This document is downloaded from DR‑NTU (https://dr.ntu.edu.sg)Nanyang Technological University, Singapore.

Power demand and supply management inmicrogrids with uncertainties of renewable energies

Wang, Ran; Wang, Ping; Xiao, Gaoxi; Gong, Shimin

2014

Wang, R., Wang, P., Xiao, G., & Gong, S. (2014). Power demand and supply management inmicrogrids with uncertainties of renewable energies. International Journal of ElectricalPower & Energy Systems, 63, 260‑269.

https://hdl.handle.net/10356/105068

https://doi.org/10.1016/j.ijepes.2014.05.067

© 2014 Elsevier. This is the author created version of a work that has been peer reviewedand accepted for publication by International Journal of Electrical Power & EnergySystems, Elsevier. It incorporates referee’s comments but changes resulting from thepublishing process, such as copyediting, structural formatting, may not be reflected in thisdocument. The published version is available at: [http://dx.doi.org/10.1016/j.ijepes.2014.05.067].

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Power Demand and Supply Management in Microgridswith Uncertainties of Renewable Energies

Ran Wang, Ping Wang, Gaoxi Xiao and Shimin GongNanyang Technological University, Singapore

Abstract

In the operation of microgrids, an important task is to match the power genera-tion and consumption at a minimum cost. Since the highly fluctuant renewableenergies constitute a significant portion of the power resources in microgrids, themicrogrid system central controller (MGCC) faces the challenge of effectivelyutilizing the renewable energies while fullfilling the requirements of customers.In this paper, a power demand and supply management framework is proposedto tackle the problem stated above. A novel uncertainty model is first developedto capture the randomness of renewable energy generation. Specifically, we in-troduce a reference distribution according to the past observations and empiricalknowledge, and then define a distribution uncertainty set to confine the uncertaintyof renewable energies. The new model allows the renewable energy to fluctuatearound the reference distribution. An optimization problem is then formulated todetermine the optimal power consumption and generation scheduling for minimiz-ing the fuel cost. A two-stage optimization approach is presented to first transformand then solve the prime problem. Numerical results indicate the properties of ourproblem formulation and provide some illuminations on the policy making for theMGCC. It is also shown that the proposed power demand and supply managementmechanism can effectively reduce the energy cost.

Keywords: microgrids, demand and supply management, renewable energyuncertainties, reference distribution, robust optimization

Email address: {wang0686,wangping,egxxiao,gong0012}@ntu.edu.sg(Ran Wang, Ping Wang, Gaoxi Xiao and Shimin Gong)

Preprint submitted to International Journal of Electrical Power & Energy Systems May 9, 2014

1. Introduction

The smart grid is the innovative future electric power system that will improvethe conventional electrical grid network to be more clean, reliable, secure, coop-erative, and efficient. The growth and evolution of the smart grid is expected tocome with the plug-and-play integration of the basic structures called microgrids.Specifically, microgrids are small-scale low voltage power supply networks de-signed to supply electrical load for a small community such as a university cam-pus, a commercial area and a trading estate, etc. Microgrids can autonomouslycoordinate local generations and demands in a dynamic manner. It can operatein either grid-connected mode or islanded mode [1]. There have been world widedeployments of pilot microgrids, such as those in US, Germany, Greece and Japan[2].

Microgrids are expected to be more robust and cost-effective than the tradi-tional approach of centralized grids. However, to achieve a stable and economicaloperation, a number of technical and regulatory issues have to be resolved be-fore the microgrid can become a commonplace. One problem that would requiredue attention is the effective management of power supply and demand loads,which amounts to matching the power generation and consumption profiles [3][4]. Specifically, the power generators or microsources employed in microgridsare usually renewable or non-conventional distributed energy resources. Whilethe incorporation of such renewable resources shall certainly bring great environ-mental benefits, it imposes new challenges as well: different from that in the tra-ditional power systems with conventional controllable electric generators, genera-tion scheduling in microgrids with fluctuant, climate-dependent renewable energysources has to cope with the nontrivial uncertainties.

The microgrids may adopt hierarchical or decentralized demand control schemes[5] [6]. The decentralized control schemes facilitate distributed control and man-agement of large complex systems. However, such control requires significantexperiments before implementation and sophisticated coordination. Also it mayintroduce new security challenges. Hierarchical control is performed by a mastercontroller which is responsible for matching the generation and load. When thedemand resources are controlled upon the occurrence of disturbance, the strategyis often known as direct load control [7] [8]. In direct control program, basedon an agreement between the central controller and customers, the controller canremotely control the operations of certain appliances in a household. This capabil-ity can be especically effecitve where there are electric devices allowing flexibleusage time and/or energy storage, such as electric water heater (EWH) equipped

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with hot water storage tank and plug-in hybrid electric vehicles (PHEVs), etc. TheKyotango microgrid project in Japan is an example of hierarchically controlledmicrogrid [2].

This paper tackles the basic problem faced by the microgrid system centralcontroller (MGCC), namely to achieve a good match between power demand andsupply subject to uncertainties of renewable energies. On the power demand side,we envision a scenario with real-time communication between the controller andenergy consumer premises. Specifically, in each time period, the operator con-troller receives consumer power demands with different power level requirements,durations and time elasticity levels. The MGCC needs to minimize the electricitygeneration cost by optimally scheduling the operation of each appliance subjectto the requirements set by the users. Here the generation fuel cost is modeled as aconvex function of instantaneous total power consumption.

On the power supply side, MGCC has to focus on how to effectively managepower generation so as to match the user load and maintain system reliability. Wepropose a novel uncertainty model to capture the fluctuant nature of renewable en-ergies. Compared with previous robust optimization based approaches which con-fine the renewable energy within a lower and upper bounds, the proposed modelprovides more statistical details in describing the underlying uncertainty (The pre-vious robust approaches will be discussed in Section 2.). Specifically, an empiricaldistribution is extracted as a useful reference, which allows the actual distributionof renewable energies to vary around it. To the best of our knowledge, this is thefirst time that the distribution uncertainty model is adopted to depict the indeter-minacy property of renewable energy generation. The load balance constraint isaptly approximated using the chance constraint representation, which allows con-venient tuning of the conservation level of the solution using a single parameter.A tractable robust optimization approach is developed for transforming the chanceconstraints into linear constraints, and some of the desired properties of the prob-lem formulation are investigated, providing illuminations for the policy makingof MGCC. It is shown that the proposed power demand and supply managementscheme can greatly reduce the energy cost for the microgrid system.

