Multi-objective optimization for sustainable groundwater resource management in a semiarid catchment

1 Copyright © 2012 by ASME

Proceedings of IMECE2012 2012 ASME International Mechanical Engineering Congress and Exposition

November 9-15, 2012, Houston, Texas, USA

IMECE2012–87950

MULTI-OBJECTIVE OPTIMIZATION APPROACH FOR THE SUSTAINABLE-RESILIENT SYNTHESIS/DESIGN/OPERATION OF A POWER NETWORK COUPLED

TO DISTRIBUTED POWER PRODUCERS VIA MICROGRIDS

Sergio Cano-Andrade, Michael R. von Spakovsky, and Alejandro Fuentes Mechanical Engineering Dept.

Virginia Tech, Blacksburg, Virginia, USA [email protected], [email protected]

Chiara Lo Prete and Benjamin F. Hobbs Geography and Environmental Engineering Dept.

Johns Hopkins University, Baltimore, Maryland, USA

Lamine Mili Electrical and Computer Engineering Dept.

Virginia Tech, Blacksburg, Virginia, USA

ABSTRACT In this paper, a multiobjective optimization approach is

proposed for evaluating the sustainable synthesis/design and operation of sets of small renewable and non-renewable energy production technologies coupled to power production/transmis-sion/distribution networks via microgrids. The optimization is conducted over a quasi-stationary twenty four-hour, winter period. Partial load behavior of the generators is included by introducing non-linear functions for efficiency, costs and emissions as a function of the electricity generated by every technology. A new index for resiliency is included in the multiobjective optimization model in order to account for the capacity of the power network system to self-recover to a new normal state after experiencing an unanticipated catastrophic event. Since, sustainability/resiliency indices are typically not expressed in the same units, a set of weighting factors are employed to calculate the value of a composite sustainability-resiliency index, applying fuzzy logic to minimize the effects of the subjectivity created in the selection of weights. Results indicate that the inclusion of microgrids into the network leads to a better overall network efficiency, a reduction in life cycle costs, and an improved network resiliency. On the other hand, total life SO2 emissions and network reliability are not improved when microgrids are included.

Keywords: Sustainability, Resiliency, Reliability, Microgrids, Multiobjective Optimization.

1. INTRODUCTION The term sustainability has been coined during the last two

decades, and is defined as “the way to meet the needs of the present without compromising the ability of future generations to meet their own needs” [1]. In order to quantify the sustainability for energy systems, four aspects are considered in the literature to be the principal pillars of sustainability, namely, the technological, environmental, economic, and social aspects of a system [2].

These four aspects are usually non-commensurable, and are commonly treated separately or by translating the units of one or more of them into those of another so that they can be treated as a set. Monetary cost has commonly been used as a base unit to sustainably design and operate a system.

At the level of individual technology development, which corresponds to the lower level of the two-tiered Sustainability Assessment Framework (SAF) under development by the authors, much effort has been expended over the last four decades to develop methodologies that can be used to determine the optimum syntheses/designs and performance of energy production technologies. Those based on the Second Law of Thermodynamics include, for example, the work by Valero and his co-workers (e.g., [3-10]) and that by Tsatsaronis and his co-workers (e.g., [11-15]). The methodologies developed quantify the resource and monetary costs needed to generate a product in an energy conversion system by identifying and quantifying the destruction of available energy


(exergy) within the components of that system. To deal with environmental aspects, von Spakovsky and Frangopoulos [16] and von Spakovsky and his co-workers [17-20] have proposed and applied an environomic approach for the optimization of energy systems in which among other things, penalty factors are assigned to the emissions released to the environment in order to drive the synthesis/design and performance of the system towards one with lower emissions. This environmental aspect is, of course, balanced with the system’s technical and economic ones. In more recent work, Tsatsaronis and his co-workers [21-24] have extended their Second Law methodology to deal with the environmental aspects of energy system synthesis/design and performance, while Diaz-Mendez et al. [25] have proposed a general thermodynamic index to account for the environmental impact of a system caused by emissions from power generators, based on the irreversibility induced in a combustion process by the pollutants from a previous combustion process. Of course, the work of many others could be mentioned but that is not the purpose of this paper.

Now, at the level of development of a network of power producing technologies, which corresponds to the upper level of the two-tiered SAF presented in this paper, the classic methodology consists of minimizing the fuel costs of several power producing technologies tied to a single node (or bus) [26]. In [27], sustainability considerations such as emissions (NOx, SO2, PM10, etc.) released to the environment and the efficiency of power producers have been added as objectives to the problem as has the capability to account for several nodes resulting in a power network (system) with flow transmission corridors in which electricity can be shared among nodes. These efforts have been extended to include fixed monetary costs proportional to the amount of electricity flowing through a corridor as a first attempt to account for (linear) transmission losses [28]. Hobbs et al. [29] have proposed a mathematical formulation of a power network system in order to solve market equilibrium models. The problem as posed maintains the convexity of the feasible region, assuring a unique solution. In this approach, a fixed number of producers compete in a power market so that the initial configuration of the network (system) is fixed. In addition, quadratic resistance losses proportional to the power flow in the high-voltage transmission corridors are included.

In addition to the work on producers tied to these high-voltage corridors, interest in the inclusion of microgrids (MGs) from an energy sustainability standpoint has increased because of their potential positive impact on the performance of the network. A MG is a localized group of small-scale electricity (and heat) producers used to satisfy the electricity (and heat) demands of a group of local customers. MGs can operate in parallel with the transmission/distribution grid or in islanded mode as might occur during an unanticipated catastrophic event which severely impacts the main grid. Lo Prete et al. [30] developed and applied the beginnings of our SAF upper-level methodology to the sustainability assessment of the Northwestern European electricity grid with MGs tied to it. For this assessment, a performance optimization of combined heat

and power (CHP) costs was developed first for scenarios in which microgrids were included in the network. This was followed by a multicriteria assessment to evaluate the technological and environmental characteristics of the optimum network configurations for the different scenarios.

