiple poles in meromorphic function field from an incomplete boundary. They investigated the accuracy of the algorithm for the open arc or the closed arc, and for the arc enclosing the poles or not enclosing the poles. El Badia and Nara [21] established the uniqueness and local stability result for the inverse source problem of the Helmholtz equation in an interior domain, assuming the source is composed of multiple point sources.

    Clerc et al. [22] applied best rational approximation techniques in the complex plane to EEG source localization and offered stability estimates. Mdimagh and Ben Saad [23] identified the point sources in a scalar problem modeled by Helmholtz equation, using reciprocity gap principle and assuming the sources are harmonic in time. They proved local Lipschitz stability by two methods: one was derived from the Gâteaux differentiability, and the other used particular test functions in the reciprocity gap functional. Vorwerk et al. [24] studies the important role of head tissue conductivity in EEG dipole reconstruction. Rubega et al. [25] estimated EEG source dipole orientation based on singular-value decomposition. Michel and Brunet provided a thorough review on EEG source imaging. There exist several reconstruction methods, such as minimum norm estimates (MNE) [26], low resolution electrical tomography (LORETA) [27,28] or multiple-signal classification algorithm (MUSIC) [29,30], etc.

    Recently, Muñoz-Gutiérrez et al. [31] managed to improve the accuracy of EEG source reconstruction by decomposing the EEG signals into frequency bands with different methods, such as empirical mode decomposition (EMD) and wavelet transform (WT). Kaur et al. [32] presented a new method of EEG source localization using variational mode decomposition (VMD) and standardized the low resolution brain electromagnetic tomography (sLORETA) inverse model. Their VMD-sLORETA model could locate EEG sources in the brain in a very accurate way. Oikonomou and Kompatsiaris [33] developed a novel Bayesian approach for EEG source localization. They incorporated a new sparse prior for the localization of EEG sources with the variational Bayesian (VB) framework and obtained more accurate localization of EEG sources than state-of-the-art approaches.

    In our study we need new methods to detect small changes in EEG source for which dipole methods have advantage. We followed the analytic dipole method by El Badia and Ha-Duong [12] and derived a new error estimate for this source localization method. We provided a mathematical proof of this estimate. We then use simulated data to validate the method. The simulation results support our error estimation, which has a different distance power than a similar error estimate in [22].

    We organize the rest of the paper as follows. In section 2, we introduce the method and its formulation. In section 3, we provide the error estimate of an inverse EEG source localization problem in a bounded domain and its mathematical proof. In section 4, we use simulated data to valid the method and error estimate. A brief conclusion and discussion is in Section 5.

    The electric field E is the negative gradient of the potential u.

    E=u. (2.1)

    The quasi-static approximation means all time derivatives in the equation are set to zero. By quasi-static approximation of Maxwell equation ×HDt=J, we have

    ×H=J

    where H is the magnetizing field, J is the total current density, and D is the displacement field.

    Since the divergence of a curl is always zero, we have

    (×H)=J=0.

    EEG problem can be modeled by a Poisson equation.

    (σu)=(σE)=(JJp)=J=0Jp=Jp=F,

    where σ is the conductivity, Jp is the primary current density, and F is the source term.

    If we assume the source is composed of a finite number of point charges, then by linear combination, we have

    F=mk=1qkδ(rrk), (2.2)

    where m is the number of point charges, qk are values of charges, and rk are the locations of the point charges.

    If we assume the source is composed of a finite number of dipoles, we have

    F=mk=1pkδ(rrk),

    where m is the number of dipoles, pk are the moments (or strengths) of the dipoles, and rk are the centers of dipoles.

    The dipolar source reconstruction problem can be viewed as a Poisson problem.

    Δu=mk=1pkδ(rrk) in Ω, (2.3)
    u=f on Γ, (2.4)
    uν=φ on Γ, (2.5)

    where f and φ are known, and ν is the outer unit normal vector.

    We will use the concept of reciprocity gap functional [34]:

    R(v)=uν,vH1/2(Γ),H1/2(Γ)u,vνH1/2(Γ),H1/2(Γ)=φ,vH1/2(Γ),H1/2(Γ)f,vνH1/2(Γ),H1/2(Γ), (2.6)

    where v is a harmonic function in Ω:

    vH(Ω)={wH1(Ω)Δw=0}. (2.7)

    By Green's formula, we have

    R(v)=mk=1pkv(rrk),vH(Ω). (2.8)

    Let m be the number of dipoles in the brain. Assume mM in our problem, i.e., there is an upper bound for the number of dipoles.

    Let us consider the harmonic polynomials

    vj(x,y)=(x+iy)j,jN.

    Then, in 2D case

    R(vj)=mk=1pkvj(rk)=mk=1[pk1pk2](xk+iyk)j=mk=1[pk1pk2][x(x+iy)jy(x+iy)j]x=xk,y=yk=mk=1[pk1pk2][j(xk+iyk)j11j(xk+iyk)j1i]=mk=1[pk1pk2][1i]j(xk+iyk)j1=jmk=1(pk1+ipk2)(xk+iyk)j1.

    We define

    βj:=R(vj)j=Mk=1(pk1+ipk2)(xk+iyk)j1,j=1,2,...,2M1. (2.9)

    Let

    ηj=[βjβj+1βj+M1]CM,1jM, (2.10)

    and

    Zi=[ηi,ηi+1,...,ηi+M1]=[βiβi+1βi+M1βi+1βi+2βi+Mβi+M1βi+Mβi+2M2],iN.

    Then,

    Z1=[η1,η2,...,ηM]=[β1β2βMβ2β3βM+1βMβM+1β2M1].

    The number m of dipoles is estimated as the rank of Z1.

    Now we can reduce the size of the matrix by recalculating βj and ηj with M replaced by m. Then, the m vectors η1,...,ηm are independent.

    To get the estimates of the positions we need to construct an m×m matrix T such that ηj+1=Tηj,j=1,...,m. Then,

    [η2,...,ηm+1]=T[η1,...,ηm].

    So,

    T=[η2,...,ηm+1][η1,...,ηm]1=[β2β3βm+1β3β4βm+2βm+1βm+2β2m][β1β2βmβ2β3βm+1βmβm+1β2m1]1=Z2Z11.

    The positions of dipoles are estimated as the eigenvalues of T.

    We now show that the eigenvalues of T are the positions of dipoles. Let us first look at an example η2=Tη1.

    Tη1=T[β1β2βm]=T[p1+p2++pmp1S1+p2S2++pmSmp1Sm11+p2Sm12++pmSm1m]=p1T[1S1Sm11]+p2T[1S2Sm12]++pmT[1SmSm1m],

    where pk=pk1+ipk2,k=1,2,...,m is the moment and Sk=xk+iyk,k=1,2,...,m is the position.

    η2=[β2β3βm+1]=[p1S1+p2S2++pmSmp1S21+p2S22++pmS2mp1Sm1+p2Sm2++pmSmm]=p1S1[1S1Sm11]+p2S2[1S2Sm12]++pmSm[1SmSm1m],

    where pk=pk1+ipk2,k=1,2,...,m is the moment and Sk=xk+iyk,k=1,2,...,m is the position.

    Since [1S1Sm11],[1S2Sm12],...,[1SmSm1m] are independent and the results are similar for ηj+1=Tηj,j=1,2,...,m, we know S1,S2,...,Sm are just the eigenvalues of T.

    Now the question is how to get T. Only η1 and η2 are not enough to determine T because vectors have no inverse. So, we use the redundant information to construct the matrices Z1 and Z2 such that T=Z2Z11, where Z1 is invertible because η1,...,ηm are independent.

    To estimate the moments of dipoles we will write Eq (2.9) in matrix form. Notice that now we use m instead of M.

    [β1β2βm]=[S01S02S0mS11S12S1mSm11Sm12Sm1m][p1p2pm], (2.11)

    where pk=pk1+ipk2,k=1,2,...,m is the moment and Sk=xk+iyk,k=1,2,...,m is the position.

    We can write Eq (2.11) in matrix form

    b=Sp, (2.12)

    where b=[β1β2βm],S=[S01S02S0mS11S12S1mSm11Sm12Sm1m], and p=[p1p2pm]. Then, the moments of dipoles in 2D are estimated as

    p=S1b. (2.13)

    Equation (2.13) works in the ideal case of no noise. In reality, due to the noise in the measurements and in the sources, we need find a linear operator L to estimate the moments, i.e.,

    ˜p=Lb (2.14)

    where ˜p represents the estimates of the moments, and b represents the quantities obtained from the measurements.

    Considering the noise accompanied in the measurements, we rewrite Eq (2.12) as

    b=Sp+n,

    where n is a random vector of mean 0. Let N be the covariance matrix of n. Also, assume that ˜p is normally distributed with mean p and its covariance matrix is P.

    Using multiple measurements and the statistical estimation theory we can find the linear operator L which minimizes the expected difference ErrL between the estimated moments ˜p and the exact moments p.

    ErrL=˜pp2=Lbp2=L(Sp+n)p2=(LSI)p+Ln2=Mp+Ln2(where M=LSI)=Mp2+Ln2(by independence of p and n)=Tr(MPMT)+Tr(LNLT).

    Setting the gradient of ErrL to 0 and solving for L, we get the optimal linear operator

    L=PST(SPST+N)1. (2.15)

    Then, by Eq (2.14) we get the best estimates of the moments.

    Theorem 2.1 (Uniqueness of solutions). Let ui,i=1,2 be the solutions of the problems

    (σui)=mik=1pk(i)δS(i)k in Ω,
    uiν=φ on Γ,

    such that

    u1=u2 on Γ,

    then

    m1=m2=m,
    pk(1)=pk(2),k=1,2,...,m,
    S(1)k=S(2)k,k=1,2,...,m.

    The solution of Poisson equation is the convolution of the fundamental solution of Laplace equation and the source function.

    w(x)=12π[m2k=1pk(xSk)|xS(2)k|2m1k=1pk(xSk)|xS(1)k|2],n=2.
    w(x)=14π[m2k=1pk(xSk)|xS(2)k|3m1k=1pk(xSk)|xS(1)k|3],n=3.