The remainder of this paper is organized as follows. Section 2 provides a briefsurvey of the related work. In Section 3, we show the mathematical depiction ofthe power demand and supply management problem and the uncertainty model ofthe renewable energies. Section 4 presents the robust approach for handling theload balance constraint, and the problem decomposition process. We provide thesimulation results and discussions in Section 5 and conclude the paper in Section6.

3

2. Related Work

The problem we are tackling can be viewed as containing two different parts.On the power demand side, we try to build a hierarchical demand control schemeso as to achieve the economic consumption scheduling and fulfill the requirementsset by energy users; on the power supply side, we need to properly model therandomness of renewable energy generation, which may account for a significantportion of power supply in microgrids. Note that load balance constraints act asthe connection between power consumption and generation.

Demand control techniques can be categorized into either price based loadcontrol techniques, referred to as demand response methods, or direct load con-trol, referred to as demand side management. Under price based load controlscheme, users are encouraged to make energy consumption decisions individuallyaccording to the price information. Demand side management strategies, however,are usually applied directly by a central controller and require consumer subscrip-tion to an economic incentive program. Several representative works have studieddemand control techniques in residential microgrids. A recent paper [9] devel-oped a real-time pricing scheme that aims at reducing the peak to average loadratio (PAR) through demand response management in smart grid systems. A two-stage optimization problem was then proposed and solved. Fathi et al. developeda stochastic model of scheduling in a local area network with the objective ofcost minimization and PAR minimization [10]. The work [11] presented a lin-ear programming formulation for minimizing the energy cost through direct loadcontrol. In [12] an optimization model was presented to adjust the hourly loadlevel of a given consumer in response to hourly electricity prices. The electricityprice uncertainty was modeled through robust optimization techniques. The un-certainties of renewable energies, however, are not considered in these studies. Assuch, the control schemes may not be readily optimal and applicable to the micro-grid scenario where renewable energies constitute a significant portion of powerresources.

There also exist some studies considering renewable energy uncertainties whenscheduling the energy generation. Such work can be categorized into two groups:the stochastic based approaches and the robust optimization based approaches.For instance, Wang et al. defined stochastic upper and lower supply curves to cap-ture a broad range of fluctuations in the power system, where energy generated byeach power source was modeled as stochastic arrivals in the queuing model [13].In [14], scenario-based stochastic operation management methods were devel-oped to tackle the fluctuant demands and renewable energies using the probability

4

distribution function (PDF) of each uncertain variable. Hidden Markov modelshave also been adopted to characterize renewable energy generation [15] [16][17]. Stimulated by observations that in practical scenarios, obtaining an accu-rate distribution function could be computationally costly and renewable energymay not follow Markov process or any simple distributions, robust optimizationhas recently received growing attention as a modeling framework for optimiza-tion under uncertainty. Instead of assuming explicit probability distribution, ro-bust optimization confines the renewable generation in a pre-defined uncertaintyset containing the worst-case scenario. For example, Zhang et al. considereda distributed economic dispatch problem for microgrid with high penetration ofrenewable energies [18]. The intrinsically stochastic properties of renewable en-ergy sources are captured by a polyhedral uncertainty set with deterministic lowerand upper bounds. Similar methods for modeling renewable energies can alsobe found in other recent work [19] [20]. Different from the existing work, ourapproach jointly considers power demand and supply management. Rather thanassuming there is available knowledge of the specific distribution of renewableenergy generation, the proposed approach describes the underlying uncertaintyin a more detailed yet flexible manner. It allows more information of renewableenergy generation to be effectively incorporated into the uncertainty model whensuch information is available.

3. Formulation of the Microgrid Demand and Supply Management Problem

In this section, a mathematical representation of the energy consumption andgeneration scheduling problem in an islanded microgrid with renewable energiesis provided. An MGCC is responsible for scheduling the operations of the micro-grid as well as performing optimization for minimizing the electricity generationcost for the microgrid system. The operations of the system and its mathematicaldepictions are introduced from the energy user side and energy generation side,respectively. The uncertainty model for describing the randomness of renewableenergies is then demonstrated.

3.1. Energy Demand SideConsider a group of energy consumers participating in this energy consump-

tion scheduling program. It is assumed that there are two-way communicationinfrastructures (e.g., a local area network (LAN)) between MGCC and energyconsumers. Let A denote the set of appliances belonging to these consumers,which may include PHEVs, dishwashers, cloth dryers, air conditioners, etc. Time

5

is divided into discrete time slots with equal length. For each appliance a that isswitched on, the active power consumed during one unit of time slot is xa. Anenergy consumption scheduling vector ya is also defined for each appliance a asfollows:

ya = [y1a, ..., yHa ] (1)

where H ≥ 1 is the scheduling horizon indicating the number of time slotsahead that are taken into account for decision making in the energy consumptionscheduling. For each coming time slot h ∈ H = [1, 2, ..., H], a binary variableyha = 0/1 denotes the state of appliance a (on/off). Under such case, the actualenergy consumption for appliance a at time slot h can be expressed as xa · yha .

