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Research article

A novel mine blast optimization algorithm (MBOA) based MPPT controlling for grid-PV systems

  • One of the most important areas in today's world is meeting the energy needs of various resources provided by nature. The advantages of renewable energy sources for many application sectors have attracted a lot of attention. The majority of grid-based enterprises use solar photovoltaic (PV) systems to collect sunlight as a reliable energy source. Due to solar PV's simple accessibility and efficient panel design, it is widely used in a variety of application scenarios. By employing the Maximum Power Point Tracking (MPPT) technique, the PV modules can typically operate at their best rate and draw the most power possible from the solar system. Some hybrid control mechanisms are utilized in solar PV systems in traditional works, which has limitations on the problems of increased time consumption, decreased efficiency, and increased THD. Thus, a new Mine Blast Optimization Algorithm (MBOA) based MPPT controlling model is developed to maximize the electrical energy produced by the PV panels under a different climatic situations. Also, an interleaved Luo DC-DC converter is used to significantly improve the output voltage of a PV system with a lower switching frequency. A sophisticated converter and regulating models are being created to effectively meet the energy demand of grid systems. The voltage source inverter is used to lower the level of harmonics and ensure the grid systems' power quality. Various performance indicators are applied to assess the simulation and comparative results of the proposed MBOA-MPPT controlling technique integrated with an interleaved Luo converter.

    Citation: I.E.S. Naidu, S. Srikanth, A. Siva sarapakara Rao, Adabala Venkatanarayana. A novel mine blast optimization algorithm (MBOA) based MPPT controlling for grid-PV systems[J]. AIMS Electronics and Electrical Engineering, 2023, 7(2): 135-155. doi: 10.3934/electreng.2023008

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  • One of the most important areas in today's world is meeting the energy needs of various resources provided by nature. The advantages of renewable energy sources for many application sectors have attracted a lot of attention. The majority of grid-based enterprises use solar photovoltaic (PV) systems to collect sunlight as a reliable energy source. Due to solar PV's simple accessibility and efficient panel design, it is widely used in a variety of application scenarios. By employing the Maximum Power Point Tracking (MPPT) technique, the PV modules can typically operate at their best rate and draw the most power possible from the solar system. Some hybrid control mechanisms are utilized in solar PV systems in traditional works, which has limitations on the problems of increased time consumption, decreased efficiency, and increased THD. Thus, a new Mine Blast Optimization Algorithm (MBOA) based MPPT controlling model is developed to maximize the electrical energy produced by the PV panels under a different climatic situations. Also, an interleaved Luo DC-DC converter is used to significantly improve the output voltage of a PV system with a lower switching frequency. A sophisticated converter and regulating models are being created to effectively meet the energy demand of grid systems. The voltage source inverter is used to lower the level of harmonics and ensure the grid systems' power quality. Various performance indicators are applied to assess the simulation and comparative results of the proposed MBOA-MPPT controlling technique integrated with an interleaved Luo converter.



    For simplicity, we consider Poisson equation with a Dirichlet boundary condition as our model problem.

    Δu=f,inΩ, (1)
    u=g,onΩ, (2)

    where Ω is a bounded polygonal domain in R2.

    Using integration by parts, we can get the variational form: find uH1(Ω) satisfying u=gonΩ and

    (u,v)=(f,v),vH10(Ω). (3)

    Various finite element methods have been introduced to solve the Poisson equations (1)-(2), such as the Galerkin finite element methods (FEMs)[2, 3], the mixed FEMs [15] and the finite volume methods (FVMs) [6], etc. The FVMs emphasis on the local conservation property and discretize equations by asking the solution satisfying the flux conservation on a dual mesh consisting of control volumes. The mixed FEMs is another category method that based on the variable u and a flux variable usually written as p.

    The classical conforming finite element method obtains numerical approximate results by constructing a finite-dimensional subspace of H10(Ω). The finite element scheme has the same form with the variational form (3): find uhVhH1(Ω) satisfying uh=IhgonΩ and

    (uh,vh)=(f,vh),vhV0h, (4)

    where V0h is a subspace of Vh that satisfying vh=0 on Ω and Ih is the kth order Lagrange interpolation operator. The FE method is a popular and easy-to-implement numerical scheme, however, it is less flexible in constructing elements and generating meshes. These limitations are mainly due to the strong continuity requirements of functions in Vh. One solution to circumvent these limitations is using discontinuous approximations. Since the 1970th, many new finite element methods with discontinuous approximations have been developed, including the early proposed DG methods [1], local discontinuous Galerkin (LDG) methods [8], interior penalty discontinuous Galerkin (IPDG) methods [9], and the recently developed hybridizable discontinuous Galerkin (HDG) methods [7], mimetic finite differences method [10], virtual element (VE) method [4], weak Galerkin (WG) method [19, 20] and references therein.

    One obvious disadvantage of discontinuous finite element methods is their rather complex formulations which are often necessary to ensure connections of discontinuous solutions across element boundaries. For example, the IPDG methods add parameter depending interior penalty terms. Besides additional programming complexity, one often has difficulties in finding optimal values for the penalty parameters and corresponding efficient solvers. Most recently, Zhang and Ye [21] developed a discontinuous finite element method that has an ultra simple weak formulation on triangular/tetrahedal meshes. The corresponding numerical scheme can be written as: find uh˜Vh satisfying uh=IhgonΩ and

    (wuh,wvh)=(f,vh),vhV0h, (5)

    where ˜Vh is the DG finite element space and w is the weak gradient operator. The notion of weak gradient was first introduced by Wang and Ye in the weak Galerkin (WG) methods [19, 20]. The WG methods allow the use of totally discontinuous functions and provides stable numerical schemes that are parameter-independent and free of locking [17] in some applications. Another key feature in the WG methods is it can be used for arbitrary polygonal meshes. The WG finite element method has been rapidly developed and applied to other problems, including the Stokes and Navier-Stokes equations [11, 18], the biharmonic [14, 13] and elasticity equations [12, 17], div-curl systems and the Maxwell's equations and parabolic problem [23], etc. The introduction of the weak gradient operator in the conforming DG methods makes the scheme (5) maintain the simple formulation of conforming finite element method while have the flexibility of using discontinuous approximations. Hence, the programming complexity of this conforming DG scheme is significantly reduced. Furthermore, the scheme results in a simple symmetric and positive definite system.

    Following the work in [21, 22], we propose a new conforming DG finite element method on rectangular partitions in this work. It can be obtained from the conforming formulation simply by replacing by w and enforcing the boundary condition strongly. The simplicity of the conforming DG formulation will ease the complexity for implementation of DG methods. We note that the conforming DG method in [21] is based on triangular/tetrahedal meshes. Then in [22], the method is extended to work on general polytopal meshes by raising the degree of polynomials used to compute weak gradient.

    In this paper, we keep the same finite element space as DG method, replace the boundary function with the average of the inner function, and use the weak gradient arising from local Raviart-Thomas (RT) elements [5] to approximate the classic gradient. Moreover, the derivation process in this paper is based on rectangular RT elements [16]. Error estimates of optimal order are established for the corresponding conforming DG approximation in both a discrete H1 norm and the L2 norm. Numerical verifications have been performed on different kinds of quadrangle finite element space. In particular, super-convergence phenomenon have been observed for Q0 elements.

