Multiple objective energy operation problem using Z utility theory

Abstract

In this paper, we introduce an energy operation model using three objectives of cost, environmental impact, and failure rate. Developed model increases the flexibility of energy systems in coping with changes in demands and energy prices while satisfying operational constraints. We use the Midwest Independent Systems Operator as a case study to show the efficient frontier for these objectives. We apply Z utility theory to choose the best alternative in our case study. Computational results and complexity analysis show that problems with extensive number of entities can be solved efficiently in reasonable time.

Appendices

Appendix 1: background for energy systems

Soylu et al. [45] introduces four building blocks for an energy system: raw material suppliers, energy producers, energy distributors, and consumers. Liu and Nagurney [32] introduce fuel supplier, power generator, power supplier, and demand market as required elements for an electricity supply chain system. Nagurney et al. [38] present electric power supply chains consisting of generators, power suppliers, power transmitters, and users. Psarras et al. [39] and Wang et al. [52] consider storage as a building block for energy systems. Psarras et al. [39] explain that introducing storage as a basic entity in energy systems brings several advantages including increasing system reliability, regulating the load and energy supply during demand fluctuations, reducing energy shortage and financial losses accompanying it, and meeting peak demand. Storages can minimize total costs by storing energy in periods with lower energy price and releasing it in period with higher price.

There is no consensus on a unique definition of distributed energy system in the literature. Though, generators in these systems are usually smaller than conventional energy plants in central energy systems and closer to users; see Alanne and Saari [3], Ackermann et al. [1]. According to Wang et al. [52], distributed energy systems have the potential to revolutionize the electricity industry. There is recently an increasing trend towards using distributed systems due to reliability concerns, efficiency, and environmental impact of centralized energy systems; see Alanne and Saari [3].

The determining factor in calling an energy system as “distributed” depends on the location of generators in the system. Terms “decentralized,” distributed, “embedded,” “autonomous,” and “dispersed” are used interchangeably in the literature to refer to distributed energy systems. However, depending on the model’s application, size, structure, and decision maker in the system, one of these terms is used predominantly. In this paper, we use the term distributed as it refers to a broader concept; see Pepermans et al. [40] and Alanne and Saari [3].

Comparison between central and distributed energy systems

Table 15 compares central energy systems with distributed energy systems and enumerates their corresponding advantages.

Since environmental friendly technologies have higher electricity supply cost, these systems are mostly operated in small scale that eventually result in lower distribution and transmission cost; see El-Khattam and Salama [17]. A clear advantage of distributed energy systems is that they are more efficient than central systems due to locality of entities; see Kirschen [27].

Table 15 Advantages of central and distributed energy systems

Variety of objective functions have been used to analyze distributed energy systems; see Alarcon-Rodriguez et al. [4], Hobbs and Horn [20], Leken [31], Martineza and Eliceche [37], and Ramanathan and Ganesh [41]. Issues of environmental concerns, reliability, and foreign oil dependence are among objectives that compete with minimizing cost.

Cost

The primary objective in all distributed energy systems is minimizing cost. The capital cost of electric power stations and storages are usually high, and energy demand is also changing frequently. Long-distant suppliers may also be cheaper than local sources.

Environmental impact

Environmental consideration is a major goal that energy systems need to accomplish. These goals are laws or international treaties which are set by governments; see Soylu et al. [45]. According to Vogel [50], cheaper energy source results in higher environmental impact and transmission and distributions cost. Eliceche et al. [16] considers the environmental impact as a life cycle associated with gaseous, liquid, and solid wastes of generators.

Reliability

Reliability of an energy system tells us the probability that the system can supply energy with a reasonable price; see Alanne and Saari [3]. According to the US Department of Energy report [15], power reliability is defined as the performance to which electricity is delivered to customers within accepted standards and in the desired amount.

