Intelligent optimization of renewable resource mixes incorporating the effect of fuel risk, fuel cost and CO₂ emission

Abstract

Power system planning is a capital intensive investment-decision problem. The majority of the conventional planning conducted since the last half a century has been based on the least cost approach, keeping in view the optimization of cost and reliability of power supply. Recently, renewable energy sources have found a niche in power system planning owing to concerns arising from fast depletion of fossil fuels, fuel price volatility as well as global climatic changes. Thus, power system planning is under-going a paradigm shift to incorporate such recent technologies. This paper assesses the impact of renewable sources using the portfolio theory to incorporate the effects of fuel price volatility as well as CO₂ emissions. An optimization framework using a robust multi-objective evolutionary algorithm, namely NSGA-II, is developed to obtain Pareto optimal solutions. The performance of the proposed approach is assessed and illustrated using the Indian power system considering real-time design practices. The case study for Indian power system validates the efficacy of the proposed methodology as developing countries are also increasing the investment in green energy to increase awareness about clean energy technologies.

Graphical abstract

Keywords

modern portfolio theory / energy policy / CO₂ emissions / multi-objective optimization / planning commission

1 Introduction

Electricity is one of the most essential infrastructure inputs for economic development of a country and therefore the growth of power sector is implicitly related to the growth of a nation. Since independence, the Indian power sector has grown many folds in size and capacity. For a better appreciation of the paradigm shift for portfolio-based planning, an overview of power system planning is depicted in Fig. 1. Figure 1 shows the hierarchy of conventional power system planning. The process of selecting the type and location of new generation and transmission equipments and scheduling their installation dates so that a good secure electricity supply is provided at an economic cost is termed as expansion planning. Additional processes include incorporation of load forecast in terms of annual peaks and energy needs for the entire utility area as well as for each region consisting of many utilities. The expansion planning can be further segregated as generation planning and network planning. Generation planning aims at bringing about most economical generation expansion schedule as a combination of required system to level reliability as ordained by the forecast of demand increase in a certain interval of time. This planning mode decides the place and time to invest for the new generating units, the type of units as well as the capacity of units to be installed. Network planning aims at bringing about optimal power system network configurations, keeping the rise in load demand and generation planning for a certain planning horizon so as to meet the necessity of delivering electrical energy cost-effectively in a valuable and safe mode [1–11].

Operation planning aims at bringing out optimal operation of the power system in a secure way along with a specified quality reflected in terms of voltage and frequency within allowable limits at economic cost using the available resources. Accordingly, different horizon planning comprises of long-term and short-term operational planning. One of the most critical steps in planning for the expansion of a modern electric utility is the generation system planning. A sound plan for generation system expansion is crucial to the success of any electric utility. In broad terms, a suitable generation expansion plan must provide the electric utility with the capability of meeting customer need for reliable, quality electric energy sources at a reasonable price. Planning commission is an institutional body of the Government of India which formulates a five year plan periodically. In 1951, the first five-year plan was commissioned. Till 1965, the two succeeding five-year plans were commissioned. In 1969, the forth five-year plan was started after two successive years of dearth. A general rise in prices and attrition of resources disrupted the planning process after three annual plans between the years 1966 to 1969. The Eighth Plan was not started in 1990 due to the political state of affairs at the capital. The years 1990–1991 and 1991–1992 were considered as yearly plans. In 1992 the Eighth Plan was commissioned. Commencement of Ninth, Tenth and Eleventh Plan were initiated on 1997, 2002 and 2007, respectively. These policies sought to address issues like agriculture, rural development, industry, services, physical infrastructure (transport, energy, urban infrastructure, communication and information technology) and so on [12–20].

In today’s scenario, development of new tools is required for management of risk associated with price volatility of different technologies due to the uncertain climatic conditions. Thus, the portfolio optimization for electricity generation planning is crucial to obtain optimal generation mix of different technologies. In this paper, a methodology based on a robust multi-objective optimization, namely NSGA-II, is developed for India’s energy portfolios. Due to the fast penetration of emerging renewable energy portfolio standards in today’s scenario, optimal planning of electricity generation mix is particularly more important. In spite of this tendency, according to the estimation of intergovernmental panel on climate change, the energy generation mix in the time-frame of 2025–2030 will essentially remain unchanged, to run the global economy with nearly 80%–85% or more of its energy delivery mostly derived from fossil fuels. For India, during the Twelfth Plan (2012–2017), among other fossil fuels, coal will chiefly stay to be the foundation of energy production mix, which provides a minimum base load power of 50%, though renewable energy share is rising progressively as authorized by the Electricity Act, 2003 [21–26].

In today’s scenario, renewable energy sources contribute approximately 14% of the total global energy consumption, which mostly involves untenable utilization of wood or hydro power with only approximately 2% of renewable and 6% of nuclear resources. Distributed generations (DGs) employing renewable energy sources has numerous advantages. Basic advantages of using renewable based DGs are limiting greenhouse gas emissions, avoiding new transmission circuits, inclusion of DGs to reduce the risk in electricity markets, improving the power quality, enhancing the reliability, and providing energy security. India’s target is, to increase the share of the renewable energy generation to more than double in this decade, which can accounts for almost 25% of the approximate entire power expenditure by the year 2020–2021 [23–26].