    As EEG imaging data are typically noisy, especially determining the rank of a near singular matrix is very unstable, the error of the numerical reconstruction method needs to be studied. Chafik et al. [12,13] proposed that when the norms of the perturbations (g=˜ff,h=˜φφ) are small in H1/2×H1/2, there exist a>0 and b>0 such that k=1,2,...,m,

    ˜SkSk2m(1Rm)dm1(1R)max{(m1j)Rj,0jm1}(agH1/2(Γ)+bhH1/2(Γ)), (3.1)

    where Sk=xk+iyk is the exact position of the kth dipole, ˜Sk=˜xk+i˜yk is the estimated position of the kth dipole, d is the minimal distance between Sk and ˜Sk, and R1 is a real number bigger than the norm of any point on Γ. However, the analysis is not given by Chafik et al.

    Here we present a new error estimate and provide a proof.

    Theorem 3.1. Suppose m dipoles are enclosed in a circular boundary of radius R. The potential f on the boundary and the gradient of the potential φ perpendicular to the boundary are known. If T is the measurements without noise, and ˜T is the measurements with noise, then the error estimate is given by

    T˜T2m(φ2R2m2πR+f2R2m2πR)(m!mm1pm1maxRm(m1)pmmindm(m1))+2m2(φ2R2m2πR+f2R2m2πR)2(m!mm1pm1maxRm(m1)pmmindm(m1))2, (3.2)

    where p is the moment of dipoles and d is the smallest distance between any two dipoles.

    Proof. We define

    Zi=[βiβi+1βi+m1βi+1βi+2βi+mβi+m1βi+mβi+2m2],iN.

    Then,

    Z1=[β1β2βmβ2β3βm+1βmβm+1β2m1].

    where

    βj=mk=1pkSj1k=mk=1(pk1+ipk2)(xk+iyk)j1,j=1,2,...,2m1.
    det(Z1)=|β1β2βmβ2β3βm+1βmβm+1β2m1|=|pkpkSkpkSm1kpkSkpkS2kpkSmkpkSm1kpkSmkpkS2m2k|=m1m2mmτ(m1,m2,...,mm)pm1pm2pmm|1Sm2Sm1mmSm1S2m2SmmmSm1m1Smm2S2m2mm|
    =m1m2mmτ(m1,m2,...,mm)pm1pm2pmm|111Sm1Sm2SmmSm1m1Sm1m2Sm1mm|S0m1S1m2Sm1mm=p1p2pm|111Sm1Sm2SmmSm1m1Sm1m2Sm1mm|(m1m2mmτ(m1,m2,...,mm)S0m1S1m2Sm1mm)=p1p2pm|111Sm1Sm2SmmSm1m1Sm1m2Sm1mm||111Sm1Sm2SmmSm1m1Sm1m2Sm1mm|=p1p2pm1i<jm(SiSj)2.

    Here, (m1,m2,...,mm) is any permutation of (1,2,...,m) and τ(m1,m2,...,mm) is the sign determined by the permutation.

    The maximum absolute row sum norm is defined by

    A=maxij|aij|,

    where A is a matrix. When A is a vector, A=maxi|ai|.

    In the following proof we will use an important inequality:

    a(x)b(x)a(y)b(y)a(x)a(y)b(x)+b(x)b(y)a(x)

    where a(x) and b(x) can be scalar, vector, or matrix.

    By Cauchy-Schwarz inequality, we have

    R(vj)=φ,vjf,vjν=ΓφvjdsΓfvjνds=Γφ(x+iy)jdsΓf(x+iy)jνds(Γφ2ds)1/2(Γ(x+iy)2jds)1/2+(Γf2ds)1/2(Γ((x+iy)jν)2ds)1/2(Γφ2ds)1/2Rj2πR+(Γf2ds)1/2jRj12πRjφ2Rj2πR+jf2Rj12πR.
    |βj|=|R(vj)j|φ2Rj2πR+f2Rj12πRφ2R2m2πR+f2R2m2πR

    where R>1.

    Let

    T=Z2Z11=Z2adj(Z1)det(Z1)

    where Z1=[β1β2βmβ2β3βm+1βmβm+1β2m1] and Z2=[β2β3βm+1β3β4βm+2βm+1βm+2β2m].

    We can view R(vj) as the measurement obtained by the "detector" vj, while βj is just a constant multiple of R(vj). So, βj is still a measurement of another form, which contains the information about the moment and the position of the dipole source. Since Z1 and Z2 are constructed by different measurements βj, T is also a matrix of measurements.

    Assume T is the measurements without noise, and ˜T is the measurements with noise. Then,

    T˜T=Z2Z11˜Z2˜Z11Z2˜Z2Z11+Z11˜Z11Z2.

    We will analyse the four norms in the above inequality one by one.

    Z2˜Z2mφ˜φ2R2m2πR+mf˜f2R2m2πR.

    To find Z11 we need to estimate adj(Z1). We first observe the results for m=3, then prove the results to the arbitrary m using mathematical induction.

    If Z1=[β1β2β3β2β3β4β3β4β5], then the absolute value of the first element of adj(Z1) would be

    abs(|β3β4β4β5|)=|β3β5β24||β3||β5|+|β24|=(p1S21+p2S22+p3S23)(p1S41+p2S42+p3S43)+(p1S31+p2S32+p3S33)2(3pmaxR2)(3pmaxR4)+(3pmaxR3)2=2(3pmaxR3)2=(31)!331p31maxR3(31)=:max(abs(|β3β4β4β5|)).

    Then,

    adj(Z1)max(abs(|β3β4β4β5|))+max(abs(|β2β4β3β5|))+max(abs(|β2β3β3β4|))3max(abs(|β3β4β4β5|))=3(31)!331p31maxR3(31)=3!331p31maxR3(31).

    Assume when m=n1, we have

    abs(|β3β4βn+1β4β5βn+2βn+1βn+2β2n1|)(n1)!nn1pn1maxRn(n1).

    In fact, this inequality is also true for other minors with matrix size (n1)×(n1).

    Then, when m=n we have

    abs(|β3β4βn+1βn+2β4β5βn+2βn+3βn+1βn+2β2n1β2nβn+2βn+3β2nβ2n+1|)max|β2n+1|max(abs(|β3β4βn+1β4β5βn+2βn+1βn+2β2n1|))++max|βn+2|max(abs(|β4β5βn+2β5β6βn+3βn+2βn+3β2n|))nmax|β2n+1|max(abs(|β3β4βn+1β4β5βn+2βn+1βn+2β2n1|))nmax|p1S2n1+p2S2n2++pnS2nn|(n1)!nn1pn1maxRn(n1)nnpmaxR2n(n1)!nn1pn1maxRn2n)=n!nnpnmaxR(n+1)nn!(n+1)npnmaxR(n+1)n.

    Then, for any m we have

    adj(Z1)=adj([β1β2βm1βmβ2β3βmβm+1βm1βmβ2m3β2m2βmβm+1β2m2β2m1])max(abs(|β3β4βm+1β4β5βm+2βm+1βm+2β2m1|))++max(abs(|β2β3βmβ3β4βm+1βmβm+1β2m2|))mmax(abs(|β3β4βm+1β4β5βm+2βm+1βm+2β2m1|))=m(m1)!mm1pm1maxRm(m1)=m!mm1pm1maxRm(m1).

    Thus,

    Z11=adj(Z1)det(Z1)m!mm1pm1maxRm(m1)p1p2pm1i<jm(SiSj)2m!mm1pm1maxRm(m1)pmmindm(m1)

    where d is the smallest distance between any two dipoles.

    Notice that

    Z1(Z11˜Z11)+(Z1˜Z1)˜Z11=0.
    Z11˜Z11=Z11(Z1˜Z1)˜Z11.
    Z11˜Z11Z11Z1˜Z1˜Z11(m!mm1pm1maxRm(m1)pmmindm(m1))2(mφ˜φ2R2m2πR+mf˜f2R2m2πR).

    Based on the above results, we have

    T˜TZ2˜Z2Z11+Z11˜Z11Z2(mφ˜φ2R2m2πR+mf˜f2R2m2πR)(m!mm1pm1maxRm(m1)pmmindm(m1))+(m!mm1pm1maxRm(m1)pmmindm(m1))2(mφ2R2m2πR+mf2R2m2πR)(mφ˜φ2R2m2πR+mf˜f2R2m2πR)2m(φ2R2m2πR+f2R2m2πR)(m!mm1pm1maxRm(m1)pmmindm(m1))+2m2(φ2R2m2πR+f2R2m2πR)2(m!mm1pm1maxRm(m1)pmmindm(m1))2. (3.3)

    We can further simplify it as

    T˜TE+E2 (3.4)

    where

    E=2mR2m2πR(f2+φ2)(m!mm1pm1maxRm(m1)pmmindm(m1)). (3.5)

    When 0<E<1, the error in the position estimate is mainly controlled by E; when E>1, the error in the position estimate is mainly controlled by E2.

    Let Ω be a circular disk centered at the origin and of radius r=1. Then, the numerical implementation can be simplified as follow.

    vjν=(x+iy)jr=(reiθ)jr=jrj1eiθj=jrjeiθjr=jvjr.
    R(vj)=f,vjν=ΓfvjνdΓ=2π0fjvjrrdθ=j2π0fvjdθ=j2π0f(reiθ)jdθ,

    where f is a function of θ on the boundary. We do not need to know the explicit form of f, but we can measure as many points as possible on the boundary to get enough discretized function values of f. Then, the above integral can be approximated by a Riemann sum.

    The measurable values we want to use in the following are

    βj=R(vj)j=2π0f(reiθ)jdθ.

    The Romberg algorithm is used to calculate the integral numerically.

    We compare the efficacy of the harmonic function method in dipolar source reconstruction when the perturbation level is 0,0.001,0.01,0.1 and the number of dipoles is 1,2,3,4,5. It is shown that as the perturbation level increases, the reconstruction error increases (see Figures 15).

    Figure 1.  The effect of the perturbation level on the reconstruction error of 1 dipole. As the perturbation level increases, the reconstruction error increases. Here, the perturbation means adding noise to the exact measurement. If the perturbation level is σ, then the perturbed measurement is the exact measurement times (1±σ), where plus or minus signs are randomly assigned to each channel. Also, the error is defined as the sum of position errors.
    Figure 2.  The effect of the perturbation level on the reconstruction error of 2 dipoles. As the perturbation level increases, the reconstruction error increases. Here, the perturbation means adding noise to the exact measurement. If the perturbation level is σ, then the perturbed measurement is the exact measurement times (1±σ), where plus or minus signs are randomly assigned to each channel. Also, the error is defined as the sum of position errors.
    Figure 3.  The effect of the perturbation level on the reconstruction error of 3 dipoles. As the perturbation level increases, the reconstruction error increases. Here, the perturbation means adding noise to the exact measurement. If the perturbation level is σ, then the perturbed measurement is the exact measurement times (1±σ), where plus or minus signs are randomly assigned to each channel. Also, the error is defined as the sum of position errors.
    Figure 4.  The effect of the perturbation level on the reconstruction error of 4 dipoles. As the perturbation level increases, the reconstruction error increases. Here, the perturbation means adding noise to the exact measurement. If the perturbation level is σ, then the perturbed measurement is the exact measurement times (1±σ), where plus or minus signs are randomly assigned to each channel. Also, the error is defined as the sum of position errors.
    Figure 5.  The effect of the perturbation level on the reconstruction error of 5 dipoles. As the perturbation level increases, the reconstruction error increases. Here, the perturbation means adding noise to the exact measurement. If the perturbation level is σ, then the perturbed measurement is the exact measurement times (1±σ), where plus or minus signs are randomly assigned to each channel. Also, the error is defined as the sum of position errors.

