A Quasi-Feed-In-Tariff policy formulation in micro-grids A ......micro-grids. Research gaps, price function, some underlying assumptions and preliminaries, and QFIT PM's and GENCO's

A Quasi-Feed-In-Tariff policy formulation in micro-grids: A bi-levelmulti-period approach

Ahmad F. Taha a, Nadim A. Hachemb, Jitesh H. Panchal c,n

a School of Electrical and Computer Engineering, Purdue University, West Lafayette, IN 47906, USAb Department of Electrical and Computer Engineering, American University of Beirut, Beirut, Lebanonc School of Mechanical Engineering, Purdue University, West Lafayette, IN 47906, USA

H I G H L I G H T S

� We present a bi-level optimization problem formulation for Quasi-Feed-In-Tariff (QFIT) policy.� QFIT dictates that subsidy prices dynamically vary over time depending on conditions.� Power grid's physical characteristics affect optimal subsidy prices and energy generation.� To maximize welfare, policy makers ought to increase subsidy prices during the peak-load.

a r t i c l e i n f o

Article history:Received 25 November 2013Received in revised form8 April 2014Accepted 10 April 2014

Keywords:Quasi Feed-In-TariffMicro-gridsBi-level optimizationSubsidy prices

a b s t r a c t

A Quasi-Feed-In-Tariff (QFIT) policy formulation is presented for micro-grids that integrates renewableenergy generation considering Policy Makers' and Generation Companies' (GENCOs) objectives assuminga bi-level multi-period formulation that integrates physical characteristics of the power-grid. The upper-level problem corresponds to the PM, whereas the lower-level decisions are made by GENCOs. Weconsider that some GENCOs are green energy producers, while others are black energy producers. Policymakers incentivize green energy producers to generate energy through the payment of optimal time-varying subsidy price. The policy maker's main objective is to maximize an overall social welfare thatincludes factors such as demand surplus, energy cost, renewable energy subsidy price, and environ-mental standards. The lower-level problem corresponding to the GENCOs is based on maximizing theplayers' profits. The proposed QFIT policy differs from the FIT policy in the sense that the subsidy price-based contracts offered to green energy producers dynamically change over time, depending on thephysical properties of the grid, demand, and energy price fluctuations. The integrated problem solves fortime-varying subsidy price and equilibrium energy quantities that optimize the system welfare underdifferent grid and system conditions.

& 2014 Elsevier Ltd. All rights reserved.

1. Introduction

The objective in the paper is to develop a bi-level multi-perioddecision making formulation in micro-grids for energy marketsthat integrates renewable energy generation considering differentPolicy Makers'(PM) and Generation Companies'(GENCOs) objec-tives, while taking into consideration the effect of power linecharacteristics and the physical constraints in the system. Thepolicy maker (or in some cases, the independent system operator)

represents the governing and monitoring body of the grid, that isconcerned with the welfare of the power grid, rather than theprofitability. In the bi-level problem, the upper-level problemcorresponds to the PMs, whereas the lower-level decisions aremade by competing market players or GENCOs. The goal of the PMis to maximize an overall social welfare (OSW) measure whichdepends on overall system reliability, price stability, supply sur-plus, percentage of renewable energy generation, and otherfactors. On the other hand, GENCOs are profit maximizing entities.GENCOs' plants are subject to physical constraints in terms of lineloading capabilities, Power Transfer Distribution Factors (PTDFs),as well as profitability considerations. We study the short-term(hourly, daily) planning and the interaction between the PM andGENCOs in a micro-grid, as well as the effect of different line

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http://dx.doi.org/10.1016/j.enpol.2014.04.0140301-4215/& 2014 Elsevier Ltd. All rights reserved.

n Corresponding author.E-mail addresses: [email protected] (A.F. Taha),

[email protected] (N.A. Hachem), [email protected] (J.H. Panchal).

Please cite this article as: Taha, A.F., et al., A Quasi-Feed-In-Tariff policy formulation in micro-grids: A bi-level multi-period approach.Energy Policy (2014), http://dx.doi.org/10.1016/j.enpol.2014.04.014i

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parameters on the optimal generation, and the eventual priceaccording to the demand. A better understanding of the interac-tion between PMs and GENCOs would enable the design of policiestowards improved grid performance.

The formulation presented in this paper takes into account theintegration of renewable energy standard (RES, also called Renew-able Portfolio Standard) into the smart-grid infrastructure(Weisenmiller et al., 2012a). Through providing incentives for thegreen energy producers (GEPs), the PM targets achieving a certainlevel of renewable energy production. Feed-In-Tariff (FIT) andTradable Green Certificate (TGC) are examples of policies thatencourage GENCOs to invest in green energy production. A FIT isan energy-based policy that provides long-term incentives forGENCOs through payment of a regular subsidy price per unit ofgreen energy generated (Couture et al., 2010). This policy accel-erates the investment in green energy production. FIT policy'sother objectives are job creation, decreasing electricity prices,growing the overall economy, building environment-friendlyplants, managing waste streams, and attracting new investments(Couture and Cory, 2009). TGC policy is different than FIT policy inthe sense that TGC GEPs are offered certificates proportional to thegreen energy produced. GENCOs utilizing TGC policy can tradethese certificates independently of electricity (Tamas et al., 2010).FIT and TGC policies' common objective is to achieve a certain levelof renewable energy production. Albeit policies like FIT and TGChave numerous advantages and benefits, some arguments couldbe raised against such policies. As discussed by Couture et al.(2010), difficulty controlling overall policy costs, near-termupward pressure on electricity prices, and distortion of wholesaleelectricity market prices are some of the disadvantages of FITpolicy. In addition, shortage of capital leading to the exclusion ofsmaller participants from the market, price instability of TGCswhen the system is near a target level and uncertainty in futureenergy prices are some potential TGC disadvantages (Poputoaiaand Fripp, 2008).

The bi-level formulation developed in this paper utilizes theintegration of these policies in the micro-grid economies. It alsoincorporates the impact of line parameters on the optimal powerflow (OPF) in the system. The Quasi-FIT (QFIT) policy proposed inthis paper is different from the usual FIT policy in the sense thatthe subsidy price-based contracts offered to green energy produ-cers are varying over time, depending on the physical properties ofthe power-network (such as line-losses and minimum and max-imum generation), demand, and energy price fluctuation, ratherthan being held constant over a long-term contract.

The paper is organized as follows. In Section 2, we present therelevant work and background in the area of computationalmodeling in energy markets. We also give an overview of OPF inmicro-grids. Research gaps, price function, some underlyingassumptions and preliminaries, and QFIT PM's and GENCO'soptimization problem formulation are presented in Section 3. InSection 4, a solution strategy for the bi-level, multi-period pro-blem is outlined. Simulation results for different system character-istics and scenarios are included in Section 5. Section 6 presentsclosing comments and conclusions.

2. Background

In this section we provide a succinct background on therelevant research areas covered throughout the paper. A simplemodel of the lower-level non-cooperative game and decisionmaking problem for GENCOs is discussed in Section 2.1. Theoptimization methodology used to model constrained non-cooperative games along with an overview of tools used to solvethe optimization problem is reviewed in Section 2.2.