Reliability of energy systems has two aspects: security and adequacy. Security is the ability of the energy system to withstand unexpected fluctuations in demand or sudden loss in the system due to natural causes. Adequacy provides enough supply of electricity at the proper voltage and frequency for customers. Generators and storages are subject to failure for a variety of reasons including overload, natural, or manmade disasters, technological failures, and physical damages. Outage can happen as a result of natural disaster, overloads, and drastic change in demand; see Billinton and Allan [9, 10], Lee [30], Lam and Li [29], and Sullivan et al. [46]. Energy providers may fail to fulfill required demand due to capacity constraints. Links are also subject to failure due to various reasons, and there is usually a failure probability associated with each link; see Billinton and Allan [10]. Failure dependencies happen when links share common equipment or when a natural disaster affects the links in the same geographical locations. However, due to the distributed nature of facilities in distributed energy systems, a malfunction in these systems affects considerably less area comparing with their central peers. Therefore, the reliability in these systems is more user-driven than system driven.

Appendix 2. Transforming energy operation problem to transportation problem

Transforming energy operation problem to transportation problem

Any single period of an EO problem can be transformed to transportation problem. The energy level in storage p, q_p,_t is determined based on the marginal energy price of the next period, Δs_p,t+1. Therefore, q_p,t is a given number. This results in reduction in number of decision variables and constraints. The reduction in number of constraints and variables is 3P and P respectively, where P is the number of storages. The remaining property of storages, hub property, makes the problem equivalent to transportation problem. There is at least one route from generator i to user j (one direct route from i to j and/or indirect routes from i to p and from p to j). The route with the least cost is always selected for energy transportation.

This section shows how to convert energy systems problem to a simpler transportation problem. It is possible to separate storing and hub properties of storages in EO problem (problem 1) and solve decomposed problems in subsequent steps. Amount of energy in storage p, q_p, can be separately determined before solving the energy optimization problem due to its large coefficient, G, in objective function. For example, for Δs = +2, the storage will be filled up to its maximum possible capacity, q_max.

There are two sets of transportation costs, Set_I and Set_II. Set_I contains all transportation costs whose minimum from i to j is through storage p, shown with c_ij,min = c_ipj, and its corresponding decision variable is x_ij,min = x_ipj. Set_II contains all transportation costs whose minimum from i to j is the direct link from i to j, which is shown with c_ij,min = c_ij and its corresponding decision variable is x_ij,min = x_ij.

Problem A.1 Modified energy system problem (transportation problem)

Minimize f ₁ (A 1)

$$ ={\displaystyle \sum_{i=1}^I{\displaystyle \sum_{j=1}^J{c}_{i j, \min }{x_{i j}}_{, \min }}}\kern.3em \mathrm{Transportation}\ \mathrm{costs}\kern1em {{\mathrm{c}}_{\mathrm{ij}, \min}}_{\in }{\mathrm{Set}}_{\mathrm{I}}\mathrm{or}{\mathrm{Set}}_{\mathrm{I}\mathrm{I}} $$

(A 1.1)

$$ +\kern0.75em {\displaystyle \sum_{i=1}^I{g}_{\mathrm{i}}{\displaystyle \sum_{j=1}^J{x}_{i j,\kern0.5em \min }}}\kern1em \mathrm{Buying}\ \mathrm{cost} $$

(A 1.2)

Subject to:

$$ {\displaystyle \sum_{i=1}^I{x}_{i j, \min }}\ge {\mathrm{D}}_j\kern1em \mathrm{for}\ \mathrm{all}\ j=1,\dots, J\ \left(\mathrm{Demand}\ \mathrm{for}\ \mathrm{user}\ \mathrm{j}\right) $$

(A 2)

$$ {\displaystyle \sum_{j=1}^J{x}_{i j,\kern0.5em \min }}\le {\mathrm{K}}_i\kern1em \mathrm{for}\ \mathrm{all}\ i=1,\dots, I\ \left(\mathrm{Capacity}\ \mathrm{of}\ \mathrm{gen}.\mathrm{i}\right) $$

