Reliability Evaluation and Financial Viability of an Electricity Power Micro-Grid System With the Incorporation of Renewable Energy Sources and Energy Storage: Case Study of KwaZulu-Natal, South Africa

The increase in the demand for a reliable electricity supply by the utilities and consumers has necessitated the evaluation of the reliability of power systems. A reliable electricity supply is characterized by no or minimal duration and frequency of supply outages. This has triggered the necessity of using renewable energy sources (RESs) with optimization methods for reliability improvement of electricity systems and reduction of greenhouse gas (GHG) emissions. The main objective of this study is to optimize micro-grid systems operations, improve reliability, reduce emissions and balance the demand and supply of energy through RESs and battery energy storage system (BESS). The adaptive model predictive control (AMPC) method is used to address the issues of micro-grid operation. The AMPC algorithm solves the optimization problem of disturbance prediction in a micro-grid with different types of RESs and BESS integration. This optimization problem considers different constraints for minimum operating costs in different case scenarios. The financial viability of the proposed method is investigated. Solar photovoltaic (Solar PV), wind plant (WP) and BESSs are used with the AMPC method to investigate the impacts of annual real interest rates on cost and emission parameters, quantification of the emission oxides from different case scenarios, reduction of the cost of electricity (Ccoe), and power system reliability improvement. A modified Roy Billinton Test System (RBTS) is used to confirm the reliability enhancement and financial feasibility of the system. Case studies are used to confirm the proposed methods using climatic data for the city of Pietermaritzburg (29.37° S and 30.23° E), South Africa. The results obtained establish that the incorporation of RESs and BESSs using the AMPC method gives satisfactory outcomes.

Nomenclature

AbbreviationExpansion

Solar PV	Solar Photovoltaic.
WP	Wind Plant.
RESs	Renewable Energy Sources.
BESSs	Battery Energy Storage Systems.
AMPC)	Adaptive Model Predictive Control.
RBTS	Roy Billinton Test System.
$C_{coe}$	Cost of Electricity ($\$$ /kW h).
$C_{amc}$	Annual Maintenance Cost ($\$$ /yr).
$C_{afc}$	Annual Fuel Cost ($\$$ /yr).
$C_{aec}$	Annual Emission Cost ($\$$ /yr).
$C_{arc}$	Annual Replacement Cost ($\$$ /yr).
$C_{acc}$	Annual Capital Cost ($\$$ /yr).
$C_{acs}$	Annual Cost of System ($\$$ /yr).
$C_{npc}$	Net Present Cost ($\$$ /yr).
$C_{toc}$	Total Outage Cost ($\$$ /yr).
$C_{eens}$	Expected Energy Not Served Cost ($\$$ /yr).
ARIR	Annual Real Interest Rate.
EENS	Expected Energy Not Served (MWh/yr).
ECOST	Expected Interruption Cost ($\$$ /yr).
SAWS	South African Weather Services.
O & M	Operation and Maintenance.
NOCT	Nominal Cell Operating Temperature (° C).
GHG	Greenhouse Gas.
SoC	State of Charge.
DoD	Depth of Discharge.
PMB	Pietermaritzburg.
EC	Emission Cost ($\$$ /hr).
CC	Capital Cost ($\$$ /hr).
RC	Replacement Cost ($\$$ /hr).
MC	Maintenance Cost ($\$$ /hr).
FC	Fuel Cost ($\$$ /hr).
SFF	Sinking Fund Factor.
CRF	Capital Recovery Factor.

SECTION I. Introduction

The ever-increasing demand for electricity due to world population growth and economic expansion has placed a massive demand on a reliable power supply. There are challenges due to the fast depleting global deposits of fossil fuels. Environmental pollution and climate change due to greenhouse gas emission coupled with unpredictable fuel pricing have given rise to the exploitation of renewable energy sources (RESs) as reliable, sustainable, and non-polluting alternatives [1]. The provision of quality and reliable power supplies to the consumers at a reasonable cost is the expectation of every utility [2]. Hence, global utilities have identified the incorporation of RESs as a reasonable option for making such provisions. It has been projected that RES penetration into power systems will increase from 25 % in 2017 to 85 % in 2050. This is mostly through the use of solar photovoltaic (solar PV) and wind plants (WP) [3]. A reliable electricity supply is a booster to the social development, economic growth, health, and physical well-being of a nation. Reliability plays a crucial role in the design, implementation, and operation of an electrical system [4]. The performance of the electrical system can be measured by utilities using reliability indices. 80% of load point outages are caused by distribution system failures rather than transmission and generation failures [5]. The effect of outages can be severe to both the utilities and the consumers. Utilities suffer economic loss and reputational damage while the consumers can suffer damage to their equipment, can experience raw materials spoilage, and suffer from loss of revenue and work. As a result, distribution system reliability deserves attention in the electrical system. Further work on improvements in the reliability of the distribution system is necessary.

Reliability serves as a benchmark for providing the regulatory bodies with necessary information in the deregulated environment. Conventional radial distribution systems can be less reliable because of its single source. A fault on any part of the network can lead to an outage of the entire system because of lack of alternative generation. Hence, the duration of outage is often longer in a single-source conventional distribution system.

When there is a fault on a lateral feeder, consumers on other lateral feeders experience voltage fluctuations due to load changes. The incorporation of a multiple-source micro-grid system has influenced the power system positively in a number of ways. The links between renewable energy sources and consumers do not require investment of hundreds of kilometers of a transmission line; hence, reducing maintenance costs, $C_{amc}$ , system costs $C_{acs}$ , and costs of energy, $C_{coe}$ as well as the installation time. The multiple-source nature of the microgrid system helps to ensure an uninterrupted supply of electricity whenever there is a fault from the main source, thereby enhancing the reliability of the system. Consideration has to be given to RESs as the most cost-effective electrification solutions for the load requirements of remote areas. These are characterized by complex terrains which require high cost installations with long installation times and safety concerns for transmission and distribution lines [6]. As of 2016, 13% of the world population is said to be without access to electricity. About one-fifth of these are said to be living in remote areas with complex and difficult terrains and unable to connect to the national grid due to many constraints [7]. RESs can be integrated into the power system to bring electricity to consumers in such remote and rural settlements. For sustainable growth, microgrid systems can be used to integrate RESs. RESs have many advantages such as low cost of emission, low maintenance and operation cost, low fuel cost, and enhancement of power system reliability [8]. Many utilities have proved their benefits and embraced the integration of RESs to meet global demand for energy and allay public fear of the environmental impacts of using fossil fuels. Wind and solar energy are now widely accepted due to their benefits both technically and economically [9].

