Research article

The probabilistic load flow analysis by considering uncertainty with correlated loads and photovoltaic generation using Copula theory

  • Received: 12 February 2018 Accepted: 01 May 2018 Published: 17 May 2018
  • In this paper, a probabilistic load flow analysis is proposed in order to deal with probabilistic problems related to the power system. Due to increasing trend of penetration of renewable energy sources in power system brought two factors: One is uncertainty, and another one is dependence. Uncertainty and dependence factor increase risk associated with power system operation and planning. In this proposed model these two factors is considered. Gaussian Copula theory is proposed to establish the probability distribution of correlated input random variables. Three sampling methods are used with Monte Carlo simulation as simple random sampling, Box-Muller sampling, and Latin hypercube sampling in order to evaluate the accuracy of the proposed method. The main advantages of this model are as: It can establish any type of correlation between input random variable with the help of Copula theory, it is free from the restrictions of Pearson coefficient of correlation, it is unconstrained by the marginal distribution of input random variables, and uncertainty is established with photovoltaic generation this is the main source of uncertainty. Additional, in order to evaluate the accuracy and efficiency of the proposed model a real load and photovoltaic generation data is adopted. For accuracy evaluation purpose two comparative test system is adopted as modified IEEE 14 and IEEE 118-bus test system.

    Citation: Li Bin, Muhammad Shahzad, Qi Bing, Muhammad Ahsan, Muhammad U Shoukat, Hafiz MA Khan, Nabeel AM Fahal. The probabilistic load flow analysis by considering uncertainty with correlated loads and photovoltaic generation using Copula theory[J]. AIMS Energy, 2018, 6(3): 414-435. doi: 10.3934/energy.2018.3.414

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  • In this paper, a probabilistic load flow analysis is proposed in order to deal with probabilistic problems related to the power system. Due to increasing trend of penetration of renewable energy sources in power system brought two factors: One is uncertainty, and another one is dependence. Uncertainty and dependence factor increase risk associated with power system operation and planning. In this proposed model these two factors is considered. Gaussian Copula theory is proposed to establish the probability distribution of correlated input random variables. Three sampling methods are used with Monte Carlo simulation as simple random sampling, Box-Muller sampling, and Latin hypercube sampling in order to evaluate the accuracy of the proposed method. The main advantages of this model are as: It can establish any type of correlation between input random variable with the help of Copula theory, it is free from the restrictions of Pearson coefficient of correlation, it is unconstrained by the marginal distribution of input random variables, and uncertainty is established with photovoltaic generation this is the main source of uncertainty. Additional, in order to evaluate the accuracy and efficiency of the proposed model a real load and photovoltaic generation data is adopted. For accuracy evaluation purpose two comparative test system is adopted as modified IEEE 14 and IEEE 118-bus test system.


    In this paper, we present a hybrid nonlinear conjugate gradient (CG) method for solving unconstrained optimization problems:

    minxRnf(x), (1.1)

    where f:RnR is continuously differentiable and its gradient g(x)=f(x) is Lipschitz continuous. CG methods are among the most prefered iterative methods for solving large-scale problems because of their simplicity in implementation, Hessian free and less storage requirements [14]. In view of these advantages, an encouraging number of CG methods were proposed (see, for example, [1,5,10,12,13,31]).

    The conjugate gradient method for solving (1.1) generates a sequence {xk} via the iterative formula

    xk+1=xk+sk,sk=αkdk,k=0,1,, (1.2)

    where dk is the search direction defined by

    dk:={gk,ifk=0,gk+βkdk1,ifk>0, (1.3)

    and for descent methods, dk is usually required to satisfy the sufficient descent property, if there exists a constant c>0 such that for all k

    gTkdkcgk2. (1.4)

    The scalar αk>0 is the step-size determined by a suitable line search rule, and βk is the conjugate gradient parameter that characterizes different type of conjugate gradient methods based on the global convergence properties and numerical performance. Some of the well-known nonlinear conjugate gradient parameters are the Fetcher-Reeves (FR) [9], Polak-Ribiére-Polyak (PRP)[25,26], Hestenes-Stiefel (HS)[15], conjugate descent (CD) [8], Liu-Storey (LS) [22] and Dai-Yuan (DY) [6]. These prameters are given by the following formulae:

    βFRk=gk2gk12,βCDk=gk2dTk1gk1,βDYk=gk2dTk1yk1, (1.5)
    βHSk=gTkyk1dTk1yk1,βPRPk=gTkyk1gk12,βLSk=gTkyk1gTk1dk1, (1.6)

    where gk=g(xk),yk1=gkgk1 and is the Euclidean norm.