The conventional power system generation planning expands the power generation system in such a way that the overall cost of power generation is optimized. But this conventional approach does not account for the risk factor that is associated with the existing generating technologies. The portfolio based power system generation planning selects the optimal or best portfolio of generating resources which includes renewable energy technologies that have the potential to often reduced cost and risk of power generation. Portfolio of generating assets may be a single generating resource or a set of different generating resources. The portfolio based power system planning focuses its attention on the selection of optimal portfolios for generating assets that are expected to generate power at minimum cost and simultaneously reduces the risk associated with generation [2–11].

Financiers mainly employ the portfolio theory to reduce risk and exploit portfolio performance with a mixture of volatile financial conclusions. On the contrary, the conventional power system generation planning only focuses on finding the least cost alternative. However, in today’s scenario, the power generation planners are faced with a range of resource options and a dynamic, complex and uncertain future. Within this uncertain environment, it is very difficult to attain the selection of only least cost alternative. Therefore, to obtain an optimal strategy, suitable financial techniques are required, which provides a solution that will be economical under a variety of system constraints [12,13].

Thus in this complex environment, electricity generation portfolios alternative is more viable for electricity production plan than its existing focus on assessing substitute technologies. A competent portfolio attempts to avoid any unnecessary risk that may be involved in its expected return. In brief, competent portfolios that consist of different alternatives can be defined as “for any given level of risk it maximize the expected return, while for every level of expected return it minimizes the risk” [5–10].

The portfolio theory is mostly used for planning strategy and evaluation of efficient optimal portfolios. Similarly, this concept is applied to energy strategy which is no different from endowing in economic securities. For a particular technology, it is imperative not to consider electrical energy production merely with respect to its cost only, rather with respect to its expected portfolio cost. The price of different technologies varies from time to time. Thus, some options in the portfolio enjoy elevated costs, whereas others suffer lower costs. Thus there is a need for smart combination of different alternatives in the portfolio such that the system possesses lower energy generation cost with respect to their risk levels.

In short, to apply the modern portfolio theory for electricity generation planning, contemporary and renewable technologies are assessed based on expected average levelized generation cost, and the expected risk associated with the portfolio [4]. To increase the energy security in the future, fossil fuel technologies such as coal, gas etc., are replaced by renewable technologies. It is so owing to the fact that renewable energy costs generally have no bearing to the variation of fossil fuel prices. This permits the renewable technologies to vary generating-mix and improve its cost-risk performance while concurrently reducing greenhouse emissions.

The Indian power system planning has several drawbacks. Fossil dominated generating resources give very high future risk. Thus, there is a need for power system planning which requires mix of different generating technologies, that will reduce the overall risk associated with the portfolio. Current Indian power system generation has no such plans which would reduce the risk factor of generating technologies. In this context, a new approach, named as portfolio based power system generation planning, is proposed for power system generation planning. California energy commission [19] have estimated levelized cost including the cost associated with the carbon emission, along with the portfolio risk to assess the impact of different generation technologies on climate change.

Several optimization algorithms have been developed that can be applied for portfolio-based planning [2,3,13,15–18]. In this paper, a portfolio-based approach for power system generation planning for the Indian power system has been proposed which reduces the overall risk, cost and emission of power generation by an intelligent choice. Thus, a generalized multi-objective genetic algorithm (NSGA-II) is developed to optimize India’s electricity generation portfolio 2012 energy mix. These portfolios include renewable energy sources, such as wind, solar, geothermal etc., that are less risky in comparison to fossil fuel-based generating technologies. An attempt has been made for optimizing the fuel risk, cost and emission including renewable energy sources using NSGA-II. The portfolio approach for power system generation planning is incorporated in this paper and the results obtained show the efficacy of proposed methodology.

2 Portfolio theory for electricity generation planning

2.1 Portfolio theory: an overview

Modern portfolio theory (MPT) was first introduced by Harry Markowitz, a Nobel Prize winner in 1952 [12]. It is a widely used financial technique for investors to deal with risk and exploit portfolio performance under multifarious uncertainties and unpredictable economic outcomes. Modern portfolio theory advocates that diversification can diminish risk within a portfolio having multiple considerations.

Portfolio-based planning is required in today’s complex energy environment that contains market risk and ignores unrelated electrical energy production costs. The portfolio theory shows the cost correlations among multiple generation alternatives. Under this uncertain environment, it is very hard to estimate the actual overall cost for modeling electricity generation planning as there is no direct relationship between the expected cost and the risk involved. Thus it leads to sensitivity analysis. Sensitivity analysis does not deal with the correlations among that predicted portfolio costs and market risks. Hence, it cannot be considered as a substitute for the portfolio-based planning approach.

To evaluate national electricity strategies, the mean variance portfolio (MVP) theory is well established and is ideal [12]. The most significant theory associated with MVP is that it always offers portfolio solutions that improve energy assortment and security and, therefore, it is considered to be more robust than random technology-mix options.

Thus MPT demonstrates that, instead of considering individual security, a portfolio with combined securities can achieve lower risk. MPT deals with a trade-off analysis among expected rate of return and risk, with respect to year-to-year variability of technology costs, to obtain efficient portfolios.

2.2 Portfolio effect for an illustrative two technology portfolio

In an uncertain environment, there are various technologies available for a single portfolio. For instance in an energy mix portfolio, a fossil dominated portfolio (coal, gas, diesel etc.) and renewable (a non-fossil fuel) technology portfolio (solar, wind, geothermal, biogas, etc.) exist. As is known, the prices of fossil fuel technologies are generally correlated with each other, thus, the energy mix portfolio using only fossil fuel technologies are normally undiversified and exposed to the fuel price risk. While inclusion of renewable portfolio in the energy mix, along with fossil fuel technologies diversify the energy mix portfolio and reduce the overall risk associated with the price volatility because the costs of renewable technologies are not correlated with the prices of fossil fuel.

