Forward-backward Sweep Method Research Articles

A Complex-Valued Three-Phase Load Flow for Radial Networks: High-Performance and Low-Voltage Solution Capability

Load flow methods for distribution networks, such as backward forward sweep (BFS) have a good computational performance and can find solutions with accuracy. However, some studies may demand the determination of low voltage solutions, and this poses a problem for these methods since they cannot find these solutions due to convergence issues. This paper presents a load flow method based on a novel complex-valued formulation developed for distribution networks, which works well on radial topologies by using an incidence matrix to avoid complicated series element models, allow high-performance and low-voltage solution capability. The formulation is solved by Newton's method via Wirtinger's calculus. To prove the low-voltage solution capability, both sides of QV curves, i.e., unstable and stable regions were traced on balanced and unbalanced networks. Performance tests in the IEEE test feeders show that the runtime is less than or equal to the runtime of the BFS method. Furthermore, the line R/X ratio and the number of controlled voltage node or volt-var functions do not affect the computational performance, yielding advantages over the classic Newton and BFS methods.

Discrete optimal capital investment and financing policies in fishery resource harvesting with reserve area

This research discusses a problem in renewable resource (fish) harvesting on single-owner fishery where the dynamics of growth are expressed by system of difference equations. The model involves three state variables (stock of equity, resource stock in reserve zone and unreserved zone) and two control variables (effort rate and dividend payout rate). Discrete Pontryagin maximum principle was applied to obtain optimality conditions which must be satisfied by the variables. Further, the forward-backward sweep method was used to determine the numerical solution of optimal control model. Three scenarios of interest rate applied to find out the response of company in managing harvesting effort and dividend payout rate were investigated.

HIV drug resistance: Insights from mathematical modelling

In 2013 the World Health Organization recommended the initiation of antiretroviral therapy (ART) to any person who tests HIV positive irrespective of his/her CD4+ count. However, implementation of the new guidelines poses a lot of challenges especially in Sub-Sahara Africa such as: drug side effects, drug resistance-mutations and significant financial burdens. Most importantly, it has been established that HIV resistance and subsequent virologic failure occur in a substantial proportion of HIV-infected patients receiving HAART. This study therefore, seeks to investigate the emergence of drug resistant HIV virus during treatment with the aim of determining the proper use of HIV therapy that would lessen drug resistance. To carry out the analysis a ten dimensional in-vivo mathematical model is proposed for HIV dynamics. The model is formulated in such away that it takes into account two virus strain, that is, the wild type as well as the naive type HIV virus. The in-vivo model is shown to be both biologically meaningful and mathematically well posed. The existence of unique infection-free equilibrium point is determined and both its local and global stability investigated. In addition, the basic reproduction number for each viral strain is computed using the next generation matrix method. An optimal control model is proposed and analysed by applying Pontryagin maximum principle, to obtain the optimal drug combination for HIV treatment. Here two drugs, that is, Reverse Transcriptase inhibitor and Protease inhibitor are used as the controls in the model. We provide an objective function for the minimisation of the number of wild type HIV virus and the drug resistant virus as well as the costs associated with the use of Reverse Transcriptase inhibitor and protease inhibitor. The forward backward sweep method is applied to numerically solve the optimality system. From the numerical simulations, it is evident that protease inhibitor is the most effective drug in controlling HIV infection. The results suggest that prolonged use of HAART leads to development of drug resistant and that people with drug-resistant infection could play a core role in the epidemic of HIV.