    In order to have some of the classical CG methods possess a descent direction and trust region as well as improving their numerical efficiency, the three-term CG and hybrid CG methods were introduced for solving (1.1). For instance, Mo et al. [23] proposed two hybrid methods for solving unconstrained optimization problems. The methods are based on the Touati-Ahmed and Storey [30] and the DY methods. Under the strong Wolfe condition, the global convergence was proved. Andrei in [2] proposed a simple three-term CG method obtained by modifying the Broyden-Fletcher-Goldferb-Shanno (BFGS) updating formula of the inverse approximation of the Hessian. The search direction satisfies both the descent and conjugacy conditions. Numerical results were given to support the theoretical results. Also in [3], Andrei proposed an eliptic CG method for solving (1.1). The search direction is the sum of the negative gradient and a vector obtained by minimizing the quadratic approximation of the objective function. In addition, the search direction satisfies both Dai-Liao (DL) cojugacy and descent conditions. Eigenvalue analysis was carried out to determine parameter which the search direction depends on. In comparison with the well-known CG_DESCENT method, the proposed method was more efficient. Liu and Li [21] proposed a new hybrid CG with it's search direction satisfying the DL conjugacy condition and the Newton direction independent of the line search. As usual the global convergence was established under the strong Wolfe line search. In [16], Jian et al. proposed a new hybrid CG method based on previous classical methods. At every iteration, the method produces a descent direction independent of the line search. Global convergence was proved and numerical experiments on medium-scale problems was conducted and the results reported. By applying a mild modification on the HS method, Dong et al. proposed a new approach that generates a descent direction. Also, the approach satisfies an adaptive conjugacy condition and has a self-restarting property. The global convergence was proved for general functions and the efficiency of the approach was shown via numerical experiments on some large-scale problems. Min Li [19] developed a three-term PRP CG method with the search direction close to the direction of the memoryless BFGS quasi-Newton method. The method collapses to the standard PRP method when an exact line search is considered. Independent of any line search, the method satisfies the descent condition. The global convergence of the method was established using an appropriate line search. Numerical results show that the method is efficient for the standard unconstrained problems in the CUTEr library. Again, Li [18] proposed a nonlinear CG method which generates a search direction that is close to that of the memoryless BFGS quasi-Newton method. Moreover, the search direction satisfies the descent condition. Global convergence for strongly convex functions and nonconvex functions was established under the strong Wolfe line search. Furthermore, an efficient spectral CG method that combine the spectral parameter and a three-term PRP method was proposed by Li et al. [20]. The method is based on the quasi-Newton direction and quasi-Newton equation. Numerical experiments reported show the superiority of the method over the three-term PRP method.

    Motivated by the works of Li [18,19] which originated from [17,27], we propose a new hybrid CG method for solving (1.1). The hybrid direction is a combination of the FR and DY directions that are both close to the direction of the memoryless BFGS quasi-Newton method. Interestingly, the hybrid direction satisfies the descent condition and is bounded independent of any line search procedure. We prove the global convergence under both the Wolfe-type and Armijo-type line searches. Numerical results are also provided to show the efficiency of the new hybrid method. Finally, application of the method in optimizing risk in portfolio selection is also considered. The remainder of this paper is organized as follows. In the next section, the hybrid method is derived together with it's convergence. In Section 3, we provide some numerical experimental results.

    We begin this section by recalling a three-term CG method for solving (1.1). From an initial guess x0, the method compute the search direction as follows:

    d0:=g0,dk:=gk+βkdk1+γkgk,k1, (2.1)

    where βk, γk are parameters. Distict choices of the parameters βk and γk correspond to distinct three-term CG methods. It is clear that, the three-term CG methods collapses to the classical ones when γk=0.

    Next, we will recall the memoryless BFGS method proposed by Nocedal [17] and Shanno [27], where the search direction can be written as

    dBFGSk:=Qkgk,
    Qk=(Isk1yTk1sTk1yk1yk1sTk1sTk1yk1+sk1yTk1yk1sTk1sTk1yk1+sk1sTk1sTk1yk1),

    sk1=xkxk1=αk1dk1 and I is the identity matrix. After simplification, dLBFGSk can be rewritten as

    dBFGSk:=gk+(βHSkyk12gTkdk1(dTk1yk1)2)dk1+gTkdk1dTk1yk1(yk1sk1). (2.2)

    Replacing βHSk with βFRk and yk12gTkdk1(dTk1yk1)2 with gk2gTkdk1gk14 in (2.2), we define a three-term search direction as

    dTTFRk:=gk+(βFRkgk2gTkdk1gk14)dk1tkgTkdk1gk12gk. (2.3)

    Again, replacing βHSk with βDYk and yk12gTkdk1(dTk1yk1)2 with gk2gTkdk1(dTk1yk1)2 in (2.2), we define another three-term search direction as

    dTTDYk:=gk+(βDYkgk2gTkdk1(dTk1yk1)2)dk1tkgTkdk1dTk1yk1gk. (2.4)

    To find the parameter tk, we require the solution of the univariate minimal problem

    mintR(yk1sk1)t.gk2.

    Let Ek=(yk1sk1)t.gk, then

    EkETk=[(yk1sk1)t.gk][(yk1sk1)t.gk]T=[(yk1sk1)t.gk][(yk1sk1)Tt.gTk]=t2gkgTkt[(yk1sk1)gTk+gk(yk1sk1)T]+(yk1sk1)(yk1sk1)T.

    Letting Ak=yk1sk1, then

    EkETk=t2gkgTkt[AkgTk+gkATk]+AkATk

    and

    tr(EkETk)=t2gk2t[tr(AkgTk)+tr(gkATk)]+Ak2=t2gk22tgTkAk+Ak2.

    Differentiating the above with respect to t and equating to zero, we have

    2tgk22gTkAk=0,

    which implies

    t=gTk(yk1sk1)gk2. (2.5)

    Hence, we select tk as

    tk:=min{ˉt,max{0,gTk(yk1sk1)gk2}}, (2.6)

    which implies 0tkˉt<1.