    The rest of this paper is organized as follows: In Section 2, we shall present the conforming DG finite element scheme for the Poisson equation on rectangular partitions. Section 3 is devoted to a discussion of the stability and solvability of the new method. In Section 4, we shall prepare ourselves for error estimates by deriving some identities. Error estimates of optimal order in H1 and L2 norm are established in Section 5. In Section 6, we present some numerical results to illustrate the theory derived in earlier sections. Finally in section 7, we conclude our major contributions in this article.

    Throughout this paper, we adopt the standard definition of Sobolev space Hs(Ω). For any given open bounded domain KΩ, (,)s,K,s,K, and ||s,K are used to denote the inner product, norm and semi-norm, respectively. The space H0(K) coincides with L2(K), and the subscripts K in the inner product, norm, and semi-norm can be dropped in the case of K=Ω. In particular, the function space H10(Ω) is defined as

    H10(Ω)={vH1(Ω):v|Ω=0},

    and the space H(div,Ω) is defined as the set of vector-valued functions q, which together with their divergence are square integrable, i.e.

    H(div,Ω)={q[L2(Ω)]d:qL2(Ω)}.

    Assume that the domain Ω is of polygonal type and is partitioned into non-overlapping rectangles Th={T}. For each TTh, denote by T0 its interior and T its boundary. Denote by Eh={e} the set of all edges in Th, and E0h=EhΩ the set of all interior edges in Th. For each TTh and eEh, denote by hT and he the diameter of T and e, respectively. h=max is the meshsize of \mathcal{T}_h .

    For any interior edge e\in\mathcal{E}^0_h , let T_1 and T_2 be two rectangles sharing e , we define the average \{\cdot\} and the jump [\![ \cdot]\! ] on e for a scalar-valued function v by

    \begin{eqnarray} \{v\}& = &\frac{1}{2}(v|_{\partial T_1}+v|_{\partial T_2}),\; \; [\![ v]\! ] = v|_{\partial T_1}\boldsymbol{n}_1+v|_{\partial T_2}\boldsymbol{n}_2, \end{eqnarray} (6)

    where v|_{\partial T_{i}},i = 1,2 is the trace of v on \partial T_i , \boldsymbol{n</italic>}_1 and \boldsymbol{n</italic>}_2 are the two unit outward normal vectors on e , associated with T_1 and T_2 , respectively. If e is a boundary edge, we define

    \begin{eqnarray} \{v\}& = &v|_e \; \; \text{and}\; \; [\![ v]\! ] = v|_e\boldsymbol{n}. \end{eqnarray} (7)

    We define a discontinuous finite element space

    \begin{eqnarray} V_h = \{v\in L^2(\Omega): v|_T \in Q_k(T), \; \forall \; T\in \mathcal{T}_h\}, \end{eqnarray} (8)

    and its subspace

    \begin{eqnarray} V^0_h = \{v \in V_h:v = 0 \; \text{on} \; \partial \Omega\}, \end{eqnarray} (9)

    where Q_k(T), \;k \geq 1 denotes the set of polynomials with regard to quadrilateral elements. The weak gradient for a scalar-valued function v\in V_h is defined by the following definition

    Definition 2.1. For a given T \in \mathcal{T}_h and a function v\in V_h , the discrete weak gradient \nabla_d v \in RT_k(T) on T is defined as the unique polynomial such that

    \begin{eqnarray} (\nabla_{d}v,\boldsymbol{q})_T: = -(v,\nabla\cdot\boldsymbol{q})_T +\langle \{v\},\boldsymbol{q}\cdot \boldsymbol{n}\rangle_{\partial T},\; \; \; \; \forall\; \boldsymbol{q}\in RT_k(T), \end{eqnarray} (10)

    where \boldsymbol{n} is the unit outward normal on \partial T , RT_k(T) = [Q_k(T)]^2+\bf{x}Q_k(T) , and \{v\} is defined in (6) and (7).

    The weak gradient operator \nabla_{d} as defined in (10) is a local operator computed at each element. It can be extended to any function v \in V_h by taking weak gradient locally on each element T. More precisely, the weak gradient of any v \in V_h is defined element-by-element as follows:

    \begin{eqnarray*} (\nabla_d v)|_T = \nabla_{d}(v|_T). \end{eqnarray*}

    We introduce the following bilinear form:

    \begin{eqnarray*} a(v,w) = (\nabla_d v, \nabla_d w), \end{eqnarray*}

    the conforming DG algorithm to solve the problems (1) - (2) is given by

    Conforming DG algorithm 1. Find u_h\in V_h satisfying u_h = I_hg \; \mathit{\text{on}} \; \partial \Omega and

    \begin{eqnarray} a(u_h, v_h) = (f, v_h), \; \; \forall \; v_h\in V^0_h, \end{eqnarray} (11)

    where I_h is the kth order Lagrange interpolation.

    We will prove the existence and uniqueness of the solution of equation (11). Firstly, we present the following two useful inequalities to derive the forthcoming analysis.

    Lemma 3.1 (trace inequality). Let T be an element of the finite element partition \mathcal{T}_h , and e is an edge or face which is part of \partial T . For any function \varphi\in H^1(T) , the following trace inequality holds true (see [20] for details):

    \begin{eqnarray} \|\varphi\|_e^2\leq C(h_T^{-1}\|\varphi\|_T^2+h_T\|\nabla\varphi\|_T^2), \end{eqnarray} (12)

    where C is a constant independent of h .

    Lemma 3.2 (inverse inequality). Let \mathcal{T}_h be a finite element partition of \Omega that is shape regualr. Assume that \mathcal{T}_h satisfies all the assumptions A1-A4 in [20]. Then, for any piecewise polynomial function \varphi of degree n on \mathcal{T}_h , there exists a constant C = C(n) such that

    \begin{eqnarray} \|\nabla\varphi\|_T\leq C(n)h_T^{-1}\|\varphi\|_T,\; \; \forall T\in \mathcal{T}_h. \end{eqnarray} (13)

    Then, we define the following semi-norms in the discontinuous finite element space V_h

    \begin{eqnarray} |\!|\!| v |\!|\!|^2& = &a(v, v) = \sum\limits_{T\in \mathcal{T}_h}\| \nabla_d v\|_T^2, \end{eqnarray} (14)
    \begin{eqnarray} \| v \|_{1,h}^2& = &\sum\limits_{T\in \mathcal{T}_h}\|\nabla v\|_T^2+\sum\limits_{e\in\mathcal{E}_h^0}h_e^{-1}\|[\![ v]\! ]\|_e^2. \end{eqnarray} (15)

    We have the equivalence between the semi-norms |\!|\!| v |\!|\!| and \|v\|_{1,h} , and it is proved in the following lemma.

    Lemma 3.3. For any v\in V_h , the following equivalence holds true

    \begin{eqnarray} C_1\|v\|_{1,h}\leq|\!|\!| v |\!|\!|\leq C_2\|v\|_{1,h}, \end{eqnarray} (16)

    where C_1 and C_2 are two constants independent of h .