(A 3)

$$ {x}_{ij \min}\ge 0\kern1em \mathrm{for}\ \mathrm{all}\ i\ \mathrm{and}\ j $$

(A 4)

Set_I = For all i and j, Min {c _ij , c _ip  + c _pj for all p} = c _ip

Set_II = For all i and j, Min{c _ij , c _ip  + c _pj for all p} = c _ij

Appendix 3: multi-objective decision making

Multi-objective decision making (MCDM) is concerned with structuring and solving decision problems considering multiple objective; see Malakooti [36] and Malakooti [35]. Two major categories have been defined in MCDM: (1) multi-objective discrete alternative problems with finite set of feasible alternatives and (2) multiple objective optimization problems with infinite set of feasible alternatives (Wallenius et al. [51]).

The complexity of multiple objective optimization problems is higher than multi-criteria discrete problems because the set of alternatives cannot be enumerated and explicitly presented. In this paper, application of multiple objective optimization in distributed energy systems has been shown.

3.1 Appendix 3.1: additive utility function

Alternatives in multi-objective optimization problem are not explicitly defined. f ₁, f ₂, and f ₃ in (A.5) show total cost, environmental impact, and failure rate, respectively. The linear additive utility function is

$$ U={w}_1{f}_1+{w}_2{f}_2+{w}_3{f}_3 $$

(A.5)

w ₁, w ₂, and w ₃ are the weights of importance of each criterion (objective) and w ₁ > 0, w ₂ > 0, w ₃ > 0. The summation of weights are one, w ₁ + w ₂ + w ₃ = 1.

Each criterion can be normalized as follows.

$$ {f_l}^{'}=\left({f}_l\hbox{--} {f_l}_{, \min}\right)/\left({f_l}_{, \max }-{f_l}_{, \min}\right)\kern1.25em \mathrm{for}\kern0.2em l=1,2,3 $$

where f _l ′ is normalized value of f _l . To be able to compare objectives with different units, they need to be normalized before being used in additive Z-utility and goal Z-utility functions; see Malakooti [36]. Therefore, additive utility function (A.5) is presented as:

$$ U'={w}_1{f}_1'+{w}_2{f}_2'+{w}_3{f}_3' $$

(A.6)

Equation (A.6) is also called Normalized Weighted Additive Utility Function (NWAUF), Malakooti [36] and Malakooti [33].

3.2 Appendix 3.2: goal-seeking MOLP approach

Problem A.2 finds the best solution which is closest to the goal even if the goal is inefficient; see Malakooti [35]. DM can select inefficient goals for optimization problem because he/she may not be aware of the whole set of alternatives (set of alternatives can be implicit). If the given goal dominates some of the efficient points, then obtained efficient will be at a minimum distance from the goal. If the given goal is dominated by some of the efficient points, then obtained efficient will be at the maximum distance from the goal. Note that z_G is a given value between −1 and 0 and it represents a risk averse DM.

Problem A.2 Z goal programming problem

$$ \mathrm{Maximize}\ \mathrm{U}=\mathrm{LV}+{\mathrm{z}}_{\mathrm{G}}{\mathrm{DV}}_{\mathrm{G}} $$

(A.7)

Subject to:

$$ \mathrm{LV}={\displaystyle \sum_{l=1}^3\left({w}_l\;{f}_l^{\hbox{'}}+{w}_l\;{f}_l^{\hbox{'}}\right)} $$

(A.8)

$$ {\mathrm{DV}}_G={\displaystyle \sum_{l=1}^3{w}_l}\left({d}_l^{+}+{d}_l^{-}\right) $$

(A.9)

$$ {f}_l'+{d_l}^{-}\hbox{--} {d_l}^{+}={f_l}_{,\mathrm{G}}'\kern1em \mathrm{for} l=1,2,3 $$

(A.10)

$$ {{\mathrm{d}}_l}^{+}\ge 0\ {{\mathrm{and}\ \mathrm{d}}_l}^{-}\ge 0\kern1em \mathrm{for} l=1,2,3 $$