There is usually an upper limit on the total energy consumption in the micro-grid in each time slot. Denoting this limit as Emax, we have:∑

a∈A

xa · yha ≤ Emax, ∀h ∈ H. (2)

Next, assume that for each appliance a ∈ A, the user indicates αa, βa ∈ H as thebeginning and end of a time interval in which the appliance a can be scheduled,respectively. Obviously, αa < βa. For instance, the user may select αa = 8 PMand βa = 6 AM (the next day) for his PHEV so that he could plug it in at nightand get it fully charged before going to work the next day. Denote the minimumnumber of time slots needed for appliance a to finish its preset work as Ta. Giventhe predetermined parameters αa, βa, Ta, the appliance scheduling is subject to thefollowing constraints:

βa∑h=αa

yha ≥ Ta, ∀a ∈ A, (3)

and

yha = 0, ∀a ∈ A, ∀h ∈ H \ [αa, βa]. (4)

Constraint (3) shows that the time length βa − αa needs to be large enough toallow finishing the normal operation of appliance a. In addition, the energy usercan choose proper αa, βa and Ta to indicate whether the operation of appliance aneeds to be started immediately (βa−αa = Ta) or can be deferred (βa−αa > Ta).

6

To reveal the ramping down and ramping up limits on load levels of each timeslot, we have:∑

a∈A

xa · yha −∑a∈A

xa · yh+1a ≤ rD, h ∈ [1, 2..., H − 1], (5)∑

a∈A

xa · yh+1a −

∑a∈A

xa · yha ≤ rU , h ∈ [1, 2..., H − 1]. (6)

In this regard, it is assumed that each household participating in this energy con-sumption scheduling program is equipped with a smart meter, which is capableof detecting the electric power level of each appliance. The energy consumer an-nounces to the MGCC his needs by selecting parameters αa, βa and Ta for eachappliance a ∈ A.

The above constraints (2) to (6) describe common characteristics of householdappliances. However, there exist some appliances of which the operation cannotbe interrupted. Such kind of loads are called as uninterruptible loads. Discussionson how such loads may be handled are presented below.

Operation of Uninterruptible Loads: Some loads are interruptible, such asPHEV, which means that it is possible to charge the battery for some time, stopcharging for some time and then switch on the charging process again. Some otherloads, however, are not interruptible, e.g., microwave oven. Appliances generatingsuch loads, once started, have to be finished in one go. For each uninterruptibleappliance a ∈ A′, where A′ represents the set of uninterruptible appliances, andeach time slot h, let zha denote an auxiliary binary variable such that zha , 1 ifappliance a starts operation at time slot h and zha , 0 otherwise. We have

βa−Ta+1∑h=αa

zha = 1 (7)

and

zha = 0, ∀h ∈ H\[αa, βa − Ta + 1]. (8)

Then we relate start time vector zha with decision variable vector yha as follows:

yha ≥ zha , yh+1a ≥ zha , ..., y

h+Ta−1a ≥ zha . (9)

From (9), if zha = 1, then yha = yh+1a = ... = yh+Ta−1

a = 1.

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Figure 1: The microgrid’s graph.

3.2. Energy Supply SideWe now turn to the energy supply side to consider the load balance constraint

in the microgrid. The microgrid may be considered as a graph consisting of threenodes as illustrated in Fig. 1. The first node represents the renewable energygeneration sources such as wind turbines, solar panels and fuel cells. At timeslot h, denote the total energy generated in this node as ξh, where ξh is a randomvariable of which the probability density function may not be known. Node 2 inFig. 1 represents the load connected through the transmission line to node 1 andnode 3. The load at time h, denoted as lh, is dependent on the energy consumptionfrom the user side which, from the above analysis, can be expressed as:

lh =∑a∈A

xa · yha . (10)

Finally the third node includes a group of controllable electricity generators,which has a total amount of generation P h

cg as commanded by MGCC. Control-lable generators in microgrid typically include gas turbines, micro-turbines, re-ciprocating internal combustion engines with generators, etc. These generatorsare powered by fossil fuels and can be controlled to compensate the mismatchbetween load and renewable power supply. A key requirement to the MGCC is toset the generation source power such that the supply could meet the demand. Thisstatement can be mathematically described as

ξh + P hcg ≥ lh. (11)

3.3. Problem FormulationThe objective function of MGCC can be defined in terms of minimizing the

energy cost of the whole microgrid system. Hence the optimal energy consump-

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tion scheduling problem is formulated as follows:

minH∑

h=1

Ch(P hcg) (12)

s.t. (2) to (11)

where Ch(·) is the cost function of electricity plant in the microgrid, which is as-sumed to be an increasing convex function. The convex property reflects the factthat each additional unit of power needed to serve the demands is provided ata higher cost. Example cases include the quadratic cost function [21] [22] andthe piecewise linear cost function [23] [3], etc. Without loss of generality, weconsider quadratic cost function Ch(P h

cg) = ahPhcg

2+ bhP

hcg + ch throughout this

paper, where ah ≥ 0, bh ≥ 0 and ch ≥ 0 are known parameters for each time sloth. In practice, the coefficient of the quadratic term is usually small. Therefore,the quadratic cost function can be reduced to a linear cost function. As to the re-newable energy cost, for typical renewable energies (e.g. solar and wind energy),capital cost dominates. The operation and maintenance costs are typically verylow or even negligible [24] [25]. In this paper, we consider the scenario whererenewable energy generators such as solar panels and wind turbines have alreadybeen installed; hence the marginal cost of renewable energy is neglected, lead-ing to its omission in the objective function [26]. The main difficulty in solvingproblem (12) is the indeterminacy of renewable energy generation ξh that existsin constraint (11). Note that to optimize over the space defined by (11) amountsto solving an optimization problem with potentially large or even infinite numberof constraints. Obviously, this realization of uncertainty is intractable. Next, apractical and flexible model will be developed to capture the uncertainty of ξh.