    In the following we show the results of source estimation, assuming there are 3 dipolar sources (m=3).

    ● Dipole 1: position (0.3,0.3) and moment (0,1).

    ● Dipole 2: position (0.6,0.2) and moment (1,1).

    ● Dipole 3: position (0.5,0.4) and moment (2,2).

    In the graphs (see Figure 6) we use a small circle and a red line segment to indicate the true value, and use a cross sign and a green line segment to indicate the reconstructed values.

    Figure 6.  The effect of the perturbation level on the reconstruction error of 3 dipoles. As the perturbation level increases, the reconstruction error increases.

    From error estimates we know that as the distance between two dipoles gets closer, the reconstruction error for the positions of dipoles gets larger (see Table 1 and Figure 7). This is verified by the numerical simulations.

    Table 1.  The effect of dipole distance on the reconstruction error. As two dipoles get closer, the mean reconstruction error in the positions of the dipoles gets larger, which is consistent with the result in the error estimate.
    Exact Dipole Distance Reconstructed Dipole Distance
    0.03 0.7429
    0.05 0.3084
    0.10 0.1200

     | Show Table
    DownLoad: CSV
    Figure 7.  The effect of dipole distance on the reconstruction error. As two dipoles get closer, the reconstruction error in the positions of the dipoles gets larger, which is consistent with the theoretical analysis in the error estimate. When dexact=0.10, ¯dest=0.1200; when dexact=0.05, ¯dest=0.3084; when dexact=0.03, ¯dest=0.7429.

    We randomly assign two dipoles with fixed distance, say 0.1, in the unit disk, then reconstruct their positions. We fix the noise level for all experiments at σ=0.001.

    Let di (i=1,2) be the distance between the ith exact dipole and the ith estimated dipole, and dmax be the largest d.

    We repeat the experiment 10 times and show their performance on average over different dipole distances.

    The above experiment also provides a numerical example to show that the estimate given by us in Theorem 3.1 provides a better error bound when the two poles are very close.

    When the number of dipoles is m=2, Chafik's estimate is bounded by C1d (see Inequality (3.1)), while our estimate is bounded by C2d2 (see Inequality (3.4) and Eq (3.5)) where d is the smallest distance between two dipoles and Ci (i=1,2) are constants independent of d. That is, when the distance is halved, the error bound will be amplified by 2 in Chafik's estimate and by 4 in our estimate.

    From the data simulation, we see that

    0.050.03=1.67<0.74290.3084=2.41<1.672=2.79.
    0.100.05=2<0.30840.1200=2.57<22=4.
    0.100.03=3.33<0.74290.1200=6.19<3.332=11.09.

    For example, when the distance between the two dipoles is reduced from 0.10 to 0.05, by Chafik's estimate the error should be amplified by 2, but in fact, the error is amplified by 2.57, which is bounded by 4 in our estimate.

    In this paper we studied a harmonic function method for the dipolar source reconstruction, derived error estimate for the harmonic function method and compared our result with Chafik's estimate. By numerical simulations it is shown that the harmonic function method can quickly and accurately locate active regions in EEG source reconstruction. In the future, we plan to extend the harmonic function method to 3D case and applied this method to some real EEG data. The brain's conductance variation in different brain regions also leads to additional challenges in source localization [35]. Although these tissue properties can be quantified through MRI methods, numerical methods such as finite element method will be needed to solve the inverse problems. Since the estimation of the number of dipoles relies on the calculation of the rank of the measurement matrix, which is significantly affected by the noise, we hope to find some way to solve or circumvent this problem. In addition, the situation that the number of exact dipoles is not equal to the estimated value could also be considered. Furthermore, when two dipoles get close enough, it may be better to regard them as an equivalent dipole to avoid increased error.

    We thank anonymous reviewers for providing us with valuable suggestions.

    All authors declare no conflicts of interest in this paper.



