1. Introduction
In developing countries, high priority is given to providing access to clean electricity and other essential utilities in remote communities [
1]. Access to such amenities is critical to enhancing the quality of life and achieving the Sustainable Development Goals (SDGs) [
2]. The seventh SDG focuses on universal access to affordable and reliable energy [
3], but as recent as 2014, more than 1 billion people (i.e., about 15% of world population) still had no access to electricity [
4]. The lack of access is especially true for geographically isolated rural communities, for which grid connectivity may not be economically or physically viable.
Efforts in providing power to off-grid communities have been pursued to address the rural electrification issue. From 2008 to 2017, the global off-grid renewable energy capacity has grown by about 300%, providing about 6.5 GW in total [
5]. The integration of various renewable energy technologies to locally integrated energy sectors (LIES) have been found to significantly reduce the resource consumption and carbon footprint associated with energy supply [
6]. Despite the dependence to climatic conditions and the inherent variability of renewable energy sources, a hybrid power system was proposed to combine renewable energy technologies with conventional energy sources while minimizing the cost and energy losses [
7], and accommodating uncertainties, especially in remote areas [
8]. The development of off-grid renewable energy systems largely varies and depend on the locally available resources and requirements of the community. In the past decade, an increase in the demand for solar energy solutions has been observed as solar photovoltaic (PV) technology matures [
5]. However, for areas where access to flowing bodies of water is possible, micro-hydropower plants are a practical choice for irrigation and to generate electricity for the community [
9]. Micro-hydropower plants are a robust and inexpensive renewable energy option [
10]. In the context of hybrid power systems, micro-hydropower plants present a viable renewable energy option, given the right geographic location with good power generation potential [
11]. A typical off-grid community will also require other utilities, such as purified water for household use. Aside from access to electricity and clean water, such communities may also require ice to preserve their produce (e.g., fish or vegetables). Polygeneration systems have been designed previously to provide such basic requirements [
12]. These systems provide an efficient, integrated means to produce multiple material and energy products, which leads to improved system-wide efficiency and reduced emissions, waste, and natural resource consumption [
13]. Polygeneration is generally defined as the co-production of various outputs along with electricity; the co-products can include heat (i.e., steam or hot water), cold energy (e.g., ice or chilled water), purified water, and chemical products [
14]. In a polygeneration system, the interdependencies among integrated process units make the whole system vulnerable to cascading failures [
10]. An operational disruption of such nature arising from any portion of the system leads to the causal operational disturbance cascading through the entire process network. Such disruption significantly affects the overall performance of the system causing divergence from the operational production state of the system [
15]. Process systems engineering (PSE) models can play an important role to minimize production loss while maximizing opportunities to operate at an abnormal state for a brief period of time. The optimization approach may then be utilized to find the optimal operational adjustment of a polygeneration system at a perturbed state. A disruptive event, which can be caused by an external (e.g., natural disaster) or internal (e.g., equipment malfunction) trigger, can be measured via a dimensionless index known as inoperability [
16]. The inoperability concept may be applied in representing operational disruption using the inoperability input-output modelling (IIM) framework [
17]. In the context of a process network such as the polygeneration system, the inoperability can be defined as the loss of capacity of a system to operate relative to its normal state, and is represented by a dimensionless number ranging from a value of 0 (fully operational) to 1 (fully non-operational). This index has been used extensively in the literature. A linear programming (LP) model has been utilized for the optimal allocation of resources aiming to minimize losses in the event of a disruption [
18]. The IIM theory was further developed in detail presenting case studies of disruption such as malevolent attacks [
19]. The IIM framework has later been used as a sequential decision support tool for bioenergy systems [
20]. It has been extended to deal with energy-related problems such as the optimization of the energy supply chain [
21] and the identification of process bottlenecks in energy systems [
22].