3.4. Probability Distribution Measure of Renewable EnergiesIt is generally difficult to characterize the renewable energy generation. In pre-

vious optimization approaches, operations on the random variable ξh is cumber-some and computationally intractable. Moreover, in practice, we may not knowthe precise distribution of ξh. Solutions based on assumed distributions hence maynot be justified. The variability of a random variable is usually measured using itsvariance or second moments which, however, may not provide sufficient details indescribing the random variable. In this paper, a reference distribution, rather thanmoment statistics is extracted from historical data that will capture the distributionproperties. Since renewable energy generation distribution is fluctuating over timeand hard to be described in a closed-form expression, empirical distribution may

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adopted as a useful reference and allow the actual distribution to fluctuate aroundit. For example, it may be assumed that the renewable energy generation distribu-tion f0(ξ

h) is shifting around a known Gaussian distribution (or other distribution)gh(ξ

h), which can be obtained based on long-term field measurement.The discrepancy between f0(ξ

h) and its reference gh(ξh) can be described by aprobabilistic distance measure, for example the Kullback-Leibler (KL) divergence[27], which is a non-symmetric measure of the difference between two probabilitydistributions. Name these two distributions as f(ξh) and g(ξh), respectively. Gen-erally, one of the distributions, say, f(ξh), represents the real distribution throughprecise modeling, while the reference g(ξh) is a closed-form approximation basedon the theoretic assumptions and simplifications. The definition of the KL diver-gence between two continuous distributions is given as follows:

DKL(f(ξh), g(ξh)) =

∫ξh∈S

[lnf(ξh)− lng(ξh)]f(ξh)dξh, (13)

where S is the integral domain. When distributions f(ξh) and g(ξh) are close toeach other, the distance measure is close to zero. Adopting the KL divergence, thedistribution uncertainty set is defined as follows:

Ur(g(ξh), D0) = {f(ξh) | Ef [lnf(ξh)− lng(ξh)] ≤ D0}, (14)

where D0 ≥ 0 represents a distance limit and is obtained from empirical dataor real-time measurement. It indicates energy generation’s variation level. If theenergy generation is very volatile, we have less confidence on the reference dis-tribution and thus may set a larger distance limit.

Considering the renewable energy generation distribution f0(ξh) with refer-

ence distribution gh(ξh) and distance limit Dh, we have the following constraints

for renewable energy generation distribution f0(ξh):

Ef0 [lnf0(ξh)− lngh(ξh)] ≤ Dh, (15)

Ef0 [1] = 1. (16)

Given (15) and (16), now it is ready to transform the load balance constraint (11)to allow efficient solution of problem (12).

Remark: Finding a proper reference distribution and obtaining an appropriatedistance limit sometimes may be difficult, especially when still in lack of historicaldata. However, there are good reasons to expect that in most cases, situation couldbe improved quickly with continuous accumulation of historical records.

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4. Optimization Algorithms

In this section, the optimization algorithms for solving the prime problem (12)is developed. First a robust approach for handling the load balance constraintis presented, and then the prime problem is decomposed into a subproblem anda main problem to allow easier solution. Finally, the possible extensions of theproposed algorithm are briefly discussed.

4.1. Robust Approach for the Load Balance ConstraintAs shown in (11), the load balance constraint is ξh + P h

cg ≥ lh. In practice,a decision criterion is to set P h

cg and lh in such a way that we are confident thatthe load balance constraint is achieved. To achieve that, we may introduce a smallvalue ϵ to control the degree of conservatism and change the above expression intoa chance constraint:

P(ξh ≤ lh − P hcg) ≤ ϵ (17)

where ϵ is the fault tolerance limit of the power grid, representing the acceptableprobability that the desirable power supply is not attained. Then its robust expres-sion can be obtained:

maxf0(ξh)∈Ur(gh,Dh)

P(ξh ≤ lh − P hcg) ≤ ϵ (18)

which is equivalent to:

maxf0(ξh)∈Ur(gh,Dh)

∫ lh−Phcg

0

f0(ξh)dξh ≤ ϵ. (19)

Defining δh = lh − P hcg as the robust renewable energy usage (REU) decision,

which equals the amount of energy dispatched to renewable energy plants at timeslot h. In addition, an auxiliary function can be introduced as follows:

h(ξh, δh) =

{1, ξh ≤ δh;

0, ξh > δh.(20)

The left part of inequality (19) then can be formulated into an optimization prob-lem:

maxf0(ξh)

∫ +∞

0

h(ξh, δh) · f0(ξh)dξh (21)

s.t. Ef0 [lnf0(ξh)− lngh(ξh)] ≤ Dh

Ef0 [1] = 1

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Define Qhf (δ

h) = maxf0(ξh)∈Ur(gh,Dh)

∫ +∞0

h(ξh, δh) · f0(ξh)dξh as the worst-casefault probability. We can then obtain a worst-case mapping Mh

ws which mapsrobust REU decision δh to Qh

f (δh):

Mhwc : δh −→ Qh

f (δh). (22)

4.2. Sub-Problem: Determine the Robust REU Decision ThresholdSince there exists a random variable ξh in the constraints, we cannot solve

energy generation and consumption scheduling problem (12) directly. As afore-mentioned, problem (12) is decomposed into a subproblem and a main problem.The goal of our sub-problem is to determine the robust REU decision thresholdδh

∗ so that the load balance constraint can be transformed into a solvable form.Proposition 1: Problem (21) is a convex optimization problem.Proof: Rewrite (21) as follows:

maxf0(ξh)

∫ +∞

0

h(ξh, δh) · f0(ξh)dξh (23)

s.t.∫ +∞

0

[lnf0(ξh)− lngh(ξh)]f0(ξh)dξh ≤ Dh (24)∫ +∞

0

f0(ξh)dξh = 1. (25)

It can be seen that the objective function (23) and equality constraint function (25)are affine with respect to f0(ξ

h). Next it is shown that the inequality constraintfunction (24) is convex.

Lemma 1: If f : Rn −→ R is convex, then the perspective of f , which isdenoted as a function g :Rn+1 −→ R that

g(x, t) = tf(x/t), (26)

with domain

dom g = {(x, t)|x/t ∈ dom f, t > 0} (27)

preserves convexity.That is to say, if f is a convex function, so is its perspective function g. Sim-

ilarly, if f is concave, so is g. This can be proved in several ways, e.g., by directverification of the defining inequality or using epigraphs and the perspective map-ping on Rn+1. Readers can refer to [28] for more detailed discussions.