    Motivated by the three-term CG directions defined in (2.3) and (2.4), we propose a hybrid three-term CG based algorithm for solving (1.1), where the search direction is defined as

    d0:=g0,dk:=gk+βHTTkdk1+γkgk,k1, (2.7)

    where

    βHTTk:=gk2wkgk2gTkdk1w2k,γk:=tkgTkdk1wk (2.8)

    and

    wk:=max{λdk1gk,dTk1yk1,gk12},λ>0. (2.9)

    Remark 2.1. Observe that, because of the way wk is defined, the parameter βHTTk is a hybrid of βFRkgk2gTkdk1gk14 and βDYkgk2gTkdk1(dTk1yk1)2 in (2.3) and (2.4), respectively. Also, the parameter γk defined by (2.8) is a hybrid of tkgTkdk1gk12 and tkgTkdk1dTk1yk1 in (2.3) and (2.4), respectively. Finally, the search direction given by (2.7) is close to that of the memoryless BFGS method when tk=gTk(yk1sk1)gk2.

    Remark 2.2. Note that, relation (2.9) is carefully defined such that the search direction possess a trust region (see Lemma 2.7) independent of the line search.

    Lemma 2.3. The search direction dk defined by (2.7) satisfy (1.4) with c=34.

    Proof. Multiplying both sides of (2.7) with gTk, we have

    gTkdk=gk2+gk2wkgTkdk1gk2w2k(gTkdk1)2tkgk2wkgTkdk1gk2+(1tk)gk2wkgTkdk1gk2w2k(gTkdk1)2=gk2+2((1tk)2gTk)gkwkgTkdk1gk2w2k(gTkdk1)2gk2+(1tk)24gk2+gk2w2k(gTkdk1)2gk2w2k(gTkdk1)2=gk2+(1tk)24gk2=(1(1tk)24)gk234gk2.

    Next, we will turn our attention to establishing the convergence of the proposed scheme by first considering the standard Wolfe line search conditions [29],

    f(xk+αkdk)f(xk)ϑαkgTkdk, (2.10)
    g(xk+αkdk)TdkσgTkdk (2.11)

    where 0<ϑ<σ<1. In addition, we will assume that

    Assumption 2.4. The level set H={x:f(x)f(x0)} is bounded.

    Assumption 2.5. Suppose H is some neighborhood of L, then f is continuously differentiable and its gradient Lipschitz continuous on H. That is, we can find L>0 such that for all x

    g(x)g(ˉx)Lxˉx,ˉxH. (2.12)

    From Assumption 2.4 and 2.5, we can deduce that for all xL there exist b1,b2>0 for which

    xb1.

    g(x)b2.

    Furthermore, the sequence {xk}L because {f(xk)} is decreasing. Henceforth, we will suppose that Assumption 2.4–2.5 hold and that the objective function is bounded below. We will now prove the convergence result.

    Theorem 2.6. Let conditions (2.10) and (2.11) be fulfilled. If

    k=01dk2=+, (2.13)

    then

    lim infkgk=0. (2.14)

    Proof. Suppose by contradiction that Eq (2.14) does not hold, then there exists a nonnegative scalar ϵ such that

    gkϵfor allkN. (2.15)

    From Lemma 2.3 and (2.10),

    f(xk+1)f(xk)+ϑαkgTkdkf(xk)ϑαkgk2f(xk)f(xk1)f(x0).

    Likewise, by Lemma 2.3, condition (2.11) and Assumption 2.5, we have

    (1σ)gTkdk(gk+1gk)Tdkgk+1gkdkαkLdk2.

    Combining the above inequality with (2.10), we obtain

    ϑ(1σ)L(gTkdk)2dk2f(xk)f(xk+1)

    and

    ϑ(1σ)Lk=0(gTkdk)2dk2(f(x0)f(x1))+(f(x1)f(x2))+f(x0)<+,

    since {f(xk)} is bounded. The above implies that

    k=0(gTkdk)2dk2<+. (2.16)

    Now, inequality (2.15) with (1.4) implies that

    gTkdk34gk234ϵ2. (2.17)

    Squaring both sides and dividing by dk20 of (2.17), yields

    k=0(gTkdk)2dk2916k=0ϵ4dk2=+ (2.18)

    This contradicts (2.16). Therefore, the conclusion of the theorem hold.

    Next, we will establish the convergence of the proposed method under the Armijo-type backtracking line search procedure. The procedure was first introduced by Grippo and Lucidi [11], where the step size αk is determined as follows: ρ,ϑ(0,1), αk=ρi, where i is the smallest nonnegative integer for which the relation

    f(xk+αkdk)f(xk)ϑα2kdk2 (2.19)

    hold.

    From (2.19) and the fact that {f(xk)} is decreasing, we can deduce that

    k=0α2kdk2<+,

    which further implies that

    limkαkdk=0. (2.20)

    Lemma 2.7. If {dk} is defined by (2.7), then there is N1>0 for which dkN1.