2.1. GENCOs' interaction as a Nash–Cournot game

Wemodel the lower-level problem as a non-cooperative, Nash–Cournot game where players make decisions independently. TheNash equilibrium of the non-cooperative games is generally usedas a solution for problems associated with several players. TheNash equilibrium is a solution that guarantees that no player canindividually improve their profits by changing their own strategies(Osborne and Rubinstein, 1994). The Nash–Cournot game (Hobbs,2001; Han and Liu, 2013) is widely used in modeling non-cooperative games in energy markets. In this paper, we assumethat the lower-level problem is modeled as Nash–Cournot game.

2.2. Bi-level problem formulation

In this paper, we develop a multi-period bi-level problemformulation in energy markets. The upper-level decision problemcorresponds to the PM, while the lower-level optimization pro-blem corresponds to the GENCOs. We first derive the equilibriumconstraints on the lower-level problem. In the problem formula-tion, we assume various constraints for both the lower- and upper-level decision problems. Formulating the First Order NecessaryConditions (FONC) of optimality is the first step in finding theoptimal solution to any non-cooperative constrained optimizationproblem. These optimality conditions are also called the Karush–Kuhn–Tucker (KKT) conditions. The FONC formulation results in acomplementarity problem (Ferris and Pang, 1997). The comple-mentarity problem is either a linear (LCP) or a nonlinear com-plementarity problem (NCP), depending on the nature of theconstraints and the objective function. The solution can be foundthrough methods developed for complementarity problems. Com-plementarity problems find the problem solution, as well as theoptimization problem's multipliers (Murty and Yu, 1997). Murtyand Yu (1997), Duan et al. (2010), and Gabriel et al. (2012) providevarious algorithms that are used to solve both LCPs and NCPs.

In this paper, we formulate the nonlinear complementarityproblem using the KKT conditions. These conditions, resultingfrom the lower-level optimization problem, are used as a set ofconstraints in the upper-level PM's decision making problem. Thisintegration of the lower-level problem's optimality conditionsresults in a mathematical problem with equilibrium constraints(MPEC) (Pieper, 2001; Hawthorne and Panchal, 2012). Due to thelack of convexity and nonlinearity of MPECs, the solution derivedfor the upper-level problem can be combinatorial (Hawthorne andPanchal, 2012). The non-linearity of the problem and the non-convexity of the feasible space make the feasible space too smalland hence the formulated problem a challenging one to solve.Efficient algorithms have been developed for solving MPEC pro-blems. Penalty interior-point algorithm (PIPA), piece-wise sequen-tial quadratic programming (PSQP), smoothing SQP, and someimplicit function based methods are all algorithms that have beendeveloped to solve such MPECs (Pieper, 2001). Gabriel et al. (2012)provided an extensive review on LCPs and MPECs, with applica-tions modeling natural gas markets. In this paper, the MPEC's mainobjective is to solve for the time-varying optimal generationquantities and subsidy prices for the QFIT bi-level policy, con-sidering price fluctuations, peak demand periods, and physicalproperties of the power-network.

2.3. Optimal power flow

The optimal power flow (OPF) problem seeks to controlgeneration/consumption to optimize certain objectives such asminimizing the generation cost or power loss in the network. It isone of the fundamental problems in power system operation (Gan

A.F. Taha et al. / Energy Policy ∎ (∎∎∎∎) ∎∎∎–∎∎∎2





and Low, 2013). In this section, we address the power flow aspectof the energy market.

2.3.1. Transmission lossesThe physical characteristics of transmission lines, represented

by the resistance and inductance, lead to electrical losses when-ever electrical power is transmitted through these lines. Transmis-sion losses are proportional to the current passing through theline, and despite the significant reduction that high-voltagetransmission achieves, these losses still impact the overall systemand the cost of energy production (Benedict et al., 1992). Thiseffect can be seen in the power balance equation that governs theoperation of the power system:

∑m

j ¼ 1PG;j ¼ PDþPLðPG;1; PG;2;…; PG;mÞ; ð1Þ

where PG;j is the power generated from the jth GENCO, PD is thepower demanded by the market and PLðPG;1;…; PG;mÞ is theaggregate lost power in the transmission lines (Benedict et al.,1992). High transmission losses imply that a higher generation isrequired to satisfy the total demand, leading to higher costs andhence higher prices on the consumer side. For that reason, the PMtries to minimize transmission losses in an attempt to increase theoverall welfare of the system.

2.3.2. Transmission lines capacity and congestionCongestion occurs in electrical power systems when power

transmission over a line is constrained by the maximum allowableflow in that line (PJM, 2010). Going above that limit can increasethe losses and may jeopardize the stability of the overall system.

2.3.3. Power transfer distribution factors (PTDF)In power systems, a transaction between nodes m and n is

defined as a power injection by the generator at node m that isremoved by the load at node n. By definition, the PTDF is thefraction of the amount of a transaction from one zone to anotherthat flows over a given transmission line (Christie et al., 2000). It isstudied in order to understand the effect of a transaction on thestate of all branches in the system, namely in terms of congestion.The power flow in line l�k due to a 1MW transaction betweennode m and node n in a power grid is given by

PTDFlk;mn ¼Xlm�Xkm�XlnþXkn

xlk; ð2Þ

where xij is the reactance on transmission line l�k and Xlm is theentry on the lth row and mth column of the reactance matrix(Huang, 2011). Since micro-grids are in fact a small-scale version ofthe larger macro-grid, we assume that the different GENCOs in themicro-grid have a single load to feed in the form of the PM, who isthen responsible for feeding energy to consumers. Hence, allpower flows in the system are between the jth GENCO and a fixedload. The PTDF can therefore be simplified to PTDFðjÞl�k, which isdefined as the power flow in line l�k due to a 1MW generation bythe jth stakeholder.

3. Problem formulation

The work presented in this paper is an extension of Hawthorneand Panchal (2012) and Taha and Panchal (2013a,b), which wasfocused on the policy design problems related to decentralizedenergy infrastructure for multiple technologies, considering theuncertainty in stakeholders' preferences and targeting long-runpolicy goals. In this paper, we focus on the short-run objectives forthe system operators such as sustaining a certain level of overallsocial welfare, while considering price and demand fluctuations

during multiple periods. Through data from the UK, Tamas et al.(2010) studied FIT and Tradable Green Certificate (TGC) policiesand derived the equilibrium expressions for the generation quan-tities and subsidy prices. In our formulation, we use a similarsocial welfare function. The formulation in Tamas et al. (2010) doesnot take into account the time-varying demand in the energymarket, nor the variation of demand in the decision problem. Thislimitation is addressed in this paper. As mentioned in Section 1,the main objective of the paper is to develop a bi-level formulationin micro-grids that integrates renewable energy generation con-sidering different PMs' and GENCOs' objectives, transmission lineparameters and the power-network topology. However, formula-tions in the area of policy decision making in micro-grids arebased on studying individual entities, one at a time (Masters,2004; Kirschen and Strbac, 2004). In this paper, we address thislimitation by studying the interaction between the two majordecision making entities in the power markets, taking into accountthe physical properties of the grid.

