Multi-objective optimal power management and sizing of a reliable wind/PV microgrid with hydrogen energy storage using MOPSO

Abstract

In this paper, a multi-objective algorithm is presented for optimal power management and design of a hybrid Wind/Photovoltaic/ generation system with hydrogen energy storage system including electrolyzer, fuel cell and hydrogen tank to supply power demand in a microgrid system. The generation units are intrinsically non-dispatchable and moreover, the major components of the system i.e. wind turbine generators, photovoltaic arrays and DC/AC converter may be subjected to failure. Also, solar radiation, wind speed and load data are assumed to be entirely deterministic. The goal of this design is to use a novel multi-objective optimization algorithm to minimize the objective functions i.e. annualized cost of the system, loss of load expected and loss of energy expected and provide optimal energy management in the microgrid. The system costs involve investment, replacement and operation and maintenance costs. Prices are all empirical and components are commercially available. The simulation results for different cases reveal the impact of components outage on the reliability and cost of the system. So, they are directly depends on component’s reliabilities, i.e. outages lead to need for a larger and more expensive generation system to supply the load with the acceptable level of reliability. In addition, an approximate method for reliability evaluation of hybrid system is presented which lead to reduce computation time. Simulation results show the effectiveness of proposed multi-objective algorithm to solve optimal sizing problem in contrast with traditional single objective methods.

Keywords

Microgrid wind energy photovoltaic fuel cell reliability optimal power management and sizing multi-objective particle swarm optimization

1 Introduction

A drawback common to Photovoltaic (PV) and Wind Generation (WG) units, is unpredictable and non-dispatchable nature of solar and wind energy sources in the microgrid systems. Moreover, the variations of these sources may not match with the time distribution of demand in the microgrid [1 –3]. These drawbacks result in serious reliability concerns in both design and operation of PV and WG systems. Although over sizing is usually an approach to overcome reliability problem, however, it may be costly and so as an alternative approach, hybrid PV/WG systems efficiently combine complementary characteristics of solar and wind sources to enhance the microgrid system’s reliability and reduce its costs [4].

Overall single line/block diagram of studied microgrid system is demonstrated in Fig. 1 including WG/PV units connected to a common DC bus and a combination of a Fuel Cell (FC) stack, an electrolyzer, and a hydrogen storage tank as the Energy Storage System (ESS). In recent years, hydrogen as a suitable storage medium in renewable energy systems has become a matter of grave challenge [5 –8].

Unlike diesel generator, hydrogen-based ESS are emission free and they do not need any supply of fuel. Moreover, hydrogen-based systems will be economically reasonable in future considering expectation of increase in fuel price and extreme reductions in FC costs [9 –11]. Also, in hybrid PV/WG/diesel systems, it is not possible to store surplus during good seasons. In contrary, in proposed hydrogen-based ESS, electrolyzer converts the excess energy into chemical form, i.e. produces hydrogen, and stores in the hydrogen tank [12, 13].

Because of intermittent characteristic of wind speed and radiation, most important challenge in design of such systems is reliable supply of demand so, the goal is optimal design of a hybrid system for reliable and economical supply of the load [7]. Literature offers a variety of methods for optimal designing of hybrid PV/WG generating systems [1–4 , 12–28].

In [15], non-Linear programming (NLP) is used to find the optimal size and location of grid-connected wind turbines. Simple iterative algorithms have been used for optimal sizing of hybrid WG/PV system with battery ESS [12 , 19–25]. Deficiency of power supply probability (DPSP) [19] and Loss of power supply probability (LPSP) [23 –25] have been used as the technical criteria in iterative solution for optimal sizing of hybrid WG/PV systems with battery ESS. In [16], Genetic Algorithm (GA) and in some other works, Particle Swarm Optimization (PSO) are successfully implemented for optimal sizing of hybrid stand-alone systems, assuming continuous and reliable supply of load [7 , 26–28]. GA and preference-inspired co-evolutionary algorithm (PICEA) have been used in [3] and [17] to find optimal size of a wind/PV/battery system subject to reliability index of LPSP. However, they do not consider outage probabilities of components such as wind turbines and PV arrays. In [14], a methodology for calculation of economic costs of power interruptions for different user sectors and interruption durations has been developed.

Reliability assessment is relevant for any engineering system [29]. But, previous researches do not consider reliability issues in depth. For instance, some very given phenomena that may extremely affect system’s reliability and cost, such as failures and outages of generating units are usually ignored.

Impacts of uncertainties in operating parameters and reliability of a Proton Exchange Membrane Fuel Cell (PEMFC) has been studied and it has been noted that, uncertainty and reliability shall be considered in designing stage of any robust and applicable system and finally, it suggests that a stochastic modeling framework should be interfaced with a numerical optimization scheme [30]. This suggestion was a great motivation to consider the impact of component reliabilities, on economical design of stand-alone renewable systems. So, in [4], the outage probabilities of PV arrays and WGs have been considered. But, further studies revealed that the availability of DC/AC converter, as the only single cut-set [29] in reliability diagram, has an extreme influence over the system’s reliability. Later, [28] investigated the problem more carefully as a Single Objective Optimization (SOO) problem.

Multi-Objective Optimization (MOO) methods have also been used to design and optimize a hybrid energy systems [11, 34]. In [11], a hybrid PV-wind-diesel-hydrogen-battery system has been designed using a MOO algorithm minimizing total net present cost, CO₂ emissions, and the unmet load as three objective functions by using Multi-Objective Evolutionary Algorithm (MOEA) and GA. Multi-objective linear Programming (MOLP) has been used to optimize economic/environmental objectives in a distributed energy system including PV, FC and gas engine for combined heat and power (CHP) plants [31].

In this paper, Multi-Objective Particle Swarm Optimization (MOPSO) algorithm is exploited to minimize costs of the system over its 20 years of operation, as well as reliability indices like Loss of Load Expected (LOLE) and Loss of Energy Expected (LOEE) subject to reliability and operational constraint. Wind speed and solar radiation data are available for Ardebil province in North West of Iran (latitude: 38_170, longitude: 48_150, altitude: 1345 m), and system costs include AC of investment, replacement, and O&M, as well as costumers’ dissatisfaction costs. Finally, an approximate method for reliability evaluations of the hybrid system is proposed. Results indicate that the approximate method provides not only acceptable accuracy, but also reduces problem complexity significantly and, consequently, needed time and computation intensity.

2 PV/WG/FC system modeling

The microgrid is simulated for one year period of operation in one hour time steps. Given yearly profile of wind speed and solar are available, generated power by PV and WG can be calculated.

2.1 Photo voltaic array

Insulation data is converted in to output power of the PV panel using the following equation [4, 7]. $P_{PV} = \frac{G}{1000} P_{PV, rated} \times η_{PV, conv}$ (1)

where, G is perpendicular radiation at array’s surface (W/m²), P_PV,rated is rated power of each PV array at G = 1000 W/m², and η_MPPT is the efficiency of PV’s DC/DC converter and Maximum Power Point Tracking (MPPT). PVs are usually equipped with MPPT systems to maximize the power output [3]. Using these systems, usually result in about 30% increase in the average amount of the extracted energy from PV arrays [2]. Thus, it is assumed that PV arrays are equipped with 95% efficient MPPT systems which provide a 48 V DC at DC bus side.

2.2 Wind Turbine Generator

Output power of WG against wind speed is depicted in Fig. 2. This curve is given by manufacturer and usually describes the real power transferred from WG to DC bus. In this paper, Bergey Wind Power’s BWC Excel R/48 [9] is considered. It has a rated capacity of 7.5 kW and provides 48 V DC as output. Here, power curve versus wind speed is defined by (Fig. 2): $P_{WG} = {\begin{matrix} 0 & ; v_{W} \leq v_{cut in}, v_{W} \geq v_{cut out} \\ P_{WG, max} \times {(\frac{v_{W} - v_{cut in}}{v_{rated} - v_{cut in}})}^{m} & ; v_{cut in} \leq v_{W} \leq v_{rated} \\ P_{WG, max} + \frac{P_{furl} - P_{WG, max}}{v_{cut out} - v_{rated}} \times (v_{W} - v_{rated}) & ; v_{rated} \leq v_{W} \leq v_{furl} \end{matrix}$ (2)

Where, P_WG,max and P_WG,furl are WG’s output power at rated and cut out speeds, respectively. In this paper, m is set to 3. Also, in above equation, v_W refers to wind speed at the height of WG’s hub. Measured data at any height can be converted to installation height through exponent law [2]: $v_{W} = v_{W}^{measure} \times {(\frac{h_{hub}}{hmeasure})}^{α}$ (3) where, α is the exponent law coefficient [3].

2.3 Power generated by renewable Units

By adding WG and PV power outputs, injected power from renewable sources to DC bus is:

$\begin{matrix} P_{ren} (n_{WG}^{fail}, n_{PV}^{fail}) = (N_{WG} - n_{WG}^{fail}) \times P_{WG} \\ + (N_{PV} - n_{PV}^{fail}) \times P_{PV} \end{matrix}$ (4)

Where, N_WG, N_PV, $n_{WG}^{fail}$ and $n_{WG}^{fail}$ are number of installed and failed WG turbines and PV panels, respectively [32]. At this point, generated power by renewable sources distributes through two streams, one stream goes to DC/AC inverter to meet the load (P_ren-inv), and other, transfers the extra power to electrolyzer for hydrogen production (P_ren-el).

2.4 Electrolyzer

Most electrolyzers produce hydrogen at pressure around 30 bars which can then be further compressed for storage [28, 33]. But, the reactant pressures within a Proton Exchange Membrane FC (PEMFC) are around 1.2 bar (a bit higher than atmosphere pressure) [34]. So, the electrolyzer output is directly injected to a hydrogen tank [8 , 35]. In this paper, the electrolyzer is directly connected to hydrogen tank, although, the developed software is flexible to handle compressor model. When the tank is fully charged, compressors pump the hydrogen into a second high-pressure tank. So, compressor does not work continuously and, it consumes lower energy. Transferred power from electrolyzer to hydrogen is: $P_{el - tank} = P_{ren - el} \times η_{el}$ (5) where, η_el is the electrolyzer efficiency [9, 28].

2.5 Hydrogen tank

The energy of hydrogen stored in the tank at time step t can be obtained by the following equation:

$\begin{matrix} E_{tan k} (t) & = & E_{tan k} (t - 1) \\ + (P_{ele - tan k} (t) - \frac{P_{tan k - FC} (t)}{η_{storage}}) \times Δ t \end{matrix}$ (6) where, P_{tank_FC} is the transferred power from the hydrogen tank to the FC. Storage efficiency (h_storage) may present losses resulted from leakage or pumping, and assumed to be equal to 95% [28, 36]. The mass of stored hydrogen, at time step t, is calculated as: $m_{tan k} (t) = \frac{E_{tan k} (t)}{{HHV}_{H_{2}}}$ (7) where, the Higher Heating Value (HHV) of hydrogen is equal to 39.7 kWh/kg [28, 37]. Hydrogen pressure drop, a small fraction of the hydrogen (here, 5%) may not be extracted. This fraction is the lower limit of the stored energy [28]. Therefore, $E_{tank, min} \leq E_{tank} (t) \leq E_{tank, max}$ (8)

2.6 Fuel cell

A PEM fuel cell has reliable performance under intermittent supply and is commercially available [33]. Its output power can be defined as a function of its input and efficiency (η_FC) which is considered to be constant (here, 50%) [9]. $P_{FC - inv} = P_{tank - FC} \times η_{FC}$ (9)

2.7 DC/AC converter

The inverter’s losses can be presented by its efficiency (η_inv). The efficiency is roughly supposed to be constant (here, 90%) [9]. $P_{inv - load} = (P_{FC - inv} + P_{ren - inv}) \times η_{inv}$ (10)

2.8 Operation strategy

Strategy of system operation is governed by the following rules:

If $P_{ren} (t) = \frac{P_{load} (t)}{η_{inv}}$ , then whole power generated by renewable sources is injected to the load through inverter.

If $P_{ren} (t) > \frac{P_{load} (t)}{η_{inv}}$ then the surplus power is transferred to electrolyzer. If the injected power exceeds electrolizer’s rated power and the surplus energy will circulate in a dump resistor.

If $P_{ren} (t) < \frac{P_{load} (t)}{η_{inv}}$ then shortage power will be supplied by fuel cell. If shortage power exceeds fuel cell’s rated power or stored hydrogen cannot afford the shortage, some fraction of the load must be shaded. This fact leads to loss of load.

Under all above conditions, components limits are regarded and Equations (1 –10) rein the system.

3 Reliability/cost assessment

The microgrid system is simulated over a year with 1-h time steps and reliability/cost assessment studies are carried out. Then, using economic factors, results are expanded to the 20-year period of a microgrid lifetime. Load growth and uncertainty in load, solar radiation and wind speed are neglected.

3.1 Reliability Indices

Several reliability indices introduced before [17 , 39]. LOLE, LOEE or Expected Energy not Supplied (EENS), LPSP, and Equivalent Loss Factor (ELF) are some of the most common used indices in the reliability evaluation of generating systems. From these, LOLE is a loss of load index, whereas others belong to category of loss of energy indices.

Loss of load expected [4, 32]

LOLE = \sum_{h = 1}^{H} E [LOL (h)]

(11) where, E [LOL (h)] is the expected value of loss of load at hth time step defined by [4, 32]:

E [LOL] = \sum_{s \in S} T (s) \times f (s)

(12)

In the above equation, f (s) is the probability of encountering the state s, and T (s) is the loss of load duration (h), given that the occurring state s and S is the set of all possible states.

Loss of energy expected (expected energy not supplied [4, 32]

LOEE = EENS = \sum_{h = 1}^{H} E [LOE (h)]

(13) where, E [LOE (h)] is the expected value of loss of energy, or energy not supplied, at time step h:

E [LOE] = \sum_{s \in S} Q (s) \times f (s)

(14)

Here, Q (s) is the amount of loss of energy (kWh) when system encounters state s.

Loss of power supply probability [4, 32]

LPSP = \frac{LOEE}{\sum_{h = 1}^{H} D (h)} = \sum_{h = 1}^{H} E [LOE (h)]

(15) where, D (h) is the load demand (kWh) at time step h.

Equivalent loss factor [4, 32]

ELF = \frac{1}{H} \sum_{h = 1}^{H} \frac{E [Q (h)]}{D (h)}

(16)

In all above equations, H is the number of time steps in which system’s reliability is evaluated (here, H = 8760). The ELF is the ratio of the effective forced outage hours to the total number of hours [33]. Therefore, ELF is chosen as the main reliability constraint of the optimization problem. In developed countries, electricity suppliers aim at ELF <0.0001. On the other hand, in rural areas and stand-alone applications (this study), ELF <0.01 is acceptable [33].

3.2 System’s reliability model

In this study, outages of PV arrays, wind turbine generators, and DC/AC converter are considered. Forced Outage Rate (FOR) of PVs and WGs is assumed to be 4% [28, 39] and they will be available with a probability of 96%. Probability of encountering each state is calculated through binomial distribution function [40]. For example, given $n_{WG}^{fail}$ out of total N_WG installed WGs, and $n_{PV}^{fail}$ out of total N_PV installed PV arrays are failed, the probability of encountering this state is calculated as follows:

$\begin{matrix} f_{ren} (n_{WG}^{fail}, n_{PV}^{fail}) \\ = [(\begin{matrix} N_{WG} \\ n_{WG}^{fail} \end{matrix}) \times A_{WG}^{N_{WG} - n_{WG}^{fail}} \times {(1 - A_{WG})}^{n_{WG}^{fail}}] \\ \times [(\begin{matrix} N_{PV} \\ n_{PV}^{fail} \end{matrix}) \times A_{PV}^{N_{PV} - n_{PV}^{fail}} \times {(1 - A_{PV})}^{n_{PV}^{fail}}] \end{matrix}$ (17) where, A_WG and A_PV are availabilities of each WG and PV array. Since, other components do not have moving parts and are installed indoors, their outage probabilities, in comparison with WGs and PV arrays, are much lower and therefore negligible. But, as simulation results will confirm the extreme influence of this component over system’s reliability, DC/AC converter plays a vital role in supplying the demand, and its failure, under any conditions, will result in loss of entire load as the only single cut-set of the system reliability diagram [29] and so, the outage probability of inverter shall be taken into reliability evaluations, although its FOR is too small [2 , 42].

Given failure rate of an inverter is equal to 2.5×10^–5 [41], since Mean Time To Failure (MTTF) of each equipment is reciprocal of its failure rate [29], MTTF of the inverter is equal to 37,037 h, which is about 4.23 years (Also, [2] assumes 5 years for Mean Time Between Failure (MTBF) of DC/AC converters). On the other hand, as it is given in [42], the Mean Time To Repair (MTTR) of each inverter is about 40 h. Steady-state reliability, or in a better word availability, of DC/AC converter can be calculated as [29]: $A_{inv} = \frac{{MTTF}_{inv}}{{MTTF}_{inv} + {MTTR}_{inv}}$ (18)

Regarding above equation, the inverter is available with a probability of 99.89%, which is indeed much more reliable than WGs and PV arrays. Finally, probability of failure of $n_{WG}^{fail} WGs$ , $n_{PV}^{fail}$ PV arrays, and $n_{inv}^{fail}$ inverters out of, respectively, N_WG, N_PV, and N_inv (here, N_inv = 1) installed components is calculated by the following equation.

$\begin{matrix} f_{system} (n_{WG}^{fail}, n_{PV}^{fail}, n_{inv}^{fail}) = f_{ren} (n_{WG}^{fail}, n_{PV}^{fail}) \\ \times [(\begin{matrix} N_{inv} \\ n_{inv}^{fail} \end{matrix}) \times A_{inv}^{N_{inv} - n_{inv}^{fail}} \times {(1 - A_{inv})}^{n_{inv}^{fail}}] \end{matrix}$ (19)

Where f_ren ( $n_{WG}^{fail},$ $n_{PV}^{fail}$ ) is the probability of failure of $n_{WG}^{fail}$ WGs and $n_{PV}^{fail}$ PV arrays calculated in Equation (17). Consideration of failures of other components as well as uncertainties in solar radiation, wind speed and load profile, in reliability assessment of the system results in numerous number of system states which leads to tedious and time-consuming computations. Accurate reliability assessment of such a system requires numerical methods like Monte-Carlo simulations [38 , 44]. In [44] a WG/PV/battery system is simulated using Monte Carlo method. Results show that this method converges to final values after more than 50 years simulation. Consequently, simulation models based on Monte Carlo-type algorithms are usually very time consuming, complex and unsuitable for a rough sizing. They, require detailed information that is often not available [44].

Figure 3 shows the availability (steady-state reliability) of a battery equipped with PEMFC over a 10-year operation period [13]. It is observed that during its useful lifetime, i.e. first 5 years, FC is almost available with a probability of more than 99%. Unfortunately, there is no information about reliabilities of electrolyzer and hydrogen tank. By the way, failure probabilities of FC, electrolyzer, and hydrogen tank are neglected. Unlike DC/AC converter, these parts, do not involve in any single cut-set and, therefore, do not directly cause loss of load. Also, they are much more reliable than WGs and PV arrays.

3.3 Cost of loss of load

Cost of electricity interruptions has been estimated in different ways. For example, looking at the costumer’s willingness to pay for expansion or at production losses at industries affected, or at the level of compensations, which makes shortages acceptable. The values found are similar in all cases: in the range of 5–40 US$/kWh for industrial users and 2–12 US$/kWh for domestic users [33]. Here, cost of customer’s dissatisfaction, caused by loss of load, as it used in [33], is equal to 5.6 US$/kWh.

3.4 Approximate method

For reliability assessments, all possible states of the system must be enumerated. Generally, Artificial Intelligence (AI) based algorithms are evolutionary or population-based and demand a number of simulations. For instance, in this study a swarm of 200 particles searches the solution spaces through 2000 iterations. Thus, system must be simulated for 200 × 2000 × 8760 = 3,504,000,000 h. Now, given system is composed of 199 PV arrays and 14 WGs, each time step must be enumerated to 2 × 15 × 200 = 6000 states and so any effort to reduce the complexity and computation time is well worth. Failure Probability Distribution Function (PDF) and equal-probability curves of 14 WGs (FOR = 4%) and 199 PV arrays (FOR = 4%) are shown in Fig. 4 (It is observed that this function is nonzero only in a small area.) In approximate method, all the possible states for outages of WGs and PV arrays to be modeled with an equivalent state. Probability of encountering the equivalent state is 100% and the generated power by renewable sources in this state is equal to the expected value of generated power in all the possible states, i.e. $P_{ren}^{eq} = E [P_{ren}] = \sum_{s \in S} P_{ren} (s) \times f_{ren} (s)$ (20)

Substituting Equations (4 –17) in above equations yields: $P_{ren}^{eq} = \sum_{n_{WG}^{fail} = 0}^{N_{WG}} \sum_{n_{PV}^{fail} = 0}^{N_{PV}} {\begin{matrix} P_{ren} (n_{WG}^{fail}, n_{PV}^{fail}) \\ \times f_{ren} (n_{WG}^{fail}, n_{PV}^{fail}) \end{matrix}}$ (21)

Ultimately, it can be proved that: $P_{ren}^{eq} = N_{WG} \times P_{WG} \times A_{WG} + N_{PV} \times P_{PV} \times A_{PV}$ (22)

However, Equation (22) could be logically anticipated. Results, presented in the following section, will validate the acceptability of the proposed approximate method. By using this method, number of all possible states for outages of 14 WGs and 199 PV arrays at each time step, reduces from 6000 to only 2 states.

4 Problem statement

Goal of this paper is reliability/cost based optimal design of a hydrogen-based hybrid WG/PV system. Optimization variables are number of wind turbine generators, number and installation angle of photovoltaic arrays, and capacities of electrolyzer, hydrogen tank, fuel cell, and DC/AC converter. System costs consist of annualized cost of investment, replacement, and operation and maintenance of components during 20 years of operation. Reliability indices used as objective functions are LOEE and LOLE. The problem is subjected to maximum allowable ELF reliability index. Besides, system simulation is subjected to some other constraints, like components’ maximum and minimum power and energy. The Annualized Cost (AC) of component i is defined as in the following [3, 18]: ${AC}_{i} = N_{i} \times {\begin{matrix} [{CC}_{i} + {RC}_{i} \times K_{i} (ir, L_{i}, y_{i})] \\ \times CRF (ir, R) + O & {MC}_{i} \end{matrix}}$ (23) where, N may be number (unit) or capacity (kW or kg), CC is capital cost (US$/unit), RC is cost of each replacement (US$/unit), and O & MC is annual operation and maintenance cost (US$/unit-yr) of the component, and R is the useful lifetime of the project (here, 20 years). ir is the real interest rate (here, 6%) which is a function of nominal interest rate (ir_nominal) and annual inflation rate (fr), defined by [16]: $ir = \frac{{ir}_{nominal} - fr}{1 + fr}$ (24)

Also, CRF and K are capital recovery factor [3] and single payment present worth [16], respectively: $CRF (ir, R) = \frac{ir \times {(1 + ir)}^{R}}{{(1 + ir)}^{R} - 1}$ (25) $K_{i} (i r, L_{i}, y_{i}) = \sum_{n = 1}^{y_{i}} \frac{1}{{(1 + i r)}^{n \times L_{i}}}$ (26) where, L and y are useful lifetime and number of replacements of the component during useful lifetime of the project, respectively [2]. Using Equation (13), annual cost of loss of load is calculated by: ${AC}_{loss} = LOEE \times C_{loss}$ (27) where, C_loss is cost of customer’s dissatisfaction (US$/kWh) described in Section III. Finally, the objective functions are defined as: $J_{Cost} = min_{X} {\sum_{i} {AC}_{i}}$ (28) $J_{Re liability 1} = min_{X} {LOEE}$ (29) $J_{Re liability 2} = min_{X} {LOLE}$ (30)

(31)

i is the component indicator, and X is a seven dimensional vector consisting of optimization variables. The optimization problem is subjected to: $E [ELF] \leq {ELF}_{max}$ (32) $N_{i} \geq 0$ (33) $0 \leq θ_{PV} \leq \frac{π}{2}$ (34) $E_{tan k} (0) \leq E_{tan k} (8760)$ (35) where, θ_PV is array’s installation angle, and constraint (34) ensures that the amount of stored energy in the hydrogen tank at the end of first year will be more than its initial amount (worst condition).

5 Multi Objective Particle swarm optimization (MOPSO)

5.1 Multi-objective optimization

A general minimization problem of M objectives can be stated as: given x = [x₁, x₂, …, x_d] ^T, where d is the dimension of the decision variable space [45, 46]: $Minimize : f (x) = [f_{1} (x), f_{2} (x), \dots, f_{M} (x)]^{T}$ (36) $Subject to : {\begin{matrix} g_{j} (x) \leq 0; i = 1, 2, \dots J \\ h_{k} (x) = 0; i = 1, 2, \dots K \end{matrix}$ (37) where, f_i (x) is the ith objective function, g_j (x) is the jth inequality constraint and h_k (x) is the kth equality constraint [45 , 48]. The constraints given by (33) define feasible region Ω and any point x in Ω defines a feasible solution. The vector f (x) is a function, which maps the set Ω into the set Λ and represents all possible values of the objective functions. The notion of an optimum solution in MOO is different than SOO, the concept of Pareto dominance formulated by Vilfredo Pareto is used as [45 , 48]: A vector u = [u₁, u₂, …, u_M] ^T is said to dominate a vector v = [v₁, v₂, …, v_M] ^T(denoted by $u \underline{≺} v$ ), for a MOO problem, if:

$\begin{matrix} \forall i \in {1, \dots, M}, u_{i} \leq v_{i} \\ \land \exists i \in {1, \dots, M} : u_{i} < v_{i} \end{matrix}$ (38) where, M is the dimension of the objective space. A solution u ∈ U, where U is the universe, is said to be Pareto Optimal if and only if, there exists no other solution v ∈ U, that u is dominated by v. Such solutions (u) are called non-dominated solutions. The set of all non-dominated solutions constitutes the Pareto set or non-dominated set [45 , 48].

5.2 Particle swarm optimization

The PSO is a population-based algorithm that exploits a population of individuals to probe promising region of the search space [45 , 50–53]. Suppose that the search space is n-dimensional, then ith particle is a n-dimensional vector, X_i = [x_i1, x_i2, …, x_in] ^T and velocity V_i = [v_i1, v_i2, …, v_in] ^T, where i = 1, 2, …, N and N is size of population. In PSO, the particle i remembers the best position it visited so far (P_best), referred to as P_i = [p_i1, p_i2, …, p_in] ^T, and best position of best particle in the swarm (G_best) is referred as G_i = [g₁, g₂, …, g_n] ^T. Each particle i adjusts its position in the next iteration t + 1, with respect to following equations [45 , 53]: $\begin{matrix} V_{i} (t + 1) & = & ω (t) \times V_{i} (t) + c_{1} \times r_{1} \\ \times (P_{i} (t) - X_{i} (t)) + c_{2} \times r_{2} \\ \times (G_{i} (t) - X_{i} (t)) \end{matrix}$ (39) $X_{i} (t + 1) = X_{i} (t) + χ \times V_{i} (t + 1)$ (40) where ω (t) is inertia coefficient. The factor χ, is used to limit velocity (here, χ = 0.7). c₁, and c₂ denote the cognitive and social parameters and r₁ and r₂ are random real numbers drawn from interval [0, 1] [45, 50]. More details have been provided in [45].

5.3 Multi-objective particle swarm optimization

To make the PSO algorithm capable of dealing with MOO problems, some modifications become necessary because PSO is an inherent SOO algorithm. In [46], the personal best performance (P_best) of each individual particle is replaced with the new solution if and only if it dominates the former P_best. Also, two major issues should be considered in the updating process of the global best performance (G_best). First, the fitness assignment and selection should be addressed such that a search can move towards the Pareto optimal set. Second, the diversity of swarm should be maintained to prevent premature convergence and obtain an evenly distributed Pareto optimal front [45, 46]. Here, the authors employ an archiving mechanism proposed in [45] to form a repository, which may contain only a limited number of solutions. The density parameter (den_i) of the solution i defined as its distance to the nearest neighbor in the archive is introduced as a measure of the diversity of archive. However, it may happen that two solutions are the closest to each other. For instance, for a two-objective (f₁ and f₂) minimization problem illustrated in Fig. 5, solutions e and f are closest to each other and therefore, will get the same value for the distance to their nearest neighbor. To prevent such a problem, the calculation for the distance of the nearest neighbor is done in an order such that, the distance of a solution to its nearest neighbor which has not already been considered, is taken. For example, if the nearest neighbor distance is computed in the order a, b, c, d, e, f, g and h then, for the nearest neighbor of solution f, solution g should be considered since, its distance to solution e has already been considered [45, 46].

5.4 Best tradeoff solution

Because none of Pareto optimal set solutions is the absolute global optimal, the designer should choose the most proper solution according. To handle this dilemma fuzzy sets are proposed [45]. In this approach, a linear membership function is defined for each objective function (J_i) as follows: $u_{i} = \frac{J_{i} - J_{i}^{min}}{J_{i}^{max} - J_{i}^{min}}$ (41)

It is evident that the lower values of the membership function indicate more degrees of achievement of the objective function. For every non-dominated solution (k), the aggregate membership function can be defined as follows: $U^{k} = \sum_{i = 1}^{M} u_{i}^{k}$ (42)

The solution with the minimum membership, U^k can be considered as the best compromise solution (Best Tradeoff I). Here, the authors propose another criterion to choose the best compromised solution. First, the distance of each non-dominated solution (k) from the origin in the fuzzified coordinate, is calculated by the following equation: $V^{k} = {[\sum_{i = 1}^{M} {(u_{i}^{k})}^{2}]}^{1 / 2}$ (43)

Then, the nearest solution (V^k) to the fuzzified origin is chosen as the best compromised solution (Best Tradeoff II).

6 Simulation results

Software is developed in MATLAB programming environment. To perform the reliability/cost assessments, it is necessary to simulate the systems through a year with 1-h time steps. The available data consist of hourly averages of wind speed, recorded at a height of 40 m, and vertical and horizontal solar radiation in one of the northeastern provinces of Iran, i.e. Ardebil (latitude: 38°17’, longitude: 48°15’, altitude: 1345 m). The load data are for IEEE Reliability Test System (IEEE RTS) [32], with a peak of 50 kW. Using Equation (3), wind speed data at 15 m (WGs’ installation height) are calculated. These data are shown in Figs. 6, 7. Origin of horizontal axis is first hour of the first day of solar year, i.e. March 21. Specifications of components of the hybrid system are summarized in Table 1 [2 , 42]. It is worth to mention that, in the following simulations it is assumed that, initial amount of the hydrogen stored in the tank, i.e. E_tank (t − 1) for t = 1 in Equation (6), is equal to half of its rated capacity. Also, for other time step (t > 1), initial amount of stored hydrogen is the expected value of stored hydrogen at the end of previous time step. Also, capital cost of hydrogen tank includes the cost of initial amount of hydrogen that is US$1.8/kg [36].

6.1 Base case

Software is run for the base case on a Pentium IV, 3.2 GHz CPU and 2 GB of RAM. For saving the computation time, states with negligible probabilities (here, 10^–16) are not enumerated. For instance, by applying this approach to a system consisting of 14 WGs and 199 PV arrays with availabilities of 96%, number of enumerated states reduces from 15 × 200 = 6000 to 252 and, as a consequence, about half of the computation time is saved. It should be noted that, this approach is not effective for small systems, because the number of low probable states is not considerable. For example in case of 3WGs and 5 PV arrays, there is no state less probable than 10^–16. By the way, it takes about 36 h that the software finds the optimal combination. Simulation Parameters have been presented in Table 2. Obtained results, are shown in Tables 3 and 4 for three systems i.e. hybrid WG/PV, only WG and only PV when WGs and PV arrays are 96% available. It is observed in Table 3 that reliability inequality constraint, i.e. Equation (31), is not activated in the base case for none of hybrid, only WG and only PV systems and because of high loss of load costs (i.e. cost of customer’s dissatisfaction), designing a reliable and expensive microgrid is economically reasonable. Pareto optimal set and best tradeoff solutions for optimal sizing of hybrid WG/PV, only WG and only PV systems have been depicted in Figs. 8 –10. It can be seen in Figs. 8 –10 that for the hybrid WG/PV and only PV systems, the Pareto frontier is evenly distributed along cost, LOLE and LOEE axes and the tradeoff criteria results into different solutions. Since reliable supply of the load at each time step, strongly depends on the amount of the stored energy, the hourly expected amounts of stored energy in the hydrogen tank, during the year, have been shown in Fig. 11 for hybrid, only WG and only PV systems.

6.2 100% available components

Literature on optimal sizing of hybrid WG/PV systems, has not considered outage probabilities of system components and as a matter of fact To understand the impact of components’ outage probabilities on the system’s costs and reliability, problem has been solved assuming 100% availability of all components. As shown in Tables 5 and 6, comparing to the base case, in this case a smaller system can supply the load with lower costs and higher reliability. In fact, as a consequence of failure probabilities of WGs, PV arrays, and DC/AC converter, annual cost of the system increases from 3,134,902 to US$3,183,155/yr, for hybrid WG/PV system, from 7077828 to US$7222327/yr, for only WG system, and from 3703402 to US$3792828/yr, for only PV system, respectively. Additionally, components’ outage probability results in increase about US$64809 for hybrid WG/PV, US$182075 for only WG, and US$89930 for only PV systems, respectively, in investment cost of the system that might be considerable. Besides, it can be observed that failure probabilities cause deterioration in system’s reliability, i.e. ELF index from 0 to 0.00005850 for hybrid WG/PV system, and from 0.00001486 to 0.00005385 for only PV system, respectively. Also, since in this case all the components are assumed 100% available, both accurate and approximate methods provide same results. In fact, at each time step, system can work only in one feasible state.

6.3 Impact of DC/AC convertor on the system’s reliability

To understand the impact of inverter on the system’s reliability, in this section the optimization problem has been subject to reliability constraint of ELF <0.0001, i.e. reliability constraint in developed countries [33]. The program has been run and actually, this huge fitness value is resulted from the heavy penalty term of fitness function assigned to reliability inequality constraint. Taking a closer look at the result reveals that the ELF of the system can fall even to zero. This fact, directly results from the vital role of the inverter, as the only single cut-set, in reliability diagram of the hybrid system. Given P_supply, A_inv, and A_others are, respectively, probability of supplying the load, availability of the inverter, and availability of all other components as well as energy resources, the upper limit of the probability of supplying the load is: $\begin{matrix} P_{sup ply} = A_{others} \times A_{inv} \end{matrix} \Rightarrow \begin{matrix} Lim (P_{sup ply}) = A_{inv} \\ A_{other \to 1} \end{matrix}$ (44)

Usually, there are two remedies for improving the reliability limit of such a hybrid system. The first, using a more reliable (more available) inverter, and the second, using two or more inverters in parallel. For example, in case of using a 100% available inverter, the software yields Tables 6 and 7.

6.4 Comparison of the accurate method with the approximate method

To compare accurate method with the approximate method, the approximate method has been applied to the problem. In this case, optimization process has been terminated in about 7 h which is considerably less than 20 h. Results are presented in Table 8. Since, the deference between approximate and accurate method is only in reliability calculations, both of them result in same investment, replacement, and operation and maintenance costs. Also, Table 8 indicates that the approximate reliability indices are acceptably close to actual indices provided by the accurate method. It is observed that, the magnitudes of the percentage errors are almost below 10%. Additionally, negative percentage errors are evidences that the approximate method estimates reliability of the base case system somehow optimistically.

6.5 Comparison of proposed MOPSO with SOO algorithm

To show the efficiency of proposed algorithm, the multi-objective particle swarm optimization presented in this paper has been compared with the conventional single objective particle swarm optimization algorithm proposed in [28]. The results have been presented in Tables 9 and 10. It can be seen that the proposed MOPSO algorithm offers better combination with lower reliability indices.

7 Conclusion

The main goal to propose optimal power management and sizing design of hybrid wind–solar generating microgrid systems is a reliable supply of the load under varying weather conditions, with minimum cost. In this paper, a hybrid WG/PV/FC microgrid system with hydrogen ESS is designed for a 20-year period of operation. Optimal combination of components is achieved by a novel MOPSO algorithm, which acceptably converges to the optimum combination. The MOO problem involves cost and reliability indices as objective functions and also subjected to reliability constraint. Results indicate that costs of the system, directly, depend on its components’ reliabilities and the outage probabilities of three major components of the system, i.e. WG, PV and DC/AC converter are taken into account. Comprehensive reliability/cost assessment of such a system, considering failures of other components as well as uncertainty in wind speed, solar radiation, and load data, demands computationally intensive and time consuming algorithms like Monte Carlo simulations which is beyond the aspects of this paper and may be subject of future studies. Versatile software, developed in MATLAB programming environment, carries out all these huge computations, including yearly simulation of the microgrid system with 1-h time steps, accurate and approximate evaluations of reliability indices, and MOO algorithm. The software is capable of integrating any component model and therefore, quite flexible to be implemented to any application. It is just needed to input the wind speed, solar radiation, and load demand data, as well as specifications of the microgrid components and then, running the software. Results of a case study, based on the empirical data, testify that component outages can extremely impact system’s reliability and economy. Moreover, this study reveals the significant impact of DC/AC converter, as the only single cut-set in the reliability diagram of the hybrid system. It is observed that the inverter’s reliability is an upper limit for the system’s reliability. Also, an approximate method for reliability evaluation of the hybrid microgrid system considerably reduces the time and computations. Since the accurate method is time-consuming, the approximate method may be useful in rough calculations like design/sensitivity analysis.

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