    Proof. From (2.8),

    |βHTTk|=|gk2wkgk2gTkdk1w2k|gk2λdk1gkgk3dk1(λdk1gk)2=(1λ+1λ2)gkdk1. (2.21)

    Also,

    |γk|=|tkgTkdk1wk|=tk|gTkdk1wk|ˉtgkdk1wkˉtgkdk1λdk1gk=ˉtλ. (2.22)

    Therefore, from (2.7), (2.21) and (2.22), we have

    dk=gk+βHTTkdk1+γkgkgk+|βHTTk|dk1+|γk|gkgk+(1λ+1λ2)gkdk1dk1+ˉtλgk=(1+1+ˉtλ+1λ2)gk (2.23)
    =(1+1+ˉtλ+1λ2)b2. (2.24)

    Letting N1=(1+1+ˉtλ+1λ2)b2, we have

    dkN1. (2.25)

    Theorem 2.8. If the step size αk is obtained via relation (2.19), then

    lim infkgk=0. (2.26)

    Proof. Suppose by contradiction equation (2.26) is not true. Then for all k, we can find an r>0 for which

    gkr. (2.27)

    Let αk=ρ1αk, then αk does not satisfy (2.19). That is

    f(xk+ρ1αkdk)>f(xk)ϑρ2α2kdk2. (2.28)

    Applying the mean value theorem together with Lemma 2.7, (1.4) and (2.12), there exist an lk(0,1) such that

    f(xk+ρ1αkdk)f(xk)=ρ1αkg(xk+lkρ1αkdk)=ρ1αkgTkdk+ρ1αk(g(xk+lkρ1αkdk)gk)Tdkρ1αkgTkdk+Lρ2α2kdk2.

    Inserting the above relation in (2.28), together with (2.25) and (2.27) we have

    αkρgk2(L+ϑ)dk2ρr2(L+ϑ)N21>0.

    This and (2.20) gives

    limkdk=0. (2.29)

    On the other hand, applying backward Cauchy-Schwartz inequality on (1.4) gives

    dk34gk.

    Thus, we have limkgk=0. This is a contradiction and therefore (2.26) holds.

    This section discusses the computational efficiency of the proposed method, namely HTT method. One way to measure the efficienecy of a method is to use the test function. An important test function is used to validate and compare among optimization methods, especially newly developed methods. We selected 34 test functions and initial points as in Table 1 mostly considered by Andrei [4]. For each function we have taken five numerical experiments with a dimension of which is among n = 10, 50, 80,100,200,400,300,500,600, 1000, 3000, 5000, 10,000, 15,000, 20,000. However, we often use dimensions that are greater than 1000. The executed methods are coded in Matlab 2019a and compiled using personal laptop; Intel Core i7 processor, 16 GB RAM, 64 bit Windows 10 Pro operating system.

    Table 1.  List of test functions and initial points.
    NumberFunctionsInitial Points
    F1Extended White & Holst(-1.2, 1, ..., -1.2, 1)
    F2Extended Rosenbrock(-1.2, 1, ..., -1.2, 1)
    F3Extended Freudenstein & Roth(0.5, -2, ..., 0.5, -2)
    F4Extended Beale(1, 0.8, ..., 1, 0.8)
    F5Raydan 1(1, 1, ..., 1)
    F6Extended Tridiagonal 1(2, 2, ..., 2)
    F7Diagonal 4(1, 1, ..., 1)
    F8Extended Himmelblau(1, 1, ..., 1)
    F9FLETCHCR(0.5, 0, 5, ..., 0.5)
    F10Extended Powel(1, 1, ..., 1)
    F11NONSCOMP(3, 3, ..., 3)
    F12DENSCHNB(10, 10, ..., 10)
    F13Extended Penalty Function U52(1/100, 2/100, ..., n/100)
    F14Hager(1, 1, ..., 1)
    F15Shallow(2, 2, ..., 2)
    F16Quadratic QF2(0.5, 0, 5, ..., 0.5)
    F17Generalized Tridiagonal 1(2, 2, ..., 2)
    F18Generalized Tridiagonal 2(1, 1, ..., 1)
    F19POWER(1, 1, ..., 1)
    F20Quadratic QF1(1, 1, ..., 1)
    F21QP2 Extended Quadratic Penalty(2, 2, ..., 2)
    F22QP1 Extended Quadratic Penalty(1, 1, ..., 1)
    F23Sphere(1, 1, ..., 1)
    F24Sum Squares(0.1, 0, 1, ..., 0.1)
    F25DENSCHNA(7, 7, ..., 7)
    F26DENSCHNC(100, -1, -1, ..., -1, -1)
    F27DENSCHNF(100, -100, ..., 100, -100)
    F28Extended Block-Diagonal BD1(1, 1, ..., 1)
    F29HIMMELBH(0.8, 0, 8, ..., 0.8)
    F30Extended Hiebert(5.001, 5.001, ..., 5.001, 5.001)
    F31ENGVAL1(2, 2, ..., 2)
    F32ENGVAL8(0, 0, ..., 0)
    F33Linear Perturbed(0, 0, ..., 0)
    F34QUARTICM(2, 2, ..., 2)

     | Show Table
    DownLoad: CSV

    As a good comparison, for NHS+ and NPRP+ all parameters are maintained according to their articles in [18] and [19], i.e, ϑ=0.1,σ=0.9. Specially, for HTT, we set the value of parameters ϑ=0.0001,σ=0.009. For all methods, we use the parameter value ˉt=0.3, λ=0.01 and the step-size αk is calculated by standard Wolfe line search. The numerical results are compared based on number of iterations (NOI), number of function evaluations (NOF), and CPU time in seconds (CPU). In this experiment, we consider a stopping condition that many researchers suggest (see [16,18,21,23]), namely that the algorithm will stop when gk106.

