P-graph Framework Research Articles

P-graph and Monte Carlo simulation approach for sustainable and risk-managed CDR portfolios

Negative emission technologies (NETs) that perform carbon dioxide removal (CDR) are now part of the decarbonization strategies to reach net-zero emissions. However, NETs will consume resources that will compete with other societal priorities. Thus, NET portfolios that provide a technology mix are better for sustainability and risk management. This study proposes a two-step approach to optimize and check the robustness of NET portfolios, particularly for industrial-scale applications where resource availability fluctuates due to variations in energy, water, and fertilizer supply. The first step involves the process graph (P-graph) framework to generate optimal and near-optimal solution structures at minimum cost. The second step measures the probability of failure of these solutions against resource availability variations through Monte Carlo simulation. By comparing the cost and probability of failure, decision-makers can select a recommended solution that strikes a balance between robustness and cost. The proposed approach is demonstrated in two case studies involving NET portfolios. The first case study investigates the performance of optimal and near-optimal solutions of NET portfolios, while the second case study focuses on bioenergy with carbon capture and storage portfolios using different feedstocks. In both cases, the results show that near-optimal solutions with higher costs but lower probabilities of failure exist, enabling decision-makers to weigh robustness against cost. This study contributes to developing and analyzing robust decarbonization decision support models, addressing the critical need for sustainable and risk-managed NET portfolios.

A P-graph Approach for Planning Sustainable Rice Straw Management Networks

The Philippines produces up to 15.2 Mt of rice straw waste annually. Unmanaged rice straw waste disposal can lead to pollution from open field burning. On the other hand, rice straw agricultural waste can be used sustainably to produce valuable products such as mushrooms, fodder, pellets, or bioenergy, in a rice straw management system. Such systems can be optimized using Process Integration tools so that the raw materials are used efficiently at the maximum profit and with a minimal carbon footprint. The P-graph framework is an efficient Process Integration tool that solves Process Network Synthesis problems. A P-graph finds the optimal and suboptimal solutions for further analysis, which is useful in decision-making. This work developed a P-graph model for a rice straw management network considering the straw collection and storage steps and the production of both bioenergy and non-bioenergy products. The model can generate optimal and sub-optimal solutions (based on profit) and can simulate raw material disruption scenarios. The model is demonstrated through a case study on three rice straw fields with a maximum total rice straw yield of 96.84 t/yr. The case study considered the operating and raw material costs but did not consider the fixed and investment costs in the calculation of the profit. The results show that mushroom production using rice straw as substrate is the optimal solution with a potential profit of US$ 14 659.60/yr, followed by pellet production with a potential profit of US$ 12 627.90/yr. Disruption scenarios at reduced diesel, manual labor, and rice straw show that mushroom production is still the optimum solution, showing the robustness of the solution. This basic model shows that P-graphs can be applied to rice straw management networks to aid with decision-making for sustainability. Caution must be exercised as the results are context- and location- specific.

The P-graph approach in optimal synthesis and planning of waste management towards achieving sustainable development goals: A systematic review.

Effective waste management remains a challenge in global environmental sustainability, underlining the urgent necessity for innovative solutions. This review explored waste management strategies, focusing on the role of P-graph frameworks in achieving sustainable development goals (SDGs). P-graphs offer a systematic approach across domains including, chemical reaction routes, carbon management networks, economic systems and resource planning to waste management synthesis and planning. Through a systematic search and analysis of relevant P-graph approaches, 28 articles meeting the inclusion criteria were selected for review. The study reveals that P-graph approach is a systematic methodology that can streamline decision-making processes, which ultimately lead to more efficient and effective waste management strategies and solutions. This research also highlighted the absence of previous studies on the application of the P-graph approach to various types of waste, underscoring its significance and originality in the field. This study seeks to advance the achievement of SDGs and promote sustainable waste management practices through the integration of the P-graph framework with waste management solutions.