With a focus on MGs, Pipattanasomporn et al. [31] developed a cost analysis of MGs based on the increase in reliability they can provide to the grid, taking into account restrictions imposed by NOx emission regulations. Their results showed that for commercial and industrial facilities where an outage cost is about $10,000/hr, it is desirable to install MG technologies in order to increase reliability.

Complementing this work is that by Costa and Matos [32], who developed an analysis of the reliability benefits of including residential and commercial MGs in a power network. The study was developed at the level of the distribution network and looked at different scenarios in order to quantify the reduction of the amount and duration of power outages in the main grid.

Finally, building on their previous work, Lo Prete et al. [33] applied their SAF methodology to quantify the effects on sustainability and reliability that the inclusion of MGs can have in a regional electricity market. In this case, a multicriteria analysis coupled with the minimization of short run costs (fuel and variable O&M costs) was used. The sustainability indices employed were those proposed in [34], i.e., those for energetic and exergetic efficiencies, capital and O&M costs, and NOx and SO2 emissions. The reliability indices implemented were the loss of load probability (LOLP) [35] and the expected loss of electricity (ELOE) [36].

In the present work, a sustainability assessment methodology is presented and applied, which employs a multiobjective optimization of a set of non-commensurable objective functions that form the basis of the sustainability indices used in [33]. In addition, a resiliency index is proposed and used as an objective function in order to account for the capacity of the network to restore itself to a new normal state after experiencing an unexpected failure. The network scenarios considered only include the transmission grid, power producers, and MGs. Results are presented for the quasi-stationary synthesis/design and performance optimization of the power network with and without MGs based on a winter day. In a quasi-stationary optimization process, a possible optimum configuration of the system as well as its performance at off-peak hours is obtained at every iteration of the optimization, until a final optimum is obtained. During the off-peak hours, part-load performance of the generators is considered, thus, impacting what the final optimal full-load performance and configuration will be. Weighting techniques obtained from a survey among various decision makers from different disciplines, and fuzzy logic analysis are applied as part of the post processing analysis to obtain the best compromise solution among the optimum solutions developed.


2. PROBLEM DEFINITION AND OPTIMIZATION

2.1. SYSTEM DESCRIPTION

General Problem A power network system is depicted in Figure 1. It can be

represented by i =1,…,I nodes connected by y = 1,…,Y high-

voltage transmission lines, setting up a = 1,…,A arcs or loops. The production and demand of electricity (and heat) can exist at a node. If the production of power exists at node i, there is a maximum number of producers, q=1,…,Q, from which power can be delivered to the network at that node.

Figure 1. Schematic representation of a power network.

Now, the problem considered in [29, 30, 33] is modified here to include multiple objectives as well as the part-load performance of the producers in a quasi-stationary optimization. The optimization is developed over a discrete period of time t = 1,…,tmax, where a maximum producer capacity, q, and a maximum number of producers, Q, at every node are available at the beginning of the optimization. An optimum configuration of energy producers is obtained through the optimization process. The producers targeted for inclusion in the most optimal configuration are operated at full or part load during the peak hour and at full or part load or not at all during the non-peak hours. The problem mathematically represents a quasi-stationary, multiobjective optimization problem with non-linear constraints. The objective of the problem is to

Minimize: T

kk FFFFF },,,,{ 121 ±±±±= −L (1)

with respect to tiqP and t

ijf , and

subject to:

( )[ ] max

11

,...,1for01 ttffSfPPJ

j

tij

tjiji

tji

Q

q

tiq

tDi

=≤−−−− ∑∑==

α (2)

( ) max

1

,...,1for0 ttffSM

m

tami

taimaim ==−±∑

=

(3)

maxminiqiqiq PPP ≤≤ (4)

maxminijijij fff ≤≤ (5)

The objective functions in Equation (1) are defined in

Section 2.2. A minus sign for the k-th element of the objective

function vector represents maximization. tiqP and t

ijf , are the

decision variables of the problem, and represent the power

produced by technology q at node i and time t, and the flow

transmitted from node i to node j at time t, respectively.

The first inequality constraint, Equation (2), is Kirchhoff’s

current law (KCL). The first term, DP , represents the load

demand of node i at time t. The second term is the total power

produced at node i and time t. The third term represents the net

power input to node i at time t, where )1( tjiji

tji fSf α− is the net

import of electricity to node i from every neighboring node j,

with Sji being the reactance of a high voltage flow line, α a

constant used to account for transmission losses [33], t

ijf the

export of power from node i to every neighboring node j, and tjif the import of power from every neighboring node j to node

i. The equality constraint, Equation (3), is Kirchhoff’s

voltage law (KVL). The assigning of a positive or negative value to the reactance depends on the direction of the power flow vector in corridor m, where m=1,…,M, and M is the total number of transmission corridors in loop a.

As to the second and third inequality constraints, Equation (4) sets restrictions on the capacity of each producer, q, at node i. The minimum capacity for every producer is set equal to zero, and the maximum capacity is set to the full-load capacity of a producer. Equation (5) sets restrictions for the power flow capacity of transmission lines. The minimum power flow capacity is equal to zero, while the maximum power flow capacity is set equal to the maximum capacity available to flow from node i to node j.

In this work, two different scenarios are considered in order to compare the synthesis/design and performance of a power network from a sustainability and resiliency standpoint, i.e., the grid only scenario (Scenario 1) and the grid with microgrids scenario (Scenario 2).

Scenario 1: Grid Only The system considered in Scenario 1 is a power network

that represents The Netherlands in the Northwest European Electricity Market. This market is commonly modeled by a Comprehensive Market Power in Electricity Transmission and Energy Simulator (COMPETES) [37]. It is composed of fifteen nodes and twenty-eight high voltage flow corridors, setting up fourteen arcs distributed among four European countries (The Netherlands, Belgium, France and Germany) in which twelve main producers compete to supply the electricity needs for the market.