    In Table 5 we list the numerical results obtained by compiling each method for completing each test function with different dimension sizes. If the number of iterations of the method exceeds 10,000 or it never reaches the optimal value, the algorithm stops and we write it as 'FAIL'.

    To compare the performance between methods, we use the performance profiles of Dolan and Moré [7] with rule as follows. Let S is the set of methods and P is set of the test problems with np,ns are the number of test problems and the number of methods, respectively.

    The performance profile ω:R[0,1] is for each sS and pP defined that ap,s>0 is NOI or NOF or CPU time required to solve problems p by method s. Furthermore, the performance profile is obtained by:

    ωs(τ)=1npsize{pP:log2rp,sτ},

    where τ>0, sizeB is the number of the elements in the set B, and rp,s is performance ratio defined as:

    rp,s=ap,smin{ap,s:pPandsS}.

    In general, the method whose performance profile plot is on the top right will win the rest of the methods or represents the best method.

    From Table 5, all the methods failed to solve the Raydan 1 and NONSCOMP with n=1,000,5,000,10,000,15,000,20,000, the NHS+ and NPRP+ methods failed to solve Extended Powel, Hager, Generalized Tridiagonal 1, Generalized Tridiagonal 2, QP2 Extended Quadratic Penalty, DENSCHNA, Extended Block-Diagonal BD1, ENGVAL1, and ENGVAL8 for all of dimension given, and NPRP+ method has more failures for the given problem compared to other methods. So, based on the numerical results in Table 5, we can say that the HTT method has the best performance. Meanwhile, from the result in performance profile in Figures 13 show that the HTT method profile is always on the top right curve, whether it's based on NOI, NOF or CPU time. The final conclusion is that the HTT method has the best performance compared the NHS+ and NPRP+ methods under the test functions given.

    Figure 1.  Performance profile using wolfe line search of all methods based on NOI.
    Figure 2.  Performance profile using wolfe line search of all methods based on NOF.
    Figure 3.  Performance profile using wolfe line search of all methods based on CPU time.

    Investment is a commitment to invest several funds carried out at this time to obtain many benefits in the future. One investment that is quite attractive is a stock investment. An investor can invest in more than one stock. Of course, the thing to consider is whether the investment can be profitable or not. One theory that discusses investment in several assets is portfolio theory. A stock portfolio is a collection of stocks owned by an investor [24].

    In this section, we present the problem of risk management in a portfolio of stocks. The main issue is how to balance a portfolio, that is, how to choose the percentage of each stock in the portfolio to minimize the risk [28]. In investing, investors expect to get a large return by choosing the smallest possible risk. Return is the income received by an investor from stocks traded on the capital market. There are several ways to calculate returns, one of which is to use arithmetic returns which can be calculated as follows:

    Rt=PtPt1Pt1,

    where Pt is the stock prices at time t and Pt1 is the stock prices at time t1. Furthermore, we may consider the mean of return, the expected value, and the variance of the return. The mean of return of a stock i is denoted by

    ˉri=1nnt=1rit,

    where n is number of returns on a stock and rit is return at time t on stock i. The expected return of stock i

    μi=E(Ri).

    The variance of return of stock i

    σ2i=Var(Ri)

    is called the risk of stock i [28]. One thing that needs to be considered is the relationship between stocks, so we must also pay attention to how stocks interact, i.e, by using the covariance of individual risk. Covariance measures the directional relationship between the returns on two stocks. If positive covariance then that stock returns move together while a negative covariance means they move inversely. Usually the covariance of return between two stocks Ri,Rj is denoted as Cov(Ri,Rj) [28].

    For our cases, we consider the seven blue chip stocks in Indonesia, namely, PT Bank Rakyat Indonesia (Persero) Tbk (BBRI), PT Unilever Indonesia Tbk (UNVR), PT Telekomunikasi Indonesia Tbk (TLKM), PT Indofood CBP Sukses Makmur Tbk (ICBP), PT Bank Mandiri (Persero) Tbk (BMRI), PT Perusahaan Gas Negara Tbk (PGAS), and PT Astra International Tbk (ASII). The stock price used is the weekly closing price which data history is taken from from the database http://finance.yahoo.com, over a period of 3 years (Jan 1, 2018 – Dec 31, 2020). Based on this data, we may see the movement of the closing price of each stock in Figure 4.

    Figure 4.  Weekly closing price of all stocks in IDR (Jan 1, 2018 – Dec 31, 2020).

    So that the portfolio returns for the seven stocks are defined as the weighted amount of returns for each stocks

    R=7i=1wiRi,

    where wi is the percentage of the value of the stock contained in the portfolio. Thus, we may define the expected return and risk on our portfolio (see [28]). The expected return μ on our portfolio is the expected value of the portfolio's return as follows:

    μ=E(7i=1wiRi)=7i=1wiμi, (4.1)

    and the risk of our portfolio is the variance of the portfolio's return as follows,

    σ2=Var(7i=1wiRi)=7i=17j=1wiwjCov(Ri,Rj). (4.2)

    Since our main objective is to determine the optimal portfolio by minimizing the risk of returns, then our problem model is

    {minimize:σ2=Var(7i=1wiRi)=7i=17j=1wiwjCov(Ri,Rj),subject to:7j=1wj=1. (4.3)

    The next step changes the problem (4.3) to unconstrained optimization model, i.e, considering w7=1w1w2w3w4w5w6, then the problem (4.3) changes into an unconstrained optimization model as follows:

    minwR77i=17j=1wiwjCov(Ri,Rj), (4.4)

    where w=(w1,w2,w3,w4,w5,w6,1w1w2w3w4w5w6).