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We consider the convex function f(x) = −ln x on R++. Its perspective is

g(x, t) = −t ln(x/t) = t ln(t/x) = t(ln t− ln x) (28)

and it is convex on R2++. The function g is called the relative entropy of t and

x. Then we have that the KL divergence∫x∈S[lnf(x) − lng(x)]f(x)dx between

distribution f(x) and g(x) is convex in f(x) (and g(x) as well). In this case, it isclaimed that the inequality constraint (24) is convex in distribution f0(ξ

h).Through Slater’s condition, strong duality holds for problem (23)-(25). Adopt-

ing the Lagrangian method, the worst-case fault probability Qhf (δ

h) can be ob-tained as follows:

Qhf (δ

h) = minτ,η

maxf0(ξh)

Ef0

[h(ξh, δh)− η − τ ln

f0(ξh)

gh(ξh)

]+ τDh + η

where τ ≥ 0 and η are Lagrangian multipliers associated with constraints (24)and (25), respectively. Let

P(δh, f0, τ, η) = Ef0

[h(ξh, δh)− η − τ ln

f0(ξh)

gh(ξh)

], (29)

the derivative of P(δh, f0, τ, η) with respect to f0 can be derived as

∂P∂f0

= limt→0

1

t

[P(f0(ξ

h) + t · g0(ξh))− P

(f0(ξ

h))]

(30)

=

∫ +∞

0

(h(ξh, δh)− τ ln

f0(ξh)

gh(ξh)− η − τ

)g0(ξ

h)dξh.

Using the Karush-Kuhn-Tucker (KKT) optimality conditions, we thus have

h(ξh, δh)− τ lnf0(ξ

h)

gh(ξh)− η − τ = 0 (31)∫ +∞

0

f0(ξh)dξh = 1 (32)

E[

lnf0(ξ

h)

gh(ξh)

]−Dh ≤ 0 (33)

τ ·(Dh − E

[lnf0(ξ

h)

gh(ξh)

])= 0 (34)

τ ≥ 0 (35)

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From (31), the optimal distribution function can be expressed as follows:

f ∗0 (ξ

h) = gh(ξh) exp

(h(ξh, δh)− η

τ− 1

). (36)

The dual variables (τ, η) in (36) should be chosen properly such that conditions(32)-(35) are satisfied. Specifically, the following results can be obtained:

Proposition 2: The choice of (τ, η) is a solution of the following nonlinearequations.

H1(τ, η) = R(δh)e−η/τ + S(δh)e(1−η)/τ − 1 = 0 (37)H2(τ, η) = S(δh)e(1−η)/τ − η − τ(1 +Dh) = 0, (38)

where S(δh) = (1−Gh(δh)) exp(−1), R(δh) = Gh(δ

h) exp(−1), and Gh(δh) =∫

ξh≥δhgh(ξ

h)dξh denotes the complementary cumulative distribution function ofreference distribution gh(ξ

h).Proof: By substituting the optimal distribution f ∗

0 (ξh) back into (32) and

f ∗0 (ξ

h), (31) into (34), we have∫ +∞

0

gh(ξh) exp

(h(ξh, δh)− η

τ− 1

)dξh = 1 (39)∫ +∞

0

(h(ξh, δh)− η − τ

)gh(ξ

h) (40)

· exp(h(ξh, δh)− η

τ− 1

)dξh −Dh · τ = 0,

which are equivalent to:

exp(−1− η

τ

)·∫ +∞

δhg(ξh)dξh (41)

+exp

(−1 +

1− η

τ

)·∫ δh

0

g(ξh)dξh − 1 = 0

(1− η − τ) exp

(−1 +

1− η

τ

)·∫ δh

0

g(ξh)dξh (42)

+(−η − τ) exp(−1− η

τ

)∫ ∞

ξhg(ξh)dξh − τDh = 0.

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Equation (37) can be easily obtained from (41) by introducing S(δh) and R(δh).Through (41), (42) can be transformed into:

(1− η − τ) exp

(−1 +

1− η

τ

)·∫ δh

0

g(ξh)dξh +

(−η − τ)

[1− exp

(−1 +

1− η

τ

)·∫ δh

0

g(ξh)dξh]− τDh = 0.

Then we have

exp

(−1 +

1− η

τ

)·∫ δh

0

g(ξh)dξh − η − τ − τDh = 0, (43)

which is equivalent to (38). Hence, Proposition 2 is proved.It is, however, still rather difficult to obtain an explicit solution from (37) and

(38). Hence we propose the Newton iterations as detailed in Algorithm 1.Once the solutions for (37) and (38) in Proposition 2 is determined, through

(31) and (34), the worst-case fault probability can be obtained as follows:

Qhf (δ

h) = Ef∗0[h(ξh, δh)] = (1 +Dh)τ + η (44)

Our next step is then to find the robust REU decision threshold δh∗ such that

Qhf (δ

h∗) = ϵ, which involves the calculation of inverse function of Qhf (δ

h) andit is not directly possible from (44). The following property of function Qh

f (δh),

however, may help us design such a search method.Proposition 3: The worst-case fault probability Qh

f (δh) is non-decreasing

with respect to the REU decision δh.The conclusion in Proposition 3 is straightforward since we have dQh

f (δh)/dδh =

dEf∗0[h(ξh, δh)]/dδh = f∗

0 (δh) ≥ 0. Though direct solution is not available, the

monotonicity of Qhf (δ) enlightens us a bisection method to search for the solution

for Qhf (δ

h) = ϵ. The main idea is to perform the search within an interval of [0, ρ],where ρ is an empirical constant such that Qh

f (ρ) > ϵ.Details of the algorithm for searching the robust REU decision threshold are

presented in Algorithm 1. Note that, from the 3rd to the 11th lines of the al-gorithm, Newton iteration is adopted to solve the equation in Proposition 2 andobtain the worst-case probability with fixed robust REU decision. Then the worst-case probability at δh− and δh

− is compared with the fault tolerant limit ϵ, respec-tively. The comparison results help shrink the search region as shown in lines12-14.