    [1] EI Hendouzi A, Bourouhou A (2020) Solar Photovoltaic Power Forecasting. J Electr Comput Eng 2020: 1–21. https://doi.org/10.1155/2020/8819925 doi: 10.1155/2020/8819925
    [2] Ürkmez M, Kallesøe C, Dimon Bendtsen J, et al. (2022) Day-ahead pv power forecasting for control applications. IECON 2022, 48th Annual Conference of the IEEE Industrial Electronics Society, Brussels, Belgium, 1–6. https://doi.org/10.1109/IECON49645.2022.9968709
    [3] Cheng S, Prentice IC, Huang Y, et al. (2022) Data-driven surrogate model with latent data assimilation: Application to wildfire forecasting. J Comput Phys 464: 111302. https://doi.org/10.1016/J.JCP.2022.111302 doi: 10.1016/J.JCP.2022.111302
    [4] Cheng S, Jin Y, Harrison SP, et al. (2022) Parameter Flexible Wildfire Prediction Using Machine Learning Techniques: Forward and Inverse Modelling. Remote Sens 14: 3228. https://doi.org/10.3390/RS14133228 doi: 10.3390/RS14133228
    [5] Zhong C, Cheng S, Kasoar M, et al. (2023) Reduced-order digital twin and latent data assimilation for global wildfire prediction. Nat Hazard Earth Sys 23: 1755–1768. https://doi.org/10.5194/NHESS-23-1755-2023 doi: 10.5194/NHESS-23-1755-2023
    [6] Gupta P, Singh R (2021) PV power forecasting based on data-driven models: a review. Int J Sustain Eng 14: 1733–1755. https://doi.org/10.1080/19397038.2021.1986590 doi: 10.1080/19397038.2021.1986590
    [7] López Santos M, García-Santiago X, Echevarría Camarero F, et al. (2022) Application of Temporal Fusion Transformer for Day-Ahead PV Power Forecasting. Energies 15: 5232. https://doi.org/10.3390/EN15145232 doi: 10.3390/EN15145232
    [8] Kanchana W, Sirisukprasert S (2020) PV Power Forecasting with Holt-Winters Method. 2020 8th International Electrical Engineering Congress (IEECON), 1–4. https://doi.org/10.1109/IEECON48109.2020.229517
    [9] Dhingra S, Gruosso G, Gajani GS (2023) Solar PV Power Forecasting and Ageing Evaluation Using Machine Learning Techniques. IECON 2023 49th Annual Conference of the IEEE Industrial Electronics Society, 1–6. https://doi.org/10.1109/IECON51785.2023.10312446
    [10] Hanif MF, Naveed MS, Metwaly M, et al. (2021) Advancing solar energy forecasting with modified ANN and light GBM learning algorithms. AIMS Energy 12: 350–386. https://doi.org/10.3934/ENERGY.2024017 doi: 10.3934/ENERGY.2024017
    [11] Hanif MF, Siddique MU, Si J, et al. (2021) Enhancing Solar Forecasting Accuracy with Sequential Deep Artificial Neural Network and Hybrid Random Forest and Gradient Boosting Models across Varied Terrains. Adv Theory Simul 7: 2301289. https://doi.org/10.1002/ADTS.202301289 doi: 10.1002/ADTS.202301289
    [12] Musafa A, Priyadi A, Lystianingrum V, et al. (2023) Stored Energy Forecasting of Small-Scale Photovoltaic-Pumped Hydro Storage System Based on Prediction of Solar Irradiance, Ambient Temperature, and Rainfall Using LSTM Method. IECON 2023 49th Annual Conference of the IEEE Industrial Electronics, 1–6. https://doi.org/10.1109/IECON51785.2023.10311982
    [13] Konstantinou M, Peratikou S, Charalambides AG (2021) Solar Photovoltaic Forecasting of Power Output Using LSTM Networks. Atmosphere 12: 124. https://doi.org/10.3390/ATMOS12010124 doi: 10.3390/ATMOS12010124
    [14] Jasiński M, Leonowicz Z, Jasiński J, et al. (2023) PV Advancements & Challenges: Forecasting Techniques, Real Applications, and Grid Integration for a Sustainable Energy Future. 2023 IEEE International Conference on Environment and Electrical Engineering and 2023 IEEE Industrial and Commercial Power Systems Europe (EEEIC/I & CPS Europe), Spain, 1–5. https://doi.org/10.1109/EEEIC/ICPSEUROPE57605.2023.10194796
    [15] Cantillo-Luna S, Moreno-Chuquen R, Celeita D, et al. (2023) Deep and Machine Learning Models to Forecast Photovoltaic Power Generation. Energies 16: 4097. https://doi.org/10.3390/EN16104097 doi: 10.3390/EN16104097
    [16] Kaushik AR, Padmavathi S, Gurucharan KS, et al. (2023) Performance Analysis of Regression Models in Solar PV Forecasting. 2023 3rd International Conference on Artificial Intelligence and Signal Processing (AISP), India, 1–5. https://doi.org/10.1109/AISP57993.2023.10134943
    [17] Halabi LM, Mekhilef S, Hossain M (2018) Performance evaluation of hybrid adaptive neuro-fuzzy inference system models for predicting monthly global solar radiation. Appl Energy 213: 247–261. https://doi.org/10.1016/J.APENERGY.2018.01.035 doi: 10.1016/J.APENERGY.2018.01.035
    [18] Zhang G, Wang X, Du Z (2015) Research on the Prediction of Solar Energy Generation based on Measured Environmental Data. Int J U e-Service Sci Technol 8: 385–402. https://doi.org/10.14257/IJUNESST.2015.8.5.37 doi: 10.14257/IJUNESST.2015.8.5.37
    [19] Peng Q, Zhou X, Zhu R, et al. (2023) A Hybrid Model for Solar Radiation Forecasting towards Energy Efficient Buildings. 2023 7th International Conference on Green Energy and Applications (ICGEA), 7–12. https://doi.org/10.1109/ICGEA57077.2023.10125987
    [20] Salisu S, Mustafa MW, Mustapha M (2018) Predicting Global Solar Radiation in Nigeria Using Adaptive Neuro-Fuzzy Approach. Recent Trends in Information and Communication Technology. IRICT 2017. Lecture Notes on Data Engineering and Communications Technologies, 5: 513–521. https://doi.org/10.1007/978-3-319-59427-9_54
    [21] Kaur A, Nonnenmacher L, Pedro HTC, et al. (2016) Benefits of solar forecasting for energy imbalance markets. Renewable Energy 86: 819–830. https://doi.org/10.1016/J.RENENE.2015.09.011 doi: 10.1016/J.RENENE.2015.09.011
    [22] Yang D, Li W, Yagli GM, et al. (2021) Operational solar forecasting for grid integration: Standards, challenges, and outlook. Sol Energy 224: 930–937. https://doi.org/10.1016/J.SOLENER.2021.04.002 doi: 10.1016/J.SOLENER.2021.04.002
    [23] Shi G, Eftekharnejad S (2016) Impact of solar forecasting on power system planning. 2016 North American Power Symposium (NAPS), 1–6. https://doi.org/10.1109/NAPS.2016.7747909
    [24] Shi J, Guo J, Zheng S (2012) Evaluation of hybrid forecasting approaches for wind speed and power generation time series. Renewable Sustainable Energy Rev 16: 3471–3480. https://doi.org/10.1016/j.rser.2012.02.044 doi: 10.1016/j.rser.2012.02.044
    [25] Mohanty S, Patra PK, Sahoo SS, et al. (2017) Forecasting of solar energy with application for a growing economy like India: Survey and implication. Renewable Sustainable Energy Rev 78: 539–553. https://doi.org/10.1016/J.RSER.2017.04.107 doi: 10.1016/J.RSER.2017.04.107
    [26] Sweeney C, Bessa RJ, Browell J, et al. (2020) The future of forecasting for renewable energy. Wiley Interdiscip Rev Energy Environ 9: e365. https://doi.org/10.1002/WENE.365 doi: 10.1002/WENE.365
    [27] Brancucci Martinez-Anido C, Botor B, Florita AR, et al. (2016) The value of day-ahead solar power forecasting improvement. Sol Energy 129: 192–203. https://doi.org/10.1016/J.SOLENER.2016.01.049 doi: 10.1016/J.SOLENER.2016.01.049
    [28] Inman RH, Pedro HTC, Coimbra CFM (2013) Solar forecasting methods for renewable energy integration. Prog Energy Combust Sci 39: 535–576. https://doi.org/10.1016/J.PECS.2013.06.002 doi: 10.1016/J.PECS.2013.06.002
    [29] Cui M, Zhang J, Hodge BM, et al. (2018) A Methodology for Quantifying Reliability Benefits from Improved Solar Power Forecasting in Multi-Timescale Power System Operations. IEEE T Smart Grid 9: 6897–6908. https://doi.org/10.1109/TSG.2017.2728480 doi: 10.1109/TSG.2017.2728480
    [30] Wang H, Lei Z, Zhang X, et al. (2019) A review of deep learning for renewable energy forecasting. Energy Convers Manage 198: 111799. https://doi.org/10.1016/J.ENCONMAN.2019.111799 doi: 10.1016/J.ENCONMAN.2019.111799
    [31] Aupke P, Kassler A, Theocharis A, et al. (2021) Quantifying Uncertainty for Predicting Renewable Energy Time Series Data Using Machine Learning. Eng Proc 5: 50. https://doi.org/10.3390/ENGPROC2021005050 doi: 10.3390/ENGPROC2021005050
    [32] Rajagukguk RA, Ramadhan RAA, Lee HJ (2020) A Review on Deep Learning Models for Forecasting Time Series Data of Solar Irradiance and Photovoltaic Power. Energies 13: 6623. https://doi.org/10.3390/EN13246623 doi: 10.3390/EN13246623
    [33] SETO 2020—Artificial Intelligence Applications in Solar Energy. Available from: https://www.energy.gov/eere/solar/seto-2020-artificial-intelligence-applications-solar-energy.
    [34] Freitas S, Catita C, Redweik P, et al. (2015) Modelling solar potential in the urban environment: State-of-the-art review. Renewable Sustainable Energy Rev 41: 915–931. https://doi.org/10.1016/J.RSER.2014.08.060 doi: 10.1016/J.RSER.2014.08.060
    [35] Gürtürk M, Ucar F, Erdem M (2022) A novel approach to investigate the effects of global warming and exchange rate on the solar power plants. Energy 239: 122344. https://doi.org/10.1016/J.ENERGY.2021.122344 doi: 10.1016/J.ENERGY.2021.122344
    [36] Gaye B, Zhang D, Wulamu A (2021) Improvement of Support Vector Machine Algorithm in Big Data Background. Math Probl Eng 2021: 5594899. https://doi.org/10.1155/2021/5594899 doi: 10.1155/2021/5594899
    [37] Yogambal Jayalakshmi N, Shankar R, Subramaniam U, et al. (2021) Novel Multi-Time Scale Deep Learning Algorithm for Solar Irradiance Forecasting. Energies 14: 2404. https://doi.org/10.3390/EN14092404 doi: 10.3390/EN14092404
    [38] Benti NE, Chaka MD, Semie AG (2023) Forecasting Renewable Energy Generation with Machine Learning and Deep Learning: Current Advances and Future Prospects. Sustainability 15: 7087. https://doi.org/10.3390/SU15097087 doi: 10.3390/SU15097087
    [39] Li J, Ward JK, Tong J, et al. (2016) Machine learning for solar irradiance forecasting of photovoltaic system. Renewable Energy 90: 542–553. https://doi.org/10.1016/J.RENENE.2015.12.069 doi: 10.1016/J.RENENE.2015.12.069