    The following table shows the mean, variance, and covariance values of the returns of UNVR, BBRI, TLKM, ICBP, BMRI, PGAS, and ASII stocks.

    According to Tables 2 and 3, we may execute the problem (4.4) by choosing some random initial points and still maintaining the same parameter values according to each method. The results obtained are stated in Table 4.

    Table 2.  Mean and variance of return for Seven Stocks.
    Stocks Mean Variance
    UNVR 0.00311 0.00127
    BBRI 0.00033 0.00273
    TLKM 0.00247 0.00166
    ICBP 0.00047 0.00142
    BMRI 0.00277 0.00309
    PGAS 0.00359 0.00667
    ASII 0.00321 0.00238

     | Show Table
    DownLoad: CSV
    Table 3.  Covariance of return for seven stocks.
    Stocks UNVR BBRI TLKM ICBP BMRI PGAS ASII
    UNVR 0.00127 0.00058 0.00053 0.00062 0.000906 0.00105 0.000744
    BBRI 0.00058 0.00273 0.00091 0.00059 0.00235 0.002341 0.001844
    TLKM 0.00053 0.00091 0.00166 0.00048 0.001101 0.001579 0.00089
    ICBP 0.00062 0.00059 0.00048 0.00142 0.000807 0.000858 0.000538
    BMRI 0.00091 0.00235 0.00110 0.00081 0.00309 0.002771 0.001888
    PGAS 0.00105 0.00234 0.00158 0.00086 0.002771 0.00667 0.002288
    ASII 0.00074 0.00184 0.00089 0.00189 0.001888 0.002288 0.00238

     | Show Table
    DownLoad: CSV
    Table 4.  Test result of NHS+, NPRP+, and HTT methods for solving problem (4.4).
    Initial PointsNHS+NPRP+HTT
    NOINOFCPUNOINOFCPUNOINOFCPU
    (0.1, 0, 2, ..., 0.6)624960.00680614970.006107830.00110
    (0.1, ..., 0.1)624910.00690604900.00610121260.00150
    (0.6, 0.5, ..., 0.1)574480.00420574550.00590111230.00130
    (0.3, ..., 0.3)584630.00680584720.006106710.00097
    (1, ..., 1)614760.00700624900.00430121270.00140
    (-0.1, ..., -0.1)645020.00630645150.00690141440.00200
    (1.2, 1, ..., 1.2, 1)634900.00670645060.006506710.00140
    (1.001, ..., 1.001)614760.01100624900.00470121270.00094
    (0.5, ..., 0.5)594640.00310604810.00570111180.00210
    (7, ..., 7)806420.00510806490.00870131430.00230

     | Show Table
    DownLoad: CSV

    From Table 4, it is clear that the HTT method more efficient than NHS+ and NPRP+ methods based on NOI, NOF, and CPU time for solving the problem (4.4). Meanwhile, the algorithm executed from each method give the same result for the value of w, i.e, w1=0.3877,w2=0.3220,w3=0.2878,w4=0.4179,w5=0.1642,w6=0.0465, and w7=0.2047. By using the value of w, (4.1), and (4.2), we obtain μ=0.00094 and σ2=0.00074. Finally, the selection of stock portfolios for our case with a minimum risk can be done by allocating each stock in the following proportions, i.e, UNVR 38.77%, BBRI 32.22%, TLKM 28.78%, ICBP 41.79%, BMRI 16.42%, PGAS 4.65%, and ASII 20.47% with the expected return is 0.00094 and the portfolio risk value is 0.00074. A negative sign in the proportion indicates that investor is short selling.

    In this work, we presented a new hybrid CG method that guarantees sufficient descent direction and boundedness of the direction independent of the line search. In addition, the gloabal convergence result is established under both the Wolfe and Armijo line searches. Based on the numerical results, it can be observed that the new hybrid method is more efficient and robust than other methods, providing faster and more stable convergence in most of the problems considered. These can be seen more clearly from Figures 13. Finally, the practical applicability of the hybrid method is also explored in risk optimization. It's efficiency in solving portfolio selection problem was outstanding as it solves the problem with less iteration, function evaluations and CPU time compared with others.

    The authors acknowledge the financial support provided by the Center of Excellence in Theoretical and Computational Science (TaCS-CoE), KMUTT. Also, the (first) author, (Dr. Auwal Bala Abubakar) would like to thank the Postdoctoral Fellowship from King Mongkut's University of Technology Thonburi (KMUTT), Thailand. Moreover, this research project is supported by Thailand Science Research and Innovation (TSRI) Basic Research Fund: Fiscal year 2021 under project number 64A306000005. The first author acknowledge with thanks, the Department of Mathematics and Applied Mathematics at the Sefako Makgatho Health Sciences University.

    The authors declare that they have no conflict of interest.