Application of Optimal Control Theory to Newcastle Disease Dynamics in Village Chicken by Considering Wild Birds as Reservoir of Disease Virus

In this study, an optimal control theory was applied to a nonautonomous model for Newcastle disease transmission in the village chicken population. A notable feature of this model is the inclusion of environment contamination and wild birds, which act as reservoirs of the disease virus. Vaccination, culling, and environmental hygiene and sanitation time dependent control strategies were adopted in the proposed model. This study proved the existence of an optimal control solution, and the necessary conditions for optimality were determined using Pontryagin’s Maximum Principle. The numerical simulations of the optimal control problem were performed using the forward–backward sweep method. The results showed that the use of only the environmental hygiene and sanitation control strategy has no significant effect on the transmission dynamics of the Newcastle disease. Additionally, the combination of vaccination and environmental hygiene and sanitation strategies reduces more number of infected chickens and the concentration of the Newcastle disease virus in the environment than any other combination of control strategies. Furthermore, a cost-effective analysis was performed using the incremental cost-effectiveness ratio method, and the results showed that the use of vaccination alone as the control measure is less costly compared to other control strategies. Hence, the most effective way to minimize the transmission rate of the Newcastle disease and the operational costs is concluded to be the timely vaccination of the entire population of the village chicken, improvement in the sanitation of facilities, and the maintenance of a hygienically clean environment.

Analysis of Distribution System with D-STATCOM by Gravitational Search Algorithm (GSA)

In order to analyze the distribution system (DS) to minimize the power losses and improve power quality, finding accurate position size of distribution static compensator (D-STATCOM) is essential. In this paper, a metaheuristic optimization technique, the gravitational search algorithm (GSA) is proposed for the analysis of DS with accurate placing and sizing of distributed static synchronous compensator (D-STATCOM) for power loss reduction, minimum voltage profile index, improvement in voltage profile and annual energy saving for distribution network operator. The effectiveness of the proposed algorithm is tested on IEEE 33 and IEEE 69 bus systems, and performance of GSA has been compared with other techniques such as immune algorithm, bat algorithm and sensitivity approach method for optimal allocation of D-STATCOM. It is found that the best performance is achieved by the proposed method. Forward–backward sweep method is used for the load flow solutions in this paper.

Modeling for heterogeneous multi-stage information propagation networks and maximizing information**Project supported by the National Natural Science Foundation of China (Grant No. 61873194).

In this paper, we propose a heterogeneous multi-stage model to study the effect of social reinforcement on information propagation. Both heterogeneity of network components and the heterogeneity of individual reinforcement thresholds are considered. An information outbreak condition is derived, according to which the outbreak scale and individual density of each state under specific propagation parameters can be deduced. Monte Carlo experiments are conducted in Facebook networks to demonstrate the outbreak condition, and we find that social reinforcement effects generally inhibit the propagation of information though it contributes to the emergence of certain hot spots simultaneously. Additionally, by applying Pontryagin’s Maximum Principle, we derive the optimal control strategy in the case of limited control resources to maximize the information propagation. Then the forward–backward sweep method is utilized to verify its performance with numerical simulation.

A Modified Backward/Forward Sweep Load Flow Method for Islanded Radial Microgrids

In this paper, a modified backward/forward sweep (BFS) load flow algorithm is proposed to solve power flow for islanded droop-controlled ac microgrids. The proposed method has been formulated such that the system frequency is one of the system’s unknown variables which is typical for an islanded network justifying further the absence of a slack bus. As a result, new equations have been systematically developed and incorporated with the conventional BFS method. The modified BFS method has been applied to the 33-bus test system and the results obtained are compared with simulation results from the direct BFS method as well as PSCAD/EMTDC. In comparison to the conventional methods, the proposed method is accurate, simple, and easy to implement thus serving as a useful tool for microgrid analysis and planning.

A Multiclass Mean-Field Game for Thwarting Misinformation Spread in the Internet of Battlefield Things

In this paper, the problem of misinformation propagation is studied for an Internet of Battlefield Things (IoBT) system, in which an attacker seeks to inject false information in the IoBT nodes in order to compromise the IoBT operations. In the considered model, each IoBT node seeks to counter the misinformation attack by finding the optimal probability of accepting given information that minimizes its cost at each time instant. The cost is expressed in terms of the quality of information received as well as the infection cost. The problem is formulated as a mean-field game with multiclass agents, which is suitable to model a massive heterogeneous IoBT system. For this game, the mean-field equilibrium is characterized, and an algorithm based on the forward backward sweep method is proposed to find the mean-field equilibrium. Then, the finite-IoBT case is considered, and the conditions of convergence of the equilibria in the finite case to the mean-field equilibrium are presented. Numerical results show that the proposed scheme can achieve a 1.2-fold increase in the quality of information compared with a baseline scheme, in which the IoBT nodes are always transmitting. The results also show that the proposed scheme can reduce the proportion of infected nodes by 99% compared with the baseline.