The system for Scenario 1 is shown in Figure 2. It consists of three nodes (i=1,…,3) connected by three high-voltage transmission lines (y=1,…,3) setting up one loop (a=1) in which demand and supply of electricity are present. It is assumed that the power network is capable of satisfying its load with no need of importing power from neighboring countries. Five main firms compete in the market. A maximum number of generators (qi=1,…,Qi) is initially available at every node.

ii Qq ,,1K= ii Qq ,,1K=

ii Qq ,,1K=

ii Qq ,,1K=

ii Qq ,,1K=

ii Qq ,,1K=


Table 1 shows the percentage of each of the main producers by fuel type available in the system.

Figure 2. Representation of the system of Scenario 1.

A sustainable optimum configuration of producers that satisfies the power demand is determined during the optimization process, while minimizing (maximizing) the objectives. A period of twenty-four hours is considered (t=1,…,24). The twenty-four hour demand of electricity from the system represents a Load Demand Profile (LDP) of a winter day in The Netherlands (Dec. 01, 2010) [38]. The LDP of a node, i, is different than the LDPs of remaining nodes in the network.

Table 1. Percentage of available capacity in the power network.

Node 1 (%) Node 2 (%) Node 3 (%) Grid (%)

Nuclear 5.21 - - 2.87 Coal 36.18 - 13.21 24.26

Natural Gas 37.48 86.18 83.70 58.56 Oil 0.18 - - 0.10

Hydro 0.16 1.89 0.07 0.34 Waste 20.80 11.93 3.02 13.88

Data for the short run costs (fuel and variable O&M) and the capacity of the centralized generators at full load are available from COMPETES and can be obtained from the Energy Research Center of the Netherlands (ECN) [39]. This data has been updated by [33] in order to account for transmission resistance losses, first and second law efficiencies, SO2 and NOx emissions, force outage rates (FOR), and an update on production capacity and costs up to 2008. In the present work, data from [33] is updated to take into account the non-linearity in performance for first law efficiency, short run costs, and SO2 emissions of the producers. The conversion factor from Euro to USD is 1.3269, corresponding to the average exchange rate for 2010 [40]. Data for non-linear energetic efficiencies, short run costs and SO2 emissions are available from the authors upon request. It is assumed that the productive life of every centralized generator is 20 years. The average hours of operation during the year is obtained from [41]. Table 2 shows the average capital costs of equipment, average full load efficiency, and average full load SO2 emissions by fuel.

In order to obtain the part load behavior of centralized and decentralized generators, full load efficiencies, fuel costs and SO2 emission factors are scaled according to standard off-design curves of every fuel technology obtained from the literature. The constants for Equations (9), (14) and (16) are obtained with a regression analysis. For each generator, the constants are the same for all the ones using the same fuel provided they are the same size.

Table 2. Characteristics of centralized technologies.

Capital Costs

($/kW) a Efficiency

(%) b SO2 Emissions

(kg/kWh) b

Nuclear 3,000 35.00 0.0000 Coal 2,000 38.14 0.0021

Natural Gas 800 41.33 1.94x10-6 Oil 350 43.00 0.0016

Hydroelectric 1,700 100.00 0.0000 Waste 2,200 52.00 0.0020

Note: a Ref. [42], b Ref. [39].

Scenario 2: Grid and Microgrids In Scenario 2, various sectors of the grid representing

residential, commercial, and industrial customers are grouped and radially connected via MGs to the main grid described in Scenario 1 (see Figure 3). A different load demand profile for the three different MG regions – i.e., residential, commercial and industrial – is considered. The total load demand of the system is equal to that for Scenario 1. Data for residential, commercial and industrial load curves is under confidentiality agreement and cannot be published.

Figure 3. Representation of a region (node) in the electric power network with MGs included for Scenario 2.

MGs represent additional generation capacity. Five residential, two commercial, and three industrial MG configurations are tied to Node 1; ten residential MG configurations are tied to Node 2; and six residential, two commercial, and one industrial MG configurations are tied to Node 3. In total, the system for Scenario 2 is composed of thirty-two nodes of which twenty-nine are MG nodes and three main grid nodes. The capacity of every MG is 20 MW for a residential configuration, 110 MW for a commercial configuration, and 200 MW for an industrial configuration. Table 3 shows the percentage of producers by fuel type available for the MGs. It is assumed that MGs are in competition with the main grid, allowing the customers to get

Node 2

Node 1

Node 3

ii Qq ,,1K=

ii Qq ,,1K=

ii Qq ,,1K=

Node 2

Node 1

Node 3

Res. MGsCom. MGs

Ind. MGs

Res. MGs

Res. MGs

Com. MGs

Ind. MGs


electricity from the main grid only, from the MGs only, or from both. If isolated from the grid, MGs can satisfy the total demand of their customers. In addition, MG generators can be operating during off-peak as well as during peak hours. Since the MG nodes are radially connected to a main node, an extra equation is added to Equation (2) for every microgrid configuration added to the network; and since they are located near to the customers, no transmission losses are considered when supplying electricity to their corresponding main grid node. The average hours of operation during the year of decentralized technologies is also taken from [41].

Table 3. Percentage of available capacity in microgrid nodes.

Node 1 (%) Node 2 (%) Node 3 (%) MGs (%)

Natural Gas 37.07 70.00 44.63 43.49 Coal 51.20 - 36.48 40.24 Oil 8.70 12.00 13.04 10.51

Wind 1.96 18.00 4.00 4.55 Solar 1.09 - 1.85 1.20

Average capital costs, average full load energetic efficiencies, and average full load SO2 emissions for MG generators are provided in Table 4. Total life of operation for these technologies is considered to be 20 years, except for SOFCs where 10 years is assumed.

Table 4. Characteristics of decentralized technologies.