In this section, we provide a bi-level mathematical formulationof the optimization problems corresponding to PMs and GENCOs.This formulation takes into consideration real-time energy pricingand multi-period decision making. In Section 3.1, the underlyingassumptions and preliminaries are stated. In Sections 3.2 and 3.3,we present upper and lower level formulations of the problem bythe PM and GENCOs, respectively.

3.1. Assumptions and preliminaries

The formulation presented in this paper corresponds to short-term planning in decentralized, deregulated energy markets. Short-term planning aims to optimize the hourly/daily operation of themicro-grid. Key issues in the short-term planning framework are:supply–demand balance, fair electricity pricing, and maintainingoverall system reliability (Kirschen and Strbac, 2004). We consider ascenario with n time periods. Index i represents a specific time-period. In this paper, we assume that a time-period is defined over1 h (the PM solves for generation dispatch quantities and GENCOsgenerate these quantities for a given price). We solve the bi-level,multi-objective optimization problem in each period i, based onhourly update signals from the PM. We assume that there are mdistinct GENCOs. Index j represents a specific GENCO. Out of the mGENCOs, k of them are Green Energy Producers (GEP), thus ðm�kÞproduce non-renewable fossil fuel based energy or Black EnergyProducers (BEP). The energy quantity supplied by the jth GENCO in

the ith time-period is denoted by ~qðjÞi for BEPs, qðjÞ

i for GEPs. Thisdistinction is important since we assume that renewable energygenerating companies receive a subsidy per unit energy generated.Let qi be defined as the total quantity generated during the ith time-period. Let the sets J and ~J be the index sets of GEPs and BEPs.

We also assume that there are o energy consumers. Theconsumer index is notated as u. The quantity demanded by uthconsumer in the ith time-period is denoted by dðuÞi . The electricityprice is updated in each time-period. Consumers adjust theirconsumption based on their needs and based on the market pricesfor electricity. In addition, GEPs are rewarded with a subsidy priceΔðjÞ

i which varies during different time-periods and among differ-ent renewable energy production types. The main objective of theformulation is to solve for the dynamically optimal subsidy pricesand energy generation quantities for each GENCO during all time-periods, whilst demand and energy market price change. Further-more, as highlighted in Tamas et al. (2010) and Newbery (1998),we assume that the electricity demand is defined by a linearinverse demand function, pi ¼ α�βQ , where pi is the price paid byconsumers and Q is the total amount of green and black energyproduced in the ith time-period.

A.F. Taha et al. / Energy Policy ∎ (∎∎∎∎) ∎∎∎–∎∎∎ 3





3.2. Upper-level QFIT policy problem formulation: PM's objectivesand constraints

In this section, we formulate the objectives of the PM alongwith the underlying constraints. A PM is a non-profit organization,hence it's main goal is to improve overall social welfare (Tamaset al., 2010), rather than overall profitability, as mentioned earlierin Section 1.

The overall social welfare (OSW) measure used in this formula-tion is adapted from Tamas et al. (2010) and is defined as

OSWi ¼ μCSCSi�μCCi�μFFi�μEEi ð3Þwhere

CSi ¼ ∑m

j ¼ 1ðqðjÞi �qLÞ� ∑

o

u ¼ 1dðuÞi ;

Ci ¼ ∑k

j ¼ 1qðjÞi ðpiÞþ ∑

m

j ¼ kþ1

~qðjÞi ðpiÞ;

Fi ¼ ∑m

j ¼ 1ΔðjÞ

i qðjÞi and

Ei ¼ ∑m

j ¼ 1κ ~qðjÞ

i :

CS is the Consumption Surplus (or the surplus in supply); C is basicenergy cost; F is the Renewable Energy Policy Price (total subsidyprice); E is the Environmental Damages Cost (which is directlyproportional to the black energy production quantities (multipliedby a constant κ).

The PM's main objective is to maximize the OSW in each time-period (4), where μfCS;C;F;Eg are weight parameters that depend onthe PM's preferences, subject to multiple constraints (4):

PM’s optimization problem :

maximize : OSWi ¼ μCSCSi�μCCi�μFFi�μEEi

subject to : 0r ∑m


o

u ¼ 1dðuÞi rUi

qL ¼ ∑m

l ¼ 1∑m

r ¼ 1ðγl� rql� r;iÞ

ql� r;i ¼ ∑m

j ¼ 1PTDFðjÞl� rq

ðjÞi rql� r;max

ρ ∑m

j ¼ 1qðjÞi r ∑

k

j ¼ 1qðjÞi

OSWminrOSWi

0r ∑m

j ¼ 1ΔðjÞ

i rΔmax

8>>>>>>>>>>>>>>>>>>>>>>>><>>>>>>>>>>>>>>>>>>>>>>>>:

:

ð4ÞThe power flow in line l�k during the ith time-period is defined asql�k;i; ql�k;max is the maximum power flow capability of branchl�k; γl�k is the power loss factor in line l�k; PTDFðjÞl�k is the powerflow in line l�k due to 1MW generation by jth stakeholder; qL arethe total losses in the transmission lines. Ui is an upper bound onthe surplus in supply; ρ is the regulatory percentage of the totalenergy produced from green energy sources (California Renew-ables Portfolio Standard (RPS) states that ρ¼0.33 by 2020,Weisenmiller et al., 2012b); OSWmin is the minimum overall socialwelfare determined by the PM; Δmax is the maximum subsidyprice that the PM is willing to pay for the GEPs. The PMs solve thisoptimization problem in each time-period i. In the next section,we formulate the GENCOs' optimization problems.

3.3. Lower level problem formulation: GENCOs' objectives andconstraints

As mentioned in Section 3.1, we assume that there are mGENCOs, out of whom k are GEPs and ðm�kÞ are BEPs. We alsoassume that the lower-level problem is modeled as Nash–Cournotgame, which is widely used in modeling non-cooperative games in

energy markets. Generators have generation and ramping constraintsthat lead to a minimum power limit below which the generatorcannot operate efficiently. First, large generating units cannot remainstable (i.e., maintain their online status) when operating at very lowoutput, hence the minimum output power below which the gen-erator is completely switched off (offline). Additionally, the start-upof large generators is a gradual and costly process, and startup cantake up to several hours (Sumbera, 2011), during which costs areincurred without any power produced. Consequently, generationcompanies tend to keep their generators at a minimum powerproduction, qmin, unless the unit is not planned to be used in theshort term future. A GENCO's objective is to maximize its net profitssubject to three constraints (5):

GENCO’s optimization problem :

maximize : V ðjÞi ¼ qðjÞi ðpiþΔðjÞ

i �CðjÞm � IðjÞÞ� f ðjÞ

subject to : gðjÞ1 : qðjÞi rqðjÞmax

gðjÞ2 : qðjÞminrqðjÞigðjÞ3 : V ðjÞ

minrV ðjÞi

8>>>>>><>>>>>>:

;

ð5Þ

where V ðjÞi is the payoff function to be maximized, qðjÞmax and V ðjÞ

min arethe generation capacity limit and the minimum profit margin for thejth stakeholder, respectively. Cm is the sum of operation and main-tenance cost per unit production; IðjÞi is the capital investment cost;f ðjÞ is the fixed cost of production for each GENCO, which isindependent on the level of energy production (such as mortgagecosts and fixed taxations).