    [40] Long H, Zhang Z, Su Y (2014) Analysis of daily solar power prediction with data-driven approaches. Appl Energy 126: 29–37. https://doi.org/10.1016/J.APENERGY.2014.03.084 doi: 10.1016/J.APENERGY.2014.03.084
    [41] Jebli I, Belouadha FZ, Kabbaj MI, et al. (2021) Prediction of solar energy guided by pearson correlation using machine learning. Energy 224: 120109. https://doi.org/10.1016/J.ENERGY.2021.120109 doi: 10.1016/J.ENERGY.2021.120109
    [42] Khandakar A, Chowdhury MEH, Kazi MK, et al. (2019) Machine Learning Based Photovoltaics (PV) Power Prediction Using Different Environmental Parameters of Qatar. Energies 12: 2782. https://doi.org/10.3390/EN12142782 doi: 10.3390/EN12142782
    [43] Kim SG, Jung JY, Sim MK (2019) A Two-Step Approach to Solar Power Generation Prediction Based on Weather Data Using Machine Learning. Sustainability 11: 1501. https://doi.org/10.3390/SU11051501 doi: 10.3390/SU11051501
    [44] Gutiérrez L, Patiño J, Duque-Grisales E (2021) A Comparison of the Performance of Supervised Learning Algorithms for Solar Power Prediction. Energies 14: 4424. https://doi.org/10.3390/EN14154424 doi: 10.3390/EN14154424
    [45] Wang Z, Xu Z, Zhang Y, et al. (2020) Optimal Cleaning Scheduling for Photovoltaic Systems in the Field Based on Electricity Generation and Dust Deposition Forecasting. IEEE J Photovolt 10: 1126–1132. https://doi.org/10.1109/JPHOTOV.2020.2981810 doi: 10.1109/JPHOTOV.2020.2981810
    [46] Massaoudi M, Chihi I, Sidhom L, et al. (2021) An Effective Hybrid NARX-LSTM Model for Point and Interval PV Power Forecasting. IEEE Access 9: 36571–36588. https://doi.org/10.1109/ACCESS.2021.3062776 doi: 10.1109/ACCESS.2021.3062776
    [47] Arora I, Gambhir J, Kaur T (2021) Data Normalisation-Based Solar Irradiance Forecasting Using Artificial Neural Networks. Arab J Sci Eng 46: 1333–1343. https://doi.org/10.1007/S13369-020-05140-Y/METRICS doi: 10.1007/S13369-020-05140-Y/METRICS
    [48] Alipour M, Aghaei J, Norouzi M, et al. (2020) A novel electrical net-load forecasting model based on deep neural networks and wavelet transform integration. Energy 205: 118106. https://doi.org/10.1016/J.ENERGY.2020.118106 doi: 10.1016/J.ENERGY.2020.118106
    [49] Zolfaghari M, Golabi MR (2021) Modeling and predicting the electricity production in hydropower using conjunction of wavelet transform, long short-term memory and random forest models. Renewable Energy 170: 1367–1381. https://doi.org/10.1016/J.RENENE.2021.02.017 doi: 10.1016/J.RENENE.2021.02.017
    [50] Li FF, Wang SY, Wei JH (2018) Long term rolling prediction model for solar radiation combining empirical mode decomposition (EMD) and artificial neural network (ANN) techniques. J Renewable Sustainable Energy 10: 013704. https://doi.org/10.1063/1.4999240 doi: 10.1063/1.4999240
    [51] Wang S, Guo Y, Wang Y, et al. (2021) A Wind Speed Prediction Method Based on Improved Empirical Mode Decomposition and Support Vector Machine. IOP Conference Series: Earth and Environmental Science, IOP Publishing. 680: 012012. https://doi.org/10.1088/1755-1315/680/1/012012
    [52] Moreno SR, dos Santos Coelho L (2018) Wind speed forecasting approach based on Singular Spectrum Analysis and Adaptive Neuro Fuzzy Inference System. Renewable Energy 126: 736–754. https://doi.org/10.1016/J.RENENE.2017.11.089 doi: 10.1016/J.RENENE.2017.11.089
    [53] Zhang Y, Le J, Liao X, et al. (2019) A novel combination forecasting model for wind power integrating least square support vector machine, deep belief network, singular spectrum analysis and locality-sensitive hashing. Energy 168: 558–572. https://doi.org/10.1016/J.ENERGY.2018.11.128 doi: 10.1016/J.ENERGY.2018.11.128
    [54] Espinar B, Aznarte JL, Girard R, et al. (2010) Photovoltaic Forecasting: A state of the art. 5th European PV-hybrid and mini-grid conference. OTTI-Ostbayerisches Technologie-Transfer-Institut.
    [55] Moreno-Munoz A, De La Rosa JJG, Posadillo R, et al. (2008) Very short term forecasting of solar radiation. 2008 33rd IEEE Photovoltaic Specialists Conference, San Diego, CA, USA. https://doi.org/10.1109/PVSC.2008.4922587
    [56] Anderson D, Leach M (2004) Harvesting and redistributing renewable energy: on the role of gas and electricity grids to overcome intermittency through the generation and storage of hydrogen. Energy Policy 32: 1603–1614. https://doi.org/10.1016/S0301-4215(03)00131-9 doi: 10.1016/S0301-4215(03)00131-9
    [57] Zhang J, Zhao L, Deng S, et al. (2017) A critical review of the models used to estimate solar radiation. Renewable Sustainable Energy Rev 70: 314–329. https://doi.org/10.1016/J.RSER.2016.11.124 doi: 10.1016/J.RSER.2016.11.124
    [58] Coimbra CFM, Kleissl J, Marquez R (2013) Overview of Solar-Forecasting Methods and a Metric for Accuracy Evaluation. Sol Energy Forecast Resour Assess, 171–194. https://doi.org/10.1016/B978-0-12-397177-7.00008-5 doi: 10.1016/B978-0-12-397177-7.00008-5
    [59] Miller SD, Rogers MA, Haynes JM, et al. (2018) Short-term solar irradiance forecasting via satellite/model coupling. Sol Energy 168: 102–117. https://doi.org/10.1016/J.SOLENER.2017.11.049 doi: 10.1016/J.SOLENER.2017.11.049
    [60] Kumari P, Toshniwal D (2021) Deep learning models for solar irradiance forecasting: A comprehensive review. J Cleaner Prod 318: 128566. https://doi.org/10.1016/J.JCLEPRO.2021.128566 doi: 10.1016/J.JCLEPRO.2021.128566
    [61] Hassan GE, Youssef ME, Mohamed ZE, et al. (2016) New Temperature-based Models for Predicting Global Solar Radiation. Appl Energy 179: 437–450. https://doi.org/10.1016/J.APENERGY.2016.07.006 doi: 10.1016/J.APENERGY.2016.07.006
    [62] Angstrom A (1924) Solar and terrestrial radiation. Report to the international commission for solar research on actinometric investigations of solar and atmospheric radiation. Q J R Meteorol Soc 50: 121–126. https://doi.org/10.1002/QJ.49705021008 doi: 10.1002/QJ.49705021008
    [63] Samuel TDMA (1991) Estimation of global radiation for Sri Lanka. Sol Energy 47: 333–337. https://doi.org/10.1016/0038-092X(91)90026-S doi: 10.1016/0038-092X(91)90026-S
    [64] Ögelman H, Ecevit A, Tasdemiroǧlu E (1984) A new method for estimating solar radiation from bright sunshine data. Sol Energy 33: 619–625. https://doi.org/10.1016/0038-092X(84)90018-5 doi: 10.1016/0038-092X(84)90018-5
    [65] Badescu V, Gueymard CA, Cheval S, et al. (2013) Accuracy analysis for fifty-four clear-sky solar radiation models using routine hourly global irradiance measurements in Romania. Renewable Energy 55: 85–103. https://doi.org/10.1016/J.RENENE.2012.11.037 doi: 10.1016/J.RENENE.2012.11.037
    [66] Mecibah MS, Boukelia TE, Tahtah R, et al. (2014) Introducing the best model for estimation the monthly mean daily global solar radiation on a horizontal surface (Case study: Algeria). Renewable Sustainable Energy Rev 36: 194–202. https://doi.org/10.1016/J.RSER.2014.04.054 doi: 10.1016/J.RSER.2014.04.054
    [67] Hargreaves GH, Samani ZA (1982) Estimating Potential Evapotranspiration. J Irrig Drain Div 108: 225–230. https://doi.org/10.1061/JRCEA4.0001390 doi: 10.1061/JRCEA4.0001390
    [68] Bristow KL, Campbell GS (1984) On the relationship between incoming solar radiation and daily maximum and minimum temperature. Agric For Meteorol 31: 159–166. https://doi.org/10.1016/0168-1923(84)90017-0 doi: 10.1016/0168-1923(84)90017-0
    [69] Chen JL, He L, Yang H, et al. (2019) Empirical models for estimating monthly global solar radiation: A most comprehensive review and comparative case study in China. Renewable Sustainable Energy Rev 108: 91–111. https://doi.org/10.1016/j.rser.2019.03.033 doi: 10.1016/j.rser.2019.03.033
    [70] Chen Y, Zhang S, Zhang W, et al. (2019) Multifactor spatio-temporal correlation model based on a combination of convolutional neural network and long short-term memory neural network for wind speed forecasting. Energy Convers Manage 185: 783–799. https://doi.org/10.1016/j.enconman.2019.02.01 doi: 10.1016/j.enconman.2019.02.01
    [71] Siddiqui TA, Bharadwaj S, Kalyanaraman S (2019) A Deep Learning Approach to Solar-Irradiance Forecasting in Sky-Videos. 2019 IEEE Winter Conference on Applications of Computer Vision (WACV), 2166–2174. https://doi.org/10.1109/WACV.2019.00234
    [72] Nie Y, Li X, Paletta Q, et al. (2024) Open-source sky image datasets for solar forecasting with deep learning: A comprehensive survey. Renewable Sustainable Energy Rev 189: 113977. https://doi.org/10.1016/j.rser.2023.113977 doi: 10.1016/j.rser.2023.113977
    [73] SkyImageNet, 2024. Available from: https://github.com/SkyImageNet.
    [74] Brahma B, Wadhvani R (2020) Solar Irradiance Forecasting Based on Deep Learning Methodologies and Multi-Site Data. Symmetry 12: 1–20. https://doi.org/10.3390/sym12111830 doi: 10.3390/sym12111830
    [75] Paletta Q, Terrén-Serrano G, Nie Y, et al. (2023) Advances in solar forecasting: Computer vision with deep learning. Adv Appl Energy 11: 100150. https://doi.org/10.1016/j.adapen.2023.100150 doi: 10.1016/j.adapen.2023.100150
    [76] Ghimire S, Deo RC, Raj N, et al. (2019) Deep solar radiation forecasting with convolutional neural network and long short-term memory network algorithms. Appl Energy 253: 113541. https://doi.org/10.1016/J.APENERGY.2019.113541 doi: 10.1016/J.APENERGY.2019.113541
    [77] Elsaraiti M, Merabet A (2022) Solar Power Forecasting Using Deep Learning Techniques. IEEE Access 10: 31692–31698. https://doi.org/10.1109/ACCESS.2022.3160484 doi: 10.1109/ACCESS.2022.3160484
    [78] Reikard G (2009) Predicting solar radiation at high resolutions: A comparison of time series forecasts. Sol Energy 83: 342–349. https://doi.org/10.1016/J.SOLENER.2008.08.007 doi: 10.1016/J.SOLENER.2008.08.007
    [79] Yang D, Jirutitijaroen P, Walsh WM (2012) Hourly solar irradiance time series forecasting using cloud cover index. Sol Energy 86: 3531–3543. https://doi.org/10.1016/J.SOLENER.2012.07.029 doi: 10.1016/J.SOLENER.2012.07.029