Capital Costs

($/kW) Efficiency

(%) SO2 Emissions

(kg/kWh)

SOFC 4,700 a 50.00 a 0.000000 f Diesel REs 350 a 34.00 a 0.000206 b

NG Microturbines 1,100 b 26.00 a 0.000003 g NGCC 1,532 c 48.00 c 0.000000 c

Coal Fired 2,043 c 36.00 c 0.000370 h IGCC 2,500 d 38.00 c 0.000304 h

Wind Turbines 900 c 18.00 e 0.000000 e Solar PV 5,884 a 81.00 a 0.000000 a

Note: a Ref. [33], b Ref. [31], c Ref. [43], d Ref. [42], e Ref. [30], f Ref. [44], g Ref. [45], h Ref. [39].

2.2. INDICES

2.2.1. TECHNICAL INDICES Factors such as system energetic efficiency, exergetic

efficiency, reliability, resiliency, etc., are commonly used to assess technological characteristics in energy sustainability. The ones used in this work are described below.

Energetic efficiency: The first objective function in Equation (2) is the energetic

efficiency of the network described as [46]:

Tot

Tot

IB

PF ==η1 (6)

where PTot is the total power produced in a certain period of time, and BTot is the total fuel needed to produce that amount of power. PTot is defined as

( ) ∑∑∑∑= = ==

=+=

maxmax

1 1 11

t

t

I

i

Q

q

tiq

t

t

tD

tL

Tot PPPP (7)

where PL represents the total transmission losses in the high-

voltage transmission lines, and DP the total demand. BTot is

defined as

∑∑∑∑∑∑∑∑∑= = == = == = =

===

maxmaxmax

1 1 11 1 11 1 1

t

t

I

i

Q

qtiq

tiq

t

t

I

i

Q

q

tiq

tiq

t

t

I

i

Q

q

tiq

TotP

LHVmBBη

& (8)

Here, m& and LHV represent the mass flow rate and lower

heating value of the fuel, B, needed to produce P. The fuel used by renewable producers is equal to zero.

The energetic efficiency of a generator in Equation (8) is written as [47]

( )2tiqiq

tiqiqiq

tiq PP γβση ++= (9)

where, σ , β , and γ are constants that represent the efficiency

characteristics of producer q in node i. This allows one to take

into account the part load behavior of the producers.

Reliability: The second objective function in Equation (2) is the

reliability of the system. Reliability is the probability that the power demand will be higher than the available power capacity in the system. The reliability index used is the LOLP, which represents the expected number of hours that the system is not able to supply the demand in a certain period of time [35]. The objective function for reliability is defined as [33]

∑=

==Z

z

LOLPzLOLP zfpFF

1

2 (10)

where pz, and zLOLPf are the probability and LOLP value of the

z-th mixture of the normal approximation (MONA) [48].zLOLPf

is given by (Equation (13) in [33])

( )

∫∞

−

=

−−=

0

)()( dxxfxf

PPPPf

tz

t

zD

zz

WPP

twind

tDLOLP

(11)

Note that in this expression wind production in the main grid is

taken into account since it is significant fraction of the total

production. For example, average wind production for the year

2010 in The Netherlands was 381 MW [49]. In Equation (11),

)(xf t

zD PP − is the normal distribution function of the production

capacity deficit, tD PPx

z−= ; and )(xf t

zW is the exponential

cumulative distribution function evaluated at x, with parameter

λ equal to the average wind production during the day. In the

absence of detailed data, it is assumed that the daily wind

production is the ratio between the yearly average production

and the number of days in the year.

Resiliency: The third objective function in Equation (2) is resiliency.

Resiliency is defined as the ability of a system to recover to a


new normal state from a failure caused by an unanticipated catastrophic event. If a power outage occurs due to a failure in the main grid resulting from, for example, a snow storm, hurricane, terroristic attack, etc. and there are no MGs present in the network, the amount of electricity not supplied to the customers within a period of time is taken into account and corresponds to one hundred percent of the power outage described by a Descriptive Resiliency Metric [50]. If, on the other hand, MGs are present, they can be isolated within a matter of seconds since MGs can operate in island mode and provide the electricity needs of at least the local customers tied to the MGs, minimizing the extent of the power outage caused by the unexpected event.

In this work, it is assumed that the resiliency of the system is directly proportional to the amount of energy produced by the MGs and is described by the MG penetration index given by

∑∑∑= = = +

==

max

1 1 1,,

,

Resiliency3

t

t

I

i

Q

qGt

iqMGt

iq

MGtiq

PP

PFF (12)

The maximum possible penetration of MGs is restricted in the present work to be only 15% of the total production of the system, since it is well known that a high penetration of MGs could cause instability problems in the network if no voltage and load frequency regulation is present [32]. Clearly, with the inclusion of state-of-the-art power electronics in the network, this problem can be overcome (and, in fact, contribute to network stability) in which case one would not need to restrict maximum MG penetration to such a low percentage.

2.2.2. ECONOMIC INDICES Capital costs and operating and maintenance costs (O&M)

are frequently considered as economic issues of energy sustainability. The economic costs considered in this work are described below.

Monetary Costs: The fourth objective function in Equation (2) is that of

monetary costs, including capital and short run costs. Short run costs account for fuel costs and variable O&M costs, while capital costs account for the initial investment in the acquisition and construction of the power plant. The objective function is defined as

( )∑∑∑= = =

+==

max

1 1 1

4

t

t

I

i

Q

q

capiq

tiqCost CCFF (13)

where capiqC is the capital cost of a producer. These costs are

amortized, accounting for interest, depreciation and taxes. The

annualization factor is: )1)1(()1( −++ nn rrr where r is the

discount rate, and n is the useful life of the producer. tiqC are

the short run costs of the producer expressed as [47]

( )2tiqiq

tiqiqiq

tiq PPC ξεδ ++= (14)

Here, δ , ε , and ξ are constants that represent the

characteristics of short run costs for producer q at node i. In this

way, the model is able to account for the part-load behavior of

the producers.