To illustrate the concept of portfolio, a simple two technology portfolio (Technology A and B) has been depicted in Fig. 2. Technology A represents the portfolio associated with the renewable energies with more cost and less risk, such as wind energy, solar energy etc. On the other hand, Technology B represents the fossil fuel dominated portfolio which possesses lower cost and higher risk alternatives. The correlation factor among the gross cost of these two technologies is assumed to be zero. This simplification is always assumed since in practice the total investment and operating cost risks of solar energies always exhibit some nonzero correlation with respect to the investment and operating costs of gas-based generation.

The main idea of the proposed two technology portfolio is explained hereafter. Suppose that Technology A represents renewable energy generation which possesses higher generation cost and low risk associated with price volatility. Technology B, which represents fossil fuel generation, possesses lower generation cost but higher risk associated with the price volatility than Technology A. Now, to simplify the portfolio approach, the correlation factor among these two technologies is neglected.

While using the portfolio theory approach, it has been found that when Technology B is clubbed to a portfolio consisting of cent percentage of Technology A, the total risk of the portfolio decreases. This is apparent as Technology B is riskier than Technology A. From Fig. 2, it can be observed that portfolio H (portfolio with least risk), depicting the scheme of modern portfolio analysis that different technologies can be combined to reduce the overall risk associated with the portfolio, rather than just consider them individually can be observed. Thus, it provides the systematic basis to the policy makers to design efficient generating mixes that minimizes the risk, maximizes the security level, and is sustainable.

3 Problem formulation

In a multi-objective framework, the most important objectives considered in the portfolio approach are portfolio cost, portfolio risk and CO₂ emission minimization.

The expected portfolio cost is given by

(1)\mathrm{E}\mathrm{x}\mathrm{p}\mathrm{e}\mathrm{c}\mathrm{t}\mathrm{e}\mathrm{d}\mathrm{\hspace{0.17em}}\mathrm{p}\mathrm{o}\mathrm{r}\mathrm{t}\mathrm{f}\mathrm{o}\mathrm{l}\mathrm{i}\mathrm{o}\mathrm{\hspace{0.17em}}\mathrm{c}\mathrm{o}\mathrm{s}\mathrm{t}\mathrm{=}E\mathrm{\left(}{C}_{\mathrm{p}}\mathrm{\right)}\mathrm{=}{\displaystyle \mathrm{\sum }_{i\mathrm{=}\mathrm{1}}^N{X}_i\mathrm{\times }{C}_i}\mathrm{,}

where X_i is the percentage contribution of technology i in the portfolio mix; C_i, the expected generation cost of technology i; C_p, the overall portfolio cost; and N, the number of different technologies.

The second objective i.e. the overall risk in terms of year-to-year price volatility associated with generating technologies is given by

(2)\mathrm{Portfolio}\mathrm{\hspace{0.17em}}\mathrm{risk}\mathrm{\left(}{\sigma }_{\mathrm{p}}\mathrm{\right)}\mathrm{=}\sqrt{{\displaystyle \mathrm{\sum }_{i\mathrm{=}\mathrm{1}}^N{X}_i^{\mathrm{2}}\mathrm{\times }{\sigma }_i^{\mathrm{2}}\mathrm{+}{\displaystyle \mathrm{\sum }_{i\mathrm{=1}}^N{\displaystyle \mathrm{\sum }_{k\mathrm{=1}}^N\mathrm{2}\mathrm{\times }{X}_i\mathrm{\times }{X}_k\mathrm{\times }{\sigma }_i\mathrm{\times }{\mathit{\sigma }}_k\mathrm{\times }{\mathit{\rho }}_{ik}}}}}\mathrm{,}

(3)\mathrm{Portfolio}\mathrm{\hspace{0.17em}}\mathrm{return}\mathrm{\hspace{0.17em}}\mathrm{variance}\mathrm{\left(}{\sigma }_{\mathrm{p}}^{\mathrm{2}}\mathrm{\right)}\mathrm{=}{\displaystyle \mathrm{\sum }_{i\mathrm{=}\mathrm{1}}^N{X}_i^{\mathrm{2}}\mathrm{\times }{\sigma }_i^{\mathrm{2}}\mathrm{+}{\displaystyle \mathrm{\sum }_{i\mathrm{=1}}^N{\displaystyle \mathrm{\sum }_{k\mathrm{=1}}^N\mathrm{2}\mathrm{\times }\mathrm{}{X}_i\mathrm{\times }\mathrm{}{X}_k\mathrm{}\mathrm{\times }\mathrm{}{\sigma }_i\mathrm{}\mathrm{\times }\mathrm{}{\sigma }_k\mathrm{}\mathrm{\times }\mathrm{}{\sigma }_{ik}}}}\mathrm{,}

where X_k is the percentage contribution of technology k in the mix; σ_i, the standard deviations of the holding period return (HPR) costs of technologies i and k; and ρ_ik, the correlation coefficient between technologies i and k.