    [80] Jaihuni M, Basak JK, Khan F, et al. (2020) A Partially Amended Hybrid Bi-GRU—ARIMA Model (PAHM) for Predicting Solar Irradiance in Short and Very-Short Terms. Energies 13: 435. https://doi.org/10.3390/EN13020435 doi: 10.3390/EN13020435
    [81] Verbois H, Huva R, Rusydi A, et al. (2018) Solar irradiance forecasting in the tropics using numerical weather prediction and statistical learning. Sol Energy 162: 265–277. https://doi.org/10.1016/j.solener.2018.01.007 doi: 10.1016/j.solener.2018.01.007
    [82] Munkhammar J, van der Meer D, Widén J (2019) Probabilistic forecasting of high-resolution clear-sky index time-series using a Markov-chain mixture distribution model. Sol Energy 184: 688–695. https://doi.org/10.1016/j.solener.2019.04.014 doi: 10.1016/j.solener.2019.04.014
    [83] Dong J, Olama MM, Kuruganti T, et al. (2020) Novel stochastic methods to predict short-term solar radiation and photovoltaic power. Renewable Energy 145: 333–346. https://doi.org/10.1016/j.renene.2019.05.073 doi: 10.1016/j.renene.2019.05.073
    [84] Ahmad T, Zhang D, Huang C (2021) Methodological framework for short-and medium-term energy, solar and wind power forecasting with stochastic-based machine learning approach to monetary and energy policy applications. Energy 231: 120911. https://doi.org/10.1016/j.energy.2021.120911 doi: 10.1016/j.energy.2021.120911
    [85] Box GE, Jenkins GM, Reinsel GC, et al. (2015) Time series analysis: Forecasting and control, John Wiley & Sons.
    [86] Louzazni M, Mosalam H, Khouya A (2020) A non-linear auto-regressive exogenous method to forecast the photovoltaic power output. Sustain Energy Techn 38: 100670. https://doi.org/10.1016/j.seta.2020.100670 doi: 10.1016/j.seta.2020.100670
    [87] Larson DP, Nonnenmacher L, Coimbra CFM (2016) Day-ahead forecasting of solar power output from photovoltaic plants in the American Southwest. Renewable Energy 91: 11–20. https://doi.org/10.1016/j.renene.2016.01.039 doi: 10.1016/j.renene.2016.01.039
    [88] Sharma V, Yang D, Walsh W, et al. (2016) Short term solar irradiance forecasting using a mixed wavelet neural network. Renewable Energy 90: 481–492. https://doi.org/10.1016/J.RENENE.2016.01.020 doi: 10.1016/J.RENENE.2016.01.020
    [89] Kumari P, Toshniwal D (2020) Real-time estimation of COVID-19 cases using machine learning and mathematical models-The case of India. 2020 IEEE 15th International Conference on Industrial and Information Systems, 369–374. https://doi.org/10.1109/ICIIS51140.2020.9342735
    [90] Ahmad MW, Mourshed M, Rezgui Y (2018) Tree-based ensemble methods for predicting PV power generation and their comparison with support vector regression. Energy 164: 465–474. https://doi.org/10.1016/J.ENERGY.2018.08.207 doi: 10.1016/J.ENERGY.2018.08.207
    [91] Wang Z, Wang Y, Zeng R, et al. (2018) Random Forest based hourly building energy prediction. Energy Buildings 171: 11–25. https://doi.org/10.1016/J.ENBUILD.2018.04.008 doi: 10.1016/J.ENBUILD.2018.04.008
    [92] Zou L, Wang L, Lin A, et al. (2016) Estimation of global solar radiation using an artificial neural network based on an interpolation technique in southeast China. J Atmos Sol-Terr Phys 146: 110–122. https://doi.org/10.1016/J.JASTP.2016.05.013 doi: 10.1016/J.JASTP.2016.05.013
    [93] Mellit A, Benghanem M, Kalogirou SA (2006) An adaptive wavelet-network model for forecasting daily total solar-radiation. Appl Energy 83: 705–722. https://doi.org/10.1016/J.APENERGY.2005.06.003 doi: 10.1016/J.APENERGY.2005.06.003
    [94] Çelik Ö, Teke A, Yildirim HB (2016) The optimized artificial neural network model with Levenberg–Marquardt algorithm for global solar radiation estimation in Eastern Mediterranean Region of Turkey. J Cleaner Prod 116: 1–12. https://doi.org/10.1016/J.JCLEPRO.2015.12.082 doi: 10.1016/J.JCLEPRO.2015.12.082
    [95] Rehman S, Mohandes M (2008) Artificial neural network estimation of global solar radiation using air temperature and relative humidity. Energy Policy 36: 571–576. https://doi.org/10.1016/J.ENPOL.2007.09.033 doi: 10.1016/J.ENPOL.2007.09.033
    [96] Gürel AE, Ağbulut Ü, Biçen Y (2020) Assessment of machine learning, time series, response surface methodology and empirical models in prediction of global solar radiation. J Cleaner Prod 277: 122353. https://doi.org/10.1016/J.JCLEPRO.2020.122353 doi: 10.1016/J.JCLEPRO.2020.122353
    [97] Díaz-Gómez J, Parrales A, Á lvarez A, et al. (2015) Prediction of global solar radiation by artificial neural network based on a meteorological environmental data. Desalin Water Treat 55: 3210–3217. https://doi.org/10.1080/19443994.2014.939861 doi: 10.1080/19443994.2014.939861
    [98] Rocha PAC, Fernandes JL, Modolo AB, et al. (2019) Estimation of daily, weekly and monthly global solar radiation using ANNs and a long data set: a case study of Fortaleza, in Brazilian Northeast region. Int J Energy Environ Eng 10: 319–334. https://doi.org/10.1007/S40095-019-0313-0/TABLES/6 doi: 10.1007/S40095-019-0313-0/TABLES/6
    [99] Rezrazi A, Hanini S, Laidi M (2016) An optimisation methodology of artificial neural network models for predicting solar radiation: a case study. Theor Appl Climatol 123: 769–783. https://doi.org/10.1007/s00704-015-1398-x doi: 10.1007/s00704-015-1398-x
    [100] Pang Z, Niu F, O'Neill Z (2020) Solar radiation prediction using recurrent neural network and artificial neural network: A case study with comparisons. Renewable Energy 156: 279–289. https://doi.org/10.1016/J.RENENE.2020.04.042 doi: 10.1016/J.RENENE.2020.04.042
    [101] Toth E, Brath A, Montanari A (2000) Comparison of short-term rainfall prediction models for real-time flood forecasting. J Hydrol 239: 132–147. https://doi.org/10.1016/S0022-1694(00)00344-9 doi: 10.1016/S0022-1694(00)00344-9
    [102] Mamoulis N, Seidl T, Pedersen TB, et al. (2009) Advances in Spatial and Temporal Databases, Springer Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-02982-0
    [103] Ren J, Ren B, Zhang Q, et al. (2019) A Novel Hybrid Extreme Learning Machine Approach Improved by K Nearest Neighbor Method and Fireworks Algorithm for Flood Forecasting in Medium and Small Watershed of Loess Region. Water 11: 1848. https://doi.org/10.3390/W11091848 doi: 10.3390/W11091848
    [104] Larose DT, Larose CD (2014) k‐Nearest Neighbor Algorithm. Discovering Knowledge in Data: An Introduction to Data Mining, Second Edition, 149–164. https://doi.org/10.1002/9781118874059.CH7
    [105] Sutton C (2012) Nearest-neighbor methods. WIREs Comput Stat 4: 307–309. https://doi.org/10.1002/WICS.1195 doi: 10.1002/WICS.1195
    [106] Chen JL, Li GS, Xiao BB, et al. (2015) Assessing the transferability of support vector machine model for estimation of global solar radiation from air temperature. Energy Convers Manage 89: 318–329. https://doi.org/10.1016/j.enconman.2014.10.004 doi: 10.1016/j.enconman.2014.10.004
    [107] Shamshirband S, Mohammadi K, Tong CW, et al. (2016) A hybrid SVM-FFA method for prediction of monthly mean global solar radiation. Theor Appl Climatol 125: 53–65.
    [108] Olatomiwa L, Mekhilef S, Shamshirband S, et al. (2015) Potential of support vector regression for solar radiation prediction in Nigeria. Nat Hazards 77: 1055–1068. https://doi.org/10.1007/s11069-015-1641-x doi: 10.1007/s11069-015-1641-x
    [109] Ramedani Z, Omid M, Keyhani A, et al. (2014) Potential of radial basis function based support vector regression for global solar radiation prediction. Renewable Sustainable Energy Rev 39: 1005–1011. https://doi.org/10.1016/J.RSER.2014.07.108 doi: 10.1016/J.RSER.2014.07.108
    [110] Olatomiwa L, Mekhilef S, Shamshirband S, et al. (2015) A support vector machine-firefly algorithm-based model for global solar radiation prediction. Sol Energy 115: 632–644. https://doi.org/10.1016/j.solener.2015.03.015 doi: 10.1016/j.solener.2015.03.015
    [111] Mohammadi K, Shamshirband S, Danesh AS, et al. (2016) Temperature-based estimation of global solar radiation using soft computing methodologies. Theor Appl Climatol 125: 101–112. https://doi.org/10.1007/s00704-015-1487-x doi: 10.1007/s00704-015-1487-x
    [112] Hassan MA, Khalil A, Kaseb S, et al. (2017) Potential of four different machine-learning algorithms in modeling daily global solar radiation. Renewable Energy 111: 52–62. https://doi.org/10.1016/j.renene.2017.03.083 doi: 10.1016/j.renene.2017.03.083
    [113] Quej VH, Almorox J, Arnaldo JA, et al. (2017) ANFIS, SVM and ANN soft-computing techniques to estimate daily global solar radiation in a warm sub-humid environment. J Atmos Sol-Terr Phys 155: 62–70. https://doi.org/10.1016/J.JASTP.2017.02.002 doi: 10.1016/J.JASTP.2017.02.002
    [114] Baser F, Demirhan H (2017) A fuzzy regression with support vector machine approach to the estimation of horizontal global solar radiation. Energy 123: 229–240. https://doi.org/10.1016/j.energy.2017.02.008 doi: 10.1016/j.energy.2017.02.008
    [115] Breiman L (2001) Random forests. Mach Learn 45: 5–32. https://doi.org/10.1023/A:1010933404324 doi: 10.1023/A:1010933404324
    [116] Fernández-Delgado M, Cernadas E, Barro S, et al. (2014) Do we need hundreds of classifiers to solve real world classification problems? J Mach Learn Res 15: 3133–3181.
    [117] Ke G, Meng Q, Finley T, et al. (2017) Lightgbm: A highly efficient gradient boosting decision tree. Adv Neural Inf Proc Syst, 30.
    [118] Wang Y, Pan Z, Zheng J, et al. (2019) A hybrid ensemble method for pulsar candidate classification. Astrophys Space Sci 364: 139 https://doi.org/10.1007/s10509-019-3602-4 doi: 10.1007/s10509-019-3602-4