2.2.3. ENVIRONMENTAL INDICES Emissions to the environment such as SOx, NOx, UHC,

PM, CO, and CO2 are considered as environmental issues in energy sustainability. In the present work, only SO2 emissions are considered.

SO2 Emissions: The fifth objective function in Equation (2) corresponds to

environmental emissions. The objective function for emissions is written as follows:

∑∑∑= = =

==

max

1 1 1

5

t

t

I

i

Q

q

tiqEmissions EFF (15)

The emissions of a producer can be expressed as [47]

( )2tiqiq

tiqiqiq

tiq PPE λψφ ++= (16)

where φ , ψ , and λ are constants that represent the emission

characteristics of producer q at node i. Again, in this way, the model is able to account for the part-load behavior of the producers.

2.2.4. SOCIAL INDICES Social issues are those that account for job creation,

increments in life quality, etc. Social indices are not considered in this work, since at least based on the sensitivity analysis found in [33], they do not substantially affect the final results of a quantitative energy sustainability assessment. Thus, as a first cut, we ignore the social indices. However, their significance will be reexamined in more detail in future work.

2.3. OPTIMIZATION STRATEGY

2.3.1. SINGLE OBJECTIVE OPTIMIZATION Two approaches for an optimization of the system are

compared below using a single objective function (see Section 3.1). For both approaches, the total costs of the system are minimized. Information on the number of power producers, fuel type used, and their performance (part-load behavior efficiencies and emissions) are obtained. The first approach is based on the optimization of the supply at the peak-hour demand only, while the second approach is based on a quasi-stationary optimization over twenty-four hours during the same day as described below. The model is written in C language and solved using a Sequential Quadratic Programming (SQP) algorithm [51].

Peak-hour optimization. In this approach, an optimum configuration is obtained by

minimizing the total life cycle costs of the system to satisfy the peak-hour load demand of the day, tpeak, shown in Figure 4. After that, the short run costs for the optimum configuration over the twenty-three off-peak hours are, as in [30], determined proportionally based on the peak-hour (in [30], the year is divided into 4 periods, i.e., summer-peak and -off peak and


winter-peak and -off peak; the minimum short run costs for the performance of the system is obtained for each single hour, and the costs are averaged to obtain the total yearly costs).

Quasi-stationary optimization. In this approach the optimum configuration of the system

and its performance over the twenty-four hours is obtained in a single optimization, i.e., minimization of the total costs over the entire day. The major disadvantage is that the problem becomes much larger, making it more expensive in terms of computational resources. Nonetheless, it is the more correct of the two approaches for determining the optimal configuration.

Figure 4. Representation of a daily load demand curve.

2.3.2. MULTIOBJECTIVE OPTIMIZATION

In the sustainability assessment of a system with a single objective optimization, the nature of the analysis is not conserved because all the objectives are grouped into a single measurement unit. In a multiobjective optimization, on the other hand, a Pareto curve of non-dominated solutions for different non-commensurable objectives is found. As part of the post-processing, the analyst can make a decision as to which optimal solution to use based on the importance assigned to each objective (weighting) [52] or based on the minimization (maximization) of an objective function derived using fuzzy logic [47].

Now, to optimize a system of non-commensurable objective functions, derivative-based [53, 54] and non-derivative-based methods [55-57] are available. A non-derivative based method, such as a Genetic Algorithm, is a robust method for optimization, in which a number of final optimum values are provided (individuals). The initial values are allowed to interchange part of their binary chain (genes) through the optimization process in order to minimize (maximize) the objective functions. The problem is converged when a certain number of generations is reached. Although it is a robust method of solution, it is very computationally expensive when used for large problems [58]. In contrast, derivative-based optimization methods, such as SQP, are less computationally expensive, although there is the possibility of getting stuck in local optima. To overcome this problem,

different initial decision variable sets are used to try and span the feasible space in order to assure that the algorithm reaches the global optimum.

It is the derivative-based method, a SQP algorithm [51],

which is used here. To optimize a set of non-commensurable

objective functions, the maximum and minimum feasible

values of the functions are obtained by optimizing each

objective function independently of all of the others. This

provides the so-called utopia point shown in Figure 5. Once this

is done, the continuous feasible space of an objective function

is discretized and the function used as a constraint in the

optimization of another objective function. This process is

repeated for all combinations of objective functions. The Pareto

set resulting from this process is represented by a kr × matrix,

where the Kk ,...,1= index the objective functions, and the

Rr ,...,1= the number of intervals into which the feasible

space is discretized.

Figure 5. Representation of the Pareto front of optimum solutions.

Fuzzy Logic. Once the Pareto set of optimum solutions is obtained,

fuzzy logic theory is applied in order to obtain the best compromise solution from among all the optimal solutions shown by the Pareto Curve. Fuzzy logic does not have the need of weighting the objective functions in order to find the best compromise solution; but instead requires the minimization (maximization) of an objective function.

To implement this, the r-th value of the k-th objective

function, krτ , is first normalized in order to eliminate the

difference in units between the objective functions This is

accomplished using Equation (17) when minimizing objectives,

i.e., emissions and costs, and Equation (18) when maximizing,

i.e., efficiencies [47]:

≥

<<−

−

≤

=

max

maxmin

minmax

max

min

0

1

rr

rrr

rr

rr

rr

kr

ττ

ττττ

ττ

ττ

µ (17)

)(2 xF

)(1 xF

( )max2

min1 ,FF

( )min2

max1 ,FF

1=r Rr =


≥

<<−

−

≤

=

max

maxmin

minmax

min

min

1

0

rr

rrr

rr

rr

rr

kr

ττ

ττττ

ττ

ττ

µ (18)

where krµ , maxrτ , and min

rτ are the normalized value of krτ ,

and the maximum and minimum values of rτ in the k-th vector,

respectively.

The next step is to take the normalized matrix, krµ , and

reduce it to a column vector according to the expression,

∑∑

∑

= =

==R

r

K

k

kr

K

k

kr

r

1 1

1

µ

µ

µ (19)

A best compromise solution is obtained by selecting the

objective function set with maximum value, maxrµ .