The CO₂ emission function can be represented in number of ways depending upon the type of the strategy employed [14]. The total system CO₂ emission can be defined as

(4)E_{\mathrm{p}}(\mathrm{total}{\mathrm{C}\mathrm{O}}_2\mathrm{emissions})\mathrm{=}{O}_{\mathrm{p}}\mathrm{\times }{\displaystyle \mathrm{\sum }_{i\mathrm{=}\mathrm{1}}^N{X}_i\mathrm{\times }{\mathrm{EF}}_i\mathrm{\left(}\mathrm{Mt}\mathrm{\right)}}\mathrm{,}

where O_p is the total electricity generation output of grid (MWh),which equals the product of Installed capacity (MW), Plant capacity (%) and 8760 h/a; and EF_i, the emission factor for specific technology and or fuel type (tCO₂/MWh).

Thus, the multi-objective framework for portfolio-based electricity generation planning can be described as

(5)\mathrm{M}\mathrm{i}\mathrm{n}\mathrm{\hspace{0.17em}}\mathit{f}\mathrm{=}\mathrm{\left[}\mathrm{p}\mathrm{o}\mathrm{r}\mathrm{t}\mathrm{f}\mathrm{o}\mathrm{l}\mathrm{i}\mathrm{o}\mathrm{\hspace{0.17em}}\mathrm{c}\mathrm{o}\mathrm{s}\mathrm{t}\mathrm{,}\mathrm{\hspace{0.17em}}\mathrm{p}\mathrm{o}\mathrm{r}\mathrm{t}\mathrm{f}\mathrm{o}\mathrm{l}\mathrm{i}\mathrm{o}\mathrm{\hspace{0.17em}}\mathrm{r}\mathrm{i}\mathrm{s}\mathrm{k}\mathrm{,}\mathrm{\hspace{0.17em}}\mathrm{C}{\mathrm{O}}_{\mathrm{2}}\mathrm{\hspace{0.17em}}\mathrm{e}\mathrm{m}\mathrm{i}\mathrm{s}\mathrm{s}\mathrm{i}\mathrm{o}\mathrm{n}\mathrm{\right]}\mathrm{.}

Subjected to the constraints:

(6)\mathrm{1})AX\mathrm{=}B\mathrm{,}

where X is an individual technology (Hydro (H), coal (C), gas (G), diesel (D), nuclear (N), renewable (R)), A= [1 1 1 ….1] and B=100;

2) Emission constraint:

(7)\mathrm{E}\mathrm{M}\mathrm{I}\mathrm{S}\mathrm{\left(}{E}_{\mathrm{p}}\mathrm{\right)}\le \psi \mathrm{,}

where ψ is the emission target in mega tones,

3) Generation limit constraint for different technologies:

For each different technology (H, C, G, D, N and R) of a portfolio, the maximum and minimum capacity limits are being represented as constraints:

(8)S_{i\min \mathrm{}}\le S_{\mathrm{cap}i}\le S_{i\max }\mathrm{,}

where S_{i\max \mathrm{}} is the maximum capacity of technology i; S_{i\min \mathrm{}}, the minimum capacity of technology i; and S_{\mathrm{cap}i}, the capacity of technology i.

4 Optimization of portfolios incorporating risk, cost and CO₂ emission using a fast elitist multi-objective genetic algorithm (NSGA-II)

The non-dominated sorting genetic algorithm (NSGA) was developed by Deb et al. [15]. It is a very effective algorithm but has certain drawbacks such as its high computational complexity, lack of elitism and choice of optimal parameter value for sharing parameter. These drawbacks have been rectified in a modified version, NSGA-II [15], which has a better sorting algorithm, incorporates elitism and does not need shared parameter to be chosen like NSGA.

In a multi-objective optimization problem, the simultaneous optimization of all the objective functions, which are conflicting in nature, considering all the system constraints is usually non-attainable. Instead of giving a single optimal solution, a multi-objective optimization provides a set of solutions called efficient frontier, which have the property that no improvement in any of the objectives is possible without sacrificing one or more of the other sub-objectives. In this paper, NSGA-II has been applied for optimizing the Indian electricity generation portfolio. The proposed NSGA-II approach [16,17] is developed on a compatible PC (CPU Intel core i5-2400 3.10 GHz and 8 GB of RAM) with a 32 bit operating system (Windows 7 professional) and the program was developed in a Matlab R2010a environment.

The step-by-step solution procedure for applying NSGA-II to the energy portfolio management problem is shown in Fig. 3 in the form of a flowchart. It can be observed that NSGA-II is an easy and straight-forward approach to implement the problem under consideration. Once the populations are initialized, each population represents one portfolio having six dimensions, where each dimension of a portfolio represents one technology and each portfolio consists of H, C, G, D, N and R. After the evaluation of objective functions, the initialized populations are sorted based on non-domination into each front. After this the usual selection, crossover and mutation operations are performed. In the ranking procedure, the non-dominated individuals are identified first from the current set of population, which is called the first front population. Thus, the first front solution is a complete non-dominated set of solution and the second front individuals are only dominated by the first front individuals and the front continues. Accordingly, all the fronts have been defined. Each individual in each front has been assigned rank values. After assignment of ranks, crowding distance is calculated for each individual. Now, only either all the first front solutions are selected or individuals are selected by employing some fair selection scheme based on rank, like tournament selection and crowding distance. Thereafter, the offspring population is generated by using the crossover and mutation operators. After that, the entire population including the current population and the current offspring populations is sorted again based on non-domination and only the best N individuals are selected (elitism is ensured), where N is the population size. The same process is continued until the maximum number of generations is reached. The structure of the multi-objective genetic algorithm, NSGA-II is shown as follows:

\begin{array}{l}{P}_{\mathrm{1}}\mathrm{\leftarrow }\mathrm{Initial}\mathrm{\hspace{0.17em}}\mathrm{population}\mathrm{\hspace{0.17em}}\mathrm{generator}\mathrm{;}\hfill \\ O_{\mathrm{1}}\mathrm{\leftarrow }\mathrm{Pareto}\mathrm{\hspace{0.17em}}\mathrm{extraction}\mathrm{\left(}{P}_{\mathrm{1}}\mathrm{\right)}\mathrm{;}\hfill \\ P_{\mathrm{0}}\mathrm{\leftarrow }\mathrm{empty}\mathrm{;}\hfill \\ \mathrm{for\; t}\mathrm{\leftarrow }\mathrm{1}\mathrm{\hspace{0.17em}}\mathrm{until}\mathrm{\hspace{0.17em}}\mathrm{maxgen}\hfill \\ R_t\mathrm{\leftarrow }{P}_t\cup P_{t\mathrm{-}\mathrm{1}}\mathrm{;}\hfill \\ \mathrm{Evaluate}\mathrm{\hspace{0.17em}}\mathrm{fitness}\mathrm{\left(}{R}_t\mathrm{\right)}\mathrm{\hspace{0.17em}}\mathrm{using}\mathrm{\hspace{0.17em}}\mathrm{fast}\mathrm{\hspace{0.17em}}\mathrm{non-dominated}\hfill \\ \mathrm{sorting}\mathrm{\hspace{0.17em}}\mathrm{and}\mathrm{\hspace{0.17em}}\mathrm{crowding-distance-assignment;}\hfill \\ R_t\mathrm{\to }\hfill \\ \mathrm{Stochastic}\mathrm{\hspace{0.17em}}\mathrm{tournamnet;}\hfill \\ \mathrm{Crossover;}\hfill \\ \mathrm{Mutation;}\hfill \\ \mathrm{\to }{P}_{t\mathrm{+}\mathrm{1}}\mathrm{;}\hfill \\ O_{t\mathrm{+}\mathrm{1}}\mathrm{\leftarrow }\mathrm{ParetoExtraction}\mathrm{\left(}{O}_t\cup P_{t\mathrm{+}\mathrm{1}}\mathrm{\right)}\mathrm{;}\hfill \end{array}

here, maxgen is the predefined number of generations to be executed. Each population P_t has N individuals; the Stochastic Tournament operator selects N individuals from its input of 2N individuals. The operator ParetoExtraction returns to the non-dominated set from the input set. The set O_{\mathrm{maxgen}} is the approximation of the pareto set of the problem.

A detailed account of each of these steps is provided herein as follows:

Step 1 Initialization process: Initialize the good acceptable initial set of chromosomes of size N where, each chromosome represents one portfolio having six dimensions, where each dimension of a portfolio represents one technology and each portfolio consists of six different technologies H, C, G, D, N and R, keeping all the system constraints into consideration.

Step 2 Fitness function: The fitness function of each chromosome in a population set is derived using Eq. (5). Genetic algorithm follows the “survival of the fittest candidate” rule and thus initiates a search. Thus, the chromosomes which have the higher fitness value will always go for the next set of iteration, to form a new generation, followed by the successive crossover and mutation operations.

Step 3 Reproduction: In this phase, the operators of the genetic algorithm start to generate new population from the initial population having different fitness values. Genetic algorithm basically consists of three basic operators, namely, mutation, crossover and reproduction.

The first operator applied on a set of chromosomes is the reproduction operator. Based on the fitness function values in a population, it selects the chromosomes and forms a mating pool. It signifies that the chromosomes which possess the higher fitness value have the higher probability, where the probability lies between 0 to 1, for evolving one or more off chromosomes for the next generation. For selection of the chromosomes in the mating pool, the cumulative probability of each string of the population is evaluated. The chromosome with a higher fitness value is copied in the mating pool. Thus, the string with a higher fitness function value will represent a larger range in the cumulative probability values and, therefore, has a higher probability of being copied in the mating pool and vice-versa.

Step 4 Crossover: No new chromosomes are evolved in the reproduction stage. In crossover operation, two chromosomes are selected at random from the mating pool, which is considered as parent chromosomes and some segment of the chromosomes are swapped among them with a high crossover probability, to produce two new chromosomes, defined as child chromosomes. The crossover probability of 0.8 is considered for the case study.

In crossover operation, normally good strings are produced, which possess the higher fitness value by exchanging information between the parent strings. If the evolved strings possess a lower fitness value from crossover operations, then in the reproduction stage, fewer copies of these strings are selected in later strings, so that such strings are eliminated shortly. Hence, it signifies that, the crossover operator may be unfavorable or may be favorable and thus every string in the mating pool is not employed in the course of the crossover.

Step 5 Mutation: Mutation operations is needed to create a point in the neighborhood of the current point by interchanging the values of 0 and 1 in certain bit positions, with a small mutation probability and, thereby, achieving a local search around the current solution. The mutation is also used to maintain diversity in the population. For each bit of a string, a random number is generated between 0 and 1 and the mutation probability (P_m): 0.02 is considered for simulation in this paper.

Step 6 Selection and elitism: The selection operator using the fast non-dominated sorting scheme, which is the optimized version of the non-dominated sorting scheme is demonstrated below [17,18]:

here, the outputs of this procedure is {\mathrm{\Omega }}_i.