This work focuses on presenting a novel application of the inoperability concept on the operational adjustment of a community-based off-grid micro-hydropower-based polygeneration plant. Micro-hydropower plants generate up to 100 kW of electricity [
23], which is enough for a small community with 20 to 30 households. Methods for the design [
24], feasibility assessment [
25], and identification of near-optimal solutions [
26] for such facilities have been conducted previously. However, few studies have been conducted for micro-hydropower-based polygeneration systems for remote communities. An off-grid polygeneration system has been designed by Ray et al. [
27] for an off-grid community in India using an LP model incorporating economic and reliability analysis. A generic decision support software was developed by Khalilpour and Vassallo [
28] for the optimal design of a polygeneration system for off-grid distributed generation and storage system. However, neither of these studies incorporated micro-hydropower plants as a technology option. Recent studies on the optimal design of a micro-hydropower based polygeneration plant have been conducted using fuzzy optimization models. Such formulations are characterized by non-sharp boundaries of feasible regions, such that constraints can be partially satisfied as defined by their membership functions [
29]. A fuzzy LP (FLP) model was developed by Ubando et al. [
30] for the optimal design of an off-grid polygeneration plant using a superstructure that includes a micro-hydropower plant, a biomass-based Stirling engine, an ice plant, a diesel engine, and a sewage treatment facility. The work was later extended to a fuzzy mixed-integer linear programming (FMILP) model by Ubando et al. [
12] considering vapor compression and absorption chillers as refrigeration options. However, these models were developed for system planning and design, and lack the capability to be used for optimal operational adjustment under abnormal conditions. An off-grid micro-hydropower-based polygeneration system is susceptible to drought which can reduce water flowrate available for the turbines; also, mechanical problems with system components may take longer to repair due to the time delays in bringing spare parts to the site. Kasivisvanathan et al. [
15] developed a mixed-integer linear programming (MILP) model which provided the solution for the optimal operational adjustment of a gas-fired polygeneration plant for commercial use. In this study, the objective function is to maximize the operating profit (or minimize economic losses) of the system while employing the inoperability in each process in cases when the system experiences a disruption. A P-graph approach to the same problem was later developed by Tan et al. [
31]. In the case of a polygeneration plant for a remote community, it is necessary to consider the demand for each product separately, depending on the needs of the inhabitants. This aspect is addressed in this work using fuzzy optimization using the approach developed by Zimmerman [
29]. This approach is known as a “symmetric” fuzzy optimization model, where both constraints and (multiple) objectives are represented by fuzzy membership functions. FLP has been previously used for the optimal design of systems such as combined heat and power plants [
32], trigeneration plants [
33], and integrated biorefineries [
34]. It was extended later to fuzzy MILP (FMILP) for the optimal planning of bioenergy parks [
35], multifunctional bioenergy systems [
36], and biofuel supply chain networks [
37]. However, these models again focused on system design rather than abnormal operations.
This study develops an FMILP model for the optimal operational adjustment of a micro-hydropower-based polygeneration system for off-grid communities. It is assumed that the likely disruption is in the form of drought with varying intensity, which can be introduced into the model as an inoperability parameter. Three case studies are used to demonstrate the efficacy of the developed model under different drought scenarios. The case studies consider different turbine configurations and the option of having a diesel generator back-up. The rest of the paper is organized as follows. The formal problem statement is given, followed by the model formulation. Three case studies then follow. Lastly, conclusions and future prospects are discussed.
3. Fuzzy Mixed Integer Linear Programming Model
The basic MILP model for optimizing abnormal operations was developed in a previous work, and is given in the
Appendix A [
15]. The weakness of such a formulation is that it places extreme preference for product streams of higher economic value, often at the expense of the other lower-value products. While applicable to many commercial or industrial applications, in the case of remote rural communities, the actual value of basic needs might not be fully reflected in their prices. Instead, a given community may have a set of subsistence-level requirements for essential supplies, all of which need to be balanced based on what are deemed as tolerable shortages. These limits can in turn be modelled as fuzzy membership functions. Thus, the basic MILP model is modified into a fuzzy model to allow for such balanced rationing of capacity during a crisis [
38]. The fuzzy MILP model is composed of the objective function and the constraints. The objective function of the model is shown in Equation (1):
where
λ is the degree of satisfaction which is represented in every membership function in the model. The
λ-value maximizes the lowest degree of satisfaction achieved among all identified fuzzy objectives. The degrees of satisfaction range from 0 to 1, with 1 signifying complete satisfaction, 0 indicating undesirable result, and values between 0 to 1 illustrating partial satisfaction of the objective. In
Figure 1, the degree of satisfaction allows the translation of a variable-value shown in the abscissa into values ranging from 0 to 1 in the ordinate following a linear relation known as the linear membership function.