Optimal weighted coefficient for heterogeneous epidemic spreading networks

In this paper, we study epidemic spreading on an arbitrary heterogeneous undirected weighted network, in which reinforcement effects among pathogens are considered to depict the characteristics of multiple infection based on a novel propagation model. In order to characterize the formation mechanism of contact weight between individuals, a new parameter defined as the weighted coefficient is introduced to coordinate each individual’s own expected weights towards neighbors. A reasonable candidate for the expected weight relating to individual sensitivity and neighbor’s infection level is also provided. Furthermore, an optimization problem is formulated to address a trade-off between individuals’ infection levels and the adjustment of their weighted coefficients. The existence of an optimal solution is analyzed and the general expression of an optimal weighted coefficient is obtained. Numerical simulations performed by adopting the forward–backward sweep method are presented to illustrate the validity and efficiency of our theoretical results.

Robust optimal control of deterministic information epidemics with noisy transition rates

In this paper the robust optimal control of deterministic information epidemics is inspected taking into consideration the noisy transition rates. Distinct from conventional works, the heterogeneous susceptible–infected–susceptible (SIS) model is adopted where both the heterogeneities in the network topology and the individual diversity are considered. In light of the commonly existing noise in the transition processes, we address the robust optimal control problem aiming at maximizing the spreading performance at the finite time instant given a fixed budget. By using the distribution analysis techniques, the inspected problem is transformed to a constrained optimal control problem and solved by the Pontryagin Maximum Principle (PMP). A novel approach combining the forward–backward sweep method and the secant method is proposed to efficiently reduce the computation burden. The performance of the robust optimal control as well as the influence of the parameters is examined by numerical experiments in real social networks.

Time Needed to Control an Epidemic with Restricted Resources in SIR Model with Short-Term Controlled Population: A Fixed Point Method for a Free Isoperimetric Optimal Control Problem

In this paper, we attempt to determine the optimal duration of an anti-epidemic control strategy which targets susceptible people, under the isoperimetric condition that we could not control all individuals of this category due to restricted health resources. We state and prove the local and global stability conditions of free and endemic equilibria of a simple epidemic compartmental model devised in the form of four ordinary differential equations which describe the dynamics of susceptible-controlled-infected-removed populations and where it is taken into account that the controlled people cannot acquire long-lived immunity to move towards the removed compartment due to the temporary effect of the control parameter. Thereafter, we characterize the sought optimal control and we show the effectiveness of this limited control policy along with the research of the optimal duration that is needed to reduce the size of the infected population. The isoperimetric constraint is defined over a fixed horizon, while the objective function is defined over a free horizon present under a quadratic form in the payoff term. The complexity of this optimal control problem requires the execution of three numerical methods all combined together at the same time, namely, the forward–backward sweep method to generate the optimal state and control functions, the secant method adapted to the isoperimetric restriction, and, finally, the fixed point method to obtain the optimal final time.

Fractional optimal control of an HIV/AIDS epidemic model with random testing and contact tracing