15

Algorithm 1 Search for robust REU decision threshold δh∗

Input: Reference distribution gh(ξh);

Distance limit Dh;Search radius ρ;Load balance fault tolerant limit ϵ;Tolerance ε.

Output: Robust REU decision threshold such that Qhf (δ

h∗) = ϵ;1: Begin2: initialize δh− = 0, δh− = ρ, and set H(τ, η) = [H1(τ, η), H2(τ, η)]

T

3: while |δh− − δh−| > ε

4: set δh = δh−+δh−

2and initiate the time iteration k = 1

5: while H(τ, η) > ε6: evaluate H(τ, η) and Jacobian matrix J(τ, η)7: solve J(τ, η)∆xk = −H(τ, η)8: update τk+1 = [τk +∆τk]

+, ηk+1 = ηk +∆ηk9: update Qh

f (δh) = (1 +Dh)τk+1 + ηk+1

10: set k = k + 111: end while12: if

(Qh

f (δh)− ϵ

)(Qh

f (δh−)− ϵ

)< 0

13: then set δh− = δh else set δh− = δh end if14: if |Qh

f (δh)− ϵ| < ε break end if

15: end while16: set δh∗ = δh

17: End

4.3. Main-Problem: Determine the Optimal Energy Consumption and Genera-tion Scheduling

Once the robust REU decision threshold δh∗ for the robust load balance con-

straint (19) is obtained, the energy generation and consumption management prob-lem can be reformulated. Specifically, the following optimization problem can be

16

tackled rather than the original Eq. (12)

minH∑

h=1

Ch(P hcg) (45)

s.t.∑a∈A

xa · yha − P hcg = δh

∗, ∀h ∈ H,

and (2) to (10),

where the optimization variables include the controllable energy generation vari-able P h

cg for all time slots h ∈ H, and the energy consumption scheduling vectorya for all appliances a ∈ A. The objective function aims at minimizing the overallenergy cost in microgrid over the whole time horizon.

It can be seen that all the constraints of (45) are linear and the objective func-tion is quadratic. This problem is a mixed integer quadratic programming prob-lem. Algorithms that can be adopted to tackle this kind of problem include thecutting plane method and the branch and bound method. This problem can also beeffectively solved by some commercial optimization softwares including CPLEX,Mosek, FortMP and Gurobi, etc.

4.4. Extensions of the Proposed Algorithm: A Brief DiscussionWith trivial or, sometimes, non-trivial extensions, the proposed algorithm may

be applied to solve some other power demand and supply problems in microgrids.A few possible extensions are briefly discussed as follows.

• The scenario that has been considered in this paper assumes that the end-users control their power consumption in accordance with the guideline thatMGCC suggests. Under such case, the uncertainties from the end-user sideare expected to be limited and can be handled by the system, since the con-trol sequences obtained by the proposed algorithm are of reasonably goodrobustness. For the cases where uncertainties from the user-side exceedthe range of tolerance (e.g., many end-users do not follow the guidelinefor whatever reasons), the proposed uncertainty model can be extended toinclude the uncertainty from the end-user side by properly integrating user-side and supply-side uncertainties. Once the integrated reference distribu-tion of the combined uncertainties is obtained, the proposed algorithm canbe used with virtually no changes. Detailed discussions on such cases, how-ever, are out of the scope of this paper. For the even worse case whereunexpected real-time changes go beyond what have been modeled, a few

17

classic approaches may be adopted, e.g., by increasing the power output ofenergy generators, turning on stand-by fast response generators, importingelectricity from the power grid, or shedding load if necessary, etc.

• The power demand and supply management framework discussed so far isan offline approach suitable for planning the energy consumption and gen-eration ahead of time. When real-time adjustment is of a big concern yetresponse time limit is not too rigid, the proposed algorithm can be easilyextended to handle such cases. One option is to adopt the model predictivecontrol (MPC) approach (also known as ”receding horizon approach”) [29][30], of which the basic idea is to calculate the optimal control sequencesyet implement only the first step of them. In other words, the power demandand supply management problem is solved at time h = τ (τ ∈ H denotesthe current time index.) for the remaining horizon [τ, τ +1, ..., H], yet onlythe solution for the current time slot τ is implemented. In the next timeslot, MGCC shall update the system information (e.g., the requirements ofend-users and robust REU decision thresholds) and re-do the calculations.For the cases where response time has to be very short, however, differentalgorithms with lower complexities and faster speed probably have to bedeveloped, in order to support real-time operation and small-step schedul-ing more efficiently. For such cases, enhancing the system fault toleranceagainst the noises of real-time data shall also be considered.

• The problem formulation can also be easily extended to handle the casewhere microgrids import electricity from outside power grid. Specifically,assume that the microgrid imports P h

E units of electricity from the powergrid, the problem formulation (45) can be modified as:

minH∑

h=1

ah · P hcg

2+ bh · P h

cg + ch + dh · P hE (46)

s.t.∑a∈A

xa · yha − P hcg = δh

∗+ P h

E, ∀h ∈ H,

and (2) to (10),

where dh is the electricity price of power grid at time slot h. The problem isessentially still a mixed integer quadratic programming problem which canbe solved by using the same algorithm. Whereas if electricity price fromthe power grid dh is time varying with non-trivial uncertainty, the problem

18

will become more complicated. A feasible option is to develop a worst-case robust optimization approach for the problem. Readers may refer to[31] [32] [12] for more exhaustive descriptions on dealing with boundeduncertainties in the coefficients of objective function. Detailed discussionsare not too difficult yet rather lengthy, and therefore have to be left to aseparate report.

5. Simulation Results and Discussions

In this section, simulation results are presented for assessing the performanceof the proposed power demand and supply management scheme and evaluatingthe effects of different system parameters. Here, an assumption is made on topof paper [33] [34], where Gaussian random process has been adopted to describethe renewable energy generation. Specifically, it is assumed that the referencedistribution is a Gaussian distribution gh(ξ

h) with mean mh and standard deviationσh. In addition, the parameters of the cost function in (45) for each time slot areset as ah > 0, bh = 0, and ch = 0.

