Joint Optimization of Hybrid Energy Storage and Generation Capacity With Renewable Energy Peng Yang,Student Member,IEEE,and Arye Nehorai,Fellow,IEEE
Abstract—In an isolated power grid or a micro-grid with a small carbon footprint,the penetration of renewable energy is usually high.In such power grids,energy storage is important to guar-antee an uninterrupted and stable power supply for end urs. Different types of energy storage have different characteristics,in-cluding their round-trip efficiency,power and energy rating,lf-discharge,and investment and maintenance costs.In addition,the load characteristics and availability of different types of renew-able energy sources vary in different geographic regions and at different times of year.Therefore joint capacity optimization for multiple types of energy storage and generation is important when designing this type of power systems.In this paper,we formulate a cost minimization problem for storage and generation planning, considering both the initial investment cost and operational/main-tenance cost,and propo a distributed optimization framework to overcome the difficulty brought about by the large size of the op-timization problem.The results will help in making decisions on energy storage and generation capacity planning in future decen-tralized power grids with high renewable penetrations.
Index Terms—Capacity planning,distributed optimization,en-ergy storage,micro-grid,renewable energy
sources.
N OMENCLATURE:
Set of different renewable generators
Types of renewable generators
Renewable generation per unit generation
capacity during time period
Renewable energy cost during time period
Renewable generation during time period
Maximum generation capacity
Set of different energy storage types
Types of energy storage
Rated power/energy ratio
One-way energy efficiency
Energy storage capacity
Energy loss ratio per unit time
Manuscript received September16,2013;revid February07,2014;ac-cepted March22,2014.Date of current version June18,2014.This work was supported by the International Center for Advanced Renewable Energy and Sus-tainability(I-CARES)at Washington University in St.Louis.Paper no.TSG-00738-2013.
The authors are with the Preston M.Green Department of Electrical and Sys-tems Engineering,Washington University in St.Louis,St.Louis,MO63130 USA(e-mail:yangp@e.wustl.edu;nehorai@e.wustl.edu).
Color versions of one or more of thefigures in this paper are available online at ieeexplore.ieee.
Digital Object Identifier10.1109/TSG.2014.2313724
Energy storage cost during time period
Charged energy during time period
Discharged energy time period
Set of different of diel generators
Types of diel generators
Diel energy cost during time period
Diel generation during time period
Maximum generation capacity
Ramp up constraint
Ramp down constraint
Set of time periods in planning horizon
Energy demand from urs during time period
Energy shortage or energy drawn from main
grid during time period检查报告范文
Amortization factor per time period
Investment cost per unit storage or generator
Operational/maintenance cost of energy storage
or generator
Objective function for planning problem
Maximal energy shortage probability allowed
Energy shortage threshold
Feasible t for the th scenario
Boundary parameters for the th scenario
Design parameters for the th scenario
Global boundary parameters一年以后
Global design parameters
Group index mapping for boundary conditions
Element-wi index mapping for boundary
conditions
非字开头的成语Dual variable update step-size
,Parameters for adaptive dual variable step-size
Maximum diel generation capacity ratio
Shortfall-to-demand ratio
I.I NTRODUCTION
R ENEWABLE energy sources[1],including solar and wind energy,provide only about3%of the electricity in the United States.However,high penetration of renew-able energy is becoming the trend for various reasons.The
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projected future shortage in fossil fuels,environmental con-cerns,and advances in smart grid[2]technologies stimulate the increasing penetration of renewable energy.Rearchers have shown that supplying all the energy needs of the United States from renewable energy is realizable in the future[3]. According to the National Renewable Energy Laboratory (NREL),renewable energy potentially will support about80% of the total electricity consumption in the U.S.in2050[4].The high penetration of renewable energy is especially common in(remote)isolated grids,or micro-grids with small carbon footprints[5].Micro-grids have many advantages,including incread reliability against power outages(which occur rarely but cau significant loss),incread u of onsite renewable e
nergy sources,reduced loss from long-distance transmission, and potential economic benefits.This is especially true with the increa of fuel costs,environmental taxes,and incentives for renewable energy.Although it is not necessary for a micro-grid to operate in isolated mode on a regular basis,they are often designed to be lf-sustained most of the time.
Most renewable energy sources,including wind and solar, are highly intermittent.The availability of such energy sources varies significantly in different geographical locations.In the same location,the amount of generation alsofluctuates de-pending on the time of day,ason,and weather conditions.A grid with high renewable energy penetration needs to build suf-ficient energy storage,to ensure an uninterrupted supply to end urs and make the best u of generated energy[6],[7].There are different types of energy storages,including super-capaci-tors,flywheels,chemical batteries,pumped hydro,hydrogen, and compresd air[8]–[13].Different types of energy storage have different ,round-trip energy efficiency, maximum capacity/power rating,energy loss over time,and investment/operational costs.For example,flywheel energy storage has high energy efficiency and charge/discharge rates, but the rate of energy loss over time is relatively high.Chemical batteries have relatively high energy efficiency and low energy loss over time,however their maintenance cost is high due to their low durability,which is quantified by cycling
capacity1. Pumped hydro and hydrogen energy storages have relatively low energy efficiency(large scale pumped hydro has higher efficiency),but their lf-discharging rate is quite small.In addi-tion,the capital cost per MWh of pumped hydro storage is low. Therefore they are often ud for longer-term energy storage. Although there has been rearch on planning and/or oper-ating a specific type of energy storage system for isolated elec-tricity grids[14]–[17],few works consider exploiting the dif-ferent characteristics of multiple types of energy storage and the different availabilities of multiple types of renewable en-ergy sources,forming a hybrid energy generation and storage system.Nevertheless,jointly planning for energy storage along with renewable generation capacity potentially results in a more economical and efficient energy system.