4. Solution methodology

In this section, we outline the methodology used to solve thebi-level, multi-period optimization problem. First, we derive theKKT conditions of optimality for the lower-level GENCOs' optimi-zation problem. We treat these conditions as constraints in theupper-level problem corresponding to the QFIT PM decisionproblem. Second, we append these conditions to the upper-levelproblem. Finally, we solve the augmented optimization problemusing existing algorithms for solving MPECs.

The PM pays no subsidy price for the black energy producersð ~qðjÞ

i Þ. As mentioned in Section 4, we solve the optimizationproblem for each time-period. Hence, for simplicity we drop thetime-index (i). We can reformulate the GENCO's payoff function asgiven in the following equation:

V ðjÞ ¼V

ðjÞ ¼ qðjÞ α�β ∑k

j ¼ 1qðjÞ þ ∑

m

j ¼ kþ1

~qðjÞ !

þΔðjÞ � CðjÞm � I

ðjÞ !

� fðjÞ; jA J

~VðjÞ ¼ ~qðjÞ α�β ∑

k

j ¼ 1qðjÞ þ ∑

m

j ¼ kþ1

~qðjÞ !

� ~CðjÞm � ~I

ðjÞ !

� ~fðjÞ; jA ~J :

8>>>>><>>>>>:

ð6Þ

Let

gðqðjÞÞ ¼ ½gðjÞ1 gðjÞ2 gðjÞ3 �> ¼ ½qðjÞi �qðjÞmax qðjÞmin�qðjÞi V ðjÞmin�V ðjÞ

i �>

be the vector of constraints formulated from (5). The GENCOoptimization problem can be rewritten as

minimize : �V ðjÞ ð7aÞ

subjectto : gðqðjÞÞr0: ð7bÞ






The KKT optimality conditions for the constrained GENCO problemcan be formulated as

KKT conditions :

μnðjÞ ¼ ½μnðjÞ1 μnðjÞ

2 μnðjÞ3 �> Z0

� ∂V ðjÞ

∂qnðjÞþμnðjÞ > ∂gðqnðjÞ Þ

∂qnðjÞ¼ 0

μnðjÞ >gðqnðjÞ Þ ¼ 0

gðqnðjÞ Þr0:

8>>>>>>><>>>>>>>:

Deriving the second KKT condition, we get the following equations:

μnðjÞ1 �μnðjÞ

2 þð1þμnðjÞ3 Þ 2βqnðjÞ �ðαþΔðjÞ � C

ðjÞm � I

ðjÞ �β ∑m

j ¼ 1;ja jqnðjÞ Þ

!¼ 0; jA J ;

ð8aÞ


2 þð1þμnðjÞ3 Þ 2β ~qnðjÞ �ðα� ~C

ðjÞm � ~I

ðjÞ �β ∑m


!¼ 0; jA ~J : ð8bÞ

The third KKTcondition can be written as in the following equations:

μnðjÞ1 ðqnðjÞ � qmaxÞþμnðjÞ

2 ðqmin� qnðjÞ ÞþμnðjÞ3 ðV min� V

ðjÞÞ ¼ 0; jA J ;

ð9aÞ

μnðjÞ1 ð ~qnðjÞ � ~qmaxÞþμnðjÞ

2 ðqmin� qnðjÞ ÞþμnðjÞ3 ð ~Vmin� ~V

ðjÞÞ ¼ 0; jA ~J :

ð9bÞThis system of equations produces m unknowns corresponding tothe equilibrium production quantities ðqðjÞÞ, and 3m unknownscorresponding to the KKT multipliers. The system of equations isundetermined with a total of 4m unknowns and 2m equations. Asmentioned in Section 4, these KKT optimality conditions (are alsoreferred to as equilibrium conditions) are added to the PM optimiza-tion problem. For a fixed time-period i, the corresponding upper-level optimization problem formulation can be rewritten as in thefollowing equation:

This formulated optimization problem, which includes optimalityconditions from lower-level problems, is called a MathematicalProgram with Equilibrium Constraints (MPEC). The decisionvariables in this MPEC problem are the subsidy price ðΔðjÞÞ and

equilibrium generation quantities ðqðjÞÞ for a given time-period andall stakeholders. The objective function to be optimized (OSW) is aquadratic function of the decision variables. Some of the con-straints in (10) are linear (such as bounds on OSW ;Δ;V ; q). Otherequality constraints, such as the equilibrium constraints, are non-linear. The formulated MPEC (10) can be rewritten as

minimize : �OSWiðxiÞ ð11aÞ

subject to : ciðxiÞr0 ð11bÞ

0rμi ? cieq ðxiÞZ0 ð11cÞ

xrxirx ; ð11dÞ

8 i¼ 1;2;…;n; ð11eÞwhere

x¼ ½ ~qð1Þi … ~qðkÞ

i qðkþ1Þi …qðmÞ

i Δð1Þi … ΔðkÞ

i �>

is the decision variable vector we are solving for, ciðxÞ is theinequality constraints vector (demand-supply balance limit andrenewable electricity standard), cieq ðxÞ represents the lower-levelequilibrium constraints, resulting in the complementarity pro-blem, and ½x ; x � are the bounds on the decision variables.

5. Simulation results

Based on the MPEC formulation (10) and fixing the market andGENCO parameters for generic values, we provide results that exploreand analyze the optimal values for the subsidy price ðΔðjÞÞ andgeneration quantities ðqðjÞi Þ for all GENCOs during each time-period,under different testing conditions: normal conditions, low capacityand high losses on a branch. Through the results, we also illustratethe feasibility of the solution and validate market scenarios.