After the fast non-dominated sorting operation, the strings of a population set is divided into different fronts {\mathrm{\Omega }}_{\mathrm{1}}\mathrm{,}\mathrm{}\mathrm{\cdots }\mathrm{,}{\mathrm{\Omega }}_{k\mathrm{-}\mathrm{1}}\mathrm{.} The first-front population ({\mathrm{\Omega }}_{\mathrm{1}}) contains only the non-dominated solutions, which cannot be dominated by any other solutions. The second-front populations ({\mathrm{\Omega }}_{\mathrm{2}}), which are dominated only by the first-front individuals and the front continues like this. All the fronts are defined accordingly.

Each individual in each front is assigned rank values (e.g. individuals in the first (second) front are assigned a rank 1(2)). After this assignment of rank, a crowding distance is calculated for each individual. Now, either all the first-front solutions only are selected or individuals are selected using a tournament selection based on rank and crowding distance. (An individual is selected if the rank is lesser than the other or if the crowding distance is greater than the other). Now, the selected population generates an offspring population from the crossover and mutation operators. And the population with the current population and the current offspring populations is sorted on the basis of non-domination and only the best N individuals are selected (ensuring elitism), where N is the population size. Continue the same process until the stopping criterion is reached (either a maximum number of iterations is reached or the objective function does not improve). Crowding distance assignment is shown as follows:

\begin{array}{l}l\mathrm{=}\left|P\right|\hfill \\ \mathit{I}{[\mathrm{\ast }]}_{\mathrm{dist}}\mathrm{=}\mathbf{0}\mathrm{;}\hfill \\ \mathrm{for}\mathrm{\hspace{0.17em}}m\mathrm{\leftarrow }\mathrm{1}\mathrm{\hspace{0.17em}}\mathrm{until}\mathrm{\hspace{0.17em}}{n}_{\mathrm{obj}}\hfill \\ \mathit{I}\mathrm{=}\mathrm{sort}\left(\mathit{I}\mathrm{,}m\right)\mathrm{;}\hfill \\ \mathit{I}{\left[1\right]}_{\mathrm{dist}}\mathrm{=}\mathit{I}{\left[l\right]}_{\mathrm{dist}}\mathrm{=}\infty \mathrm{;}\hfill \\ \mathrm{for}\mathrm{\hspace{0.17em}}i\mathrm{\leftarrow }\mathrm{1}\mathrm{\hspace{0.17em}}\mathrm{until}\mathrm{\hspace{0.17em}}l\mathrm{-}\mathrm{1}\hfill \\ \mathit{I}{\left[i\right]}_{\mathrm{dist}}\mathrm{=}\mathit{I}{\left[i\right]}_{\mathrm{dist}}\mathrm{+}\frac{\mathit{I}{\left[i\mathrm{+}1\right]}_m\mathrm{-}\mathit{I}{\left[i\mathrm{-}1\right]}_m}{f_m^{\max \mathrm{}}\mathrm{-}{f}_m^{\min \mathrm{}}}\mathrm{;}\hfill \end{array}

here, l is the number of individuals in the population. \mathit{I} is the matrix with objective function values. The function \mathrm{sort}\mathrm{\left(}\mathit{I}\mathrm{,}m\mathrm{\right)} sorts \mathit{I} by the mth objective and \mathit{I}{\mathrm{\left[}i\mathrm{\right]}}_m returns to the value of objective m of individual i. \mathit{I}{\mathrm{\left(}i\mathrm{\right)}}_{\mathrm{dist}} is the crowding value of individual i.

5 Results and discussion

To illustrate the application of MPT, only six different technologies for the planning problem have been considered. Through MPT, an efficient frontier can be achieved that provides a number of optimal portfolios based on expected generation cost and technology risks for decision makers. Assumption has been made that all the input data are available for evaluation of generation costs, the risks associated with price volatility of different technologies, the correlation between costs of different technologies and CO₂ emission. Using Indian forecasted generation technologies mix for the year 2012, the generation system depends 66.27% on fossil fuel technology to meet electricity supplies. In fact, such a high reliance on fossil-fuel-based technologies has a high level of risk involved. Moreover, such a high reliance on fossil fuel technologies makes these forecasted fossil fuel prices increase in the future. As a result the cost of such portfolio could be expected to increase.

A case study of portfolio generation planning for Indian power system is conducted from the end of the Sixth Plan to the end of the Eleventh Plan, including recent annual plans as shown in Fig. 4. It represents the growth of installed generating capacity since the beginning of the Sixth Plan to the end of the Eleventh Plan in the form of a bar chart. Table 1 shows the growth of installed generating capacity, since the Sixth plan in MW. The detailed data are obtained from Ref. [26]. Table 2 lists the growth of installed generating capacity of the Sixth Plan in percentage along with portfolio cost, risk and CO₂ emission (Mt). Table 3 gives the correlation factors between different technologies. The Indian data set consists of five different generating technologies, namely, coal, nuclear, gas, oil, and renewable (mainly solar and wind) power between 1995 and 2003. No external cost data for India was available, therefore for calculating the correlation factors between different technologies, the data from the UK were used [11]. Correlation factors between different technologies have been estimated based on the available data set from 1995 to 2003 for fuel price [11]. Normally, with increasing diversity, portfolio risk falls, as has been demonstrated by the absence of correlation between different components of portfolio. Including fixed cost technology components to a risky generating mix, results in a lower expected portfolio cost at any level of risk, even if the costs of the fixed cost technology is more. A fixed cost technology component has a standard deviation value of zero. This lowers portfolio risk, which in turn, allows other higher-risk/lower-cost technologies into the optimal mix. Table 4 presents the standard deviation of different technologies, estimated in Ref. [19]. Table 5 shows the emission factor associated with different fossil fuel types estimated in Ref. [19].