The benefits of reducing the emission of greenhouse gases has encouraged utilities to harness the wind and solar potential. The use of more than one RES in a system enables the weakness of one source to be overcome by the strength of other and the disadvantage of intermittent supply is overcome by the use of a battery system. A battery energy storage system (BESS) can store energy during normal operation which is used to supplement the supply during deficit or peak period. Many studies have explored the methodologies and impacts of BESSs and RESs on the reliability and financial viability of electrical systems.

Keshavarzi and Ali presented a control strategy under different operating conditions for BESSs for minimization of power fluctuations due to intermittent supplies from RESs [10]. Hidalgo-Leon et al. made a holistic review of the BESS, encompassing its technology, practical implementation, financial viability, and environmental effects [11]. They inferred that integration of a BESS into a RES could effectively mitigate major RES issues. Graditi et al. proposed a model for regulating and controlling a BESS in a RES micro-grid to avoid frequency instability and ensure energy restoration in the network [12]. Farias and Canha analyzed the US Department of Energy (DOE) database to provide cases of battery technologies, including their services, applications, and benefits with respect to RES [13]. They explored the technical characteristics of different battery types to provide insights into their energy flexibility, life cycle, and energy-storing capacity. Hassanzadeh et al. proposed a multi-term signal feedback technique for regulating constant power and frequency deviation of RESs and BESSs, thereby improving microgrid performance [14]. Montoya et al. addressed the problem of optimal dispatch of DC micro-grids with penetration of RESs and BESSs using an exponential load model [15]. Gil-González et al. developed a mathematical model for the optimal operation of dc micro-grids. Second-order cone programming was used to convert non-convex into a convex model of economic dispatch and applied it to a system with high penetration of BESSs and RESs to realize their objectives [16]. Brogan et al. used the ramp time and delayed time of a BESS to improve the inertial and frequency of a power micro-grid system [17]. Reihani et al. proposed optimization techniques for the control of how a BESS charges and discharges in a micro-grid system [18]. Effective voltage regulation, power curve smoothing, and peak load shaving were achieved. Kiptoo et al. considered a cost-benefit analysis in harnessing the benefits of the demand-side in optimal capacity sizing of micro-grid components [19].

Ganesan et al. proposed a hybrid power controller to manage the intermittent nature of the power supply in a RES by sharing power between diesel generators, solar PV, and BESS [22]. To realize the power flow control, active power from the diesel generator and solar PV were managed with respect to the system frequency. Alhejaj and Gonzalez-Longatt demonstrated that the response from BESS inertia can cause a variation in the degree of change of frequency; thereby, enhancing frequency response and offer frequency support to the system [21]. Ganesan et al. discussed an analytical method for identifying suitable rating of voltage source that can act as frequency and voltage references for a BESS [22]. This voltage source-based BESS was found to be suitable in supplying reactive and real power to the load during grid outage. Marchi et al. proposed a model covering the life cycle cost analysis of a BESS considering the operation cost components of the system, such as cost of maintenance, decommissioning, and disposal [23].

Badal et al. presented the benefits of RESs, integration complexities, and control problems [24]. They investigated different control methods in different scenarios. Adefarati et al. proposed a cost-effective, optimized micro-grid system using solar PV, diesel generator, methanol generator and BESS; implemented by the HOMER application tool [4]. The performance of the system was investigated using fuel cost, inflation rate, load demand and solar radiation. Çelikbilek and Tüysüz presented a model for RES evaluation using Multi-criteria Optimization, Analytic Network Process, Decision Making Trial and Evaluation Laboratory [25]. Ranking of RESs was further performed using the proposed method. Kasturi and Nayak proposed a model for optimally allocating RESs using mathematical methods in distribution systems [26]. The optimal allocation problem formulated was solved using a multi-objective optimization approach. Karanki and Xu presented an optimal location and sizing for BESSs to achieve loss reduction in distribution system using the particle swarm optimization method [27]. Tan et al. presented a risk and cost model for optimal scheduling of a hybrid energy system using Latin hypercube optimization technique [28]. The results not only reduced the intermittency of RESs, but also smoothed the tie-line power. Ovaskainen et al. explored the simultaneous use of a BESS as an active harmonic filter for improving power quality and voltage stability of a micro-grid [29]. Hussain et al. developed a coordinated control strategy in a hybrid power system for maintaining system frequency thereby ensuring satisfactory power system stability [30]. It was proved from the simulation results that the load demands can be met by power generated from the RES and stored in the BESS. Khalili et al. [31] investigated how voltage reduction (VR) and a demand response program (DRP) affects the operation of a distribution system (DS). The reliability of the network was evaluated using the energy not supplied (ENS) index. The ENS was minimized by reducing the voltage level of the network using the load. The DS reliability was effectively improved through the combination of DRP and VR methods. [32] explored the optimal scheduling of microgrids (MGs) containing a fossil fuel generator and RESs with a DRP. A multi-objective model using a weighted sum technique was used to obtain Pareto optimal solutions which minimized the unused energy of the implemented scenarios. The costs of electricity and generators were minimized, and the MG DRP profit was maximized to achieve an optimal economic status.

In view of the studies reviewed, it can be seen that more work is needed to incorporate RESs into power systems in order to realize the objectives of reliability improvement and cost reduction. The studies reviewed did not carry out a simultaneous evaluation of system reliability, i.e., the system costs $C_{acs}$ , energy costs $C_{coe}$ and net present cost $C_{npc}$ . This was carried out in this study. The impacts of the annual real interest rate (ARIR) on the cost and emission parameters are also investigated. This gap is identified in Table 1 which compares the studies reviewed. The models proposed in past studies did not consider the stochastic features of the vital components of the micro-grid system. These are essential for evaluating the reliability of a power system. This study has the goal of exploring the stochastic features of vital components of the micro-grid system to evaluate the reliability of the power system with respect to cost analysis considering $C_{acs}$ , $C_{npc}$ and $C_{coe}$ . Case studies for the reliability evaluation of the micro-grid system are explored. These estimate the expected energy not served $(EENS)$ and expected interruption cost $(ECOST)$ using the proposed model. The contributions from this work are:

1. Evaluation of the reliability, financial viability, and eco-friendly impacts of RESs incorporated into a micro-grid system.

2. Development of a model used for reliability evaluation of a power system with the integration of RESs.

3. Development of a model used for reducing $C_{acs}$ and $C_{coe}$ and increasing the use of RESs in a power system.

4. Integration of a model which will help in estimating the costs of supply interruptions by utilities, thereby improving power system reliability.