5.1. The Impacts of Distribution Uncertainty SetWe first set the fault tolerant limit ϵ = 10−3 and try to investigate the relations

between robust REU decision threshold δh∗ and distance limit Dh for different

values of mh and σh. The results are plotted in Fig. 2 and Fig. 3. It is shownthat the robust REU decision threshold decreases with the increase of the distancelimit. This observation is intuitive since a larger distance limit defines a largerdistribution set which allows the renewable energy output to fluctuate more in-tensively. Given the required fault tolerant limit, REU decision threshold has tobe set at a lower value so as to rely less on the more uncertain renewable energyand guarantee the system reliability. Note that when Dh = 0, the renewable en-ergy follows exactly the reference distribution gh(ξ

h). In this special case, renew-able energy generation is a random variable with determinate distribution gh(ξ

h).While Dh > 0, our reference model considers a more general case which allowsdiscrepancy between actual distribution and its reference. The discrepancy how-ever is limited and confined by a probabilistic distance measure. Simply put, thereference model allows the actual renewable energy generation to follow a differ-ent distribution function from the reference distribution, but not be too disparatebased on historical data or empirical knowledge.

From Fig. 2 and Fig. 3, the following statement can also be proposed: whenthe reference distribution is Gaussian, the robust REU decision threshold δh

∗ lin-

19

0 0.005 0.01 0.015 0.0220

21

22

23

24

25

26

27

28

29

30

Distance to Reference Distribution Dh

Rob

ust R

EU

Dec

isio

n T

hres

hold

δh*

mh = 34, σh = 2

mh = 35, σh = 2

mh = 36, σh = 2

Figure 2: Robust REU Decision Threshold δh∗ with Distance Limit Dh for differ-

ent mh

0 0.005 0.01 0.015 0.0220

21

22

23

24

25

26

27

28

29

30

Distance to Reference Distribution Dh

Rob

ust R

EU

Dec

isio

n T

hres

hold

δh*

mh = 35, σh = 1.9

mh = 35, σh = 2

mh = 35, σh = 2.1

Figure 3: Robust REU Decision Threshold δh∗ with Distance Limit Dh for differ-

ent σh

20

1 1.2 1.4 1.6 1.8 2

x 10−3

19

20

21

22

23

24

25

26

27

28

29

30

Fault Tolerant limit ε

Rob

ust R

EU

Dec

isio

n T

hres

hold

δh*

Dn = 0

Dn = 0.005

Dn = 0.01

Dn = 0.015

Dn = 0.02

Figure 4: Robust REU Decision Threshold δh∗ with Fault Tolerant Limit ϵ for

different Dh

early increases with the mean of reference distribution mh and linearly decreaseswith the standard deviation of reference distribution σh. This statement can beexplained analytically as follows. Gh(δ

h) in (37) and (38) is first transformedinto:

Gh(δh) =

∫ξh≥δh

gh(ξh)dξh =

∫ +∞

δh−mhσh

n(x)dx, (47)

where n(x) is the probability density function of the standard Gaussian distribu-tion. Since fault tolerant limit ϵ is of a relatively small value, we have that δh∗ isless than mh. As mh and σh vary, in order to preserve the same worst-case prob-ability Qh

f (δh∗), the solutions (η, τ ) of the equations (37) and (38) need to remain

unchanged, indicating that S(δh∗), R(δh∗) and G(δh

∗) also need to be constants.

In this regard:

δh∗ −mh

σh

= C =⇒ δh∗= Cσh +mh,

where C is a negative constant. Thus, δh∗ linearly increases with mh and linearlydecreases with σh.

5.2. Effects of Fault Tolerant Limit ϵWe set mh = 36 and σh = 2 and investigate how the robust REU decision

threshold varies when fault tolerance limit increases. Fig. 4 plots the mapping

21

from fault tolerant limit ϵ to robust REU decision threshold δh∗ under different

values of the distance limit Dh. The figure indicates that a larger fault tolerantlimit permits a higher reliance on renewable energy (a larger robust REU decisionthreshold), which is straightforward to understand. Also note that the worst-casefault probability is an increasing function of the REU decision threshold. Thus itis justified to adopt the bisection method as presented in Algorithm 1 to searchfor the REU decision threshold which satisfies the fault tolerant limit requirement.Note that in this figure, the red triangle line is the special case where renewableenergy follows reference distribution exactly. It is also observed that robust REUthreshold δh

∗ is more sensitive to ϵ when Dh increases.

5.3. The Impacts of Uninterruptible LoadsFor the experiment studied in this part, we set the power consumption schedul-

ing horizon |H| = 12 h. That is, the MGCC solves optimization problem (45) todecide on the operations of each appliance for the next 12 hours. In this paper,30 household appliances including electric cookers (EC), air conditioners (AC),electric water heaters (EWH), cloth dryers (CD), dish washers (DW) and plug-inhybrid electric vehicles (PHEVs) are considered to study the optimal power con-sumption scheduling with a mixed integer quadratic programming approach. Thedetailed operation data used in this paper is modified based on the informationfrom [35] [36] [37] [38] [39], and is shown in Table 1. Note that a user elasticityindex is also introduced as follows

γa =Ta

βa − αa + 1

to describe the scheduling flexibility of appliance a. Obviously, γa ∈ (0, 1], and alarger γa implies a more inflexible arrangement property. The operation window[αa, βa] of each appliance is chosen according to the preferences of different users.In this paper, the values of αa and βa are not enumerated one by one due to limitedspace, instead, the ranges of γa for each kind of appliance are listed, which ispresented in the last column of Table 1.