    Proof. It follows from the definition of \nabla_d v , integration by parts, the trace inequality, and the inverse inequality that

    \begin{eqnarray} \|\nabla_d v\|_{T_1}^2 & = & (\nabla_d v, \nabla_d v)_{T_1} = -(v, \nabla\cdot\nabla_d v)_{T_1} + \langle\{v\}\boldsymbol{n}, \nabla_d v\rangle_{\partial {T_1}}\\ & = & (\nabla v, \nabla_d v)_{T_1} - \langle (v-\{v\})\boldsymbol{n}, \nabla_d v\rangle_{\partial {T_1}}\\ &\leq& \|\nabla v\|_{T_1}\|\nabla_d v\|_{T_1}+\|(v-\{v\})\boldsymbol{n}\|_{\partial {T_1}}\|\nabla_d v\|_{\partial {T_1}}\\ &\leq&\|\nabla_d v\|_{T_1}(\|\nabla v\|_{T_1} + h_{T_1}^{-\frac{1}{2}}\|(v-\{v\})\boldsymbol{n}\|_{\partial {T_1}}). \end{eqnarray} (17)

    For any e\subset\partial T_1 , e = \partial T_1\cap\partial T_2 , we have

    \begin{eqnarray*} (v-\{v\})|_e\boldsymbol{n}_1 & = & v|_{\partial T_1}\boldsymbol{n}_1-\frac{1}{2}(v|_{\partial T_1}+v|_{\partial T_2})\boldsymbol{n}_1\\ & = & \frac{1}{2}(v|_{\partial T_1}\boldsymbol{n}_1+v|_{\partial T_2}\boldsymbol{n}_2)\\ & = & \frac{1}{2}[\![ v]\! ]_e. \end{eqnarray*}

    Then we can get

    \begin{eqnarray} \|(v-\{v\})\boldsymbol{n}\|_{\partial T_1}^2\leq\frac{1}{2}\sum\limits_{e\in\partial T_1}\|[\![ v]\! ]\|_e^2. \end{eqnarray} (18)

    Substituting (18) into (17) gives

    \begin{eqnarray*} \|\nabla_d v\|_{T_1}^2 &\leq& C_2\|\nabla_d v\|_{T_1}(\|\nabla v\|_{T_1}+\sum\limits_{e\in\partial T_1}h_e^{-\frac{1}{2}}\|[\![ v]\! ]\|_e), \end{eqnarray*}

    this completes the proof of the right-hand of (16).

    To prove the left-hand of (16), we consider the subspace of RT_k(T) for any T\in\mathcal{T}_h

    \begin{eqnarray*} D(k,T): = \{\bf{q}\in RT_k(T):\; \boldsymbol{q}\cdot\boldsymbol{n} = 0\; \text{on}\; \partial T\}. \end{eqnarray*}

    Note that D(k,T) is a dual space of [Q_{k-1}(T)]^2 [13]. Thus, for any \nabla v\in[Q_{k-1}(T)]^2 , we have

    \begin{eqnarray} \|\nabla v\|_T = \sup\limits_{\boldsymbol{q}\in D(k,T)}\frac{(\nabla v,\boldsymbol{q})_T}{\|\boldsymbol{q}\|_T}. \end{eqnarray} (19)

    Using the integration by parts, Cauchy-Schwarz inequality, the definition of D(k,T) and \nabla_d v , we get

    \begin{eqnarray*} (\nabla v,\boldsymbol{q})_T & = & -(v,\nabla\cdot\boldsymbol{q})_T + \langle v, \boldsymbol{q}\cdot\boldsymbol{n}\rangle_{\partial T}\\ & = & (\nabla_d v, \boldsymbol{q})_T-\langle \{v\},\boldsymbol{q}\cdot \boldsymbol{n}\rangle_{\partial T}\\ & = & (\nabla_d v, \boldsymbol{q})_T\\ &\leq& \|\nabla_d v\|_T\cdot\|\boldsymbol{q}\|_T, \end{eqnarray*}

    where we have used the fact that \boldsymbol{q</italic>}\cdot \boldsymbol{n}|_{\partial T} = 0 in the definition of D(k,T) . Combining the above result with (19), one has

    \begin{eqnarray} \|\nabla v\|_T \leq \|\nabla_d v\|_T. \end{eqnarray} (20)

    We define the space D_e(k,T) as the set of all \boldsymbol{q</italic>} \in RT_k(T) such that all degrees of freedom, except those for \boldsymbol{q</italic>} \cdot\boldsymbol{n}|_e , vanish. Note that D_e(k,T) is a dual space of [Q_k(e)]^2 [13]. Thus, we know

    \begin{eqnarray} \|[\![ v]\! ]\|_e = \sup\limits_{\boldsymbol{q}\in D_e(k,T)}\frac{\langle [\![ v]\! ], \boldsymbol{q}\cdot\boldsymbol{n}\rangle_e}{\|\boldsymbol{q}\cdot\boldsymbol{n}\|_e}. \end{eqnarray} (21)

    Following the integration by parts and the definition of \nabla_d , we can derive that

    \begin{eqnarray*} (\nabla_d v,\boldsymbol{q})_T = ( \nabla v, \boldsymbol{q} )_T - \langle v,\boldsymbol{q}\cdot\boldsymbol{n}\rangle_e + \langle \{v\}, \boldsymbol{q}\cdot\boldsymbol{n}\rangle_e. \end{eqnarray*}

    Together with (20), we obtain

    \begin{eqnarray*} |\langle [\![ v]\! ], \boldsymbol{q}\cdot\boldsymbol{n}\rangle_e| & = &2|(\nabla_d v,\boldsymbol{q})_T - (\nabla v, \boldsymbol{q})_T|\\ &\leq& 2|(\nabla_d v,\boldsymbol{q})_T|+2|(\nabla v, \boldsymbol{q})_T|\\ &\leq& C(\|\nabla_d v\|_T\|\boldsymbol{q}\|_T+\|\nabla v\|_T\|\boldsymbol{q}\|_T)\\ &\leq& C\|\nabla_d v\|_T\|\boldsymbol{q}\|_T. \end{eqnarray*}

    Substituting the above inequality into (21), by the scaling argument [13], for such \boldsymbol{q} \in D_e(k,T) , we have \|\boldsymbol{q}\|_T \leq h^{\frac{1}{2}}\|\boldsymbol{q}\cdot\boldsymbol{n}\|_e , then

    \begin{eqnarray} \|[\![ v]\! ]\|_e \leq C\frac{\|\nabla_d v\|_T\|\boldsymbol{q}\|_T}{\|\boldsymbol{q}\cdot\boldsymbol{n}\|_e} \leq Ch^{\frac{1}{2}}\|\nabla_d v\|_T. \end{eqnarray} (22)

    Combining (20) and (22) gives a proof of the left-hand of (16).

    Lemma 3.4. The semi-norm |\!|\!| \cdot |\!|\!| defined in (14) is a norm in V_h^0 .

    Proof. We shall only verify the positivity property for |\!|\!| \cdot |\!|\!| . To this end, assume |\!|\!| v |\!|\!| = 0 for some v\in V_h^0 . By Lemma 3.3, it follows that \| v \|_{1,h} = 0 for all T\in\mathcal{T}_h , which means that \nabla v = {\mathit{\boldsymbol{0}}} for all elements T\in\mathcal{T}_h and [\![ v]\! ] = 0 for all edges e\in\mathcal{E}_h^0 . We can derive from \nabla v = {\mathit{\boldsymbol{0}}} for all T\in\mathcal{T}_h that v is a constant in each T . [\![ v]\! ] = 0 on each e\in\mathcal{E}_h^0 implies v is a continuous function. This two conclusions and v = 0 on \partial \Omega show that v = 0 , which completes the proof of the lemma.

    The above two lemmas imply the well posedness of the scheme (11). We prove the existence and uniqueness of solution of the conforming DG method in Theorem 3.1.