5.1. System modeling

The model of the studied system is shown in Fig. 1. The systemis composed of four GENCOs (nodes 1, 2, 3 and 4) and a single load

maximize : OSWi ¼ μCSCSi�μCCi�μFFi�μEEi

subject to : 0r ∑m


o

u ¼ 1dðuÞi rUi

qL ¼ ∑n

l ¼ 1∑n

k ¼ 1ðγl�kql�k;iÞ

ql�k;i ¼ ∑m

j ¼ 1PTDFðjÞl�kq

ðjÞi rql�k;max

ρ ∑m

j ¼ 1qðjÞi r ∑

k

j ¼ 1qðjÞi

OSWminrOSWi

0r ∑m

j ¼ 1ΔðjÞ

i rΔmax

μnðjÞ ¼ ½μnðjÞ1 μnðjÞ

2 μnðjÞ3 �> Z0


2 þð1þμnðjÞ3 Þ 2βqnðjÞ �ðαþΔðjÞ � C

ðjÞm � I

ðjÞ �β ∑m


!¼ 0; jAJ


2 þð1þμnðjÞ3 Þ 2β ~qnðjÞ �ðα� ~C

ðjÞm � ~I

ðjÞ �β ∑m


!¼ 0; jA ~J

μnðjÞ1 ðqnðjÞ � qmaxÞþμnðjÞ

2 ðqmin� qnðjÞ ÞþμnðjÞ3 ðV min� V ðjÞÞ ¼ 0; jAJ

μnðjÞ1 ð ~qnðjÞ � ~qmaxÞþμnðjÞ

2 ðqmin� qnðjÞ ÞþμnðjÞ3 ð ~Vmin� ~V

ðjÞÞ ¼ 0; jA ~J

:

8>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>><>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>:

ð10Þ






(node 5) that represents the PM, encapsulating the demand side ofthe system. These simulation results are assumed for a micro-gridthat supplies energy to a small city with a population of 10,000–12,000. As mentioned in Section 5.1, we assume that there are fourmain GENCOs (2 BEPs, 2 GEPs). GENCOs 1 and 2 are assumed to beGEPs while GENCOs 3 and 4 are assumed to be BEPs. As seen in thefigure, all GENCOs are directly connected to the load, whileGENCOs of similar energy production type are also interconnected.Furthermore, a link between GENCOs 1 and 3 ensures the cohe-siveness of the system and the ability to overcome any outage insome parts of it. The single load represents the consumer side ofthe system, ultimately embodied by the PM. The model structureis meant to represent an energy system where similar generationunits are in close proximity to one another.

The line characteristics are taken into consideration in the formof line loss factors (γl�k), the line loading capability ðql�k;maxÞ andthe system PTDF matrix parameters ðPTDFðjÞl�kÞ. Loss factors (γ)indicate the percentage of the power flow in a certain branch thatis lost in transmission. For instance, a 5% loss factor of branch 1–5means that only 95 kW out of 100 kW sent through that line arriveto the load. The line loading capability ðql�k;maxÞ specifies themaximum power that can flow through a particular branch in thesystem. The entry pij of the PTDF matrix is the proportion of powerflowing along line i as the result of an injection at node j (e.g.,p11 ¼ 0:25 states that 0.25 units of power will flow along line 1–2at the result of a 1 unit injection at node G1, to be withdrawnat the load node). Table 1 displays the PTDFs for the adapted5-node model.

5.2. Demand curve

An essential characteristic in any power flow calculation is theadopted demand curve. In the simulations, we use the NationalGrid New York electricity company's posted demand curve toestimate the hourly demand for the demand from the StandardService in New York (National Grid, 2013). We use the hourlyprojected load profile for the year 2013 to compute the averagehourly demand for the year (averaged value through all days andhours for a year). Fig. 2 shows the resultant demand curve.

5.3. Simulation methodology

As mentioned in Section 3.2, the overall optimization problemis an MPEC (10) which solves for the QFIT policy parameters: theoptimal time-varying generation quantities and subsidy prices. Tosolve for the derived MPEC (11) decision variables ðq;ΔÞ, we usethe Network-Enabled Optimization System (NEOS) server onlinesolver. Different solvers on NEOS server were tested. In thesimulations, we use the KNITRO nonlinear complementarity opti-mization problem solver. KNITRO solver provided optimal solu-tions compared to other optimization problem solvers such asCONOPT, MINOS, and filter. The coding language used to model ourMPEC and simulate the problem through KNITRO is AMPL. Othersimulation packages such as MATLAB's fmincon has been alsotested, providing unfeasible solutions.

Table 2 shows the simulation results for a standard case undernormal conditions that illustrates the feasibility of some of NEOSsolver's optimization problem solvers. Table 4 includes the pro-blem's parameters for this simulation. To test for different micro-grid scenarios, we change some parameters from Table 4 inSection 5.4. We list the changed parameters in the relevantsubsection of Section 5.4.

The variables listed in Table 2 ðqj;Δ j;OSW ;CS;REÞ are theaveraged outputs of the optimization problem solver for each solver

(i.e., q1 ¼∑T ¼ 24i ¼ 1 qðjÞj =T). Under normal conditions, PATH solver pro-

duces a higher total subsidy price than the one allowed

ðΔ1þΔ2 ¼ 17:32g4Δmax ¼ 10gÞ, as well as a higher averagesupply surplus (consumption surplus) than the upper bound on CSðCS ¼ 564 kW4U ¼ 500 kWÞ. In addition, CONOPT and filter solvers

Table 1PTDF matrix (Huang, 2011).

PTDF values

Branch G1 G2 G3 G4

1–2 0.2381 �0.381 0.0952 0.04761–3 0.2857 0.1429 �0.2857 �0.14291–5 0.4762 0.2381 0.1905 0.09522–5 0.2381 0.6191 0.0952 0.04763–4 0.0952 0.0476 0.2381 �0.3813–5 0.1905 0.0952 0.4762 0.23814–5 0.0952 0.0476 0.2381 0.6191

Fig. 2. Demand curve from National Grid NY Electricity Company (National Grid,2013).Fig. 1. 5-node system model.






produce negative values for the OSW function for many time-periodsand a significantly high Renewable Energy percentage (RE).

Table 3 shows the optimal generation quantities for all time-periods with the variation of demand. As shown in Table 3, KNITROprovides the most feasible solutions. It is observed that thefluctuations in the generation quantities between two consecutivetime-periods is lower than the fluctuations for the generationquantities produced by filterMPEC. Precisely, between the 8th and9th time-periods, q3 increases from 300 kW to 2384 kW. Thiscould cause stability problems for GENCO 3. Using KNITRO solver,energy generation is distributed evenly among the GENCOs,satisfying all the constraints and bounds of the formulated MPEC(10). Thus, we use KNITRO solver to simulate the overall problemfor different grid and system conditions.

5.4. Sample results

To analyze the effect of transmission line parameters on thegeneration quantities and optimal subsidy prices for the QFITpolicy and the overall social social welfare function, we considerdifferent test cases to simulate the overall problem. In Section5.4.1, we simulate the formulated model for standard transmissionline parameters. Section 5.4.2 includes simulation results when

changing the maximum capacity of one of the transmission lines.This allows us to test the impact of line loading capabilityconstraints on generation quantities and the social welfare mea-sure. In Section 5.4.3, we study the effect of high transmission line

Table 2Problem solution feasibility using different NEOS solvers.

SolversAveraged variables for all time-periods i:e:; q1 ¼∑T ¼ 24

i ¼ 1qðjÞjT

� �Feasibility

q1 q2 q3 q4 Δ1 Δ2 Δ1þΔ2 OSW CS RE ð%Þ

filterMPEC 5992 3967 1559 3482 2.09 2.13 4.22 42 506 66.39 NonfeasibleKNITRO 3169 2858 3940 4999 2.38 1.84 4.22 50 500 40.27 FeasiblePATH 3636 3530 3912 3910 10.20 7.12 17.32 63 564 47.81 NonfeasibleCONOPT 6000 3614 300 300 1.62 1.71 3.33 �4397 �4090 94.12 Nonfeasiblefilter 6000 4890 3559 753 2.10 2.14 4.24 �5 461 71.63 Nonfeasible

Table 3Optimal generation quantities for different solvers under normal conditions (red color illustrates unfeasibility).