Weighting Process. In the weighting post-processing approach, an interaction

between the analyst and the decision makers is required. Decision makers such as scientists, investors, general population, etc., are polled to assess the different objectives considered in the analysis based on their perceptions or interests. The polls are then used to determine an assignment of weight, Φ , for each objective function. The best compromise solution is then obtained based on the following relation [52]:

∑=

⋅Φ=K

kkkr F

1

µ (20)

where

11

=Φ∑=

K

kk (21)

3. RESULTS AND DISCUSSION

3.1. SINGLE-OBJECTIVE OPTIMIZATION

For the peak-hour optimization, the total costs of the system for Scenario 1 are minimized. In order to obtain the performance of the system, the short run costs of every non-peaking hour are proportionally determined relative to the load demand profile using the peak-hour value as the basis. The sum of the twenty four-hour short run costs plus the capital costs determined at the peak hour represents the total life cycle costs for the system over the twenty-four hour period.

Table 5 shows the results for the total life cycle costs of the peak-hour optimization strategy versus the total life cycle costs of the quasi-stationary optimization strategy. The minimum costs for the peak-hour optimization approach are 84,035.31 Million USD (MM-$) and those obtained from the quasi-stationary optimization approach are 75,114.25 Million USD, a difference of about 12 %. Table 5 also shows the results for total life SO2 emissions, efficiency, and LOLP. As can be seen, a higher efficiency for that of the quasi-stationary approach is

obtained, while total life SO2 emissions are better for the peak-hour strategy. Reliability is similar for both approaches. The differences with the peak-hour optimization results are attributable to the fact that the part-load performance of the system for the peak-hour strategy is not determined directly but are instead based on averages. Another approach used to obtain the part-load performance of the system, when optimizing only the peak-hour, is that of using the peak-load configuration obtained to optimize the twenty-three non peaking hours separately. Using this approach, the optimum values are almost identical with those obtained by the quasi-stationary approach, but the optimum configuration and its part-load performance are different. Thus, although the quasi-stationary optimization is more computationally expensive, it does make a difference in the correctness of the results obtained. This difference becomes even more important when optimizing the synthesis/design performance of a transient system [59].

Table 5. Optimization results for the peak-hour averaging approach versus the quasi-stationary approach.

Peak-hour Averaging

Quasi-stationary

Difference (%)

Total Life Costs (MM-$) 84,035.31 75,114.21 11.88

Total Life SO2 Emissions (Ton)

264,313.56 324,832.54 -18.63

Network Efficiency (%) 43.36 43.71 0.79

LOLP (Outage hr/10-yr) 3.21 3.24 -1.07

3.2. MULTIOBJECTIVE OPTIMIZATION

Limits of the objective functions The minimum and maximum values for the objective

function in both scenarios are obtained in order to find the feasible synthesis/design/operation space of the system. Table 6 shows the results for Scenario 1 and Table 7 for Scenario 2.

As seen, the feasible space is larger for Scenario 2. This is because of the addition of renewable-based production technologies such as PV and wind turbines as well as more fuel-expensive reciprocating engines and natural gas (NG) microturbines.

Table 6. Feasible space for Scenario 1.

Minimum Maximum

Total Life Cycle Costs (MM-$) 75,126.39 104,144.03

Total Life SO2 Emissions (Ton) 96,400.58 324,832.50

Network Efficiency (%) 41.40 44.31

LOLP (Outage hr/10-yr) 2.7809 3.2742

Resiliency (% MGs) - -


Table 7. Feasible space for Scenario 2.

Minimum Maximum

Total Life Cycle Costs (MM-$) 71,113.32 108,911.72

Total Life SO2 Emissions (Ton) 77,087.83 368,935.78

Network Efficiency (%) 40.81 44.43

LOLP (Outage hr/10-yr) 4.53 23.44

Resiliency (% MGs) 4.41 12.17

The next sections present Pareto sets for two objective functions at a time. Visualizing more than two or three at a time is difficult, since a k-dimensional space is required for k objective functions. When using more than three objectives, the Pareto sets are shown with numerical values.

Monetary Costs Figure 6 shows a view of the Pareto set for capital costs

versus total life cycle short run costs (variable O&M and Fuel). Capital Life Costs denote the investment made for the total capacity of each optimum configuration, assuming that each network configuration is fully built with no capacity expansion. It can be seen that as the capital cost of the grid decreases, the total short run costs increase. This trend is observed because coal and waste plants are replaced by natural gas technologies in the most expensive optimum configurations, as can be seen in Figure 7 for Scenario 1 and Figure 8 for Scenario 2. Coal plants are expensive to build but relatively cheap to operate, while natural gas technologies are relatively cheap to build but more expensive to operate.

Figure 6. Optimal capital costs versus optimal life cycle O&M costs.

Comparing both scenarios, it is seen that Scenario 2 is

more expensive than Scenario 1 for all possible optimum configurations in the synthesis/design/operation space. This is because of the higher unit capital cost of distributed generation technologies such as fuel cells, photovoltaics, and small coal plants, as well as the high operating costs of diesel reciprocating engines, NG microturbines, and solid oxide fuel cells.

Figure 7. Sizes of the optimum network configuration in the Pareto set

for Scenario 1.

(a)

(b)

Figure 8. Sizes of the optimum network configuration in the Pareto set for Scenario 2: (a) grid, and (b) MGs.

Costs – Efficiency Figure 9 shows a view of the Pareto set for the network

efficiency. An optimum point is shown at about 77,000 MM-USD for Scenario 1 and about 83,000 MM-USD for scenario 2. For both scenarios, the efficiency of the system decreases as the total life cycle costs increase. This is because smaller less efficient technologies begin to play a role in the production of


electricity, while the base-load technologies, such as coal, decrease their participation in the optimum configurations.