    Theorem 3.1. The conforming DG scheme (11) has and only has one solution.

    Proof. To prove the scheme (11) is uniquely solvable, it suffices to verify that the homogeneous equation has zero as its unique solution. To this end, let u_h\in V_h be the solution of the numerical scheme 11 with homogeneous data f = {0} , {g} = 0 . Letting v_h = u_h , we obtain

    \begin{eqnarray*} a(u_h, u_h) = 0, \end{eqnarray*}

    which leads to u_h = 0 by using Lemma 3.4. This completes the proof of the theorem.

    In this section, we will derive an error equation which will be used for the error estimates. For any \boldsymbol{q</italic>}\in H(div, \Omega) , we assume that there exist an interpolation operator \Pi_h satisfying \Pi_h\boldsymbol{q}\in H(div, \Omega)\cap RT_k(T) on each element T\in\mathcal{T}_h and

    \begin{eqnarray} (\nabla\cdot\boldsymbol{q},v)_T = (\nabla\cdot\Pi_h\boldsymbol{q}, v)_T, \; \; \forall v \in Q_k(T). \end{eqnarray} (23)

    For any w\in H^{1+k}(\Omega) with k\geq 1 , from Lemma 7.3 in [20], we have the estimate of \Pi_h as follows.

    \begin{eqnarray} \|\Pi_h(\nabla w)-\nabla w\|\leq Ch^k\|w\|_{1+k}. \end{eqnarray} (24)

    Moreover, it is easy to verify the following property holds true.

    Lemma 4.1. For any \boldsymbol{q}\in H(div, \Omega) ,

    \begin{eqnarray} \sum\limits_{T\in\mathcal{T}_h}(-\nabla\cdot\boldsymbol{q}, v)_T = \sum\limits_{T\in\mathcal{T}_h}(\Pi_h\boldsymbol{q}, \nabla_d v)_T, \; \; \forall v\in V_h^0. \end{eqnarray} (25)

    Proof. \Pi_h\boldsymbol{q}\in H(div, \Omega) implies that \Pi_h\boldsymbol{q} is continuous across each interior edge. Since v\in V_h^0 , we know that \{v\} = v = 0 on \partial \Omega . Then

    \begin{eqnarray} \sum\limits_{T\in\mathcal{T}_h}\langle\{v\}, \Pi_h\boldsymbol{q}\cdot\boldsymbol{n}\rangle_{\partial T} = 0. \end{eqnarray} (26)

    By the definition of \Pi_h and \nabla_d and the equation (26), we have

    \begin{eqnarray*} \sum\limits_{T\in\mathcal{T}_h}(-\nabla\cdot\boldsymbol{q}, v)_T & = & \sum\limits_{T\in\mathcal{T}_h}(-\nabla\cdot\Pi_h\boldsymbol{q}, v)_T\\ & = & \sum\limits_{T\in\mathcal{T}_h}(-\nabla\cdot\Pi_h\boldsymbol{q}, v)_T+ \sum\limits_{T\in\mathcal{T}_h}\langle\{v\}, \Pi_h\boldsymbol{q}\cdot\boldsymbol{n}\rangle_{\partial T}\\ & = & \sum\limits_{T\in\mathcal{T}_h}(\Pi_h\boldsymbol{q}, \nabla_d v)_T. \end{eqnarray*}

    This completes the proof of the lemma.

    Before establishing the error equation, we define a continuous finite element subspace of V_h as follows

    \begin{eqnarray} \tilde{V}_{h} = \{v \in H^{1}(\Omega) :v|_T \in Q_k(T), \; \forall T \in \mathcal{T}_{h}\}. \end{eqnarray} (27)

    so as a subspace of \tilde{V}_h

    \begin{eqnarray} \tilde{V}_h^0: = \{v\in\tilde{V}_h:v|_{\partial\Omega} = 0\}. \end{eqnarray} (28)

    Lemma 4.2. For any v\in\tilde{V}_h , we have

    \begin{eqnarray*} \nabla_d v = \nabla v. \end{eqnarray*}

    Proof. By the definition of \nabla_d and integration by parts, for any \boldsymbol{q</italic>}\in RT_k(T) , we have

    \begin{eqnarray*} (\nabla_d v, \boldsymbol{q})_T & = & -(v, \nabla\cdot\boldsymbol{q})_T + \langle\{v\}, \boldsymbol{q}\cdot\boldsymbol{n}\rangle_{\partial T}\\ & = & -(v,\nabla\cdot\boldsymbol{q})_T + \langle v, \boldsymbol{q}\cdot\boldsymbol{n}\rangle_{\partial T}\\ & = & (\nabla v, \boldsymbol{q})_T, \end{eqnarray*}

    which gives

    \begin{eqnarray*} (\nabla_d v-\nabla v, \boldsymbol{q})_T = 0, \; \; \forall \boldsymbol{q}\in RT_k(T). \end{eqnarray*}

    Letting \boldsymbol{q} be \nabla_d v-\nabla v in the above equation yields \|\nabla_d v-\nabla v\| = 0 , which completes the proof of the lemma.

    Let e_h = I_h u - u_h , where I_h is the kth order Lagrange interpolation, u\in H^{k+1}(\Omega) with k \geq 1 is the exact solution of the Poisson equations (1) - (2), and u_h\in V_h is the numerical solution of the scheme (11). The following estimate of the Lagrange interpolation operator I_h holds true.

    \begin{eqnarray} \|I_hu-u\|\leq Ch^{k+1}\|u\|_{k+1}, \end{eqnarray} (29)
    \begin{eqnarray} \|\nabla I_hu-\nabla u\|\leq Ch^k\|u\|_{k+1}. \end{eqnarray} (30)

    It is obvious that e_h\in V_h^0 and I_h u\in\tilde{V}_h . We have the following lemma:

    Lemma 4.3. Denote e_h = I_h u - u_h the error of conforming DG method arising from (11). For any v_h\in V_h^0 , we have

    \begin{eqnarray} a(e_h, v_h) = l_u( v_h), \end{eqnarray} (31)

    where

    \begin{eqnarray} l_u(v_h) = \sum\limits_{T\in{\mathcal{T}_h}}(\nabla I_{h} u-\Pi_h\nabla u, \nabla_d v_h). \end{eqnarray} (32)

    Proof. Since I_h u\in\tilde{V}_h , we have \nabla_d I_h u = \nabla I_h u . Using the property (25), we can derive

    \begin{eqnarray*} \sum\limits_{T\in{\mathcal{T}_h}}(\nabla_d I_h u, \nabla_d v_h)_T & = & \sum\limits_{T\in{\mathcal{T}_h}}(\nabla I_h u, \nabla_d v_h)_T\\ & = & \sum\limits_{T\in{\mathcal{T}_h}}(\nabla I_h u -\Pi_h\nabla u + \Pi_h\nabla u, \nabla_d v_h)_T\\ & = & \sum\limits_{T\in{\mathcal{T}_h}}(\nabla I_h u -\Pi_h\nabla u, \nabla_d v_h)_T +\sum\limits_{T\in{\mathcal{T}_h}}(\Pi_h\nabla u, \nabla_d v_h)_T\\ & = & l_u(v_h) - \sum\limits_{T\in{\mathcal{T}_h}}(\nabla\cdot\nabla u, v_h)_T\\ & = & l_u(v_h) + (f,v_h). \end{eqnarray*}

    By the definition of the scheme (11), we have

    \begin{eqnarray*} \sum\limits_{T\in{\mathcal{T}_h}}(\nabla_d I_h u - \nabla_d u_h, \nabla_d v_h)_T = l_u(v_h). \end{eqnarray*}

    This completes the proof of the lemma.