The optimal operational adjustment of the off-grid polygeneration system is subject to the following constraints. The technology matrix is introduced to define the optimal product output of the polygeneration system together with the process scaling vector as shown in Equation (2):
s.t.
where
A is the technology matrix consisting of elements a
ij where
i represents the product stream and
j represents the technology considered. The technology matrix
A is defined by a steady-state operational material and energy balance which are assumed to be scale invariant. In cases where the material or energy stream is identified as a raw material to a process, the value in matrix
A would be represented by a negative value. Similarly, the positive values in matrix
A represent the product or by-product streams generated from a process. The process scaling vector
x with all elements equal to 1, indicates that all processes
j in the system are operating in their optimal capacity. As the drought scenario sets in, the optimal operation adjustment in the off-grid polygeneration will be reflected in the change in values of
x. The product output
y of the off-grid polygeneration system represents the optimal production level of each product stream
i. The product demand limits utilize membership functions which maximize the product output levels of the polygeneration system given a threshold demand range described in Equation (3). The application of the linear membership function for maximization was introduced by Zimmermann [
39] using a fuzzy set framework:
where
yil is the lower threshold limit of each material or energy stream
i, and
yiu is the upper threshold limit of each material or energy stream
i. Equation (3) represents the fuzzy linear membership function for maximizing a product stream. In Equation (3), any value less than
yil is considered undesirable while any value greater than
yiu is highly desirable for a product stream y
i. The values between
yil and
yiu are considered as partially acceptable where degrees of satisfaction are described by a linear function with a slope of
λ. Equation (4) represents the fuzzy linear membership function for minimizing a product stream which in this case is the fuel. For minimizing the fuel stream in Equation (4),
yl is the highly desired value,
yu is the undesirable value, and the values in between are treated as partially acceptable values. The diagram for the fuzzy linear membership function for maximization used in Equation (3) and for minimization used in Equation (4) are shown in
Figure 1. The introduction of the drought scenario is described by Equation (5):
where
D is the drought intensity factor with values ranging from 0% to 100%. A drought intensity factor of
D = 0 represents a no drought scenario signifying full operational flow level of the river water during normal condition, while a drought intensity factor of
D = 1 implies that there is no river water flowing. The
yi* is the normal product output vector which represents the state of fully operational flow levels for all products
i in the subset of
R. The subset of
R identifies all products which utilize the river water as raw material. The introduction of the binary variable vector is shown in Equations (6)–(8):
The operational capacity limit of each technology
j is described by the lower limit operational capacity
xjl and the upper limit operational capacity
xju. Equation (7) defines whether a technology
j should operate (
bj = 1) or not (
bj = 0) during the chosen scenario. The topological binary constraint for the simultaneous selection of power generating technologies within a subset of
E is shown in Equation (8) where the subset
E represents all power generating technologies. The number of allowable operating technologies is limited by the minimum and maximum number defined by f
l and f
u, respectively. The introduction of the binary variable vector
b in Equations (6)–(8), enables the developed model in becoming a mixed-integer linear programming which provided the selection of the appropriate technology for the off-grid polygeneration system. Lastly, Equation (9) ensures that the
λ-value falls within the range of 0 to 1:
It should be noted that equipment capital cost and stream prices are not included in the model. In the context of operational optimization, all equipment are assumed to have sunk costs, and there is no longer an option to modify them within any given scenario (Kasivisvanathan et al., [
15]). Also, as the model deals with crisis conditions for remote rural communities, it is assumed that actual physical requirements for basic needs take precedence over cost considerations during the emergencies envisioned in this work. The model is solved utilizing Lingo 12.0, linked with MS Excel using the object linking and embedding (OLE) function of Lingo. The computer hardware used is powered by an Intel Core i7 processor with 8 GB of random access memory (RAM). Computational time for the examples in this work was negligible.