In addition, it is shown that Scenario 2 has a better network efficiency than Scenario 1 for almost all the optimum points, except for a range between 91,000–97,000 MM-USD. In this range, coal reduces significantly its role in the network for both scenarios. Approximately 1.2 GW of centralized NG is replaced by distributed generation in Scenario 2. Nuclear, oil, waste and hydro technologies produce almost the same amount of power in both scenarios. In addition, transmission losses are reduced in Scenario 2, because of the local power generation, improving the performance of the power network system.

Figure 9. Optimal network efficiency versus optimal total life cycle

costs.

Costs – SO2 Emissions Figure 10 shows the total life SO2 emissions versus total

life costs for the optimal configurations. For both scenarios, it is seen that emissions decrease while the cost of production increases. This is because technologies using cheaper fuels with higher emissions of SO2, such as coal and waste, decrease production costs, while cleaner and more expensive technologies such as natural gas increase production costs. Technologies fueled with nuclear and hydro maintain the same production costs. Oil does not play a role in any configuration for the centralized system; it is required for the MGs only. In MG configurations, natural gas and oil technologies increase their penetration in the system for the most expensive cases.

Figure 10 also shows that the optimal total life SO2 emissions are lower for Scenario 2, because PV, wind turbines, and fuel cells play a role in the optimum configurations. For the most expensive cases, the emissions are lower for scenario 1 because oil starts to increase its contribution in the MG configurations.

Costs – Reliability Figure 11 shows a side view of the Pareto set for optimal

reliability versus optimal total life cycle costs. The LOLP is almost independent of costs for Scenario 1, so that little variation in the reliability of the optimal system occurs. This is because, although the configurations are changing at every

point in the Pareto set, they always include a big amount of base- and intermediate-load power plants.

The figure also shows that reliability is better for Scenario 1 than for Scenario 2. A high penetration of wind and PV results in uncertainties in the power available to the grid due to the intermittent nature of these technologies. Thus, their FOR is low. There is an optimum point for Scenario 2 at about 84,000 MM-USD, when the MG penetration is the lowest of all the possible optimum configurations.

Figure 10. Optimal total life SO2 emissions versus optimal total life

cycle costs.

Figure 11. Optimal LOLPs versus optima total life cycle costs for

Scenarios 1 and 2.

Costs – Resiliency Figure 12 shows the view of the Pareto set for resiliency

versus total life costs for the optimal configurations. Since resiliency is defined to be directly proportional to microgrid penetration, Scenario 1 has a resiliency of zero for all the possible optimum solutions. On the other hand, Scenario 2 shows the highest resiliency at the lowest optimal total life costs available, when small coal technologies provide their maximum contribution to the network.

Table 8 and Table 9 show the values for the Pareto front of optimum solutions for Scenarios 1 and 2, respectively.


3.3. POST-PROCESSING ANALYSIS

Table 10 and Table 11 show the normalized values for Scenarios 1 and 2, respectively. Table 12 shows the best compromise solution among the optimum solutions in the Pareto set using fuzzy logic, while Table 13 lists the composite sustainability-resiliency index for each of these optimum solutions with the best indicated in bold. As can be seen in Table 12, the best compromise solution for Scenario 2 shows that the inclusion of MGs into the network leads to a better overall efficiency, a reduction in monetary costs, and an improved network resiliency. On the other hand, total life SO2 emissions and network reliability are not improved, i.e., they are slightly better in Scenario 1. Figure 13 shows the normalized indices of the best compromise solution for both scenarios.

Figure 12. Optimal resiliency (penetration of MGs) versus optimal

total life cycle costs.

Table 8. Optimal objective function values for Scenario 1.

Total Life Cycle Costs

(MM-$) η (%)


LOLP (hr/10-yr)

Resiliency (% MGs)

75,126.39 43.50 324,832.50 3.1781 - 79,627.47 44.31 266,434.56 3.1730 - 84,825.51 44.18 220,529.50 2.7809 - 89,561.29 43.48 183,274.87 3.2027 - 94,024.09 43.27 154,562.07 3.2742 - 98,907.32 42.50 124,239.55 3.0988 -

104,144.03 41.40 96,400.58 3.1867 -

Table 9. Optimal objective function values for Scenario 2.

Total Life Cycle Costs

(MM-$) η (%)


LOLP (hr/10-yr)

Resiliency (% MGs)

71,113.32 42.77 368,935.78 19.2158 12.17 77,533.19 44.33 288,398.01 6.8360 9.00 83,725.21 44.43 226,373.55 4.5277 4.41 90,177.43 43.58 173,952.60 7.2091 5.07 96,308.14 42.73 128,959.63 12.2217 5.97

103,236.06 42.82 114,170.12 21.5413 6.66 108,911.72 40.81 77,087.83 23.4428 6.48

Table 10. Normalized indices for Scenario 1.

Economic Index

Efficiency Index

Environmental Index

Reliability Index

Resiliency Index

0.8938 0.7431 0.1511 0.9806 0.0000 0.7747 0.9669 0.3512 0.9811 0.0000 0.6372 0.9309 0.5085 1.0000 0.0000 0.5119 0.7376 0.6362 0.9797 0.0000 0.3939 0.6796 0.7345 0.9763 0.0000 0.2647 0.4669 0.8384 0.9845 0.0000 0.1261 0.1630 0.9338 0.9802 0.0000

Table 11. Normalized indices for Scenario 2.

Economic Index

Efficiency Index

Environmental Index

Reliability Index

Resiliency Index

1.0000 0.5414 0.0000 0.2043 1.0000 0.8302 0.9724 0.2760 0.8035 0.7395 0.6663 1.0000 0.4885 0.9153 0.3624 0.4956 0.7652 0.6681 0.7856 0.4166 0.3334 0.5304 0.8223 0.5431 0.4906 0.1502 0.5552 0.8729 0.0920 0.5472 0.0000 0.0000 1.0000 0.0000 0.5325

Table 12. Best compromise solution using Fuzzy Logic.