    The goal of this section is to derive the error estimates in H^1 and L^2 norms for the conforming DG solution u_h .

    Theorem 5.1. Let u\in H^{k+1}(\Omega) with k \geq 1 be the exact solution of the Poisson equation (1) - (2), and u_h\in V_h be the numerical solution of the scheme (11). Let e_h = I_h u - u_h , there exists a constant C independent of h such that

    \begin{eqnarray} |\!|\!|e_h|\!|\!| \leq Ch^k|u|_{k+1}. \end{eqnarray} (33)

    Proof. Letting v_h = e_h in (31), and by the definition of |\!|\!|\cdot|\!|\!| , we have

    \begin{eqnarray} |\!|\!|e_h|\!|\!|^2 = l_u(e_h). \end{eqnarray} (34)

    From the Cauchy-Schwarz inequality, the triangle inequality, the definition of |\!|\!|\cdot|\!|\!| , (24), and (30), we arrive at

    \begin{eqnarray*} l_u(v_h)& = & \sum\limits_{T\in\mathcal{T}_h}(\nabla I_hu-\Pi_h(\nabla u), \nabla_d v_h)_T\\ &\leq& \sum\limits_{T\in\mathcal{T}_h}\|\nabla I_hu-\Pi_h(\nabla u)\|_T\|\nabla_d v_h\|_T\\ &\leq& \left(\sum\limits_{T\in\mathcal{T}_h}\|\nabla I_hu-\Pi_h(\nabla u)\|_T^2\right)^{\frac{1}{2}}\left(\sum\limits_{T\in\mathcal{T}_h}\|\nabla_d v_h\|_T^2\right)^{\frac{1}{2}}\\ & = & \left(\sum\limits_{T\in\mathcal{T}_h}\|\nabla I_hu-\nabla u+\nabla u-\Pi_h(\nabla u)\|_T^2\right)^{\frac{1}{2}}|\!|\!|v_h|\!|\!|\\ &\leq& \left(\sum\limits_{T\in\mathcal{T}_h}\|\nabla I_hu-\nabla u\|_T^2+\|\nabla u-\Pi_h(\nabla u)\|_T^2\right)^{\frac{1}{2}}|\!|\!|v_h|\!|\!|\\ &\leq& Ch^k|u|_{k+1}|\!|\!|v_h|\!|\!|. \end{eqnarray*}

    Then, we have

    \begin{eqnarray} l_u(e_h) \leq Ch^k|u|_{k+1}|\!|\!|e_h|\!|\!|. \end{eqnarray} (35)

    Substituting (35) to (34), we obtain

    \begin{eqnarray*} |\!|\!|e_h|\!|\!|^2 \leq Ch^k|u|_{k+1}|\!|\!|e_h|\!|\!|, \end{eqnarray*}

    which completes the proof of the lemma.

    It is obvious that \tilde{V}_h^0\subset V_h^0 . Let \tilde{u}_h\in\tilde{V}_h be the finite element solution for the problem (1)-(2) which satisfies \tilde{u}_h = I_hg on \partial\Omega and

    \begin{eqnarray} (\nabla\tilde{u}_h, \nabla v) = (f, v), \; \; \forall v\in\tilde{V}_h^0. \end{eqnarray} (36)

    For any v\in\tilde{V}_h^0\subset\tilde{V}_h , we have \nabla_d v = \nabla v , i.e.

    \begin{eqnarray} (\nabla_d u_h - \nabla\tilde{u}_h, \nabla v) = 0, \; \; \forall v\in\tilde{V}_h^0. \end{eqnarray} (37)

    In the rest of this section, we derive an optimal order error estimate for the conforming DG approximation (11) in L^2 norm by adopting the duality argument. To this end, we consider the following dual problem that seeks \Phi\in H_0^1(\Omega) satisfying

    \begin{eqnarray} -\nabla\cdot(\nabla\Phi) = u_h-\tilde{u}_h, \; \; in\; \Omega. \end{eqnarray} (38)

    Assume that the dual problem satisfies H^2 -regularity, which means the following priori estimate holds true

    \begin{eqnarray} \|\Phi\|_2 \leq C\|u_h-\tilde{u}_h\|. \end{eqnarray} (39)

    In the following of this paper, we note \varepsilon_h = u_h-\tilde{u}_h for simplicity.

    Theorem 5.2. Assume u\in H^{k+1}(\Omega) with k \geq 1 is the exact solution of the Poisson equation (1) - (2), and u_h\in V_h is the numerical solution obtained with the scheme (11). Furthermore, assume that (39) holds true. Then, there exists a constant C independent of h such that

    \begin{eqnarray} \|u-u_h\|\leq C h^{k+1}|u|_{k+1}. \end{eqnarray} (40)

    Proof. First, we shall derive the optimal order for \varepsilon_h in L^2 norm. Consider the corresponding conforming DG scheme defined in (11) and let \Phi_h\in V_h^0 be the solution satisfying

    \begin{eqnarray} a(\Phi_h, v) = (\varepsilon_h, v), \; \; \forall v\in V_h^0. \end{eqnarray} (41)

    Since I_h\Phi\in\tilde{V}_h , it follows from (37) that

    \begin{eqnarray*} (\nabla_d u_h-\nabla\tilde{u}_h, \nabla I_h\Phi) & = & 0,\\ \nabla_d I_h\Phi & = & \nabla I_h\Phi, \end{eqnarray*}

    which gives

    \begin{eqnarray} (\nabla_d u_h-\nabla\tilde{u}_h, \nabla_d I_h\Phi) = 0. \end{eqnarray} (42)

    Setting v = \varepsilon_h in (41), then by the definition of \varepsilon_h and (42), we have

    \begin{eqnarray*} \|\varepsilon_h\|^2& = &a(\Phi_h, \varepsilon_h) = \sum\limits_{T\in\mathcal{T}_h}(\nabla_d \Phi_h, \nabla_d\varepsilon_h)_T\\ & = & \sum\limits_{T\in\mathcal{T}_h}(\nabla_d (\Phi_h-I_h\Phi), \nabla_d u_h -\nabla\tilde{u}_h)_T\\ &\leq&|\!|\!| \Phi_h-I_h\Phi |\!|\!|(|\!|\!|u_h-I_h u|\!|\!|+\|\nabla(I_h u-\tilde{u}_h)\|). \end{eqnarray*}

    Then, by the Cauchy-Schwarz inequality, (33) and (39), we obtain

    \begin{eqnarray*} \|\varepsilon_h\|^2&\leq& Ch|\Phi|_2h^k|u|_{k+1} \leq Ch^{k+1}|u|_{k+1}\|\varepsilon_h\|, \end{eqnarray*}

    which gives

    \begin{eqnarray} \|\varepsilon_h\| \leq Ch^{k+1}|u|_{k+1}. \end{eqnarray} (43)

    Combining the error estimate of finite element solution, the triangle inequality and (43) yields (40), which completes the proof of the theorem.

    In this section, we shall present some numerical results for the conforming discontinuous Galerkin method analyzed in the previous sections.