In the case of renewable technology components, the fuel cost standard deviation is zero and its correlation with fossil fuel costs is also zero. Based on the data given in Tables 1 through 5 and using Eqs. (1), (2) and (4), the portfolio cost, risk and CO₂ emission have been calculated and shown in Table 2.

Figure 4 shows energy mixes of Indian power system from the end of the Sixth Plan till the target year, 2012. During that period, projected kW demand in the region is likely to rise by more than 300%. Energy forecasts indicate that this increased demand will be met primarily through capacity increases in fossil fuel technologies, mainly gas and coal. The output of hydro and renewable energy shares is also larger in 2012. Figure 5 depicts the portfolio cost (blue line circle) vs. risk (red circle) for the overall Indian power system scenario from the end of the Sixth plan to the end of the Tenth plan. Figure 5 indicates that from the end of the Seventh Plan to the end of the Ninth Plan, the portfolio cost, risk and emission rise continuously due to the fact that the ratio at which the fossil fuel technologies (mainly coal, gas, diesel) is rising as compared to the ratio of renewable energy share is about 220%. But energy forecasts indicate that from the end of the Ninth Plan to the end of Eleventh Plan, the ratio of rise of fossil fuel technologies as compared to renewable share is just about 33.83%. It implies that the focus is on renewable energy generation but the ratio at which load demand is still increasing, fossil fuel technology is a dominating component in today’s scenario. Figure 6 shows an inclination of CO₂ emission vs. the end of different plan/year.

To verify the performance of the proposed multi-objective algorithm, a case study of the Indian power system scenario between the end of the Sixth Plan to the end of the Eleventh plan was considered. Values of different parameters were assumed as follows: maximum number of generations (100), total independent runs (20); population size (75); crossover value (0.8), mutation value (0.02) and the plant load factors for different technologies are considered as 0.70 for H, C, G, and D, 0.61 for N and, 0.30 for R.

A total of 25 non-dominated solutions were obtained (Table 6) starting from Mix 1 (maximum risk portfolio) compounded mainly by higher ratio of fossil fuel technology and ends with Mix 25 representing portfolio with lowest risk. Figures 7 and 8 depict the optimal Pareto fronts of portfolio cost vs. CO₂ emission and portfolio cost vs. risk respectively.

As observed from the optimization result (Table 6), it can be concluded that, as the cost expected of a portfolio increases, the risk and CO₂ emission associated to the securities of volatility held in the portfolio decreases. Besides, it is possible to reduce the risk, cost and CO₂ emission simultaneously through the result of diversification of portfolios.

The projected 2012 electricity generation portfolio mainly comprises of 66.3% of fossil fuel and 33.7% of renewable technologies. Then MPT demonstrates that this portfolio is not efficient because for the same level of risk, a lower generation cost can be achieved by selecting a portfolio that includes more renewable technology represented by Mix 16 (30% hydro, 45.3% coal, 10.7% gas, 1% diesel, 1% nuclear, 12% renewable) as shown in Table 6. On the other hand, Mix 24 (30% hydro, 40.6% coal, 7.5% gas, 4% diesel, 1% nuclear, 16.9% renewable) represents an efficient portfolio with the same expected generation cost as the target portfolio (2012 energy mix); however, Mix 24 achieves a lower risk by the inclusion of more hydro and renewable technology. Now, from the obtained set of Pareto optimal solution, a suitable choice can be made based on practical considerations. Various scenarios to study the impacts associated with portfolio cost, the risk and the carbon emission were simulated. The impact of carbon emission and risk is mainly dependent on the coal, natural gas, nuclear and diesel projects. Carbon emission has a very marginal impact on the proportion of renewable power in view of its high cost. It is important to note that in order to obtain optimal portfolios, more share of renewable technologies need to be included, reducing the risk for the portfolio having the same expected cost as portfolios depend highly on fossil fuels. This outcome could have not been pointed out with least-cost methodologies. Another important outcome is that starting from Mix 1 and to the efficient frontier Mix 25 as shown in Table 6, it is observed that as the risk and CO₂ emission increase, the portfolio cost decreases, which means that a lower expected generation cost cannot be achieved without taking more and more risk as well as CO₂ emission and vice versa. This illustrates the trade-off between portfolio cost, risk and CO₂ emission.

6 Conclusions

A multi-objective planning to obtain optimal portfolio for Indian electricity generation planning using a robust multi-objective approach called, NSGA-II is proposed which considers simultaneous optimization of cost, risk and CO₂ emission associated with different fuel sources. The solution obtained using NSGA-II is an efficient frontier, which provides the number of optimal portfolio and is helpful to system planners for quickly obtaining the typical portfolio based on their experiences on real-time practice.

The ongoing trend toward green energy has brought about a paradigm shift in the deployment of renewable sources of energy. Thus, portfolio-based generation planning events that accommodate the risk associated with market uncertainty and de-emphasizes stand alone generating costs in order to incorporate prevailing dynamic and uncertain energy environment have been prominent. Mean variance portfolio techniques helps the energy analysts and energy policy makers to set renewable targets and portfolio standards that make economic and policy sense in this uncertain environment. It also provides the systematic basis for energy policy makers to formulate efficient generating mixes that minimize the risk, maximizes the security level and sustainability. Mean variance portfolio analysis shows that, contrary to widespread belief, attaining these objectives do not need to increase cost.