Modeling of a Biomass-Based Energy Production Case Study Using Flexible Inputs with the P-Graph Framework

In this work, a modeling technique utilizing the P-Graph framework was used for a case study involving biomass-based local energy production. In recent years, distributed energy systems gained attention. These systems aim to satisfy energy supply demands, support the local economy, decrease transportation needs and dependence on imports, and, in general, obtain a more sustainable energy production process. Designing such systems is a challenge, for which novel optimization approaches were developed to help decision making. Previous work used the P-Graph framework to optimize energy production in a small rural area, involving manure, intercrops, grass, and corn silage as inputs and fermenters. Biogas is produced in fermenters, and Combined Heat and Power (CHP) plants provide heat and electricity. A more recent result introduced the concept of operations with flexible inputs in the P-Graph framework. In this work, the concept of flexible inputs was applied to model fermenters in the original case study. A new implementation of the original decision problem was made both as a Mixed-Integer Linear Programming (MILP) model and as a purely P-Graph model by using the flexible input technique. Both approaches provided the same optimal solution, with a 31% larger profit than the fixed input model.

Retrofit heat exchanger network optimization via graph-theoretical approach: Pinch-bounded N-best solutions allows positional swapping

Retrofit heat exchanger network (HEN) optimization is a fundamentally unique problem which requires the consideration of existing structures, compared to grassroots design problems. The optimization of retrofit HENs is particularly difficult due to the integration of both existing and newly acquired equipment. The re-routing of existing equipment can lead to various network topologies, increasing the complexity of considerations. In this work, we exploit the P-graph framework to solve retrofit HEN problems, guaranteeing to find the topology of optimal solutions within the constrained space of the HEN retrofit problem. The P-graph framework has additional advantages that allows topologically-efficient search space, simplifies additional unit placement, considers unit positional swapping (re-sequencing and re-piping within search constraints), considers stream splitting, and n-best solution visualization. The pinch minimum utility constraint also provides a bound for the maximum number of modifications in the HEN, significantly reducing search space. The proposed P-graph-based approach is demonstrated using a real refinery case study to show its capability in obtaining the topology of the optimal HEN, highlighting the economic and energy benefits. Further extensions to other retrofit process integration problems (e.g. retrofit water network, hydrogen network etc.) will be enabled via the proposed P-graph approach.

Optimal reaction pathways of carbon dioxide hydrogenation using P-graph attainable region technique (PART)

The attainable region interpretation of the thermodynamic principles has indicated that carbon dioxide (CO2) can be either hydrogenated directly to form dimethyl ether (DME) or gasoline. The process that converts CO2 to DME is more thermodynamically favourable at lower temperature. A certain thermodynamic temperature range (25 to 300 °C) is suggested for the conversion of CO2 to DME via a methanol intermediate pathway without addition of work. Optimal synthesis routes derived from P-graph's mutual exclusion solver were compared with reactions reported in literature and showed great correlation. The reactions collectively possess Gibbs free energy of less than zero, and negative enthalpy of reaction. With P-graph attainable region technique, the case studies have demonstrated that the synthesis of DME and gasoline using CO2 hydrogenation via methanol intermediate and carbon monoxide intermediate from Fischer–Tropsch synthesis is feasible with no work and heat requirement. Both case studies have demonstrated visual advantage of P-graph and data-driven applications. The benefit of integrating the P-graph framework with machine learning model like decision tree classifier was also demonstrated in the second case study as it solves topological optimisation problems without scaling constraints.