Time, demand Optimization problem solver option (NEOS)

filterMPEC KNITRO PATH CONOPT

qð1Þi qð2Þi qð3Þi qð4Þi qð1Þi qð2Þi qð3Þi qð4Þi qð1Þi qð2Þi qð3Þi qð4Þi qð1Þi qð2Þi qð3Þi qð4Þi

00:00–01:00, 11047 6000 3615 300 2299 3000 2085 2101 5000 3233 3408 3316 3500 6000 3615 300 30001:00–02:00, 10646 6000 3615 300 1882 2142 1398 3261 5000 2762 2938 2902 3092 6000 3615 300 30002:00–03:00, 10465 6000 3615 300 1677 2982 2035 1535 5000 2617 2798 2772 2965 6000 3615 300 30003:00–04:00, 10464 6000 3615 300 1677 3158 2099 1302 5000 2730 2813 2873 3066 6000 3615 300 30004:00–05:00, 10627 6000 3615 300 1882 3168 2116 1482 5000 2780 2825 2957 3150 6000 3615 300 30005:00–06:00, 11033 6000 3615 300 2299 3000 2085 2101 5000 2848 2883 3042 3233 6000 3615 300 30006:00–07:00, 12209 6000 3615 300 3582 3038 2190 3226 5000 3742 4637 2576 2487 6000 3615 300 30007:00–08:00, 13289 6000 3615 300 4757 3075 2294 4248 5000 4091 4631 2980 2892 6000 3615 300 30008:00–09:00, 14597 5907 4185 2384 3488 3363 2616 5000 5000 3664 4112 4082 3994 6000 3615 300 30009:00–10:00, 15348 6000 4278 2700 3719 2960 3725 5000 5000 3816 4344 4300 4213 6000 3615 300 30010:00–11:00, 16089 6000 4405 3125 4002 3470 4044 5000 5000 3903 4432 4623 4536 6000 3615 300 30011:00–12:00, 16470 6000 4451 3275 4120 3736 4094 5000 5000 4188 4432 4633 4547 6000 3615 300 30012:00–13:00, 16673 6000 4486 3388 4180 3789 4243 5000 5000 4171 4405 4803 4718 6000 3615 300 30013:00–14:00, 16509 6000 4781 4322 2954 3725 4203 5000 5000 3952 4108 4984 4907 6000 3615 300 30014:00–15:00, 16461 6000 4704 4082 3157 2859 5000 5000 5000 3917 4070 4958 4882 6000 3615 300 30015:00–16:00, 16116 6000 4405 3125 4002 3470 4044 5000 5000 3993 3904 4841 4771 6000 3615 300 30016:00–17:00, 15600 6000 4326 2860 3825 3151 3845 5000 5000 4100 3625 4773 4711 6000 3615 300 30017:00–18:00, 14680 5903 4188 2395 3478 3376 2603 5000 5000 3775 3297 4463 4408 6000 3615 300 30018:00–19:00, 14350 6000 3780 932 5000 3222 2447 5000 5000 3821 3223 4400 4347 6000 3615 300 30019:00–20:00, 14277 6000 3758 850 5000 3151 2414 5000 5000 4039 2966 4280 4230 6000 3615 300 30020:00–21:00, 13997 6000 3715 687 5000 3098 2363 4894 5000 4169 2790 4205 4159 6000 3615 300 30021:00–22:00, 13504 6000 3615 300 4971 3082 2313 4433 5000 4145 2543 4079 4040 6000 3615 300 30022:00–23:00, 12515 6000 3615 300 3902 3045 2218 3508 5000 3668 2841 3702 3676 6000 3615 300 30023:00–00:00, 11801 6000 3615 300 2727 3011 2119 2479 5000 3161 2714 3344 3327 6000 3615 300 300

Average 5992 3968 1559 3483 3170 2858 3940 5000 3637 3531 3912 3910 6000 3615 300 300

Table 4Overall problem parameters.

Parameter Value Parameter Value

α $0.1 U 500 kWβ $1e�6 Ið1Þ $0.05

μcs 1 Ið2Þ $0.05

μC 0.25 Ið3Þ $0.002

μF 0.1 Ið4Þ $0.002

μE 0.1 Cð1Þm

$0.0005

K 0.01 Cð2Þm

$0.0005

γl�k 0.05 Cð3Þm

$0.0005

qðjÞmin300 kW Cð4Þ

m$0.0005

qðjÞmax6000 kW V ð1Þ

min$45

ql�k;max 7000 kW V ð2Þmin

$40

OSWmin 5 V ð3Þmin

$20

Δmax $0.1 V ð4Þmin

$20






losses in couple branches on the optimal generation quantities andmarket price.

5.4.1. Standard case, normal conditionsThe standard case will be used as a benchmark for all simula-

tions. Table 4 contains the adopted values for most problemparameters used in the simulations. The only changed parameterfor this simulation is Δmax ¼ $0:05 .

Transmission losses are 5% per line across the system. Thisassumption represents normal operating conditions and takes intoconsideration various losses throughout transmission. The max-imum subsidy price (Δmax) is equal to $0.1, α and β are set to $0.1and $0.000001, respectively. The line loading capabilities are set to7000 kW across the system. We assume that GENCOs 1 and 2 havehigher investment, operation and maintenance costs, and conse-quently higher required profitability margin. This assumption ismeant to give the optimization problem a renewable energyprioritization aspect that we will observe later on. Finally, themaximum and minimum generation levels for the GENCOs are setto be 5000 kW and 300 kW, respectively, except for GENCO 1ðqð1Þmax ¼ 6000 kWÞ, for reasons explored in the next section. Theminimum generation capacity is set to 300 kW for all units. Inaddition, the renewable energy generation percentage (ρ) is setto 30%.

Fig. 3 shows the optimal generation quantities for all GENCOsand the PM's objective function (OSW) under normal conditions asa function of time. The inverse relationship between the highdemand and OSW function can be seen. As the demand increases,the overall social welfare decreases due to tighter constraints onGENCOs and the PM. GENCOs produce more energy while operat-ing within their generation capacity limits, transmission lineconstraints, and profitability margins, while the PM has to payhigher subsidy prices for GEPs, and higher environmental damageis caused due to high energy production from BEPs.

Moreover, GENCO 1 generates over 50% of the supplieddemand, while operating at maximum capacity, as the optimalsolution shows. Generated power is transmitted through trans-mission line 1–5 and through the following paths: 1–2–5, 1–3–5,and 1–3–4–5. GENCO 2 produces roughly 32% of the demand,ensuring that the minimum renewable energy standard is satis-fied. GENCOs 1 and 2 produce more energy due to the higherminimum profitability value (V ð1Þ

min ¼ 45, while V ð4Þmin ¼ 20), hence a

need to produce more which is reflected by the solver's optimalsolution). GENCO 3 is operated at the minimum generation leveland GENCO 4 is limited to just 20% of the generation. As demandincreases, the reliance on the BEPs (GENCOs 3 and 4) increases asseen in Fig. 4.