In this paper, we present a fractional model for transmission dynamics of HIV/AIDS with random testing and contact tracing and incorporate into the model the control parameters of condom use and antiretroviral therapy (ART) aimed at controlling the spread of HIV/AIDS epidemic. The stability of equilibria of the model is discussed using the stability theorem and using the fractional La-Salle invariance principle for fractional differential equations. The effect of the fractional derivative order $$\alpha $$ ( $$ 0.6\le \alpha <1 $$ ) on the HIV/AIDS epidemic model is investigated. The numerical results show that the derivative order $$\alpha $$ can play the role of precautionary measures against infection transmission, treatment of infection and delay in accepting HIV test. We give a general formulation for a fractional optimal control problem, in which the state and co-state equations are given in terms of the left fractional derivatives. We develop the Forward–Backward sweep method using the Adams-type predictor–corrector method to solve the fractional optimal control of the model. The results show that implementing the ART treatment control of diagnosed HIV-infected people together with contact tracing results in a significant decrease in the number of undiagnosed HIV-infected population and AIDS people. Also, implementing all the control efforts decreases significantly the number of diagnosed and undiagnosed HIV-infected population and AIDS people. In addition, the values of the controls are reduced when the fractional derivative order $$\alpha $$ limits to 1 ( $$ 0< \alpha < 1$$ ).

Optimal control of a fractional-order model for the HIV/AIDS epidemic

In this paper, we present a general formulation for a fractional optimal control problem (FOCP), in which the state and co-state equations are given in terms of the left fractional derivatives. We develop the forward–backward sweep method (FBSM) using the Adams-type predictor–corrector method to solve the FOCP. We present a fractional model for transmission dynamics of human immunodeficiency virus/acquired immunodeficiency syndrome (HIV/AIDS) with treatment and incorporate three control efforts (effective use of condoms, ART treatment and behavioral change control) into the model aimed at controlling the spread of HIV/AIDS epidemic. The necessary conditions for fractional optimal control of the disease are derived and analyzed. The numerical results show that implementing all the control efforts increases the life time and the quality of life those living with HIV and decreases significantly the number of HIV-infected and AIDS people. Also, the maximum levels of the controls and the value of objective functional decrease when the derivative order [Formula: see text] limits to 1 ([Formula: see text]). In addition, the effect of the fractional derivative order [Formula: see text] ([Formula: see text]) on the spread of HIV/AIDS epidemic and the treatment of HIV-infected population is investigated. The results show that the derivative order [Formula: see text] can play the role of using ART treatment in the model.

Three-Phase Distribution Power Flow Calculation for Loop-Based Microgrids

This paper introduces an efficient method for calculating the three-phase power flow in a loop-based microgrid. The proposed method incorporates the conventional Newton-Raphson (NR) iterative approach in a backward/forward sweep (BFS) algorithm for power distribution network analyses. Conventional compensation-based approaches are commonly used to account for loop breakpoints (LBPs) and PV nodes. However, the efficiency and the convergence of traditional solutions deteriorate as the number of loops and PV nodes increases. In this paper, we convert microgrid loops into radial structures by breaking up LBPs, when PV nodes connected to distributed generators (DGs) are regulated with scheduled constant voltage magnitudes. Then, we apply a three-phase BFS-based power flow method with an acceptable convergence for radial distribution networks. Next, we use the NR method for power mismatch corrections at LBPs and PV nodes. Finally, the proposed method is extended to islanded microgrids by introducing the system frequency as a variable. We label the proposed loop-based method an NR-BFS power flow calculation scheme, which combines NR and BFS methods for microgrid solutions. The solution of the proposed algorithm, which signifies the application of the improved BFS method, is applicable to active distribution systems with several loops and DGs. The simulation results demonstrate the efficiency of the proposed method in the loop-based microgrid applications.

Compensation in Complex Variables for Microgrid Power Flow

The solution of the distribution network power flow in practical applications is based either on the forward–backward sweep method for radial networks, or the current injection method for meshed networks. While the power flow in microgrids that operate in grid-connected mode could be resolved using the above-mentioned approaches, their operation in island mode would require simulating local generation droop controllers for sharing the complex power load and network loss among the generators. This letter proposes a complex power compensation approach, which is based on Wirtinger calculus, for extending the applicability of practical distribution power flow methods to microgrids operating in island mode. Supporting numerical results are reported on microgrids with up to 3139 nodes.