A novel hybrid method for constructing resilient microalgae supply chain: Integration of n-1 contingency analysis with stochastic modelling

Microalgae biomass is currently gaining the limelight due to its flexible characteristics such as easily adaptable in cultivation environment, high lipid content that makes it suitable for biofuel processing. However, one of the main challenges faced is the resiliency of the microalgae supply chain network (MSCN) when facing external factors that can lead to supply and demand disruptions. In this study, the integration of n-1 contingency analysis with Monte Carlo simulation model is proposed to conduct resilient supply chain planning when there is supply and demand disruption (i.e., the developed supply chain is robust to fulfil all the demand even when any of the processing hubs is malfunctioning). The n-1 contingency analysis is conducted via the P-graph model framework with the integration of sustainability index —economic and environmental. Subsequently, Monte Carlo simulation model is performed for the extracted distribution network to evaluate the financial probability profile of the optimized distribution network. The collaboration between separate processing hubs is proposed to assist in meeting demand when one of the processing hubs malfunctions with the consideration of the lowest additional cost and carbon dioxide emissions pathway. The results obtained highlighted that inter-plant collaboration is able to increase the mean Net Present Value (NPV) value by up to four times and percentage standard deviation is able to be reduced by up to 49% when inter-plant collaboration is being implemented. Thus, it can be said that the inter-plant collaboration can provide a solution when production plant malfunction and also increases the cash flow in.

Optimization of vertical farms energy efficiency via multiperiodic graph-theoretical approach

A systematic method is proposed for enhancing the energy efficiency of vertical farms when integrated into urban infrastructure. The method relies on the optimization of a multiperiod model that determines the system's best operation plan considering variations in its parameters throughout the day. These parameters include factors such as resources availability from the urban infrastructure and fluctuations in electricity prices. The model is formulated and solved via the tools of the P-graph framework, which permits the automatic formulation of the rigorous superstructure and the efficient optimization of the integrated system's operation. The method is illustrated via a case study comprising three scenarios. The most efficient case demonstrates up to 40% electricity cost saving potential for a particular operation day, and up to 31% for the entire year. The method presented constitutes an optimization tool that can use real data on electricity prices and forecasts of weather conditions to reduce operating costs and enhance the energy efficiency and sustainability of vertical farming systems.

A comprehensive, semi-automated systematic literature review (SLR) design: Application to P-graph research with a focus on sustainability

Because of the vast number of academic products with focus on sustainability being currently generated, it is becoming exceedingly difficult and time-consuming to perform accurate literature reviews that have actual relationship with such a topic and correctly reflect its state of the art. Thus, this work proposes a comprehensive Systematic Literature Review (SLR) design as a novel approach for collecting the body of contributions for any given research field in general and sustainability in particular. This design is guided by a semi-automated procedure that aims at incorporating the best automation features of earlier SLRs and involves available scientific databases, e.g., Scopus and Web of Science. The results of the proposed SLR design include bibliometric analyses, a list of researcher and institution networks, and a qualitative assessment of practical applications of the field of interest. Moreover, the proposed semi-automated approach allows for the inclusion of subject categorization based on the generation of author-density maps and keyword co-occurrence networks. For illustration, the semi-automated SLR design is implemented by thoroughly reviewing the contributions pertaining to the P-graph (Process Graph) framework with a focus on sustainability. The P-graph framework has been applied to multiple areas, such as network design, optimization, and system integration; in particular, the framework has proven to be useful for the development of sustainable systems. Upon execution, the semi-automated SLR design collected 284 contributions related to P-graph, of which 139 were classified as sustainability-related, and further partitioned into specific categories. It is expected that the proposed semi-automated SLR design will help researchers in the sustainability community accelerate the generation of works in that area by rapidly generating accurate literature reviews.

Enabling technology models with nonlinearities in the synthesis of wastewater treatment networks based on the P-graph framework

Designing effective wastewater treatment networks is challenging because of the large number of treatment options available for performing similar tasks. Each treatment option has variability in cost and contaminant removal efficiency. Moreover, their mathematical models are highly nonlinear, thus rendering them computationally intensive. Such systems yield mixed-integer nonlinear programming models which cannot be solved properly with contemporary optimization tools that may result in local optima or may fail to converge. Herein, the P-graph framework is employed, thus generating all potentially feasible process structures, which results in simpler, smaller mathematical models. All potentially feasible process networks are evaluated by nonlinear programming resulting in guaranteed global optimum; furthermore, the ranked list of the n-best networks is also available. With the proposed tool, better facilities can be designed handling complex waste streams with minimal cost and reasonable environmental impact. The novel method is illustrated with two case studies showing its computational effectiveness.