    [119] Si Z, Yang M, Yu Y, et al. (2021) Photovoltaic power forecast based on satellite images considering effects of solar position. Appl Energy 302: 117514. https://doi.org/10.1016/j.apenergy.2021.117514 doi: 10.1016/j.apenergy.2021.117514
    [120] Chung J, Gulcehre C, Cho K, et al. (2014) Empirical Evaluation of Gated Recurrent Neural Networks on Sequence Modeling. arXiv preprint arXiv: 1412.3555.
    [121] Wang Y, Liao W, Chang Y (2018) Gated Recurrent Unit Network-Based Short-Term Photovoltaic Forecasting. Energies 11: 2163. https://doi.org/10.3390/EN11082163 doi: 10.3390/EN11082163
    [122] Pazikadin AR, Rifai D, Ali K, et al. (2020) Solar irradiance measurement instrumentation and power solar generation forecasting based on Artificial Neural Networks (ANN): A review of five years research trend. Sci Total Environ 715: 136848. https://doi.org/10.1016/j.scitotenv.2020.136848 doi: 10.1016/j.scitotenv.2020.136848
    [123] Wang F, Xuan Z, Zhen Z, et al. (2020) A day-ahead PV power forecasting method based on LSTM-RNN model and time correlation modification under partial daily pattern prediction framework. Energy Convers Manage 212: 112766. https://doi.org/10.1016/j.enconman.2020.112766 doi: 10.1016/j.enconman.2020.112766
    [124] Zhang J, Yan J, Infield D, et al. (2019) Short-term forecasting and uncertainty analysis of wind turbine power based on long short-term memory network and Gaussian mixture model. Appl Energy 241: 229–244. https://doi.org/10.1016/j.apenergy.2019.03.044 doi: 10.1016/j.apenergy.2019.03.044
    [125] Liu H, Mi X, Li Y, et al. (2019) Smart wind speed deep learning based multi-step forecasting model using singular spectrum analysis, convolutional Gated Recurrent Unit network and Support Vector Regression. Renewable Energy 143: 842–854. https://doi.org/10.1016/j.renene.2019.05.039 doi: 10.1016/j.renene.2019.05.039
    [126] Tealab A (2018) Time series forecasting using artificial neural networks methodologies: A systematic review. Future Comput Inf J 3: 334–340. https://doi.org/10.1016/j.fcij.2018.10.003 doi: 10.1016/j.fcij.2018.10.003
    [127] Dong N, Chang JF, Wu AG, et al. (2020) A novel convolutional neural network framework based solar irradiance prediction method. Int J Electr Power Energy Syst 114: 105411. https://doi.org/10.1016/j.ijepes.2019.105411 doi: 10.1016/j.ijepes.2019.105411
    [128] Hinton GE, Srivastava N, Krizhevsky A, et al. (2012) Improving neural networks by preventing co-adaptation of feature detectors.
    [129] Han Z, Zhao J, Leung H, et al. (2021) A Review of Deep Learning Models for Time Series Prediction. IEEE Sens J 21: 7833–7848. https://doi.org/10.1109/JSEN.2019.2923982 doi: 10.1109/JSEN.2019.2923982
    [130] Shi X, Chen Z, Wang H, et al. (2015) Convolutional LSTM Network: A Machine Learning Approach for Precipitation Nowcasting. Adv Neural Inf Proc Syst, 28.
    [131] Oord A van den, Dieleman S, Zen H, et al. (2016) WaveNet: A Generative Model for Raw Audio. arXiv preprint arXiv: 1609.03499. https://doi.org/10.48550/arXiv.1609.03499
    [132] Bai S, Kolter JZ, Koltun V (2018) An Empirical Evaluation of Generic Convolutional and Recurrent Networks for Sequence Modeling. https://doi.org/10.48550/arXiv.1803.01271
    [133] Vaswani A, Brain G, Shazeer N, et al. (2017) Attention Is All You Need. arXiv preprint arXiv: 1706.03762.
    [134] Zang H, Liu L, Sun L, et al. (2020) Short-term global horizontal irradiance forecasting based on a hybrid CNN-LSTM model with spatiotemporal correlations. Renewable Energy 160: 26–41. https://doi.org/10.1016/j.renene.2020.05.150 doi: 10.1016/j.renene.2020.05.150
    [135] Qu J, Qian Z, Pei Y (2021) Day-ahead hourly photovoltaic power forecasting using attention-based CNN-LSTM neural network embedded with multiple relevant and target variables prediction pattern. Energy 232: 120996. https://doi.org/10.1016/j.energy.2021.120996 doi: 10.1016/j.energy.2021.120996
    [136] Schmidhuber J, Hochreiter S (1997) Long Short-Term Memory. Neural Comput 9: 1735–1780. https://doi.org/10.1162/neco.1997.9.8.1735 doi: 10.1162/neco.1997.9.8.1735
    [137] Venkatraman A, Hebert M, Bagnell J (2015) Improving Multi-Step Prediction of Learned Time Series Models. Proceedings of the AAAI Conference on Artificial Intelligence, 29. https://doi.org/10.1609/aaai.v29i1.9590 doi: 10.1609/aaai.v29i1.9590
    [138] Muhammad, Kennedy J, Lim CW (2022) Machine learning and deep learning in phononic crystals and metamaterials—A review. Mater Today Commun 33: 104606. https://doi.org/10.1016/J.MTCOMM.2022.104606 doi: 10.1016/J.MTCOMM.2022.104606
    [139] Yao G, Lei T, Zhong J (2019) A review of Convolutional-Neural-Network-based action recognition. Pattern Recogn Lett 118: 14–22. https://doi.org/10.1016/J.PATREC.2018.05.018 doi: 10.1016/J.PATREC.2018.05.018
    [140] Akram MW, Li G, Jin Y, et al. (2019) CNN based automatic detection of photovoltaic cell defects in electroluminescence images. Energy 189: 116319. https://doi.org/10.1016/J.ENERGY.2019.116319 doi: 10.1016/J.ENERGY.2019.116319
    [141] Bejani MM, Ghatee M (2021) A systematic review on overfitting control in shallow and deep neural networks. Artif Intell Rev 54: 6391–6438. https://doi.org/10.1007/s10462-021-09975-1 doi: 10.1007/s10462-021-09975-1
    [142] McCann MT, Jin KH, Unser M (2017) Convolutional neural networks for inverse problems in imaging: A review. IEEE Signal Proc Mag 34: 85–95. https://doi.org/10.1109/MSP.2017.2739299 doi: 10.1109/MSP.2017.2739299
    [143] Qian C, Xu B, Chang L, et al. (2021) Convolutional neural network based capacity estimation using random segments of the charging curves for lithium-ion batteries. Energy 227: 120333. https://doi.org/10.1016/J.ENERGY.2021.120333 doi: 10.1016/J.ENERGY.2021.120333
    [144] Liu Y, Guan L, Hou C, et al. (2019) Wind Power Short-Term Prediction Based on LSTM and Discrete Wavelet Transform. Appl Sci 9: 1108. https://doi.org/10.3390/APP9061108 doi: 10.3390/APP9061108
    [145] Husein M, Chung IY (2019) Day-Ahead Solar Irradiance Forecasting for Microgrids Using a Long Short-Term Memory Recurrent Neural Network: A Deep Learning Approach. Energies 12: 1856. https://doi.org/10.3390/EN12101856 doi: 10.3390/EN12101856
    [146] Zhao Z, Chen W, Wu X, et al. (2017) LSTM network: a deep learning approach for short-term traffic forecast. IET Intell Transp Syst 11: 68–75. https://doi.org/10.1049/IET-ITS.2016.0208 doi: 10.1049/IET-ITS.2016.0208
    [147] Suresh V, Janik P, Rezmer J, et al. (2020) Forecasting Solar PV Output Using Convolutional Neural Networks with a Sliding Window Algorithm. Energies 13: 723. https://doi.org/10.3390/EN13030723 doi: 10.3390/EN13030723
    [148] Zameer A, Jaffar F, Shahid F, et al. (2023) Short-term solar energy forecasting: Integrated computational intelligence of LSTMs and GRU. PLoS One 18: e0285410. https://doi.org/10.1371/journal.pone.0285410 doi: 10.1371/journal.pone.0285410
    [149] Bommasani R, Hudson DA, Adeli E, et al. (2021) On the Opportunities and Risks of Foundation Models. arXiv preprint arXiv: 2108.07258. https://doi.org/10.48550/arXiv.2108.07258
    [150] Devlin J (2018) BERT: Pre-training of Deep Bidirectional Transformers for Language Understanding. arXiv preprint arXiv: 1810.04805.
    [151] Mann B, Ryder N, Subbiah M, et al. (2020) Language Models are Few-Shot Learners. arXiv preprint arXiv: 2005.14165, 1.
    [152] Radford A, Kim JW, Hallacy C, et al. (2021) Learning Transferable Visual Models from Natural Language Supervision. International conference on machine learning. PMLR.
    [153] Child R, Gray S, Radford A, et al. (2019) Generating Long Sequences with Sparse Transformers. arXiv preprint arXiv: 1904.10509. https://doi.org/10.48550/arXiv.1904.10509
    [154] Kitaev N, Kaiser Ł, Levskaya A (2020) Reformer: The Efficient Transformer. arXiv preprint arXiv: 2001.04451. https://doi.org/10.48550/arXiv.2001.04451
    [155] Beltagy I, Peters ME, Cohan A (2020) Longformer: The Long-Document Transformer. arXiv preprint arXiv: 2004.05150. https://doi.org/10.48550/arXiv.2004.05150
    [156] Wang S, Li BZ, Khabsa M, et al. (2020) Linformer: Self-Attention with Linear Complexity. arXiv preprint arXiv: 2006.04768. https://doi.org/10.48550/arXiv.2006.04768
    [157] Rae JW, Potapenko A, Jayakumar SM, et al. (2020) Compressive Transformers for Long-Range Sequence Modelling. arXiv preprint arXiv: 1911.05507. https://doi.org/10.48550/arXiv.1911.05507
    [158] Dai Z, Yang Z, Yang Y, et al. (2019) Transformer-XL: Attentive Language Models Beyond a Fixed-Length Context. Proceedings of the 57th Annual Meeting of the Association for Computational Linguistics, 2978–2988, Florence, Italy. Association for Computational Linguistics. https://doi.org/10.18653/v1/p19-1285
    [159] Zhou H, Zhang S, Peng J, et al. (2021) Informer: Beyond Efficient Transformer for Long Sequence Time-Series Forecasting. Proceedings of the AAAI Conference on Artificial Intelligence, 35: 11106–11115. https://doi.org/10.1609/AAAI.V35I12.17325 doi: 10.1609/AAAI.V35I12.17325
    [160] Hanif MF, Mi J (2024) Harnessing AI for solar energy: Emergence of transformer models. Appl Energy 369: 123541. https://doi.org/10.1016/J.APENERGY.2024.123541 doi: 10.1016/J.APENERGY.2024.123541
    [161] Hussain A, Khan ZA, Hussain T, et al. (2022) A Hybrid Deep Learning-Based Network for Photovoltaic Power Forecasting. Complexity. https://doi.org/10.1155/2022/7040601 doi: 10.1155/2022/7040601
    [162] Vennila C, Titus A, Sudha TS, et al. (2022) Forecasting Solar Energy Production Using Machine Learning. Int J Photoenergy 2022: 7797488. https://doi.org/10.1155/2022/7797488 doi: 10.1155/2022/7797488
    [163] So D, Oh J, Leem S, et al. (2023) A Hybrid Ensemble Model for Solar Irradiance Forecasting: Advancing Digital Models for Smart Island Realization. Electronics 12: 2607. https://doi.org/10.3390/electronics12122607 doi: 10.3390/electronics12122607