5. Use of a model which will help in monitoring the efficiency of a power system with the integration of RESs.

6. Implementation of a modified Roy Billinton Test System (RBTS) model which is verified using the adaptive model predictive control AMPC) method.

7. Quantification of the emission parameters (oxides) from different case scenarios.

8. Investigation of the impacts of annual real interest rate on the cost and emission parameters of the micro-grid.

TABLE 1 Comparison of Studies

To the best of the authors’ knowledge, the reliability problems in micro-grid systems with integrated RESs, and the investigation of the impacts of ARIR on the cost and emission parameters, have not been previously addressed using an AMPC optimization algorithm. This work proves, through its results, that the reliability of a power system can be improved by the incorporation of RESs using the proposed method, and $C_{acs}$ and $C_{coe}$ can simultaneously be reduced. This outcome can serve as a measure for making investment decisions by utilities and the governments with respect to renewable energy policies.

SECTION II. Background

In this study, RESs (in the form of a micro-grid) using AMPC method are adopted for reliability and economic improvements. The micro-grid is described as a small entity of an electricity network with a local supply that can function independently (islanded mode) or in conjunction with the central grid. This can also be stand-alone which is useful for use in remote areas which are isolated from the central grid due to operational barriers such as distance and cost of linking [37]. Islanded operation saves utilities from economic loss as well as improving system reliability and environmental perspective. Economic advantages of the micro-grid are that they do not require long transmission line investment if stand-alone and its operation and maintenance costs are low; in addition and there is emission reduction. A micro-grid network comprises of the loads, control units, sources (in the form of wind plant, mini gas turbine, solar PV, hydropower, diesel generator or other generator), and BESSs. The micro-grid system proposed in this study is made up of diesel generators, solar PV, wind plant, and a BESS. They are designed with system constraints and load requirements arranged to reduce costs (total outage cost, $C_{toc}$ , $C_{acs}$ , and, $C_{coe}$ ) and improve reliability. Case studies are used to confirm the proposed methods using climatic data of the city of Pietermaritzburg, PMB (29.37° S and 30.23° E), South Africa. The climatic data is obtained from the South African Weather Services (SAWS) and consists of the wind speed, solar irradiance, and average daily ambient temperature. The proposed micro-grid system is presented in Figure 1. The expression for total power generated $P_{T}$ by the proposed microgrid is obtained using [38]:

$$\begin{equation*} P_{T}(i,t) = P_{SPV}(i,t)+P_{WP}(i,t)+P_{GEN}(i,t)\tag{1}\end{equation*}$$ and power demand by consumers is expressed as: $$\begin{equation*} P_{D}(i,t) = P_{GEN}(i,t) \pm P_{BESS}(i,t)\tag{2}\end{equation*}$$ where $P_{SPV}$ , $P_{WP}$ , $P_{GEN}$ , and $P_{BESS}$ are the power from solar PV, wind plant, diesel generator, and BESS, respectively; $t$ and $i$ are the time period and node under analysis.

FIGURE 1.

Configuration of the micro-grid network under study.

SECTION III. Modelling of the System Under Study

The system dynamics of a micro-grid network, using a diesel generator with RESs and a BESS, are modeled in the MATLAB/Simulink environment in this work. The reliability of the energy system is investigated in the micro-grid network using the proposed technique. The performance of four case studies are investigated: the diesel generator and BESS; the diesel generator, solar PV and BESS; the diesel generator, wind plant and BESS; and the diesel generator, solar PV, wind plant, and BESS. It can be noted that the RES-generated energy during the normal operation of the micro-grid does not meet demand; hence, the inclusion of a diesel generator in all cases. This serves as back-up to the BESS. The BESS stores excess energy which is used during a generation deficit. The units are connected to the DC buses using power electronic components. The diesel generator, RESs and the BESS have their respective local controllers. These controllers carry out the power conversion commands. Therefore the BESS absorbs any network unbalance, thereby improving reliability and minimizing costs [39], [40].

SECTION IV. Power Scheduling of the Adaptive Model Predictive Controller

AMPC uses common ideas in addressing complex micro-grid problems and utilizes comprehensive structures in organized forms. This work adopts the technique of controlling the micro-grid adaptively in order to guarantee an improvement in the reliability of consumer power supply. The adaptive controller harmonizes the power in the network, thereby allowing the optimal generation of power supply from each micro-grid unit. AMPC offers a solution by forming an optimal design of generation, demand, and energy storage for every optimization sample instance. The next sample instance offers a new optimization solution using the output from the previous solution as the new input. In theory, the feedback mechanism generates an optimal design that takes care of the disturbances acting on the micro-grid. The main sources of uncertainty or disturbances in the micro-grid system are the RES-generated energy (caused by wind speed and solar irradiation) and energy demanded. The conventional model predictive controller (MPC) is unable to manage the variations in RESs; hence, the AMPC is more suitable. This operates by updating the system with changes to its internal operating conditions. The AMPC architecture and algorithm flowchart are shown in Figures 2 and 3. The state-space expressions often used for AMPC modeling is given by [41]:

$$\begin{align*} x(t+1)=&Ax(t) + Bu(t) \tag{3}\\ y(t)=&Cx(t)\tag{4}\end{align*}$$ where $x(t)$ , $u(t)$ and $y(t)$ are the BESS charging state, generating units vector variables and the output vector of the system state, respectively.

FIGURE 2.

Control architecture of the study. ($Pgen$ , $Pdem$ , and $Pnet$ are the generated, demanded, and net power, respectively).

FIGURE 3.

AMPC algorithm flowchart.

The generated and demanded powers that are causes of disturbance in micro-grids during normal operations are difficult to predict, vary with time, and cannot be manipulated by the controller because they are external inputs into the system. Hence, the disturbance is a problem that the controller has to overcome. The effects of disturbances on the output can be incorporated into the dynamic model to allow the controller to predict their impact on the performance of the system. The effect of the disturbance, $d(t)$ , can be factored into the AMPC state-space design. The dynamic system equation can be expressed as [39]:

$$\begin{align*} x(t+1)=&Ax(t) + Bu(t)+E_{d}d(t) \tag{5}\\ y(t)=&Cx(t)\tag{6}\end{align*}$$ where $E_{d}$ is the matrix that computes the impact of disturbances on the states. By discretizing (5) and (6) with a sample time $T_{s}$ , the discrete-time space model is: $$\begin{align*} x(t+1)=&A_{d}x(t) + Bu(t)+E_{d}d(t) \tag{7}\\ y(t)=&Cx(t)\tag{8}\end{align*}$$ where the discrete-time expressions are $x(t+1)$ , $x(t)$ , $d(t)$ , $u(t)$ , and $y(t)$ . Also, $A_{d}$ , $B_{d}$ and $E_{d}$ are $e^{AT_{s}}$ , $\int _{0}^{T_{s}} e^{AT_{s}} \,Bdt$ and $\int _{0}^{T_{s}} e^{AT_{s}} \,Edt$ respectively. Since (7) and (8) increment, they become: $$\begin{align*} \triangle x(t+1)=&A_{d}\triangle x(t) + B\triangle u(t)+E_{d}\triangle d(t) \tag{9}\\ \triangle y(t)=&C\triangle x(t)\tag{10}\end{align*}$$ where the incremental expressions are $\triangle x(t+1)$ , $\triangle x(t)$ , $\triangle d(t)$ , $\triangle u(t)$ , and $\triangle y(t)$ . Since existing plant data is required when moving horizon control for control and prediction purposes, $u(t)$ is assumed to be implicitly unable to affect output y(t); but at the $t^{th}$ instant, $u(t-1)$ can affect the output $y(t)$ . So (9) and (10) can be re-arranged to [42]: $$\begin{align*} \begin{bmatrix} \triangle x(t+1)\\ y(t+1) \end{bmatrix}=&A\begin{bmatrix} \triangle x(t)\\ y(t) \end{bmatrix} + Bu(t) + Ed(t) \tag{11}\\ y(t)=&C\begin{bmatrix} \triangle x(t)\\ y(t) \end{bmatrix}\tag{12}\end{align*}$$ where, A $\begin{aligned} = \begin{bmatrix} A_{d} & 0_{d}^{T}\\ C_{d}A_{d} & 1 \end{bmatrix} \end{aligned}$ ; B $\begin{aligned} = \begin{bmatrix} B_{d}\\ C_{d}B_{d} \end{bmatrix} \end{aligned}$ ; E $\begin{aligned} = \begin{bmatrix} E_{d}\\ C_{d}E_{d} \end{bmatrix} \end{aligned}$ ; C $= \begin{bmatrix} 0_{d}^{T} & 1 \end{bmatrix}$ ; $0_{d} = \begin{bmatrix} 0 & 0 & 0 & 0 \end{bmatrix}$ where the augmented state space model matrices A, B and C are used for the predictive control design.

SECTION V. Modeling of the Renewable Energy Sources, Diesel Generator, and Load

The RESs, the BESS, the diesel generator, and the load used in this study are modeled in this section.

A. Modelling of the Solar Photovoltaic

Solar PV has good qualities such as no fuel cost, no carbon emission cost, and low O & M cost. The energy generated by solar PV depends on the ambient temperature, solar irradiance, and the sun’s position in the sky. The power outputs of the MonoXTH PV module can be expressed using [43]:

$$\begin{equation*} P_{SPV}(s(t)) = n_{cells} \times FF \times V \times I\tag{13}\end{equation*}$$ where $s(t)$ is the random irradiance, $n_{cells}$ is the number of functioning photovoltaic cells, and $FF$ is the fill factor: $$\begin{equation*} FF = \frac {V_{mp} \times I_{mp}}{V_{oc} \times I_{sc}}\tag{14}\end{equation*}$$ where $V_{mp}$ , $V_{oc}$ , $I_{mp}$ , and $I_{sc}$ are the voltage at maximum power [V], open-circuit voltage [V], current at maximum power [A], and short circuit current [A], respectively. $$\begin{equation*} V = (V_{oc}+K_{vt} \times T_{ct})\tag{15}\end{equation*}$$ where $V_{oc}$ , $K_{vt}$ , and $T_{ct}$ are the open-circuit voltage [V], the voltage temperature coefficient [mV/° C], and the current temperature [° C], respectively. $$\begin{equation*} I = s(t) \times [I_{sc}+K_{ct} \times (T_{ct}-25)]\tag{16}\end{equation*}$$ where $K_{ct}$ is the current temperature coefficient [mA/° C]. $$\begin{equation*} T_{cell} =T_{amp} + s(t) \times \left ({\frac {NOCT - 20}{0.8}}\right)\tag{17}\end{equation*}$$ where $T_{cell}$ , $T_{amp}$ , and $NOCT$ are the cell temperature [° C], ambient temperature [° C], and nominal cell operating temperature [° C], respectively.

B. Modelling of the Wind Energy

Wind energy, which is a clean and environmentally friendly RES. It has many advantages such as relatively low cost of production, low cost of O & M, free from Greenhouse gas (GHG) emission, sustainable energy source, no fossil fuel costs, and advanced technologies [9], [44]. The power generated by a wind plant can be represented by [43], [45]:

$$\begin{align*} P_{W}(v_{i}) = \begin{cases} 0 & v_{i} < v_{ci} \\ P_{r}^{W}\times \frac {v_{i} - v_{ci}}{v_{r} - v_{ci}} & v_{ci}\le v_{i} < v_{r} \\ P_{r}^{W} & v_{r}\le v_{i} < v_{co} \\ 0 & v_{co}\le v_{i} \end{cases}\tag{18}\end{align*}$$ where $P_{r}^{W}$ is the rated power output; and $v_{i}$ , $v_{ci}$ , and $v_{r}$ stand for the cut-in, cut-out, and the rated wind speeds.

C. Modelling of the Battery Energy Storage System

RESs are characterized by an intermittent generation. This problem is solved using a BESS. Excess energy is stored in the BESS, which is used when the RESs experience intermittency, thereby avoiding power fluctuation and enhancing the reliability of the system [11]. The durability and performance of the battery depend on its rate of charge, rate of discharge, state of charge, voltage effect, and ambient temperature. The state of charge (SoC) of the battery, whose operation capacity must be between the minimum and maximum allowable, is [38]:

$$\begin{align*} SoC^{min}\le &SoC(t)+ \eta _{c}\sum _{t=1}^{k} P_{i}(t) \\& \,\, -\,\eta _{d}\sum _{t=1}^{k}P_{i}(t) \le SoC^{max}, ~for 1 \le t \le k\tag{19}\end{align*}$$ and $$\begin{equation*} SoC^{min} = (1-DoD) SoC^{max}\tag{20}\end{equation*}$$ where $SoC^{min}$ and $SoC^{max}$ are the minimum and maximum allowable states of charge, $\eta _{c}$ and $\eta _{d}$ are the battery charging and discharging efficiencies, and DoD is the depth of discharge.

D. Modelling of the Diesel Generator

The operation of a diesel generator can be as the prime source, as standby, as a stand-alone system, or connected to the grid. They can be reliable, mobile, fuel-flexible, easy, and quick to start but have high O & M costs and GHG emissions. The operating parameters of the CAT 3512B diesel generator, such as power output, fuel consumption, and fuel cost, are used in modeling the generator in this study. The power output of the diesel generator $P_{GEN}$ can be represented by [38]:

$$\begin{equation*} P_{GEN} = P_{n} \times N_{GEN} \times \eta _{GEN}\tag{21}\end{equation*}$$ where $P_{n}$ , $\eta _{GEN}$ and $N_{GEN}$ are the nominal power generated, efficiency, and number of the diesel generators, respectively.

A diesel generator operates within power constraints which are:

$$\begin{equation*} P_{GEN}^{min}(i,t) \le P_{GEN}(i,t) \le P_{GEN}^{max}(i,t)\tag{22}\end{equation*}$$

The generator fuel cost $FC$ is

$$\begin{equation*} FC_{i} = a_{i}P_{GEN}^{2}(i,t) + b_{i}P_{GEN}(i,t) + c_{i}\tag{23}\end{equation*}$$ where $a$ , $b$ and $c$ are the respective coefficients of the fuel cost for the $i^{\text {th}}$ diesel generator.

A conventional power generator produces emissions. Emission Cost (EC) is introduced as a penalty for GHG emissions. The EC is an economic benchmark used by the environmental regulators to control the emission of GHG pollutants. It is applied as taxes to discourage the generation of GHGs. The GHG Emission factor and costs for diesel generation is shown in Table 2.

TABLE 2 GHG Emission Factor and Costs for Diesel Generation [38]

E. Comparison of Microgrid Component Costs and Characteristics

The Cost and technical characteristics of micro-grid system components are shown in Table 3. This is important for the design of a system so that they can be built within budget constraints.

TABLE 3 Cost and Technical Characteristics of the Micro-Grid System [38]

F. Modelling of the Load

Some loads are critical while others are curtailable. Critical loads are classified as essential and have to be met. The AMPC controller makes necessary load forecasting decisions. The load is predicted at time-steps using the preceding data for future projections. The process continuously estimates and updates the model parameters to minimize errors. The load demand for the micro-grid is [41]:

$$\begin{equation*} P_{load}(i,t) = P_{load-curt}(i,t)(1- \theta (i,t)) P_{load-crit}(i,t)\tag{24}\end{equation*}$$ where $\theta (i,t)$ is the curtailment ratio of the curtailable load; the $P_{load-curt}(i,t)$ and $P_{load-crit}(i,t)$ are the curtailable and the critical loads, respectively.

SECTION VI. Economic Modeling of the Renewable Energy Sources

The economic model of the micro-grid under study is made up of the cost of energy $C_{coe}$ , the annual cost of the system $C_{acs}$ , the net present cost $C_{npc}$ , and the total outage cost $C_{toc}$ [38], [46]. The cost of energy is:

$$\begin{equation*} C_{coe} = \frac {C_{acs}}{P_{i}} {\$/\text{yr}}\tag{25}\end{equation*}$$ where $C_{acs}$ is the annual cost of the system in ($\$$ /kWh) and $P_{i}$ is the annual energy production of the generating units in (kWh).

The annual cost can be represented by

$$\begin{equation*} C_{coe} = (C_{amc}+C_{aec}+C_{acc}+C_{arc}+C_{afc}) {\$/\text{yr}}\tag{26}\end{equation*}$$ where $C_{acc}$ is the annual capital cost, $C_{arc}$ is the annual replacement cost, $C_{amc}$ is the annual maintenance cost, $C_{afc}$ is the annual fuel cost and $C_{aec}$ is the annual emission cost.

The annual capital cost for a micro-grid system consisting of the diesel generator, solar PV, wind plant, and BESS is a simple summation:

$$\begin{align*} C_{acc} = &C_{acc,i} \sum _{i=1}^{n} (P_{GEN}+P_{SPV} \\& \qquad \qquad \qquad \qquad \quad +\,P_{WP}+P_{BESS}) {\$/\text{yr}}\tag{27}\end{align*}$$ where $C_{acc,i}$ is the annual capital cost of each component of the micro-grid system. These can be obtained from: $$\begin{align*} C_{acc,i}=&CC + CRF(i,n) \tag{28}\\ C_{arc,i}=&RC + SFF(i,n) \tag{29}\\ C_{amc,i}=&MC + (1+f)^{n} \tag{30}\\ C_{afc,i}=&FC + (1+f)^{n} \tag{31}\\ C_{aec,i}=&EC + (1+f)^{n}\tag{32}\end{align*}$$ where $CC$ , $RC$ , $MC$ , $FC$ and $EC$ are the capital, replacement, maintenance, fuel and emission costs respectively. The $SFF$ , $f$ and $n$ are the sinking fund factor, annual inflation rate and lifetime of each micro-grid system component.

The sinking fund factor is:

$$\begin{equation*} SFF(i,n) = \left ({\frac {i}{(1+i)^{n}-1}}\right)\tag{33}\end{equation*}$$

The maintenance cost for generating unit $i$ is:

$$\begin{equation*} MC_{i} = \sum _{i=1}^{n} (C_{i}P_{i}+CR_{i} \times FCR_{i}) {\$/\text{hr}}\tag{34}\end{equation*}$$ where $C_{i}$ , $CR_{i}$ , $FCR_{i}$ and $P_{i}$ are the proportionality factor, the capacity rating, the fixed charge rate, and the generating unit output power, respectively.

The fuel cost is:

$$\begin{equation*} FC_{i} = \sum _{i=1}^{n} (a_{i}+b_{i}P_{i}+c_{i}P_{i}^{2}) {\$/\text{hr}}\tag{35}\end{equation*}$$ where $a$ , $b$ and $c$ are the diesel generator cost coefficients.

The emission cost is:

$$\begin{align*} EC_{i} = &\sum _{i=1}^{n} (\alpha _{i}P_{i}EF_{COx}+\alpha _{j}P_{i}EF_{SOx} \\& \qquad \qquad \qquad \qquad \qquad \quad \,\,+\,\alpha _{k}P_{i}EF_{NOx}) {\$/\text{hr}}\tag{36}\end{align*}$$ where $\alpha _{i}P_{i}$ , $\alpha _{i}P_{j}$ , $\alpha _{i}P_{k}$ and $EF_{COx}$ , $EF_{SOx}$ , $EF_{NOx}$ are the externality emission costs (in $\$$ /kg) and the emission factors for COx, SOx, and NOx gases respectively.

The net present cost $C_{npc}$ can be expressed as:

$$\begin{align*} C_{npc}=&\left ({\frac {C_{acs}}{CRF(i,n)}}\right) {\$/\text{yr}} \tag{37}\\ i=&\left ({\frac {i'-f}{1+f}}\right) \tag{38}\\ CRF(i,n)=&\left ({\frac {i(1+i)^{n}}{(1+i)^{n}-1}}\right)\tag{39}\end{align*}$$ where $i$ , $i'$ , $f$ , $n$ and $CRF$ represent the real interest rate, the nominal interest rate, the annual inflation rate, the number of years (i.e., the lifetime of each micro-grid system component) and the capital recovery factor respectively.

The expected energy not served cost $(C_{eens})$ and expected interruption cost $(ECOST)$ are used to obtain the total outage cost:

$$\begin{align*} C_{toc}=&(C_{eens} + ECOST) {\$/\text{yr}} \tag{40}\\ C_{eens}=&(k_{e} \times EENS) {\$/\text{yr}}\tag{41}\end{align*}$$ where $EENS$ is the total annual energy not served (kWh) and the lost load value $(k_{e})$ is in ($\$$ /kWh).

SECTION VII. Modelling of the Adaptive MPC Optimization Problem

The focus of the proposed method is the economic feasibility of integrating RESs into the power system while taking into consideration the relevant operational constraints. In this section, the cost function formulation is discussed, i.e., the operational and functional constraints.

A. Formulation of the Cost Function

The proposed method is designed to achieve the objectives of improving the reliability of electricity supply, reducing the operating cost of energy, preventing the battery from deep overcharge and discharge, and guaranteeing energy efficiency by balancing supplies from the diesel generator, RESs, and the BESS. Variables such as the power rates and values, which relate to lifespan and utility cost of the generating units, are incorporated into the cost function. Quadratic cost functions related to different generating units in the micro-grid are harnessed to minimize the total cost of the system and solved by the AMPC algorithm. The cost function can be expressed as [47]

$$\begin{align*} \min J=&\sum _{k=1}^{N_{c}} \Big [\alpha _{1}P_{GEN(t+k)}^{2} + \alpha _{2}P_{SPV(t+k)}^{2} \tag{42}\\&+\,\alpha _{3}P_{WP(t+k)}^{2} +\alpha _{4}P_{BESS(t+k)}^{2}\tag{43}\\&+\,\beta _{1} \triangle P_{GEN(t+k)}^{2}+\beta _{2} \triangle P_{SPV(t+k)}^{2} \tag{44}\\&+\,\beta _{3} \triangle P_{WP(t+k)}^{2}+\beta _{4} \triangle P_{BESS(t+k)}^{2} \Big]\tag{45}\\&+\, \sum _{k=1}^{N_{p}}\Big [\gamma _{1}(SoC_{t+k}-SoC_{ref})^{2}\Big]\tag{46}\end{align*}$$ where $N_{c}$ is the control horizon, $N_{p}$ is the prediction horizon, and $\alpha _{i}$ , $\beta _{i}$ and $\gamma _{i}$ are the respective weights of the variables. In the first four terms of the cost function, the weight of the manipulated variables is used; hence, the optimal solution will apply weighted means in satisfying the objectives. The subsequent four and one terms penalize the rate and confine the stored energy within an operating point respectively.

B. Power Balance Constraints

It is necessary to include the energy balance constraints to achieve stability of the power system. The energy demand and production balance for a reliable and effective operation of the micro-grid network need to be met. The equality constraint is

$$\begin{align*}& \sum _{i=1}^{n}P_{GEN}(i,t)+\sum _{g=1}^{n}P_{WP}(g,t) \\& \qquad \,\,+\, \sum _{h=1}^{n}P_{SPV}(h,t) + \sum _{j=1}^{n}P_{BESS}(j,t) \\&\qquad \,\,-\, \sum _{k=1}^{n}P_{BESS}(k,t) = \sum _{l=1}^{n}P_{D}(l,t)\tag{47}\end{align*}$$ where $P_{GEN}(i,t)$ , $P_{WP}(g,t)$ and $P_{SPV}(h,t)$ are the power output of the diesel generator, wind plant, and solar PV. $P_{BESS}(j,t)$ and $P_{BESS}(k,t)$ are the charge and discharge power of the battery while $P_{D}(l,t)$ is the consumer load point power demand. $n$ denotes the number of occurrences in time $t$ in the respective instances.

C. Inequality Constraints

The inequality constraints are the generating limits of each source as specified by the OEM. To prevent overcharging and undercharging, the SoC of the battery bank is operated within the minimum and maximum limits. This will improve the lifespan of the battery. The constraints are

$$\begin{align*} \begin{cases} P_{GEN}^{min}(i,t)\le P_{GEN}(i,t) \le P_{GEN}^{max}(i,t)\\ P_{WP}^{min}(g,t)\le P_{WP}(g,t) \le P_{WP}^{max}(g,t)\\ P_{SPV}^{min}(h,t)\le P_{SPV}(h,t) \le P_{SPV}^{max}(h,t) \\ P_{BESS}^{min}(j,t)\le P_{BESS}(j,t) \le P_{BESS}^{max}(j,t) \\ P_{BESS}^{min}(k,t)\le P_{BESS}(k,t) \le P_{BESS}^{max}(k,t) \\ SoC^{min}(m,t)\le SoC(m,t) \le SoC^{max}(m,t) \end{cases}\tag{48}\end{align*}$$

Variation in power can lead to energy losses. Therefore power rates constraints are formulated. The micro-grid units and BESS are assumed to be electrically strong in responding to fast power rates so that

$$\begin{align*} \begin{cases} \triangle P_{GEN}^{min}(i,t)\le \triangle P_{GEN}(i,t) \le \triangle P_{GEN}^{max}(i,t)\\ \triangle P_{WP}^{min}(g,t)\le \triangle P_{WP}(g,t) \le \triangle P_{WP}^{max}(g,t)\\ \triangle P_{SPV}^{min}(h,t)\le \triangle P_{SPV}(h,t) \le \triangle P_{SPV}^{max}(h,t) \\ \triangle P_{BESS}^{min}(j,t)\le \triangle P_{BESS}(j,t) \le \triangle P_{BESS}^{max}(j,t) \\ \triangle P_{BESS}^{min}(k,t)\le \triangle P_{BESS}(k,t) \le \triangle P_{BESS}^{max}(k,t) \\ \triangle SoC^{min}(m,t)\le \triangle SoC(m,t) \le \triangle SoC^{max}(m,t) \end{cases}\tag{49}\end{align*}$$

SECTION VIII. Results and Discussions

The reliability and economic impacts of integrating RESs into the power system are explored using a modified Roy Billinton system (Figure 4). The system is made up of 26 circuit breakers; 1 transformer each for the solar PV, the wind plant and BESS, and 22 distribution transformers. The repair rates, failure rates, customer and feeder details of the major system components are found in [48]. The system is modeled with a 4 MW diesel generator, 2 MW solar PV, 2 MW wind plant, and 0.3 MW BESS installed capacities. The network performance is measured by utilities using the reliability indices. The integration of RESs using AMPC is necessitated for reliability improvement and cost reduction. Case studies were used to assess the effects of the proposed method. RESs 1, 2, and 3 are solar PV, BESS and wind plant. The case studies are:

1. Power system containing the diesel generator and RES 2.

2. Power system containing the diesel generator, RESs 1 and 2.

3. Power system containing the diesel generator, RESs 2 and 3.

4. Power system containing the diesel generator, RESs 1, 2, and 3.

FIGURE 4.

Modified Distribution System for RBTS Bus.

Table 4 shows the impacts on the reliability of the system. Table 5 shows the results for the system costs for different case studies. The annual maintenance cost $C_{amc}$ , annual fuel cost $C_{afc}$ , annual emission cost $C_{aec}$ , annual replacement cost $C_{arc}$ , annual capital cost $C_{acc}$ , annual cost of system $C_{acs}$ , net present cost $C_{npc}$ , and cost of energy, $C_{coe}$ were considered for the different case studies. Table 6 shows the impacts of incorporating solar PV, wind plant, and BESS on the annual emissions and fuel consumption.

TABLE 4 Cost Impacts on Reliability Using Renewable Energy Sources Integration

TABLE 5 Cost Saving Impacts of Renewable Energy Sources Integration

TABLE 6 Impacts of Incorporating RESs and BESS on the Annual Emissions and Fuel Consumption

Figures 5 to 10 show the results of EENS (MWh/yr.), $ECOST$ , $C_{eens}$ or $K_{e}EENS$ , $C_{toc}$ , $C_{amc}$ , $C_{afc}$ , $C_{aec}$ , $C_{acs}$ , $C_{arc}$ , $C_{acc}$ , $C_{npc}$ , and $C_{coe}$ , respectively, in the four case studies. Generally, it can be observed that the results in Case 4 supersede the other case studies, thereby demonstrating the benefits of the integration of RES in reliability improvements.

FIGURE 5.

Change in $EENS$ .

FIGURE 6.

Change in $ECOST$ , $C_{eens}$ and $C_{toc}$ .

FIGURE 7.

Change in $C_{amc}$ , $C_{afc}$ , $C_{aec}$ and $C_{acs}$ .

FIGURE 8.

Change in $C_{arc}$ and $C_{acc}$ .

FIGURE 9.

Change in $C_{npc}$ .

FIGURE 10.

Change in $C_{coe}$ .

Figure 11 shows the improvements in $EENS$ (MWh/yr.), of Cases 2 to 4 from Case 1. Cases 2, 3 and 4 increased by 422.97, 359.14 and 506.89 respectively. In Figure 12, Cases 2, 3 and 4 of $ECOST$ ($\$$ /yr) improved by 536.9, 704.3 and 804.5 over Case 1; Cases 2, 3 and 4 of $C_{eens}$ (million $\$$ /yr) improved by 2.326, 1.975 and 2.788 over Case 1; while for $C_{toc}$ (million $\$$ /yr), Cases 2, 3 and 4 improved by 2.327, 1.976 and 2.789 over Case 1 respectively. Figure 13 shows that Cases 2, 3 and 4 of $C_{amc}$ (thousand $\$$ /yr) improved by 82.83, 146.65 and 146.73 respectively over Case 1 (hence, the integration of RESs results in spending less on maintenance costs); Cases 2, 3 and 4 of $C_{afc}$ (million $\$$ /yr) improved by 1.811, 1.836 and 1.866 over Case 1 (resulting in a significant drop in spending on diesel fuel); Cases 2, 3 and 4 of $C_{aec}$ (thousand $\$$ /yr) improved by 139.03, 140.22 and 142.27 over Case 1 while Cases 2, 3 and 4 of $C_{acs}$ (million $\$$ /yr) improved by 1.815, 1.896 and 1.715 over Case 1 respectively. Figure 14 shows that there is an increase in $C_{arc}$ (thousand $\$$ /yr) in Cases 2, 3 and 4 by 18.0, 18.9 and 23.6 respectively over Case 1 and an increase in $C_{acc}$ (thousand $\$$ /yr) in Cases 2, 3 and 4 by 207.7, 234.9 and 410.3 respectively over Case 1. These two increases are reasonable and understandable. Although there are increases in these two costs in Cases 2 to 4 as compared to Case 1, the overall cost of the system, $C_{acs}$ , shows improvement.

FIGURE 11.

Improvements on $EENS$ .

FIGURE 12.

Improvements on $ECOST$ , $C_{eens}$ , and $C_{toc}$ .

FIGURE 13.

Improvements in $C_{amc}$ , $C_{afc}$ , $C_{aec}$ , and $C_{acs}$ .

FIGURE 14.

Improvements in $C_{arc}$ and $C_{acc}$ .

Figure 15 shows $C_{npc}$ (million $\$$ /yr) in Cases 2, 3 and 4 with an improvement of 3.946, 4.123 and 4.155 respectively over Case 1. All these improvements in Figures 11–​15 lead to the improvement in $C_{coe}$ ($\$$ /kWh) in Cases 2, 3 and 4 by 0.0107, 0.0114 and 0.0115 respectively over Case 1, i.e., cheaper costs of electricity units. This is illustrated in Fig. 16.

FIGURE 15.

Improvement in $C_{npc}$ .

FIGURE 16.

Improvements in $C_{coe}$ .

Figures 17 to 19 illustrate the percentage improvements in Cases 2 to 4 when compared to Case 1. The percentage improvements of $EENS$ , $C_{npc}$ and $C_{coe}$ are shown in Figure 17. $EENS$ improved by 6.15 %, 5.22 %, and 7.37 % in Cases 2, 3, and 4, respectively, when compared to Case 1. The percentage improvement in $C_{npc}$ ranges from 14.3 % to 15.0 %, while $C_{coe}$ ranges from 14.2 % to 15.3 %. In Figure 18, $ECOST$ , $C_{eens}$ and $C_{toc}$ have percentage improvements ranging from 6.1 % to 9.1 %, 6.1 % to 7.4 %, and 2.9 % to 7.4 % respectively. Figure 19 shows the $C_{amc}$ , $C_{afc}$ , $C_{aec}$ and $C_{acs}$ with percentage improvements ranging from 9.9 % to 17.6 %, 27.0 % to 27.8 %, 27.3 % to 27.9 % and 13.5 % to 14.9 % respectively. The percentage increase of $C_{arc}$ and $C_{acc}$ in Figure 20 range from 0.9 % to 1.2 % and 7.4 % to 14.7 % respectively. These increases in $C_{arc}$ and $C_{acc}$ are projected to reduce overtime as investments in RESs intensify.

FIGURE 17.

Percentage improvement in $EENS$ , $C_{npc}$ and $C_{coe}$ .

FIGURE 18.

Percentage improvements in $ECOST$ , $C_{eens}$ , and $C_{toc}$ .

FIGURE 19.

Percentage improvement in $C_{amc}$ , $C_{afc}$ , $C_{aec}$ , and $C_{acs}$ .

FIGURE 20.

Percentage increase in $C_{arc}$ and $C_{acc}$ .

SECTION IX. Impacts of the Annual Real Interest Rate

The effects of the annual real interest rates (ARIRs) on the various economic and emission parameters are presented in this section. The economic parameters considered in this study are $C_{acs}$ , $C_{coe}$ , $C_{npc}$ and $C_{toc}$ . The emission parameters are COx, SOx, and NOx are also addressed.

A. Relationship Between Annual Real Interest Rate and Annual Cost of System, Net Present Cost and Total Outage Cost

The plot in Figure 21 shows the relationships between the ARIR and $C_{acs}$ , $C_{npc}$ , and $C_{toc}$ . Initially, $C_{acs}$ is approximately $\$$ 10.9 million/yr between an ARIR of 2% to 5.2% before increasing to $\$$ 11.17 million/yr at 6 % and then steadily increasing to $\$$ 11.88 million/yr at 12 %. This shows that higher ARIR means higher $C_{acs}$ . This cost is directly or indirectly borne by the consumers. $C_{npc}$ is shown to be inversely proportional to the ARIR. This illustrates that higher ARIR means lower $C_{npc}$ for RES goods and services. By implication, higher ARIR erodes the present values of RES goods and services. At first, $C_{toc}$ maintains an approximate value of $\$$ 35.11 million/yr between an ARIR of 2 % to 4.5 % before increasing to $\$$ 35.23 million/yr at 6 %, and then steadily increasing to $\$$ 36.33 million/yr at 12 %. By implication, higher ARIR means higher $C_{toc}$ and higher consumer cost burden.

FIGURE 21.

Effect of annual real interest rate on $C_{acs}$ , $C_{npc}$ and $C_{toc}$ .

B. Relationship Between Annual Real Interest Rate and Annual Cost of Energy

Figure 22 shows that the relationship between ARIR and the $C_{coe}$ is almost linear and slightly rising. When the ARIR is high, $C_{coe}$ . This implies that policymakers and government should make efforts to reduce the annual real interest rate on RES-related goods and services in order to encourage mass deployment of RESs. This will lead to cheaper electricity for consumers.

FIGURE 22.

Effect of annual real interest rate on $C_{coe}$ .

C. Relationship Between Annual Real Interest Rate and Emission Parameters

Figures 23 to 25 show the relationships between the emission oxides (COx, SOx, and NOx), the ARIR, and the RESs. Generally, at lower RES input, the COx, SOx, and NOx emissions were high because the diesel generator operates for more hours. Therefore, the ARIR must be kept low in order to keep the emission low and maintain a high penetration of RESs.

FIGURE 23.

Effect of annual real interest rate on COx.

FIGURE 24.

Effect of annual real interest rate on SOx.

FIGURE 25.

Effect of annual real interest rate on NOx.

SECTION X. Comparison With Other Reliability Optimization Methods

Table 7 presents the comparison of AMPC with other reliability optimization methods for the cost function formulation of Case 4. The acronymns in Table 7 are defined as follows: DS-differential search; PSO-particle swarm optimization; DE-differential evolutionary algorithm; GA-genetic algorithm; ACO-ant colony optimization; GSA-gravitational search algorithm; and AMPC-adaptive model predictive control. The results show that the simulation times for AMPC are much less than that of the other methods when calculating $C_{eens}$ , $ECOST$ , $COx$ , and $C_{acs}$ . The results also show much smoother characteristics and superior computational efficiency of the AMPC algorithm.

TABLE 7 Comparison With Other Reliability Optimization Methods

SECTION XI. Conclusion and Future Study

In this study, the economic, environmental, and reliability impacts of fossil fuel generators, Solar PV, WP, and BESS in a micro-grid power system are investigated using the meteorological data of Pietermaritzburg, KZN, South Africa. The objective functions considered are reliability improvement, cost and emission minimization. The variables $C_{toc}$ , $C_{amc}$ , $C_{afc}$ , $C_{aec}$ , $C_{arc}$ , $C_{acc}$ , $C_{npc}$ , and $C_{coe}$ are carefully chosen as operational costs for the network. The oxides selected for emissions are $COx$ , $SOx$ , and $NOx$ . $EENS$ and $ECOST$ are the reliability indices used for the evaluation of the network. The optimization problem is solved using an AMPC algorithm. The verification of the proposed approach is done using a modified RBTS test system. Simulation results show that the integration of a fossil fuel generator, RESs and a BESS using AMPC can improve reliability, reduce emissions and minimize operational costs of the micro-grid. AMPC implementation in a micro-grid system containing a fossil fuel generator, RESs and a BESS decreases the $C_{coe}$ by decreasing the associated operational costs. Decreasing the emissions level is done by reducing the operational hours of the fossil fuel generator. By introducing different sources of renewable energy into the micro-grid, $EENS$ and $ECOST$ are minimized leading to an improvement in the reliability of the system. The results show that for policymakers, government, and investors to embark on a system that makes economic and environmental sense, the ARIR must be kept as low as possible. The economic, environmental, and reliability improvement of the power system is expected to impact the performance of the system. The RESs are projected to dominate power supply worldwide in the future.

However, this study acknowledges the limitation that all the standards, costs and data used in this research are assumed to be accurate at the time this study was conducted. In this study, the physical implementation of the proposed method on the field is not included.

Further research works considering other optimization methods could be conducted to explore further reliability improvement of power systems. While optimal solutions have been determined using AMPC algorithm, similar data and objective functions can be subjected to other algorithms and the output and efficiency compared. Superior results and reduced computation time of the algorithm are possible if a combination of the methods are used.

ACKNOWLEDGMENT

The authors acknowledge the South African Weather Service (SAWS) for the provision of real-life data for this research. They also acknowledged the Eskom Power Plant Engineering Institute (EPPEI) Specialization Centre in HVDC and FACTS, University of KwaZulu-Natal, for the provision of research facilities.