    We solve the following Poisson equation on the unit square domain \Omega = (0,1)\times (0,1) ,

    \begin{align} -\Delta u & = 2 \pi^2 \sin(\pi x)\sin(\pi y) &&\hbox{in }\Omega && \end{align} (44)
    \begin{align} u& = 0 &&\hbox{on }\partial\Omega. \end{align} (45)

    The exact solution of the above problem is u = \sin(\pi x)\sin(\pi y) . Uniform square grids as shown in Figure 1 are used for computation.

    Figure 1.  The first three grids used in the computation.

    We first use the P_k conforming discontinuous Galerkin spaces (8) to compute the test case (44)-(45), where P_k denotes the set of polynomials of 2 variables of degree less than or equal to k. The weak gradient is computed locally using rectangular RT_k polynomials. The errors and the order of convergence of the conforming DG approximations are listed in Table 1. Optimal order of convergence is achieved in every case, which is consistent with our theory. In particular, a superconvergence of order \mathcal{O}(h^2) was observed in the discrete H^1 norm for P_0 elements. Furthermore, the results obtained with P_0 elements seems to be slightly better than that obtained with P_1 elements.

    Table 1.  Error profiles and convergence rates for test case (44)-(45) obtained with uniform grids and P_k conforming DG spaces.
    level \|u_h- Q_h u\|_0 rate {|\!|\!|} u_h- Q_h u{|\!|\!|} rate \#Dof
    by P_0 conforming discontinuous Galerkin elements
    6 0.1996E-02 1.97 0.8887E-02 1.98 1024
    7 0.5013E-03 1.99 0.2228E-02 2.00 4096
    8 0.1255E-03 2.00 0.5574E-03 2.00 16384
    by P_1 conforming discontinuous Galerkin elements
    6 0.2427E-02 1.97 0.1027E+00 1.02 3072
    7 0.6100E-03 1.99 0.5105E-01 1.01 12288
    8 0.1527E-03 2.00 0.2546E-01 1.00 49152
    by P_2 conforming discontinuous Galerkin elements
    5 0.1533E-03 3.00 0.2042E-01 2.03 1536
    6 0.1915E-04 3.00 0.5061E-02 2.01 6144
    7 0.2394E-05 3.00 0.1260E-02 2.01 24576
    by P_3 conforming discontinuous Galerkin elements
    5 0.7959E-05 4.00 0.1965E-02 3.00 2560
    6 0.4971E-06 4.00 0.2451E-03 3.00 10240
    7 0.3140E-07 3.98 0.3059E-04 3.00 40960
    by P_4 conforming discontinuous Galerkin elements
    4 0.1055E-04 4.97 0.1421E-02 4.05 960
    5 0.3314E-06 4.99 0.8735E-04 4.02 3840
    6 0.1057E-07 4.97 0.5417E-05 4.01 15360
    by P_5 conforming discontinuous Galerkin elements
    2 0.2835E-02 6.24 0.1450E+00 5.49 84
    3 0.4532E-04 5.97 0.4718E-02 4.94 336
    4 0.7115E-06 5.99 0.1478E-03 5.00 1344

     | Show Table
    DownLoad: CSV

    The same test case is also computed using the Q_k conforming DG finite element space, where Q_k denotes the set of polynomials of 2 variables defined on \Omega , and for each variable, the degree of the variable is at most k. Table 2 illustrates the numerical performance of the corresponding conforming DG scheme. It can be seen from numerical computing that, in this case, the results obtained with the Q_1 element are more accurate than those obtained with Q_0( = P_0) elements (see Table 1). All numerical results converge at the corresponding optimal order, which is consistent with the theory.

    Table 2.  Error profiles and convergence rates for test case (44)-(45) obtained with uniform grids and Q_k conforming DG spaces.
    level \|u_h- Q_hu\|_0 rate {|\!|\!|} u_h- Q_h u{|\!|\!|} rate \#Dof
    by Q_1 conforming discontinuous Galerkin elements
    6 0.4006E-03 1.99 0.2389E-02 1.99 4096
    7 0.1003E-03 2.00 0.5982E-03 2.00 16384
    8 0.2510E-04 2.00 0.1496E-03 2.00 65536
    by Q_2 conforming discontinuous Galerkin elements
    6 0.2360E-04 2.99 0.3186E-02 1.99 9216
    7 0.2953E-05 3.00 0.7976E-03 2.00 36864
    8 0.3692E-06 3.00 0.1995E-03 2.00 147456
    by Q_3 conforming discontinuous Galerkin elements
    5 0.1413E-04 4.08 0.1650E-02 2.97 4096
    6 0.8676E-06 4.03 0.2072E-03 2.99 16384
    7 0.5398E-07 4.01 0.2593E-04 3.00 65536
    by Q_4 conforming discontinuous Galerkin elements
    3 0.2226E-02 4.59 0.5414E-01 3.52 400
    4 0.9610E-04 4.53 0.3723E-02 3.86 1600
    5 0.3279E-05 4.87 0.2392E-03 3.96 6400

     | Show Table
    DownLoad: CSV

    To test the superconvergence of P_0 DG element, we solve the following 2nd order elliptic equation on the unit square domain \Omega = (0,1)\times (0,1) ,

    \begin{align*} -\Delta u +u & = f &&\hbox{in }\Omega &&\\ u& = 0 &&\hbox{on }\partial\Omega, \end{align*}

    where f is chosen so that the exact solution is not symmetric,

    \begin{align} u = (x-x^2)(y-y^3). \end{align} (46)

    Uniform square grids as shown in Figure 1 are used for numerical computation. The numerical results are listed in Table 3. Surprising, for this problem, the H^1 -like norm of error superconverges at 1.5 order, and the L^2 error has one order of superconvergence. But we do not yet know if such a superconvergence exists in general.

    Table 3.  Error profiles and convergence rates for test case (46) obtained with uniform grids and P_0 conforming DG spaces.
    level \|u_h- Q_h u\|_0 rate {|\!|\!|} u_h- Q_h u{|\!|\!|} rate \#Dof
    by P_0 conforming discontinuous Galerkin elements
    3 0.8265E-02 1.06 0.4577E-01 1.14 16
    4 0.2772E-02 1.58 0.1732E-01 1.40 64
    5 0.7965E-03 1.80 0.6331E-02 1.45 256
    6 0.2142E-03 1.90 0.2290E-02 1.47 1024
    7 0.5564E-04 1.94 0.8213E-03 1.48 4096
    8 0.1419E-04 1.97 0.2928E-03 1.49 16384

     | Show Table
    DownLoad: CSV

    To test further the superconvergence of P_0 DG element, we solve the following 2nd order elliptic equations on the unit square domain \Omega = (0,1)\times (0,1) ,

    \begin{align*} -\nabla(a \nabla u ) & = f &&\hbox{in }\Omega &&\\ u& = 0 &&\hbox{on }\partial\Omega, \end{align*}

    where a = 1+x+y and f is chosen so that the exact solution is not symmetric,

    \begin{align} u = (x-x^3)(y^2-y^3). \end{align} (47)

    Uniform square grids as shown in Figure 1 are used for computation. The numerical results are listed in Table 4. Surprising, again, the H^1 -like norm of error superconverges at 1.5 order, and the L^2 error has one order of superconvergence for this problem.

    Table 4.  Error profiles and convergence rates for test case (47) obtained with uniform grids and P_0 conforming DG spaces.
    level \|u_h- Q_h u\|_0 rate {|\!|\!|} u_h- Q_h u{|\!|\!|} rate \#Dof
    by P_0 conforming discontinuous Galerkin elements
    3 0.4929E-02 0.97 0.5371E-01 0.80 16
    4 0.1917E-02 1.36 0.2401E-01 1.16 64
    5 0.6004E-03 1.67 0.9407E-02 1.35 256
    6 0.1682E-03 1.84 0.3507E-02 1.42 1024
    7 0.4457E-04 1.92 0.1275E-02 1.46 4096
    8 0.1148E-04 1.96 0.4576E-03 1.48 16384

     | Show Table
    DownLoad: CSV

    In this paper, we establish a new numerical approximation scheme based on the rectangular partition to solve second order elliptic equation. We derived the numerical scheme and then proved the optimal order of convergence of the error estimates in L^2 and H^1 norms of the conforming DG method. Numerical experiments are then present to verify the theoretical analysis, and all numerical results converging at the corresponding optimal order. Comparing with existing numerical methods, the confoming DG method has the following two characteristics: 1. The formulation is relatively simple. The stabilizer s(\cdot\; , \; \cdot) is no longer needed, and the boundary function u_b is omitted, which is replaced by the average of internal function u_0 ; 2. The projection operator Q_h used in the traditional WG method is replaced by the Lagrange interpolation operator I_h , which makes the theoretical analysis much easier. As can be seen from the numerical examples in Section 6, this method reduces the programming complexity while ensuring the optimal order of convergence.