    Table 5.  The numerical results of all methods using standard wolfe line search.
    FunctionDimNHS+NPRP+HTT
    NOINOFCPUNOINOFCPUNOINOFCPU
    F11000FAILFAILFAILFAILFAILFAIL482110.1035
    5000291440.3053FAILFAILFAIL452050.4427
    10,000FAILFAILFAIL281330.5752492140.8004
    15,000FAILFAILFAILFAILFAILFAIL452051.0857
    20,000291441.0183311431.0605452051.4329
    F21000321680.049FAILFAILFAIL826130.1332
    5000321680.1757FAILFAILFAIL594960.3751
    10,000351760.354391850.3593876281.0267
    15,000FAILFAILFAIL452010.6693594961.1967
    20,000321680.56381820.6532594961.4625
    F3100013670.0401FAILFAILFAIL27960.0456
    5000FAILFAILFAILFAILFAILFAIL27960.1473
    10,000FAILFAILFAILFAILFAILFAIL27960.5049
    15,00013670.2733FAILFAILFAIL28990.6252
    20,000FAILFAILFAILFAILFAILFAIL28990.6011
    F4100020660.3053FAILFAILFAIL802571.0738
    500020660.1586FAILFAILFAIL802570.5459
    10,00020660.3257FAILFAILFAIL802571.1544
    15,00020660.4333FAILFAILFAIL812601.5616
    20,00020660.5468361130.9527852722.1239
    F51000FAILFAILFAILFAILFAILFAILFAILFAILFAIL
    5000FAILFAILFAILFAILFAILFAILFAILFAILFAIL
    10,000FAILFAILFAILFAILFAILFAILFAILFAILFAIL
    15,000FAILFAILFAILFAILFAILFAILFAILFAILFAIL
    20,000FAILFAILFAILFAILFAILFAILFAILFAILFAIL
    F610007300.0239FAILFAILFAIL612060.0917
    5000FAILFAILFAILFAILFAILFAIL742550.5254
    10,000FAILFAILFAILFAILFAILFAIL622240.9381
    15,000FAILFAILFAILFAILFAILFAIL933271.9179
    20,000FAILFAILFAILFAILFAILFAIL802712.0203
    F710006160.00816160.005914390.0141
    50006160.04476160.028414390.0546
    10,0006160.05086160.05914390.1035
    15,0008220.08468220.085914390.1331
    20,0009250.13028220.110714390.1755
    F810008280.02688280.013215540.0125
    50009310.05878280.034716570.0585
    10,0009310.09158280.076916570.1157
    15,0009310.10948280.096216570.1837
    20,0009310.1358280.122716570.21
    F91000231010.0668241040.032422990.0418
    5000301470.1632281370.13421960.0972
    10,000FAILFAILFAILFAILFAILFAIL21960.2199
    15,000FAILFAILFAILFAILFAILFAIL21970.278
    20,000FAILFAILFAILFAILFAILFAIL21970.3312
    F101000FAILFAILFAILFAILFAILFAIL65661974910.365
    5000FAILFAILFAILFAILFAILFAIL90272713259.7488
    10,000FAILFAILFAILFAILFAILFAILFAILFAILFAIL
    15,000FAILFAILFAILFAILFAILFAILFAILFAILFAIL
    20,000FAILFAILFAILFAILFAILFAILFAILFAILFAIL
    F111000FAILFAILFAILFAILFAILFAILFAILFAILFAIL
    5000FAILFAILFAILFAILFAILFAILFAILFAILFAIL
    10,000FAILFAILFAILFAILFAILFAILFAILFAILFAIL
    15,000FAILFAILFAILFAILFAILFAILFAILFAILFAIL
    20,000FAILFAILFAILFAILFAILFAILFAILFAILFAIL
    F12100010420.037511450.019819710.0153
    500010420.071711450.053620740.0798
    10,00010420.095311450.107120740.1711
    15,00010420.142311450.151920740.2083
    20,00010420.159811450.165921770.2929
    F131000FAILFAILFAIL321590.04775918470.3214
    5000FAILFAILFAILFAILFAILFAILFAILFAILFAIL
    10,000FAILFAILFAILFAILFAILFAILFAILFAILFAIL
    15,000FAILFAILFAILFAILFAILFAILFAILFAILFAIL
    20,000FAILFAILFAILFAILFAILFAILFAILFAILFAIL
    F1450FAILFAILFAILFAILFAILFAIL20730.0071
    100FAILFAILFAILFAILFAILFAIL241100.013
    200FAILFAILFAILFAILFAILFAIL311710.031
    300FAILFAILFAILFAILFAILFAILFAILFAILFAIL
    500FAILFAILFAILFAILFAILFAILFAILFAILFAIL
    F15100023640.0224FAILFAILFAIL24750.0274
    500027740.0825FAILFAILFAIL26800.0886
    10,000FAILFAILFAILFAILFAILFAIL27830.2039
    15,000FAILFAILFAILFAILFAILFAIL27830.2489
    20,000FAILFAILFAILFAILFAILFAIL29880.3151
    F1650652140.0109622510.0111123820.014
    100943610.0339FAILFAILFAIL1555340.0319
    2001415180.05521415190.04892227770.074
    300FAILFAILFAIL1786000.07932609240.1116
    5002288180.13832438120.122631711510.1591
    F1750FAILFAILFAILFAILFAILFAIL26840.0086
    100FAILFAILFAILFAILFAILFAIL26840.009
    200FAILFAILFAILFAILFAILFAIL26840.0244
    300FAILFAILFAILFAILFAILFAIL26870.0216