    [164] He Y, Liu Y, Shao S, et al. (2019) Application of CNN-LSTM in Gradual Changing Fault Diagnosis of Rod Pumping System. Math Probl Eng 2019: 4203821. https://doi.org/10.1155/2019/4203821 doi: 10.1155/2019/4203821
    [165] Huang CJ, Kuo PH (2018) A Deep CNN-LSTM Model for Particulate Matter (PM2.5) Forecasting in Smart Cities. Sensors 18: 2220. https://doi.org/10.3390/S18072220 doi: 10.3390/S18072220
    [166] Cao K, Kim H, Hwang C, et al. (2018) CNN-LSTM Coupled Model for Prediction of Waterworks Operation Data. J Inf Process Syst 14: 1508–1520. https://doi.org/10.3745/JIPS.02.0104 doi: 10.3745/JIPS.02.0104
    [167] Swapna G, Soman KP, Vinayakumar R (2018) Automated detection of diabetes using CNN and CNN-LSTM network and heart rate signals. Procedia Comput Sci 132: 1253–1262. https://doi.org/10.1016/j.procs.2018.05.041 doi: 10.1016/j.procs.2018.05.041
    [168] Jalali SMJ, Ahmadian S, Kavousi-Fard A, et al. (2022) Automated Deep CNN-LSTM Architecture Design for Solar Irradiance Forecasting. IEEE Trans Syst Man Cybernetics Syst 52: 54–65. https://doi.org/10.1109/TSMC.2021.3093519 doi: 10.1109/TSMC.2021.3093519
    [169] Lim SC, Huh JH, Hong SH, et al. (2022) Solar Power Forecasting Using CNN-LSTM Hybrid Model. Energies 15: 8233. https://doi.org/10.3390/EN15218233 doi: 10.3390/EN15218233
    [170] Covas E (2020) Transfer Learning in Spatial-Temporal Forecasting of the Solar Magnetic Field. Astron Nachr 341: 384–394. https://doi.org/10.1002/ASNA.202013690 doi: 10.1002/ASNA.202013690
    [171] Sheng H, Ray B, Chen K, et al. (2020) Solar Power Forecasting Based on Domain Adaptive Learning. IEEE Access 8: 198580–198590. https://doi.org/10.1109/ACCESS.2020.3034100 doi: 10.1109/ACCESS.2020.3034100
    [172] Ren X, Wang Y, Cao Z, et al. (2023) Feature Transfer and Rapid Adaptation for Few-Shot Solar Power Forecasting. Energies 16: 6211. https://doi.org/10.3390/EN16176211 doi: 10.3390/EN16176211
    [173] Zhou S, Zhou L, Mao M, et al. (2020) Transfer Learning for Photovoltaic Power Forecasting with Long Short-Term Memory Neural Network. 2020 IEEE International Conference on Big Data and Smart Computing (BigComp), Busan, Korea (South), 125–132. https://doi.org/10.1109/BIGCOMP48618.2020.00-87
    [174] Soleymani S, Mohammadzadeh S (2023) Comparative Analysis of Machine Learning Algorithms for Solar Irradiance Forecasting in Smart Grids. arXiv preprint arXiv: 2310.13791. https://doi.org/10.48550/arXiv.2310.13791
    [175] Sutarna N, Tjahyadi C, Oktivasari P, et al. (2023) Machine Learning Algorithm and Modeling in Solar Irradiance Forecasting. 2023 6th International Conference of Computer and Informatics Engineering (IC2IE), Lombok, Indonesia, 221–225. https://doi.org/10.1109/IC2IE60547.2023.10330942
    [176] Bamisile O, Oluwasanmi A, Ejiyi C, et al. (2022) Comparison of machine learning and deep learning algorithms for hourly global/diffuse solar radiation predictions. Int J Energy Res 46: 10052–10073. https://doi.org/10.1002/ER.6529 doi: 10.1002/ER.6529
    [177] Sahaya Lenin D, Teja Reddy R, Velaga V (2023) Solar Irradiance Forecasting Using Machine Learning. 2023 14th International Conference on Computing Communication and Networking Technologies (ICCCNT), Delhi, India, 1–7. https://doi.org/10.1109/ICCCNT56998.2023.10307660
    [178] Syahab AS, Hermawan A, Avianto D (2023) Global Horizontal Irradiance Prediction using the Algorithm of Moving Average and Exponential Smoothing. JISA 6: 74–81. https://doi.org/10.31326/JISA.V6I1.1649. doi: 10.31326/JISA.V6I1.1649
    [179] Aljanad A, Tan NML, Agelidis VG, et al. (2021) Neural Network Approach for Global Solar Irradiance Prediction at Extremely Short-Time-Intervals Using Particle Swarm Optimization Algorithm. Energies 14: 1213. https://doi.org/10.3390/EN14041213 doi: 10.3390/EN14041213
    [180] Mbah OM, Madueke CI, Umunakwe R, et al. (2022) Extreme Gradient Boosting: A Machine Learning Technique for Daily Global Solar Radiation Forecasting on Tilted Surfaces. J Eng Sci 9: E1–E6. https://doi.org/10.21272/JES.2022.9(2).E1 doi: 10.21272/JES.2022.9(2).E1
    [181] Cha J, Kim MK, Lee S, et al. (2021) Investigation of Applicability of Impact Factors to Estimate Solar Irradiance: Comparative Analysis Using Machine Learning Algorithms. Appl Sci 11: 8533. https://doi.org/10.3390/APP11188533 doi: 10.3390/APP11188533
    [182] Reddy KR, Ray PK (2022) Solar Irradiance Forecasting using FFNN with MIG Feature Selection Technique. 2022 International Conference on Intelligent Controller and Computing for Smart Power (ICICCSP), Hyderabad, India, 01–05. https://doi.org/10.1109/ICICCSP53532.2022.9862335
    [183] Chandola D, Gupta H, Tikkiwal VA, et al. (2020) Multi-step ahead forecasting of global solar radiation for arid zones using deep learning. Procedia Comput Sci 167: 626–635. https://doi.org/10.1016/j.procs.2020.03.329 doi: 10.1016/j.procs.2020.03.329
    [184] Yang Y, Tang Z, Li Z, et al. (2023) Dual-Path Information Fusion and Twin Attention-Driven Global Modeling for Solar Irradiance Prediction. Sensors 23: 7649. https://doi.org/10.3390/S23177469 doi: 10.3390/S23177469
    [185] Meng F, Zou Q, Zhang Z, et al. (2021) An intelligent hybrid wavelet-adversarial deep model for accurate prediction of solar power generation. Energy Rep 7: 2155–2164. https://doi.org/10.1016/J.EGYR.2021.04.019 doi: 10.1016/J.EGYR.2021.04.019
    [186] Kartini UT, Hariyati, Aribowo W, et al. (2022) Development Hybrid Model Deep Learning Neural Network (DL-NN) For Probabilistic Forecasting Solar Irradiance on Solar Cells To Improve Economics Value Added. 2022 Fifth International Conference on Vocational Education and Electrical Engineering (ICVEE), Surabaya, Indonesia, 151–156. https://doi.org/10.1109/ICVEE57061.2022.9930352
    [187] Singla P, Duhan M, Saroha S (2022) A dual decomposition with error correction strategy based improved hybrid deep learning model to forecast solar irradiance. Energy Sources Part A 44: 1583–1607. https://doi.org/10.1080/15567036.2022.2056267 doi: 10.1080/15567036.2022.2056267
    [188] Marinho FP, Rocha PAC, Neto ARR, et al. (2023) Short-Term Solar Irradiance Forecasting Using CNN-1D, LSTM and CNN-LSTM Deep Neural Networks: A Case Study with the Folsom (USA) Dataset. J Sol Energy Eng 145: 041002. https://doi.org/10.1115/1.4056122 doi: 10.1115/1.4056122
    [189] Kumari P, Toshniwal D (2021) Long short term memory-convolutional neural network based deep hybrid approach for solar irradiance forecasting. Appl Energy 295: 117061. https://doi.org/10.1016/j.apenergy.2021.117061 doi: 10.1016/j.apenergy.2021.117061
    [190] Elizabeth Michael N, Mishra M, Hasan S, et al. (2022) Short-Term Solar Power Predicting Model Based on Multi-Step CNN Stacked LSTM Technique. Energies 15: 2150. https://doi.org/10.3390/EN15062150 doi: 10.3390/EN15062150
    [191] Srivastava RK, Gupta A (2023) Short term solar irradiation forecasting using Deep neural network with decomposition methods and optimized by grid search algorithm. E3S Web Conf 405. https://doi.org/10.1051/E3SCONF/202340502011 doi: 10.1051/E3SCONF/202340502011
    [192] Ziyabari S, Zhao Z, Du L, et al. (2023) Multi-Branch ResNet-Transformer for Short-Term Spatio-Temporal Solar Irradiance Forecasting. IEEE Trans Ind Appl 59: 5293–5303. https://doi.org/10.1109/TIA.2023.3285202 doi: 10.1109/TIA.2023.3285202
    [193] Carneiro TC, De Carvalho PCM, Dos Santos HA, et al. (2022) Review on Photovoltaic Power and Solar Resource Forecasting: Current Status and Trends. J Sol Energy Eng 144: 010801. https://doi.org/10.1115/1.4051652 doi: 10.1115/1.4051652
    [194] Chaibi M, Benghoulam ELM, Tarik L, et al. (2021) An Interpretable Machine Learning Model for Daily Global Solar Radiation Prediction. Energies 14: 7367. https://doi.org/10.3390/EN14217367 doi: 10.3390/EN14217367
    [195] Mason L, González AB de, García-Closas M, et al. (2023) Interpretable, non-mechanistic forecasting using empirical dynamic modeling and interactive visualization. PLoS One 18: e0277149. https://doi.org/10.1101/2022.10.21.22281384 doi: 10.1101/2022.10.21.22281384
    [196] Rafati A, Joorabian M, Mashhour E, et al. (2021) High dimensional very short-term solar power forecasting based on a data-driven heuristic method. Energy 219: 119647. https://doi.org/10.1016/J.ENERGY.2020.119647 doi: 10.1016/J.ENERGY.2020.119647
    [197] Wang H, Cai R, Zhou B, et al. (2020) Solar irradiance forecasting based on direct explainable neural network. Energy Convers Manage 226: 113487. https://doi.org/10.1016/J.ENCONMAN.2020.113487 doi: 10.1016/J.ENCONMAN.2020.113487
    [198] Theocharides S, Makrides G, Livera A, et al. (2020) Day-ahead photovoltaic power production forecasting methodology based on machine learning and statistical post-processing. Appl Energy 268: 115023. https://doi.org/10.1016/J.APENERGY.2020.115023 doi: 10.1016/J.APENERGY.2020.115023
  • This article has been cited by:

    1. Xincai Zhu, Hanxiao Wu, Existence and Limit Behavior of Constraint Minimizers for a Varying Non-Local Kirchhoff-Type Energy Functional, 2024, 12, 2227-7390, 661, 10.3390/math12050661
    2. Xincai Zhu, Chunxia He, Blow-Up Analysis of L2-Norm Solutions for an Elliptic Equation with a Varying Nonlocal Term, 2024, 13, 2075-1680, 336, 10.3390/axioms13050336
    3. Xincai Zhu, Yajie Zhu, Existence and limit behavior of constraint minimizers for elliptic equations with two nonlocal terms, 2024, 32, 2688-1594, 4991, 10.3934/era.2024230
  • Reader Comments
  • © 2024 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
通讯作者: 陈斌, bchen63@163.com
  • 1. 

    沈阳化工大学材料科学与工程学院 沈阳 110142

  1. 本站搜索
  2. 百度学术搜索
  3. 万方数据库搜索
  4. CNKI搜索

Metrics

Article views(2936) PDF downloads(170) Cited by(5)

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog