With the ever increasing concerns on environmental issues and the deple­tion of fossil fuels, the photovoltaic (PV) technology has drawn great at­tention and remarkable investments in the past decade [1]. This is due to the fact that the PV technology shows many advantages over other renew­able energy technologies in terms of modularity, expandability, mainte­nance and reliability. In recent years, the contribution of the PV power generation to the grid has been rapidly increasing; at the current growth rate, it is expected to reach 2% of the world electricity generation by 2020 and up to 5% by 2030 [2, 3]. During the next ten years, up to 15% of elec­tricity in European Union will be produced by solar energy resources [4].

Economical Evaluation of Large-Scale Photovoltaic Systems Using Universal Generating Function Techniques. © Ding Y, Shen W, Levitin G, Wang P, Goel L and Wu Q. Journal of Modern Power Sys­tems and Clean Energy 1,2 (2013); DOI 10.1007/s40565-013-0017-z. Licensed under the Creative Commons Attribution 2.0 Generic License, http://creativecommons. org/licenses/by/2.0/.

Подпись: 21 8 Solar Energy: Application, Economics, and Public Perceptio

Grid connection Grid connection


String 1 String n


Grid connection


String 1 String 2 String 3 String 4

String n


String 1 String 2


(a) Centralized inverter system (b) String inverter system (c) Combination of centralized and string inverters


FIGURE 1: Configurations ofPV system


An important question for investors, planners and regulators is the re­turn and cost of a PV project. The cost structure of PV systems is differ­ent from that of conventional generation system using fossil fuels such as coal, oil or natural gas. The initial capital cost is higher: basic components of a PV system—solar panels are quite expensive. However prices of solar panels are dropping fast: the average one-off installation cost of solar pan­els has already dropped from more than $2 per unit of generating capacity in 2009 to about $1.50 in 2011 [5]. On the other hand, there are no fuel cost and greenhouse gas emissions during the lifespan operation of 20-30 years. The maintenance cost of PV system is also relatively low.

PV systems are complicated engineering systems. A PV system is mainly composed of many solar panels and DC/AC inverters. The trend of the fast growing PV systems is to adopt large-scale PV systems, which may require tens or hundreds of solar panels. Depending on input voltage ranges, maximum input currents and capacities of inverters, several solar panels are connected in series to form a string and a few strings are paral­leled and tied up to a centralized-inverter or each of the strings is directly interfaced by a separate string-inverter or a combination of both, which are illustrated in Fig. 1. Different configurations have their own performance efficiencies for electricity production. When the performances of those configurations of PV systems are evaluated, it is assumed that the systems work without interruption. Although PV systems are relatively reliable, they may fail occasionally. Ignoring the effects of those failures may result in an optimistic estimation of energy production, which also decreases ac­curacy of cost assessment.

The approaches for improving the engineering system reliabilities are to increase the redundancy or/and reliability of the components in the sys­tem. For example, the use of multiple inverters in PV systems can increase system reliabilities. These approaches can improve the reliability of the PV systems and hence its energy production, but they may result in higher system cost. The reliability based cost assessment for renewable energy systems (RESs) and restructured generation systems has been studied in some recent research. Reference [6] provided a comprehensive analysis of the reliability and its cost implications on various choices of installation sites and operating policies as well as energy types, sizes and mixes in ca­pacity expansion of the RESs. The genetic algorithm was used to optimize the offshore wind farms considering both energy production cost and sys­tem reliability [7]. A framework for analyzing the adequacy uncertainties of distributed generation systems was proposed in [8]. However reliability based economical evaluation of large-scale PV systems has not been com­prehensively studied, which may be a useful analytical tool for assisting stakeholders in making optimal decision.

The large-scale PV system can be modeled as a typical multi state sys­tem (MSS). The UGF technique provides a systematic method for the per­formance and reliability assessment of MSS, which can replace extremely complicated combinational algorithms and reduce the computational bur­den [9-11]. Moreover, the UGF technique provides a flexible approach for representing reliability models of various energy systems. The UGF technique and genetic algorithm were used to determine the optimal struc­ture of power systems subject to reliability constraints [12]. The reliability of flow transmission system was analyzed by using the combination of the UGF technique and extended block diagram methods [13]. The re­dundancy analysis of inter-connected generating systems was discussed in [14]. In [15], the UGF technique was used to determine the reserve expan­sion for maintaining the reliability level of power systems with high wind power penetration.

In this paper, the UGFs representing probabilistic performance distri­butions of solar panel arrays, PV inverters and energy production units (EPUs) are developed. The expected energy production models for PV systems under different configurations are also developed. The life cycle cost and annualized life cycle cost are evaluated to conduct economical assessment of a PV project. Moreover, a new economical index for PV systems—expected unit cost of electricity (EUCE) is developed for pro­viding useful information.

Section 2 presents reliability models of large-scale PV systems. The developed UGFs are used to evaluate expected energy production. Cost analysis of PV systems is conducted in Section 3. Section 4 proposes a methodology to identify the feasible configurations of PV systems and determine the optimal one at the minimum EUCE. The proposed methods are used to assess the PV system configurations for providing electricity for a water purification process in Section 5. The conclusions are summa­rized in Section 6.

Подпись: String 1 image121
Подпись: Stnng
Подпись: DC/AC
Подпись: DC AC
Подпись: Stnng n'm
Подпись: String n
Подпись: String 1*m

FIGURE 2: Generalized configuration of PV system


A large-scale PV system basically consists of two major parts: solar panels and DC/AC inverters. Figure 2 shows a generalized configuration of the PV system. In the following, the reliability models of solar panel arrays, PV inverters and EPUs, and expected energy production calculation are discussed.


image129Solar panels are the key components of the PV systems. Solar panels can fail due to the degradation of mechanical properties of encapsulants, the

adhesional strength, the presence of impurities, metalization, solder bond integrity and breakage, corrosion, and aging of backing layers, etc.

Given the failure rate T and repair rate pi of the solar panel i, the cor­responding availability Ai can be calculated by

Подпись:A = h, / О, +

where T is the failure rate referring to the rate of departure from a com­ponent up-state (successful state) to its down-state (failure state) and ц, is the repair rate referring to the rate of departure from the down-state to the up-state.

Some strings consist of several solar panels and a blocking diode in series. Any failure of a solar panel or a diode leads to the total failure of the string. Therefore, the availability of the string s can be evaluated by:

Подпись: (2)■ Ad


where n is a number of solar panels in the string and Ad is the availability of the diode:

Ad = pd / (Ы + pd) (3)

where and pd are the failure rate and the repair rate of the diode, respec­


The available capacity W s of the string s can be calculated as:

Ws =£ Wt – Wd


image131 Подпись: (5)

where Wi is the available capacity of the solar panel i; Wd is the capacity loss caused by the blocking diode, which can be determined as

where Ud is the voltage drop across the blocking diode and Is is the current of the string.

The UGF technique is proved to be very convenient for numerical re­alization and requires small computational resources [9-11] for perfor­mance and reliability evaluation of engineering systems [9]. Therefore, the UGF technique is used to evaluate the expected energy production of the PV system. The UGF representing the capacity distribution of a string s can be defined as a polynomial:


Подпись: (6)US(Z) = £ ps, ks ■ Zwsxs


where ps, ks and ws, ks are the probability and the capacity level of state ks for the string s, Us(Z) represents the capacity distribution of the string s, Z represents the Z-transform of any discrete random variables that has the probability mass function taking the form shown in (6) [10].

The string s has two states: failure state and successful state. For the failure state, the capacity level and unavailability are 0 and (1-As), respec­tively. For the working state, the capacity level and availability are Ws and As, respectively.

A few strings are also arranged in parallel to form solar array and con­nected to a string inverter. Failure of any string in the array is tolerated without the loss of an entire array. However, the failure of a string de­grades the available capacity of the array, leading to several de-rated states of the array. As a result, the solar array in the PV system can be regarded as a MSS. The parallel operator fi is applied for the parallel MSS by us­ing associative and commutative properties. The parallel operator is a kind of composition operator to calculate the UGF for the parallel MSS, which
strictly depends on the properties of the parallel structure function [9, 10]. For example, if elements are connected in parallel, its capacity level for the state ks is the sum of the corresponding capacities wsks (s = 1, 2,…, N)

Подпись: of its elements, and the structure function for such a subsystem takes the form:


The capacities of elements unambiguously determine the capacity of the subsystem or system. The transform, which maps the space of the ele­ment capacities into the space of the system capacity, is the system struc­ture function [9, 10].


For a solar array with N strings in parallel, its UGF can be obtained based on the UGFs for the arrays using the parallel composition operator over UGF representations of N strings:


where pa, ka and wa, ka are the probability and the available capacity of the array in the state ka, respectively. Equation (8) represents the capacity dis­tribution of the solar array [8]: the coefficients of the terms in the polyno-


FIGURE 3: Structure of inverter

mial (8) represent the probabilities of the array states while the exponents represent the corresponding capacities. The array has 2 N states.