General formulation of resilience for designing process networks

Herein, the formula proposed for quantifying the resilience of engineering systems, including processing systems, is general in several aspects. The structure of a system can be highly complex where the numbers of loops, raw materials, and products are not limited. The mathematical models of the operating units can be either linear or nonlinear for simulating the effect of the failures. The damage caused by an unexpected event can result in various levels of operation for the operating units. The proposed formula is also general in the sense that all possible combinations of failures are considered. The problem formulation, the related structure representation, the enumeration and evaluation of possible failures are based on the P-graph framework and its algorithms. The proposed formula for resilience is applicable to any complex engineering system whose behavior is primarily determined by its structure, including supply chains, information systems, municipal infrastructures, and electrical transmission networks.

Heat Integrated Water Regeneration Network Synthesis via Graph-Theoretic Sequential Method

The integration of multiple resources conservation networks is necessary to attain the ever-stringent sustainable goals. This work takes initiatives to develop a heat integrated water network via a proposed P-graph-based sequential methodology. In the first step, a set of feasible water regeneration networks is generated using the conventional P-graph framework. Then, the obtained feasible networks will be used as the inputs in the second stage which aims to generate various sets of feasible heat exchanger networks. It is worth noting that the second model is solved by an extended P-graph framework (P-HENS) for combinatorial process network optimization. The solutions are then ranked based on the total network cost. To demonstrate the effectiveness of the proposed method, a typical water regeneration network (three sources and three sinks) with multi-contaminants is used. The results show a total of 103 feasible water network structures (water network cost ranging from 0.76 M$/y to 1.18 M$/y). Thereafter, a list of feasible HIWRN can be determined using P-HENS. The top four HIWRNs which offer similar total network cost (~1.639 M$/y) are demonstrated. This proposed method provides valuable insights that allow decision-makers to further select the optimal solution which may be more beneficial as compared to the one obtained via conventional methods.

Accelerating the Search for Optimal Solution of Integrated Process-Network Synthesis and Scheduling

Scheduling is crucial for effective implementations of production, manufacturing, and logistics. With rare exceptions of extremely simple cases, only mathematical programming, e.g., Mixed Integer Linear Programming (MILP), can guarantee the optimal answer to practical scheduling problems. Unfortunately, the applicability of general-purpose MILP solvers is limited by the required computational time. To achieve a sufficiently fast answer in practice by mathematical programming, a highly flexible solution procedure is needed that can be tailored to the problem under investigation. Solution methods utilizing graph theory in parallel with algebraic operations give more room for customization. Process Network Synthesis (PNS) and the P-graph framework were originally developed to design and optimize chemical engineering process structures with continuous operation. Time Constrained Process Network Synthesis (TCPNS) has made it capable to handle batch processes with time constraints and storage strategies. P-graph algorithms extended to TCPNS can solve the precedence-based MILP model formulation of scheduling problems and are highly customizable as well. The aim of the current research is to examine and find the most suitable decision variable selection strategy for P-graph framework’s optimization method to gain possible accelerations for different classes of scheduling problems.

Systematic Design and Evaluation of Energy-Efficient Alternatives of Heterogeneous Azeotropic Distillation: Furfural Case Study