(A.11)

$$ \begin{array}{l}{f_1}^{'}=\hbox{--} \kern0.5em \left({f}_1\hbox{--} {f}_{1, \max}\right)/\left({f}_{1, \max}\hbox{--} {f}_{1, \min}\right)\ \hfill \\ {}{f_2}^{'}=\hbox{--} \kern0.5em \left({f}_2\hbox{--} {f}_{2, \max}\right)/\left({f}_{2, \max}\hbox{--} {f}_{2, \min}\right)\ \hfill \\ {}{f_3}^{'}=\kern0.5em \left({f}_3\hbox{--} {f}_{3, \min}\right)/\left({f}_{3, \max}\hbox{--} {f}_{3, \min}\right)\ \hfill \end{array} $$

and all energy operation constraints

$$ {\mathrm{f}}_l\ \mathrm{is}\ \mathrm{unrestricted}\ \mathrm{in}\ \mathrm{sign}\ {{\mathrm{and}\ \mathrm{d}}_l}^{+}\ge 0\ {{\mathrm{and}\ \mathrm{d}}_l}^{-}\ge 0\kern1em \mathrm{for} l=1,2,3 $$

(A.12)

A polynomial goal convergent function can be used instead of Equation (A.7). Therefore, decision maker can consider the interaction between criteria f ₁ and f ₂, f ₁*f ₂, in utility function, see Equation (A.13).

$$ \mathrm{U}=\mathrm{LV}+{\mathrm{z}}_{1*}{\mathrm{DV}}_{\mathrm{G}}+{\mathrm{z}}_2{{\mathrm{DV}}_{\mathrm{G}}}^2+{\mathrm{z}}_3{{\mathrm{DV}}_{\mathrm{G}}}^3+.....+{\mathrm{z}}_{2\mathrm{k}}{{\mathrm{DV}}_{\mathrm{G}}}^{2\mathrm{k}} $$

(A.13)

This function is concave when −1 < z₁ ≤ 0, −1 < z₂ ≤ 0,… and 1 < z_2k ≤ 0 and −1 < z₁ + 2z₂ + 3z₃ + … + 2kz_2k ≤ 0.

Appendix 4: parameters for energy operation problem

For case study, the loss coefficients for all links in case study are assumed to be 0.9 and loss coefficients for storages are 1. Also, reliability index for generators (1, 2, 3) and storages (1,2) are (0.9, 0.8,0.7) and (0.7,0.8), respectively, and environmental impact factors for these generators are 0.4, 0.6, and 0.5, respectively. Parameters for case study are shown in Tables 16 and 17.

Table 16 Parameters for EO Case Study

We used generators capacities and users demands from MISO data available through the following link:

https://www.misoenergy.org/Library/MarketReports/Pages/MarketReports.aspx

For generator offers, we used “Day-Ahead Cleared Offers” CSV file with publication date 1/17/2013 after selecting “Offers” report type. Definition of data in this file is as follows:

LMP shows the market clearing price (or locational marginal price), and Price-MW pairs are the unit’s energy incremental offer prices (equivalent to the energy generation costs). MISO allows up to 10 monotonically increasing segments. We assume that there is a piecewise constant curve for all price-mw pairs. All costs and prices are in terms of US dollars/megawatt hour and all generators’ capacities are in terms of MW.

For demand information, we use “Cleared Bids” CVS file with publication date 1/17/2013 after selecting “Bid” report type. These reports are available by downloading Cleared Bids file after selecting the “Bids” report type in MISO website. MW represents the load demand of a user (often downstream from a distribution substation). LMP is the market clearing price for the load (locational marginal clearing price). Price-MW Pairs are the bid costs of the load.

Capacities and demands of generators and users for the case study is selected from day ahead offers list on 1/17/2013.

Table 17 Parameters for EO case study with 3 generators, 2 storages, and 3 users