Objective Scenario 1 Scenario 2

Total Life Cycle Costs (MM-$)

84,825.51 77,533.19

η (%) 44.18 44.33 Total Life SO2 Emissions

(Ton) 220,529.50 288,398.01

LOLP (hr/10-yr) 2.7809 6.8360 Resiliency (% Penetration) 0 9.00

Table 13. Composite sustainability-resiliency index using Fuzzy Logic.

Composite sustainability-resiliency Index for scenario 1

Composite sustainability-resiliency Index for scenario 2

0.07149 0.07090 0.07937 0.09351

0.07945 0.08863 0.07399 0.08085 0.07189 0.07023 0.06596 0.05726 0.05689 0.03957

Figure 13. Normalized indices for the best compromise solution in the Pareto set for Scenarios 1 and 2 using fuzzy logic.


Now, if instead of fuzzy logic the weighting post-processing approach proposed by [34] is used, sustainability sub-indices and a composite sustainability-resiliency index for each of the optimum solutions in the Pareto set are obtained. In order to determine the weights for the different objectives, a total of six decision makers, all of them involved in academia but in different disciplines, were asked to provide their weighting scheme. The fuzziness of the responses is minimized using the methodology in [60]. Table 14 shows the results for Scenario 1 and Table 15 for Scenario 2. The best compromise solution for Scenario 1 has a composite sustainability-resiliency index of 0.5562, while that for Scenario 2 has one of 0.6446. These two best solutions are also the ones found using Fuzzy logic although the values of the composite sustainability-resiliency indices are different.

Table 14. Sustainability-resiliency sub-index and composite sustainability-resiliency index for Scenario 1.

Economic Sub-Index

Environmental Sub-Index

Technological Sub-Index

Composite Index

0.8938 0.1511 0.4696 0.5194 0.7747 0.3512 0.5144 0.5553 0.6372 0.5085 0.5134 0.5562

0.5119 0.6362 0.4682 0.5379 0.3939 0.7345 0.4556 0.5231 0.2647 0.8384 0.4159 0.4975 0.1261 0.9338 0.3539 0.4586

Table 15. Sustainability-resiliency sub-index and composite sustainability-resiliency index for Scenario 2.

Economic Sub-Index

Environmental Sub-Index

Technological Sub-Index

Composite Index

1.0000 0.0000 0.6477 0.5659 0.8302 0.2760 0.8069 0.6446

0.6663 0.4885 0.6707 0.6106 0.4956 0.6681 0.6070 0.5867 0.3334 0.8223 0.5157 0.5489 0.1502 0.8729 0.3995 0.4622 0.0000 1.0000 0.2518 0.4019

24-hr performance Figure 14 shows the twenty-four hour performance of the

best compromise solution for Scenario 1. Coal, nuclear, waste, and hydro technologies serve as base-load technologies, while NG provides intermediate-load and peak-shaving. This is expected, since coal, nuclear, hydro, and waste are typically large power plants that are cheap to operate, although more capital intensive.

Figure 15 shows the performance of the best fuzzy logic compromise solution for Scenario 2. Compared with Scenario 1, the main grid for Scenario 2 is producing about 1,000 MW less electricity from NG technologies. Nuclear, coal, waste, and hydro produce the same electricity in both scenarios. In the

MGs, NG technologies such as fuel cells and microturbines produce a constant amount of electricity throughout the day, while small coal plants produce most of the electricity, especially for industrial applications. Diesel reciprocating engines operate during peak hours only.

Figure 14. Power production throughout the day for the best fuzzy logic compromise solution for Scenario 1.

(a)

(b)

Figure 15. Power production throughout the day for the best fuzzy logic compromise solution for Scenario 2: (a) Grid, (b) MGs.


4. CONCLUSIONS In this work, a multiobjective, quasi-stationary non-linear

optimization analysis for the synthesis/design and operation of a three-node network system with and without MG configurations is presented. Results show that decentralized power generation brings improvements to the network, especially relative to the technological and monetary aspects of sustainability-resiliency. Coal, nuclear, hydro, and waste fueled technologies serve as base- and intermediate-load technologies, while NG serves as a peak-load technology only in both scenarios. The inclusion of MGs replaces part of the NG production in the main grid. Industrial MGs play an important role in the synthesis/design/performance of the network, because the load demand of this specific set of microgrid configurations has small variations throughout the day and is supplied by small coal technologies in most cases.

Although social sustainability considerations are not included, since they do not significantly alter the final quantitative results, it is nonetheless important to know for sake of implementation the perceptions of different stake-holders towards the inclusion of decentralized technologies into the main grid. Furthermore, although fuzzy logic and a weight assignment process are good tools for obtaining a best compromise solution among the optimum solutions in the Pareto set, a detailed analysis applying multiple multi-criteria decision-making methods to analyze the trend of the Pareto sets is necessary when dealing with more complex systems.

NOMENCLATURE

P Power (MW)

f Power flow (MW)

α Constant for transmission losses S Reactance

F Objective function

F Objective function vector

B Fuel (MW)

Iη Energetic efficiency

m& Mass flow rate (kg/s) LHV Lower heating value (MJ/kg)

E Emissions C Monetary costs p Probability τ Element in the kr × matrix µ Normalize value of τ

Φ Weight

Subscripts k Number of the objective function i Node

j Neighboring node to node i y Transmission line q Generation technology a Voltage loop m Transmission line in loop a

L Transmission losses

D Demand

MO & Operation and maintenance LOLP Loss of load probability

z Mixture of normal approximation W Wind

r Interval in the discrete feasible space.

Superscripts t Discrete time

max Maximum value min Minimum value Tot Total T Transpose of a vector

G Grid MG Microgrid

ACKNOWLEDGMENTS The present study was supported by the U.S. National

Science Foundation under grant NSF–EFRI 0835879 and the Mexican National Council of Science and Technology (CONACyT) under assistantship ID 213158.

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