Individual-based optimal weight adaptation for heterogeneous epidemic spreading networks

We study epidemic spreading on arbitrary weighted, directed, and heterogeneous complex networks. We propose an individual-based weight adaptation mechanism in which individuals’ contact strength is adaptable depending on the level of contagion spreading over the network. An optimal control formulation is presented to address the trade-off between the global infection level and the local weight adaptation cost corresponding to the topology of the underlying contact network. We prove the existence of a solution to the optimal control problem and obtain its explicit expression from a rigorous mathematical analysis. Additionally, we derive an approximation to the optimal solution and analyze its uniqueness with a specific cost function. In a series of numerical experiments, the forward-backward sweep method is adopted to verify the obtained theoretical results. Our findings uncover novel insights into the relationship between epidemic spreading on heterogeneous complex networks, the resulting dynamics of behavioral responses, and the associated control strategy costs, with important consequences for the future development of adaptive public health policies.

A Chain-table Storage Structure Based Forward/Backward Sweep Method for the Power Flow Calculation in a Distribution Network with DFIG Wind Generators and Battery Energy Storage Systems

In this paper, a chain-table storage structure based forward/backward sweep (FBS) method is proposed to calculate the power flow distribution of a distribution network (DN) with DFIG wind generators and battery energy storage systems (BESSs). It is one of the key techniques for solving the problem of optimal charging/discharging scheduling of BESSs. With the consideration of DFIG wind generators and BESSs, the general node model is presented and the chain-table based storage structure is used to storage radial DNs with less memories. Further, the reactive power compensation strategy of PV wind generators is proposed and integrated into the primary FBS method. Finally, the modified PG&E 69 node distribution system is used to conduct the simulation experiments. Experimental results prove that the proposed method owns excellent performances.

Optimal Control on Model of SARS Disease Spread with Vaccination and Treatment

The spread of SARS (Severe Acute Respiratory Syndrome) disease in a human population is one of the phenomena that can be mathematically modeled. The exposed period of SARS disease underlies the formation of the SVEIR epidemic model which is a modification of the SVIR epidemic model by adding subpopulation E (exposed). In the SVEIR model, there are two control variables in the form of vaccination and treatment which aimed to minimize exposed subpopulation, infected subpopulation, and control implementation cost. The Pontryagin’s minimum principle is used to obtain optimal control and system, thus minimizing objective functional as the objective to be achieved. Furthermore, the forward-backward sweep method is used for numerical simulation in order to determine the most appropriate control strategy in a finite time. The simulation results show that implementation of both vaccination and treatment is the most effective decision making to control the spread of SARS disease. Keywords: optimal control, Pontryagin’s minimum principle, SARS.

Sensitivity analysis of convergence characteristics in power flow methods for distribution systems

This paper presents an evaluation of the main convergence characteristics of multiphase power flow methods in unbalanced distribution systems. Two widely used methods for solving unbalanced power flows are analyzed: methods based on Newton–Raphson (NR) and methods based on backward forward sweep (BFS). The limits and robustness of the BFS method and NR method are tested in the following aspects: (i) variation of the X/R ratio, (ii) load’s increase up to the convergence limit, (iii) load model impacts (ZIP model), and (iv) voltage regulators modeling impacts. Some models presented in the literature were implemented and tested. Tests are performed on the IEEE 34 and IEEE 123 systems. Analytical explanations are also presented.

Discrete mathematical modeling and optimal control of the marital status: the monogamous marriage case

In this paper, we consider a discrete model of the marital status of the family dynamics. It is assumed that individuals in the society can be classed in one of the eight compartments: virgin men, virgin women, married men, married women, divorced men, divorced women, widowed men and widowed women. The objective of this work is to treat the modeling and control the system that describes the case of monogamous marriage. We determine two controls which allow to reduce the number of virgin, divorced individuals and increase the number of married individuals. The first control is the benefits of an awareness campaign to educate virgin men and women about the benefits of marriage for the individual and for the society, the second control characterizes the legal procedures, administrative complications and the heavy financial and social consequences of divorces. The optimal control problems are procured based on a discrete version of Pontryagin’s maximum principle and determined numerically using a progressive-regressive discrete schema that converges following a convenient test related to the Forward-Backward Sweep Method (FBSM) on the optimal control.