Separation of azeotropic mixtures receives special attention for their impact on various significant industrial processes. Because of the non-ideal behaviour of these mixtures, it is impossible to separate them by conventional distillation. Instead of a single distillation unit, a system of multiple operations is to be employed. Heterogeneous azeotropic distillation (HAD) is an example of this kind of systems, where entrainers are applied to modify the behaviour of the mixture. The selection of the best separation system is a key objective during the synthesis of the process network. However, synthesis of HAD is especially difficult because of the complex interaction between its continuous and discrete features. Therefore, traditional separation network synthesis tools are incapable of solving this problem. In this work, the properties of the ternary vapor-liquid-liquid equilibrium diagram are exploited for systematically identifying plausible operating units that perform the separation of the azeotrope. Subsequently, energy consumption of the entire network is estimated through rigorous simulation. The P-graph framework is employed to represent the system’s structure. Additionally, its combinatorial algorithms generate a rigorous superstructure for the synthesis problem, and the set of n-best designs that minimize energy consumption. The method is illustrated by synthesizing the dehydration of furfural through HAD. The results demonstrate that it constitutes a valuable tool for the designer by being effective in the systematic identification and assessment of HAD alternatives.

VIKOR - P-graph Method for Optimal Synthesis of Philippine Agricultural Waste-Based Sustainable Integrated Biorefinery

A sustainable integrated biorefinery (SIBR) is a biomass processing facility that converts agricultural wastes or residues into a wide range of valuable products where both economic sustainability and environmental sustainability are optimized. This work proposed a hybrid method that incorporates Vlsekriterijumska Optimizacija I Kompromisno Resenje (VIKOR), a widely used multi-criteria decision-making (MCDM) tool with the P-graph (process-graph) framework. The proposed VIKOR - P-graph model can (i) generate a set of combinatorically feasible solutions, and (ii) rank the solution sets using VIKOR method, simultaneously in the same P-graph model. In other words, all the drawbacks attributed to the sequential optimization methods can, therefore, be avoided. To demonstrate the effectiveness of the proposed hybrid methodology, a case study in the Philippines is presented in this paper. The hybrid P-graph model generated a total of 7 feasible solutions with different configuration for the SIBR based on the overall profit (i.e., economic goal) and carbon emissions (i.e., environmental goal). Results show that the best compromised solution can be obtained when majority (70 %) of the rice husk is used to produce bioethanol, where the required power is supplied by combusting the remaining rice husk and by importing external power. It offers an hourly profit of 161 $/h with a lower (~38.6 %) carbon emissions as compared to the most profitable option (3.504 tCO2-eq/h). This research is essentially a guide for policymakers to make informed decisions that can maximise the benefits of SIBR on a national scale.

Modeling Renewable Energy Systems in Rural Areas with Flexible Operating Units

Designing energy systems becomes more and more important as distributed and renewable energy production gains attention. As distributed systems have quite different challenges than centralized ones, novel solution methods are in demand and have been proposed. Sustainability is an ever increasingly important factor for decision makers. The aim is to utilize renewable energy sources like wood, grass, manure, etc., while keeping in mind that their transportation for long distances is impractical. Revitalization of local and regional economy is also a priority. Previous work applied the P-Graph methodology to design an energy system based on biomass in a rural region. A more recent extension of the P-Graph framework is the utilization of operating units with flexible input materials, which makes more accurate equipment models possible. This flexibility is important because certain operating units can tolerate some change in the ratio of their input materials and others cannot. In this work, a method is proposed to design an energy system in a rural area. The notion of flexible operating units is applied to a case study, giving us the possibility to identify optimal composition of biomass feeds and plant size. Our model includes the collection and transportation of local renewable resources, fermenters and combined heat and power (CHP) plants, in order to optimize profit from generated heat and electricity. The case study shows that this modeling technique results in 31% more profit with a considerably lower computational effort.

Optimization of photovoltaic-based microgrid with hybrid energy storage: A P-graph approach

Renewable energy is the key to decarbonize energy use despite the growing global energy demand. However, energy storage is required to tackle the supply-demand mismatch caused by the intermittent nature of renewable energy sources. As each type of energy storage has a distinct discharge duration, a hybrid energy storage system can be more cost-effective than a single energy storage system. While various process integration tools have been employed for the optimization of microgrid with hybrid energy storage, a graph theoretic algorithm known as P-graph allows the identification of optimal and near-optimal solutions for practical decision making. P-graph involves modelling by graphs and the embedded accelerated branch-and-bound algorithm enables efficient determination of optimal solution. This study proposes a multi-period P-graph optimization framework for the optimization of photovoltaic-based microgrid with battery-hydrogen energy storage and the proposed approach is demonstrated through two case studies. Results showed that the optimal cost of microgrid with hybrid battery-hydrogen storage is 704,990 USD/y, a carbon price of 1000 USD/t and above is required to make it more economical than conventional electricity use. Replacing the hydrogen storage system with grid support reduces the total cost and required carbon price to 262,334 USD/y and 300 USD/y respectively.