From the simulation results, it is observed clearly that an investor does not have to invest in all the available assets, but rather in a few assets to explore a wide risk-return area. The portfolio manager has the option to make a tradeoff between risks, return and the number of assets, to decide on the portfolio according to his requirement. The case studies pertaining to India have a vital significance for developing countries as a whole, since renewable energy technologies have begun to penetrate in these countries quickly due to increasing awareness of the global mission of green energy. The results reported in this paper corroborate the impact assessment of such technologies for overall reduction of risk as well as emissions and certainly would be beneficial for augmenting power system generation planning.

The limitation of the MVP theory in the perspective of a developing nation like India, is the barriers associated with the implementation of projects which is as important as the cost and risk assessment while planning capacity augmentation. The MVP theory considers only the risks associated with each cost component. Thus there should be a development of comprehensive risk barrier index to indicate the combined impact of risks and implementation barriers associated with each portfolio. The following barrier profiles in the context of developing countries exist:

1) Barrier linked with land availability;

2) Barriers associated with public policy;

3) Barriers associated with environmental clearance;

4) Barriers associated with resource and infrastructure schemes.

Future research work includes the incorporation of risk and barrier indices to form the comprehensive risk barrier index using a suitably weighted risk-barrier combination function. Attention should be focused on advanced local search operators into the proposed algorithm model, which is expected to allow better exploration and exploitation of the search space. The performance of the proposed algorithm using other realistic data can also be conducted to validate its potentiality. Nevertheless, the method presented in this paper can be effective and helpful to system planners for quickly obtaining the optimal portfolios for effective power system generation planning.

References

[1] Wang X, McDonald J R. Modern Power System Planning. McGraw Hill, 1994

[2] Mishra S K, Panda G, Majhi R. Constrained portfolio asset selection using multiobjective bacteria foraging optimization. Operations Research, 2014, 14(1): 113–145

[3] Liu S T. A fuzzy modeling for fuzzy portfolio optimization. Expert Systems with Applications, 2011, 38(11): 13803–13809

[4] Bazilian M, Roques F. Analytical Methods for Energy Diversity and Security. Elsevier Science, 2008

[5] Awerbuch, S. Getting it right: the real cost impacts of a renewables portfolio standard. Public Utilities Fortnightly, 2000-February-15.

[6] Awerbuch S. New economic cost perspectives for valuing solar technologies. In: Böer K W ed. Advances in Solar Energy: an Annual Review of Research and Development. Boulder: ASES. 1995

[7] Awerbuch S. Market-Based IRP: It’s Easy! Electricity Journal, 1995, 8(3): 50–67

[8] Awerbuch S, Berger M. Applying portfolio theory to EU electricity planning and policy-making. International Energy Agency Report number EET/2003/03. 2003

[9] Awerbuch S, Dillard J, Mouck T, Preston A. Capital budgeting, technological innovation and the emerging competitive environment of the electric power industry. Energy Policy, 1996, 24(2): 195–202

[10] Stirling A. On the economics and analysis of diversity. SPRU Electronic Working Paper No. 28. 1998

[11] Krey B, Zweifel P. Efficient electricity portfolios for the United States and Switzerland: an investor view. 2008-10

[12] Awerbuch S. The Cost of geothermal energy in the western us regions: a portfolio based approach. Sandia Report SAND2005–5173, Sandia National Laboratories, TN, USA, 2005

[13] Anagnostopoulos K P, Mamanis G. The mean–variance cardinality constrained portfolio optimization problem: an experimental evaluation of five multiobjective evolutionary algorithms. Expert Systems with Applications, 2011, 38(11): 14208–14217

[14] Gent M R, Lamont J W. Minimum emission dispatch. IEEE Transactions on Power Apparatus and Systems, 2007, PAS-90(60): 2650–2660

[15] Deb K, Pratap A, Agarwal S, Meyarivan T. A fast and elitist multiobjective genetic algorithm: NSGA-II. IEEE Transactions on Evolutionary Computation, 2002, 6(2): 182–197

[16] Carrano E G, Soares L A E, Takahashi R H C, Saldanha R R, Neto O M. Electric distribution network multiobjective design using a problem-specific genetic algorithm. IEEE Transactions on Power Delivery, 2006, 21(2): 995–1005

[17] Srinivas N, Deb K. Multiobjective optimization using nondominated sorting in genetic algorithms. Evolutionary Computation, 1994, 2(3): 221–248

[18] Kumar D, Samantaray S R, Kamwa I, Sahoo N C. Reliability-constrained optimal placement of multiple distributed generators in power distribution network using cat-swarm-optimization. Electric Power Components and Systems, 2014, 42(2): 149–164

[19] California Energy Commission. Comparative costs of California central station electricity generation technologies (Draft Staff Report). 2007

[20] Planning Commission of India. Mid-term appraisal for eleventh five year plan (2007–2012). 2010

[21] Planning Commission of India. Approach paper to the twelfth five year plan (2012–2017). 2011-10

[22] Corporate Catalyst India. A report on Indian power and energy industry. 2007

[23] Ministry of New and Renewable Energy. India. Annual Report 2010–2011.

[24] Lazard. Levelized cost of energy analysis. 2009

[25] Central Electricity Authority. India. Cost of generation for the year 2008–2009. 2010

[26] Central Electricity Authority. India. Growth of electricity sectors in India from 1947–2013. 2013-07

RIGHTS & PERMISSIONS

Higher Education Press and Springer-Verlag Berlin Heidelberg