    [1] Kumar SP, Agyekum EB, Kumar A, Velkin VI (2023) Performance evaluation with low-cost aluminum reflectors and phase change material integrated to solar PV modules using natural air convection: An experimental investigation. Energy 266: 126415. https://doi.org/10.1016/j.energy.2022.126415 doi: 10.1016/j.energy.2022.126415
    [2] Praveenkumar S, Agyekum EB, Kumar A, Velkin VI (2023) Thermo-enviro-economic analysis of solar photovoltaic/thermal system incorporated with u-shaped grid copper pipe, thermal electric generators and nanofluids: An experimental investigation. J Energy Storage 60: 106611. https://doi.org/10.1016/j.est.2023.106611 doi: 10.1016/j.est.2023.106611
    [3] Essa ME-SM, Hussian OS, Hassan MM (2021) Intelligent Fractional Control Design of MPPT for a Standalone PV System Based on Optimization Technique. 2021 17th International Computer Engineering Conference (ICENCO), 107‒111. IEEE. https://doi.org/10.1109/ICENCO49852.2021.9698966
    [4] Subramanian A, Jayaparvathy R (2021) Performance comparison of modified elephant herding optimization tuned MPPT for PV based solar energy systems. Circuit World 48: 309‒321. https://doi.org/10.1108/CW-11-2020-0316 doi: 10.1108/CW-11-2020-0316
    [5] Subramanian A, Raman J (2021) Grasshopper optimization algorithm tuned maximum power point tracking for solar photovoltaic systems. J Amb Intel Hum Comp 12: 8637‒8645. https://doi.org/10.1007/s12652-020-02593-9 doi: 10.1007/s12652-020-02593-9
    [6] Kihal A, Krim F, Laib A, Talbi B, Afghoul H (2019) An improved MPPT scheme employing adaptive integral derivative sliding mode control for photovoltaic systems under fast irradiation changes. ISA T 87: 297‒306. https://doi.org/10.1016/j.isatra.2018.11.020 doi: 10.1016/j.isatra.2018.11.020
    [7] Mirza AF, Mansoor M, Ling Q, Yin B, Javed MY (2020) A Salp-Swarm Optimization based MPPT technique for harvesting maximum energy from PV systems under partial shading conditions. Energ Convers Manage 209: 112625. https://doi.org/10.1016/j.enconman.2020.112625 doi: 10.1016/j.enconman.2020.112625
    [8] Mirza AF, Mansoor M, Ling Q, Khan MI, Aldossary OM (2020) Advanced variable step size incremental conductance MPPT for a standalone PV system utilizing a GA-tuned PID controller. Energies 13: 1‒25. https://doi.org/10.3390/en13164153 doi: 10.3390/en13164153
    [9] Karrag A, Messalti S (2019) PSO‐based SMC variable step size P & O MPPT controller for PV systems under fast changing atmospheric conditions. Int J Numer Model El 32: e2603. https://doi.org/10.1002/jnm.2603 doi: 10.1002/jnm.2603
    [10] Mahesh PV, Meyyappan S, Alla RKR (2022) A new multivariate linear regression MPPT algorithm for solar PV system with boost converter. ECTI Transactions on Electrical Engineering, Electronics, and Communications 20: 269‒281. https://doi.org/10.37936/ecti-eec.2022202.246909 doi: 10.37936/ecti-eec.2022202.246909
    [11] Ebrahim M, Osama A, Kotb KM, Bendary F (2019) Whale inspired algorithm based MPPT controllers for grid-connected solar photovoltaic system. Energy Procedia 162: 77‒86. https://doi.org/10.1016/j.egypro.2019.04.009 doi: 10.1016/j.egypro.2019.04.009
    [12] Aly M, Rezk H (2022) An improved fuzzy logic control-based MPPT method to enhance the performance of PEM fuel cell system. Neural Computing and Applications 34: 4555‒4566. https://doi.org/10.1007/s00521-021-06611-5 doi: 10.1007/s00521-021-06611-5
    [13] Chauhan U, Singh V, Kumar B, Rani A (2020) An improved MVO assisted global MPPT algorithm for partially shaded PV system. J Intell Fuzzy Syst 38: 6715‒6726. https://doi.org/10.3233/JIFS-179749 doi: 10.3233/JIFS-179749
    [14] Gupta AK, Pachauri RK, Maity T, Chauhan YK, Mahela OP, Khan B, et al. (2021) Effect of various incremental conductance MPPT methods on the charging of battery load feed by solar panel. IEEE Access 9: 90977‒90988. https://doi.org/10.1109/ACCESS.2021.3091502 doi: 10.1109/ACCESS.2021.3091502
    [15] Wasim MS, Amjad M, Habib S, Abbasi MA, Bhatti AR, Muyeen S (2022) A critical review and performance comparisons of swarm-based optimization algorithms in maximum power point tracking of photovoltaic systems under partial shading conditions. Energy Reports 8: 4871‒4898. https://doi.org/10.1016/j.egyr.2022.03.175 doi: 10.1016/j.egyr.2022.03.175
    [16] Dagal I, Akın B, Akboy E (2022) MPPT mechanism based on novel hybrid particle swarm optimization and salp swarm optimization algorithm for battery charging through simulink. Scientific reports 12: 1‒17. https://doi.org/10.1038/s41598-022-06609-6 doi: 10.1038/s41598-021-99269-x
    [17] González-Castaño C, Restrepo C, Kouro S, Rodriguez J (2021) MPPT algorithm based on artificial bee colony for PV system. IEEE Access 9: 43121‒43133. https://doi.org/10.1109/ACCESS.2021.3066281 doi: 10.1109/ACCESS.2021.3066281
    [18] Yap KY, Sarimuthu CR, Lim JM-Y (2020) Artificial intelligence based MPPT techniques for solar power system: A review. J Mod Power Syst Cle 8: 1043‒1059. https://doi.org/10.35833/MPCE.2020.000159 doi: 10.35833/MPCE.2020.000159
    [19] Mirza AF, Mansoor M, Ling Q (2020) A novel MPPT technique based on Henry gas solubility optimization. Energ Convers Manage 225: 113409. https://doi.org/10.1016/j.enconman.2020.113409 doi: 10.1016/j.enconman.2020.113409
    [20] Khan FU, Gulzar MM, Sibtain D, Usman HM, Hayat A (2020) Variable step size fractional incremental conductance for MPPT under changing atmospheric conditions. Int J Numer Model El 33: e2765. https://doi.org/10.1002/jnm.2765 doi: 10.1002/jnm.2765
    [21] Ali AIM, Mohamed HRA (2022) Improved P & O MPPT algorithm with efficient open-circuit voltage estimation for two-stage grid-integrated PV system under realistic solar radiation. Int J Elec Power 137: 107805. https://doi.org/10.1016/j.ijepes.2021.107805 doi: 10.1016/j.ijepes.2021.107805
    [22] Bahari MI, Tarassodi P, Naeini YM, Khalilabad AK, Shirazi P (2016) Modeling and simulation of hill climbing MPPT algorithm for photovoltaic application. 2016 International Symposium on Power Electronics, Electrical Drives, Automation and Motion (SPEEDAM), 1041‒1044. https://doi.org/10.1109/SPEEDAM.2016.7525990
    [23] Hamouda N, Babes B, Kahla S, Boutaghane A, Beddar A, Aissa O (2020) ANFIS controller design using PSO algorithm for MPPT of solar PV system powered brushless DC motor based wire feeder unit. 2020 International Conference on Electrical Engineering (ICEE), 1‒6. https://doi.org/10.1109/ICEE49691.2020.9249869
    [24] Laxman B, Annamraju A, Srikanth NV (2021) A grey wolf optimized fuzzy logic based MPPT for shaded solar photovoltaic systems in microgrids. Int J Hydrogen Energy 46: 10653‒10665. https://doi.org/10.1016/j.ijhydene.2020.12.158 doi: 10.1016/j.ijhydene.2020.12.158
    [25] Mohammed SS, Devaraj D, Ahamed TI (2021) GA-optimized fuzzy-based MPPT technique for abruptly varying environmental conditions. Journal of The Institution of Engineers (India): Series B 102: 497‒508. https://doi.org/10.1007/s40031-021-00552-2 doi: 10.1007/s40031-021-00552-2
    [26] Divyasharon R, Banu RN, Devaraj D (2019) Artificial neural network based MPPT with CUK converter topology for PV systems under varying climatic conditions. 2019 IEEE International Conference on Intelligent Techniques in Control, Optimization and Signal Processing (INCOS), 1‒6. https://doi.org/10.1109/INCOS45849.2019.8951321
    [27] VanDeventer W, Jamei E, Thirunavukkarasu GS, Seyedmahmoudian M, Soon TK, Horan B, et al. (2019) Short-term PV power forecasting using hybrid GASVM technique. Renew Energy 140: 367‒379. https://doi.org/10.1016/j.renene.2019.02.087 doi: 10.1016/j.renene.2019.02.087
    [28] Sobri S, Koohi-Kamali S, Rahim NA (2018) Solar photovoltaic generation forecasting methods: A review. Energ Convers Manage 156: 459‒497. https://doi.org/10.1016/j.enconman.2017.11.019 doi: 10.1016/j.enconman.2017.11.019
    [29] Chang JF, Dong N, Ip WH, Yung KL (2019) An ensemble learning model based on Bayesian model combination for solar energy prediction. J Renew Sustain Ener 11: 043702. https://doi.org/10.1063/1.5094534 doi: 10.1063/1.5094534
    [30] Siwakoti YP, Blaabjerg F (2017) Common-ground-type transformerless inverters for single-phase solar photovoltaic systems. IEEE T Ind Electron 65: 2100‒2111. https://doi.org/10.1109/TIE.2017.2740821 doi: 10.1109/TIE.2017.2740821
    [31] Beena V, Jayaraju M, Davis S (2018) Active and reactive power control of single phase transformerless grid connected inverter for distributed generation system. Int J Appl Eng Res 13: 150‒157.
    [32] Yadeo D, Chaturvedi P, Suryawanshi HM, Atkar D, Saketi SK (2021) Transistor clamped dual active bridge DC‐DC converter to reduce voltage and current stress in low voltage distribution network. Int T Electr Energy 31: e12665. https://doi.org/10.1002/2050-7038.12665 doi: 10.1002/2050-7038.12665
    [33] Lakshmi M, Hemamalini S (2019) Coordinated control of MPPT and voltage regulation using single-stage high gain DC–DC converter in a grid-connected PV system. Electr Pow Syst Res 169: 65‒73. https://doi.org/10.1016/j.epsr.2018.12.011 doi: 10.1016/j.epsr.2018.12.011
    [34] Prasad V, Jayasree P, Sruthy V (2018) Active power sharing and reactive power compensation in a grid-tied photovoltaic system. Materials Today: Proceedings 5: 1537‒1544. https://doi.org/10.1016/j.matpr.2017.11.243 doi: 10.1016/j.matpr.2017.11.243
    [35] Somalinga SS, Santha K (2021) Modified high-efficiency bidirectional DC–DC converter topology. J Power Electron 21: 257‒268. https://doi.org/10.1007/s43236-020-00160-1 doi: 10.1007/s43236-020-00160-1
    [36] Rahmani B, Li W, Liu G (2015) An Advanced Universal Power Quality Conditioning System and MPPT method for grid integration of photovoltaic systems. Int J Elec Power 69: 76‒84. https://doi.org/10.1016/j.ijepes.2014.12.031 doi: 10.1016/j.ijepes.2014.12.031
    [37] Yang B, Yu T, Shu H, Zhu D, An N, Sang Y, et al. (2018) Perturbation observer based fractional-order sliding-mode controller for MPPT of grid-connected PV inverters: Design and real-time implementation. Control Eng Pract 79: 105‒125. https://doi.org/10.1016/j.conengprac.2018.07.007 doi: 10.1016/j.conengprac.2018.07.007
    [38] Sadollah A, Bahreininejad A, Eskandar H, Hamdi M (2012) Mine blast algorithm for optimization of truss structures with discrete variables. Comput Struct 102: 49‒63. https://doi.org/10.1016/j.compstruc.2012.03.013 doi: 10.1016/j.compstruc.2012.03.013
    [39] Yıldız BS (2020) The mine blast algorithm for the structural optimization of electrical vehicle components. Mater Test 62: 497‒502. https://doi.org/10.3139/120.111511 doi: 10.3139/120.111511
    [40] Jothimani G, Palanichamy Y, Natarajan SK, Rameshkumar T (2021) Single‐phase front‐end modified interleaved Luo power factor correction converter for on‐board electric vehicle charger. Int J Circ Theor App 49: 2655‒2669. https://doi.org/10.1002/cta.3017 doi: 10.1002/cta.3017
    [41] Singh B, Kushwaha R (2021) Power factor preregulation in interleaved Luo converter-fed electric vehicle battery charger. IEEE T Ind Appl 57: 2870‒2882. https://doi.org/10.1109/TIA.2021.3061964 doi: 10.1109/TIA.2021.3061964
    [42] Chauhan U, Rani A, Kumar B, Singh V (2019) A multi verse optimization based MPPT controller for drift avoidance in solar system. J Intell Fuzzy Syst 36: 2175‒2184. https://doi.org/10.3233/JIFS-169929 doi: 10.3233/JIFS-169929
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