This change is paralleled with a decrease in the output ofGENCO 1, which despite being the most economical source ofenergy at low loads, cannot satisfy high demands single handedly,and is therefore phased out to allow for the other GENCOs todistribute the required generation amongst them. This can beobserved between 9h00 and 17h00 when the demand goes abovethe 15,000 kW mark. The decrease in generation from GENCO 1 isalso met with an increase with the output of GENCO 2 that ensuresthe Green Energy Generation is still above the required level.

Fig. 5 shows the time-varying optimal subsidy prices ðΔ1;Δ2Þfor the QFIT policy. The first observation is that Δ1 and Δ2 arehighly correlated with the nature of the demand curve. For theQFIT policy proposed, the subsidy prices for both generatorsincrease from 1.8 to 1:9g for low demand time-periods, andincrease to 2:4g at peak loads. This increase in subsidy prices is aresponse to system's attempt to incentivize GEPs into more energyproduction, leading to the satisfying of the formulated MPEC'sconstraints in (10): satisfying the increased demand, and main-taining the decreasing green energy production above the mini-mum RES, as seen in Fig. 4. Another observation is the fact that Δ2

exceeds Δ1 for all time-periods when the load is lower than14,600 kW. This is reflected by a dominance of q1 over q2 in theseFig. 3. Generation quantities and upscaled OSW under normal conditions.

Fig. 4. Black and green energy generation quantities: normal conditions. Fig. 5. Optimal subsidy prices under normal conditions.






same time-periods. In fact, with q1 already at the maximumgeneration limit, a high QFIT subsidy price (Δ2) must be paid forq2 in order to ensure that both the demand and the RES constraintare met simultaneously.

In order to test the feasibility of the solution under differentsystem conditions, we vary some parameters in different testcases. This allows us to observe the impact of these parameterson the system performance, generation quantities and transmis-sion lines loading, as well as the GENCOs' profitability and marketprices.

5.4.2. Low capacity on line 1–5In order to test the impact of line loading capability constraints

on system performance, we change the capacity of line 1–5 from7000 kW to 4000 kW while keeping all other system parametersunchanged. The optimal generation quantities for all GENCOs as afunction of time is plotted in Fig. 6.

Decreasing the capacity constraint of line 1–5 impacts thegeneration of all units. Under low-load conditions, the relianceon GENCO 1 as the main generator decreases from 9 time-periods

Fig. 6. Generation quantities under low capacity conditions.

Fig. 7. Generation quantities comparison between normal and low capacity conditions. (a) q1 and q2. (b) q3 and q4.

Fig. 8. Black and green energy production: low capacity.

Fig. 9. Subsidy price comparison for GENCO 1 under normal and low capacityconditions.






to 7 time-periods (a time-period is defined as 1 h). This is due tothe fact that the power generated from GENCO 1 and transmittedthrough the line is decreased due to the lower maximum capacityon that line. Consequently, starting at 7h00, using the three otherunits for generation becomes more economical than relying onGENCO 1 as the main provider. GENCO 1 is therefore phased out,with its output going below the 5500 kW mark by 7h00, as seen inFig. 7a below.

GENCOs 3 and 4 increase their output to the maximum of5000 kW, leading to an increase in the power flow in lines 3–4,3–5 and 4–5. This increase is due to the increase in load and thedecrease in the reliance on GENCO 1 and line 1–5. In fact,compared to the standard case, GENCO 4 generates over the1000 kW mark for four additional time-periods in the low linecapacity case, as seen in Fig. 7b. The need to maintain a minimumof 30% green energy production results in increasing GENCO 2'sgeneration to its maximum possible output, particularly between10h00 and 17h00 when peak load conditions occur. The Renew-able Energy Standard (RES) is thus satisfied despite the dominanceof black energy producers during peak hours, as illustrated inFig. 8.

In addition, the decrease in the generation of GENCO 1 leads toan increase in the subsidy price paid to that unit: the PM is willingto offer a higher subsidy price (Δ1) to give GENCO 1 an incentive toproduce more, and hence satisfying the RES. This is mostlyobserved during peak-load conditions, as shown in Fig. 9. Further-more, decreasing the capacity of a line imposes additional con-straints on the solution. The overall system solution is no lessfavorable to the PM compared to normal conditions. The PM has topay higher subsidy prices for the GEPs to ensure that both thedemand and the minimum green energy production constraint arecontinuously met as previously discussed. This leads to an overalldecrease in the OSW, particularly at peak demand time-periods.

5.4.3. High lossesIn order to understand the impact of faulty conditions and the

high line losses on the overall system, we increase the losscoefficient of lines 1–2 and 2–5 from 0.05 to 0.4 ðγ1�2 ¼ γ2�5¼ 0:4Þ. This allows us to understand the impact of high losses onthe optimal generation quantities of the GENCO that is mostaffected by these changes (in this case, GENCO 2). Since no feasiblesolution could be found using the original generation upper bound

constraint, the maximum power generation is changed to8000 kW for all generating units.

The chosen high-loss lines are both connected to the two GEPs,who are therefore the most affected by this high line losscondition. As seen in Fig. 11, the output of GENCO is approximately6700 kW throughout the day. This value is high enough to ensurethat the GEP percentage constraint is met for all demand values,regardless of all the variations in q2, as observed in Fig. 11.

Furthermore, simulation results show that as in the case forlow capacity on line 1–5 in the previous section, the optimal QFITpolicy subsidy prices increase compared to the subsidy prices forthe standard case. GENCOs 2 and 4 start the day generating4000 kW. However, as demand increases, GENCO 2 is phasedout, while GENCO 4 dominates the generation (producing6000 kW by 9h00). As shown in Fig. 11, GENCO 2 is practicallyshut down during the peak load conditions, operating at minimumgeneration between 13h00 and 21h00. The latter, combined with aconstant q1, leads to the domination of BEPs under high loadconditions, and a subsequent decrease in the OSW. In fact, anincreased reliance on GENCO 4 that makes up for the simultaneousphasing out of GENCO 2 can be seen in Fig. 10. GENCO 4 is

Fig. 10. Generation quantities comparison between normal and high losses condition. (a) q1 and q2. (b) q3 and q4.

Fig. 11. Generation quantities under high line loss conditions.






complementing GENCO 2's production level by increasing energyproduction when q2 is low, despite the absence of a physicalconnection between the two units. This decreased reliance onGEPs throughout the day, and particularly during peak demand,leads to a decrease in the OSW.