    500FAILFAILFAILFAILFAILFAIL244312567615.9497
    F1850FAILFAILFAILFAILFAILFAIL31930.0206
    100FAILFAILFAILFAILFAILFAIL321120.0126
    200FAILFAILFAILFAILFAILFAIL321170.0197
    300FAILFAILFAILFAILFAILFAIL394990.0556
    500FAILFAILFAILFAILFAILFAIL32970.0349
    F1950661980.0098651958.40E-03661987.30E-03
    1001414230.0351404202.98E-021414233.58E-02
    2002978910.08392968887.57E-022978917.60E-02
    30045313590.158245413620.175945513650.1431
    50076823040.321677023100.2792302290661.0477
    F2050381140.0068381140.0108381140.0307
    100561680.0174561680.0223561680.0247
    200812430.0373812430.0368812430.0373
    3001013030.04641013030.04351013030.0523
    5001313930.07061313930.06931313930.0693
    F21100272430.0411473170.02820826710.1393
    500FAILFAILFAILFAILFAILFAIL43146820.6918
    1000FAILFAILFAILFAILFAILFAIL74259921.5017
    3000FAILFAILFAILFAILFAILFAIL2402120757.9879
    5000FAILFAILFAILFAILFAILFAIL43452111420.7249
    F221011410.209610380.023718640.0022
    5012490.011511450.009353830.2059
    80FAILFAILFAILFAILFAILFAIL18720.0054
    10011450.004FAILFAILFAIL5716350.0704
    300FAILFAILFAILFAILFAILFAIL354850.2809
    F231000130.0127130.0027131.60E-03
    5000130.0168130.0089130.0061
    10,000130.0165130.0134130.0129
    15,000130.0217130.0198130.0183
    20,000130.028130.0296130.0219
    F2410001364080.09781364080.09691364080.0976
    50003119330.76943119330.756340612180.975
    10,00044313292.475744313292.357244313292.3631
    15,00054416324.671854416324.403454416324.264
    20,00063018906.856263018908.61863018906.5118
    F251000FAILFAILFAILFAILFAILFAIL21770.0481
    5000FAILFAILFAILFAILFAILFAIL21770.1557
    10,000FAILFAILFAILFAILFAILFAIL23820.3301
    15,000FAILFAILFAILFAILFAILFAIL23820.4332
    20,000FAILFAILFAILFAILFAILFAIL23820.5819
    F26100031550.0804FAILFAILFAILFAILFAILFAIL
    500031550.3545FAILFAILFAILFAILFAILFAIL
    10,00031550.6368FAILFAILFAILFAILFAILFAIL
    15,00031550.8296FAILFAILFAILFAILFAILFAIL
    20,00031551.0828FAILFAILFAILFAILFAILFAIL
    F27100014910.047214940.032412870.0278
    500016970.335614940.084413900.0859
    10,00014910.206314940.20513900.1935
    15,000FAILFAILFAIL14940.250313900.2522
    20,000FAILFAILFAIL14940.355913900.2913
    F281000FAILFAILFAILFAILFAILFAIL111630.05
    5000FAILFAILFAILFAILFAILFAIL111640.1523
    10,000FAILFAILFAILFAILFAILFAIL121730.3325
    15,000FAILFAILFAILFAILFAILFAIL121660.4909
    20,000FAILFAILFAILFAILFAILFAIL142280.7552
    F29200FAILFAILFAILFAILFAILFAIL7310.2276
    400FAILFAILFAILFAILFAILFAIL7230.0132
    600FAILFAILFAILFAILFAILFAIL7210.008
    8005150.00835150.02727260.0115
    1000FAILFAILFAILFAILFAILFAIL7360.2199
    F301000351990.2817FAILFAILFAIL783760.1087
    5000351990.1804FAILFAILFAIL853970.404
    10,000372050.3754FAILFAILFAIL853970.6769
    15,000FAILFAILFAILFAILFAILFAIL853971.0886
    20,000FAILFAILFAILFAILFAILFAIL853971.3642
    F3150FAILFAILFAILFAILFAILFAIL254040.0131
    100FAILFAILFAILFAILFAILFAIL244080.0248
    200FAILFAILFAILFAILFAILFAIL776393782.1743
    300FAILFAILFAILFAILFAILFAIL5619030.156
    500FAILFAILFAILFAILFAILFAIL1236633036.2995
    F3250FAILFAILFAILFAILFAILFAIL14490.0224
    100FAILFAILFAILFAILFAILFAIL14690.0112
    200FAILFAILFAILFAILFAILFAIL192770.027
    300FAILFAILFAILFAILFAILFAILFAILFAILFAIL
    500FAILFAILFAILFAILFAILFAIL16879131.0719
    F3310001404200.10271404200.09651404200.0993
    50003209600.83343209600.76893209600.8576
    10,00045613682.587945613582.665145613582.5723
    15,00056216864.469156216866.065556116834.3236
    20,00065119536.6436651195313.698665019507.2173
    F341000766250.2802766250.27113270.0206
    5000816951.645816951.50163270.0844
    10,000837242.8429837242.97673270.1274
    15,000847394.2936847395.87323270.2009
    20,000857546.17788575410.21593270.2372

     | Show Table
    DownLoad: CSV
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