Multiple-solution heat exchanger network synthesis using P-HENS solver

Analysis on the alternative designs on top of the optimal network has proven valuable and meaningful for the decision-makers in determining the most suitable options which fulfill a wide range of objective functions. On the basis of an extension of the P-graph framework, a procedure was developed previously for multiple-solution heat exchanger network (HEN) synthesis. This procedure is capable of generating the n-best HENs depending on predefined structural constraints, for example, the maximum number of heat exchangers used for the entire system, the maximum number of sequential heat exchangers on each stream, the maximum number of stream splittings per stream. Since the choice of a parameter influences the effect of other parameters on the result, it is difficult to find the proper set of parameters for the solver that result in all plausible solutions to the original problem. Naturally, this issue emerges in any HEN synthesis problem, its systematic treatment is essential. The purpose of the current work was to develop a heuristic framework for determining the most reasonable parameters of the HEN generation algorithm and guiding the designer through the optimization process. The outcome of this study not only benefits the researchers and industrial practitioners of this field but may also be extended for educational purposes.

Conceptual Design of a Negative Emissions Polygeneration Plant for Multiperiod Operations Using P-Graph

Reduction of CO2 emissions from industrial facilities is of utmost importance for sustainable development. Novel process systems with the capability to remove CO2 will be useful for carbon management in the future. It is well-known that major determinants of performance in process systems are established during the design stage. Thus, it is important to employ a systematic tool for process synthesis. This work approaches the design of polygeneration plants with negative emission technologies (NETs) by means of the graph-theoretic approach known as the P-graph framework. As a case study, a polygeneration plant is synthesized for multiperiod operations. Optimal and alternative near-optimal designs in terms of profit are identified, and the influence of network structure on CO2 emissions is assessed for five scenarios. The integration of NETs is considered during synthesis to further reduce carbon footprint. For the scenario without constraint on CO2 emissions, 200 structures with profit differences up to 1.5% compared to the optimal design were generated. The best structures and some alternative designs are evaluated and compared for each case. Alternative solutions prove to have additional practical features that can make them more desirable than the nominal optimum, thus demonstrating the benefits of the analysis of near-optimal solutions in process design.

P-graph Causality Maps

A new class of bipartite causality maps is developed in this work to model influences in socio-technological systems. Unlike earlier forms of causality maps such as fuzzy cognitive maps, this novel approach uses two kinds of elements represented as different types of graph nodes. Based on the P-graph framework, “objects” and their “mechanisms” (or actions) are represented by O-type and M-type nodes, respectively. Objects consist of both tangible (e.g., technologies) and intangible (e.g., policies) elements, while mechanisms are the roles that objects play within the system. Objects in the network influence each other through these mechanisms as indicated by arcs. Two P-graph algorithms are used for this application. First, the maximal structure generation (MSG) is used to generate a maximal causality map based on specifications of local links of individual objects; the causality map assembly is automatic and does not require human understanding of the global problem topology. Next, the solution structure generation (SSG) algorithm is used to algorithmically generate all structurally feasible causality maps for the problem. The alternative networks can be evaluated to gauge practically viable candidate policy and technology combinations. Criticality of different objects can be quantified based on the frequency of occurrence in the enumerated causality maps. The methodology is illustrated using a case study on a policy framework for negative emissions technologies (NETs) for CO2 removal. The potential of this technique to determine how to sustain “virtuous networks” or to deactivate “vicious networks” is also discussed.