Transmission lines 1–2 and 2–5 become too costly under highline loss conditions. Hence, the optimal solution under high-lossconditions avoids these lines as much as possible, and focuses thegeneration and transmission on GENCOs 1 and 4 and paths 1–5,4–5, and 4-3-5. This hypothesis is confirmed by looking at acomparison of the power flow in the system between normal andhigh-loss conditions during peak hours, as shown in Fig. 12.Furthermore, increasing the loss factors in lines 1–2 and 2–5 leadsto a decrease in q2. The 40% decrease in the power flow in paths2–5 highlights the decreased reliance on GENCO 2 as a directenergy supplier in the system. As depicted in Fig. 12, the powerflow direction in line 1–2 changes under high loss conditions, withthe increase in q1, causing power to be transmitted from node 1 tonode 2. In addition, the increase in power flow for BEPs (GENCOs3 and 4) highlights an increase in generation and transmission,as can be seen by the 6800 kW and 5600 kW transmitted in lines4–5 and 3–5. The PM relies more on GENCOs 1 and 4 for energyproduction.

6. Future work and conclusions

The second lower-level problem in our formulation corre-sponds to large-scale consumers, represented by manufacturingfirms and commercial entities. In this paper, we assumed that thedemand is static. In the future work, we are interested in studyingdemand response: the variability and deviation in the consumers'energy consumption in response to the change in the market priceof electricity over time. According to the Federal Energy RegulatoryCommission (FERC), demand response is defined as the variabilityand deviation in the consumers' energy consumption in responseto the change in the market price of electricity over time. Inaddition, demand response also incorporates changing the con-sumers' behavior (decreasing energy consumption), throughincentivising end-users to consume less whenever system relia-bility is under jeopardy. It is estimated that a 5% lowering of

demand would result in a 50% price reduction during the peakhours of the California electricity crisis in 2000–2001. The marketalso becomes more resilient to intentional withdrawal of offersfrom the supply side.

6.1. Optimal demand backlogging problem

In this section, a formulation is presented for the optimalenergy backlogging problem developed in Roozbehani et al.(2011) for the low-level decision problem of the large-scaleconsumers. The formulation incorporates this model into the bi-level optimization problem proposed in this paper. This will leadto a better response by consumers during peak-demand periods,which will cause overall social welfare to increase. In the followingformulation, we assume that the demand from any consumer u intime-period i is defined as

dðuÞi ¼ dðusÞi þdðuf Þi ; i¼ 1;…;n; and u¼ 1;…; o;

where dðusÞi is the shiftable demand and d

ðuf Þi is the firm demand

which cannot be postponed. The firm demand dðuf Þi needs to befulfilled in the ith time-period, whereas shiftable demand can besatisfied in subsequent time-periods (any period t, where

tA ½iþ1;…;n0�). Let eðuÞi be the amount of electricity the consumer

allocates to fulfilling some or all of their shiftable demands ðdðusÞi Þ

at time i. Our goal is to find the optimal eðuÞi for all time-periods.

Denote by xðuÞi the amount of backlogged accumulated demand.

Define yðuÞi as the total consumed energy. We also define a penalty

function JiðxðuÞi Þ which represents a quantification for the incon-venience caused by the backlogging of the user demand. Hence,the dynamic optimization problem for the uth consumer can beformulated as

Consumer’s optimization problem

:

minimize : ∑n

i ¼ 1ðJiðxðuÞi ÞþpiðyðuÞi ÞÞ

subject to : xðuÞiþ1 ¼ xðuÞi þðeðuÞi �dðusÞi Þ

yðuÞi ¼ dðuf Þi þeðuÞi

8>>>>><>>>>>:

ð12Þ

Fig. 12. Line power flow under normal and high line loss conditions.






Given a price signal pi and information from consumers about

their preferences ðdðuf Þi ; dðusÞi ; JiÞ, the goal of the dynamic optimalbacklogging problem is to solve for the optimal allocated demand

ðeðuÞi Þ. After solving for the optimal demand, this quantity isincorporated in the PM optimization problem (4) by setting

dðuÞi ¼ eðuÞi þdðuf Þi for all time-periods.

6.2. Summary and conclusions

Micro-grids and decentralized energy systems are complexlarge-scale systems that use the information supplied by differentinteracting blocks, such as producers' and consumers' behaviorand preferences, to autonomously improve the overall socialwelfare and reliability. A better interaction between the differentcooperating blocks (PMs, GENCOs) leads to a better sustainabilityof micro-grid energy resources. Hence, it is crucial to formulateand analyze the interaction between the different stakeholders inthese energy systems, while implementing a certain renewableenergy policy or standard.

In this paper, we present a multi-period bi-level problemformulation of the decisions made by the PM and GENCOs,resulting in non-linear constrained MPEC, that formulates aQuasi-Feed-In-Tariff model. We assume that some GENCOs areGEPs, while others are BEPs. PM's main objective is to maximize anoverall social welfare utility function that includes meeting acertain renewable energy production standard and supply–demand balancing, as well as satisfying the physical constraintsof the transmission lines. The GENCOs' goal is to maximize theirprofits under certain capacity constraints. The necessary condi-tions of optimality for the lower-level optimization problem arealso derived. The problem's objective is to solve for optimal time-varying generation quantities and QFIT policy subsidy per unitunder different conditions such as peak-demand and high dis-turbance/losses in the power-network.

Simulation examples are shown to analyze and comparethe results under different conditions, in order to understandthe effect of line characteristics and system parameters on theperformance of the overall grid, as well as the impact on thegeneration levels for the GENCOs, the optimal subsidy prices, andthe energy price. We consider a simplistic market with two GEPs,two BEPs, and a load. The primary objective of the formulatedoptimization problem is to solve for the optimal generationquantities and dynamically varying subsidy prices (for GEPs),assuming a generic demand function. In order to solve theformulated bi-level MPEC, we simulate the system using differentoptimization problem solvers. Simulation results show the effectof the solvers used on the feasibility of the proposed solution.KNITRO provided feasible solutions for all time-periods. Further-more, results illustrate that the OSW function decreases duringpeak demand time-periods due to the increase in the QFIT policysubsidy price and generation quantities, and thus the environ-mental risks associated with the BEP. Similarly, tightening theconstraints on the GENCOs by line losses or low line capacities leadto less favorable conditions that decrease the OSW of the system.This subsequently results in an increase in the price. The resultsshow that GENCOs must receive a higher subsidy price duringpeak-demand time-periods. These results illustrate the appropri-ateness of the welfare function choice under different conditionsand constraints. In addition, the implementation of a time-varyingsubsidy price for GEPs (that depends on the grid and demandconditions) through the QFIT policy would increase the socialwelfare of the micro-grid.

In future work, our goal is to integrate a real-time analysis forthe demand side of a micro-grid as formulated in Section 6.1,where the consumers adjust their consumption optimally

according to a time varying price function. In addition, we willconsider constrained OP problems for GENCOs and PMs thatinclude power-flow constraints, voltage stability analysis, andother technical constraints. We also aim to study the stochasticnature of the demand-side to achieve a better understanding ofthe interaction between the consumers and other smart-gridblocks, as well as the uncertainty in the bi-level problemparameters.

Acknowledgments

The authors gratefully acknowledge the financial support fromthe National Science Foundation through NSF CMMI Grants1201114 and 1265622.

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