The mean mh and standard deviation σh of the reference distribution, togetherwith the distance limits for the next 12 time slots are given in Table 2. Based onthese data and adopting Algorithm 1, the robust REU threshold δh

∗, representingthe amount of energy dispatched to renewable energy plants for each time slot, isobtained. The results are demonstrated in the last column of Table 2 and are usedto solve the main problem (45). Our experiments utilize MOSEK optimizationtoolbox 6.0 on an Intel-P4 2.4-GHz personal computer. To investigate the impacts

22

Table 1: Operation Data for Appliances in the microgrid

Type of Appliance Power Level (KW) Ta γa

EC 2 1 0.3-0.4AC 3.5 10 0.9-1.0

EWH 4.5 3 0.5-0.7CD 5 3 0.3-0.4DW 0.85 2 0.3-0.4

PHEV 7.3 7 0.6-0.7

Table 2: Parameters of Distribution Uncertainty Set and Corresponding RobustREU Threshold

Time Slot mh σh Dh δh∗

1 14.678 0.9571 0.0162 8.4192 14.757 0.4853 0.0181 11.4533 14.743 0.8002 0.0025 11.5314 14.392 0.1418 0.0182 13.4235 14.655 0.4217 0.0126 12.1376 14.171 0.9157 0.0019 10.6307 14.706 0.7922 0.0055 10.9958 14.031 0.9594 0.0109 8.5779 14.276 0.6557 0.0191 9.716

10 14.046 0.0357 0.0192 13.79711 14.097 0.8491 0.0031 10.56512 14.823 0.9339 0.0194 8.294

23

Case 1 Case 2 Case 3 Case 4 Case 5 Case 623.5

24

24.5

25

25.5

26

26.5

27

27.5

28

Different Cases

Ene

rgy

Cos

t of t

he M

icro

grid

($)

23.9886 24.080624.2482

24.628724.7356

27.1694

Figure 5: The impacts of uninterruptible loads on the energy cost of the microgrid

of uninterruptible loads, the following cases are studied:

• Case 1: Only electric cookers are classified into the uninterruptible appli-ance set A′, i.e., A′ = {EC}.

• Case 2: On top of Case 1, air conditioners are added to the uninterruptibleappliance set A′, i.e., A′ = {EC,AC}.

• Case 3: On top of Case 2, electric water heaters are added to the uninter-ruptible appliance set A′, i.e., A′ = {EC,AC,EWH}.

• Case 4: On top of Case 3, cloth dryers are added to the uninterruptibleappliance set A′, i.e., A′ = {EC,AC,EWH,CD}.

• Case 5: On top of Case 4, dish washers are added to the uninterruptibleappliance set A′, i.e., A′ = {EC,AC,EWH,CD,DW}.

• Case 6: On top of Case 5, PHEVs are added to the uninterruptible applianceset A′, i.e., A′ = {EC,AC,EWH,CD,DW,PHEV}.

Figure 5 demonstrates the energy cost for each case. Obviously, the energycost goes up when the uninterruptible appliance set A′ is scaled up. We comparethe costs of adjacent cases, and the difference between these costs is called costgap. The largest cost gap is shown between Case 5 and Case 6 due to PHEVs’ highelectric power consumption (P = 7.3 KW) and relatively considerable schedulingelasticity (γa = 0.6− 0.7).

24

−6 −4 −2 0 2 420

25

30

35

40

45

50

55

60

65

Number of Time Slots Extended

Ene

rgy

Cos

t of t

he M

icro

grid

($)

58.2328

52.2923

39.3850

27.169324.7936 24.6649

Figure 6: The Impact of User Elasticity on the Energy Cost of the Microgrid

5.4. The Price of User ElasticityIn this section, the effects of user elasticity on the energy cost of the microgrid

system is explored. First it is assumed that all the appliances are uninterruptible.Since the minimum running time of each appliance Ta is fixed, we extend orshrink the operation window [αa, βa] to relax or tighten the user elasticity. Notethat, at one time, the operation window [αa, βa] of each appliance a will expandor shrink one unit of time slot from both sides, i.e., the operation window willscale up or down two time slots. If one side of the operation window cannotbe extended due to the finite length of time horizon, the operation window willonly scale up on one side until it covers the whole time horizon. The operationwindow of each appliance is kept on extending or shrinking until all the operationwindows cannot be changed. Then, how energy cost changes when operationwindows vary is demonstrated. Note that the cases when operation windows areshrunk 6, 4, 2 time slots and extended 0, 2, 4 time slots are selected respectively.The results are presented in Fig. 6. In this figure, it is observed that when userelasticities are tightened, energy cost increases rapidly. Such phenomenon canbe interpreted that the user elasticity can make a significant impact on the energycost of the microgrid system. Compared with the the effects of interruptibilityproperty, user elasticity has stronger influences on the expenditure of the wholesystem. This result may give MGCC an inspiration that it is worthy to providemore rewards to customers who agree to have more time flexibility than those whoallow interruptions to some appliances. Moreover, it is also shown that when the

25

operation windows are shrunk by 6 time slots, nearly all the appliances’ elasticityreaches 1. This approximates the case when all the appliances operate at theirdesired time with no flexibility. Compared with this benchmark case, we observethat the proposed power consumption management scheme can reduce the energycost significantly.

6. Conclusion

In this paper, a fundamental problem of using a microgrid system central con-troller to optimally schedule the demand and supply profiles so as to minimize thefuel consumption costs during the whole time horizon is studied. We focus ona scenario where the end-users control their power consumptions in accordancewith the guideline that MGCC suggests. To tackle the randomness of renewableenergy, a reference distribution is introduced and then a distribution uncertaintyset is defined to confine the uncertainty. Such a novel model allows convenienthandling of fluctuating renewable generation as long as the renewable energy gen-eration profile is not too drastically different from the past observation or empir-ical knowledge. An optimization formulation of the problem was proposed anda two-stage algorithm approach was developed, to first transform and then solvethe problem. Numerical results indicated that the proposed energy consumptionmanagement scheme can significantly cut down energy expenses. Effects of afew factors, including the reference distribution, the fault tolerant limit, the typesand amount of uninterruptible loads, and the user elasticity etc. were carefullyevaluated. Such evaluations, as we believe, help provide some useful insights forMGCC to develop more effective payback policies for their customers.

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