Since the future grid is becoming decentralized,we consider the scenario of an isolated grid,or a micro-grid with a small carbon footprint,who energy is generated mainly from renew-able energy sources.To make the scenario more practical,we assume the grid also has traditional diel generators.The diel 1The maximum number of charging cycles(full charge and discharge).generator on its own is insufficient to supply the demand of the grid,as its generation capacity is significantly less than the peak load.We formulate an optimization problem with the objective of minimizing the investment cost and operational/maintenance cost of energy storage and
generators,byfinding an optimal com-bination of different energy storages and generators(which we refer to as design parameters)and optimizing their operations. The renewable generation and ur demands change with time,and have different characteristics at different times of day and different days of the year.It is often difficult to obtain an accurate probability density function to reflect the complex characteristics.Therefore,veral years of historical data may be needed to obtain better optimization results.As the size of the historical databa increas,the design horizon of the optimiza-tion problem increas,and the problem becomes increasingly difficult to solve.To resolve this problem,we reformulate the original problem as a connsus problem.The entire design horizon is divided into multiple shorter horizons,and thus the design parameters become the connsus parameters,which should be consistent across all sub-problems.This framework can also be extended to the ca of solving chance-constrained optimization using scenario approximations,as we will elab-orate later.We propo to solve the connsus problem in a parallel distributed manner bad on the alternating direction method of multipliers(ADMM)[18],which mitigates the cur of dimensionality due to incread number of scenarios.
The rest of this paper is organized as follows.In Section II we briefly review some relevant works.In Section III we describe the system model,including the energy storage and generators. In Section IV
we formulate the optimization problem and solve it in a distributed manner.We provide numerical examples in Section V,and conclude the paper in Section VI.
Notations
We u italic symbols to denote scalars,bold italic symbols to denote vectors,calligraphic symbols to denote ts,
to denote the cardinality of a t,and superscript to denote matrix or vector transpo.We u to denote a collection of all’s for,to denote the th element of vector ,and to denote its norm.The concatenation of two vec-tors is equivalent to.We u to denote“is a member of”,to denote“for all”,to denote t union,and to denote t interction.
II.R ELATED W ORK
There have been veral works on optimization with energy storages and renewable generation,and we briefly review some of them here.In[8],the authors investigated the combined op-timization of a wind farm and a pumped storage facility from the perspective of a generation company,using a two-step sto-chastic optimization approach.The optimization produces op-timal bids for the day-ahead sp
ot market,and optimal operation strategies of the facilities.The optimal planning of generation and energy storage capacity was not considered.Zhou et al.[10] propod a composite energy storage system that contains both high energy density storage and high power density storage.The propod power converter configuration enables actively dis-tributing demands among different energy storages.Brown et al.
[15]provided an economical analysis of the benefits of having pumped storage in a small island system with abundant renew-able energy,and propod tofind the optimal pumped storage capacity through linear programming.In[16],the authors con-sidered optimizing the rating of energy storage in a wind-diel isolated grid,and demonstrated that high wind penetration po-tentially results in significant cost savings in terms of fuel and operating costs.
The main contributions of our work are two fold.First,in-stead of a single type of energy storage or renewable energy source,we consider a hybrid system with multiple types of en-ergy storage and renewable energy sources,and jointly optimize their capacities and operation.This joint optimization exploits the benefits from each individual element,and therefore is more cost efficient.Second,we propo a distributed optimization framework,so that the capacity design problem becomes scal-able when the number of scenarios increas.
III.S YSTEM M ODEL
A.Energy Storage Model
Assume there is a t of different types of energy storages. We u superscript to denote the type of the storage.Each type of energy storage is characterized by a group of parameters. We u to denote the one-way energy efficiency of energy storage type;to denote the ratio between the rated power and rated energy;and to denote the lf-discharging rate per unit time period.The cost of energy storage includes the initial investment cost and operational/maintenance cost.We u to denote the amortization factor.
Let denote the energy in storage at the beginning of time period,satisfying the following equation:
if
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(1)骆组词拼音
where positive denotes discharge from storage during time period,and negative denotes charge to the storage.Make the following substitution:
(2) and we can then rewrite(1)as
(3)
An interpretation of(3)is that the energy stored in a specific energy storage type equals the stored energy at the beginning of the previous time point,minus(plus)the discharge(charge) during the previous time period,minus the energy loss due to the nature of the storage.
The amount of stored energy and the charge/discharge power is constrained by the capacity of the ,
(4)
(5)In this work we u to denote the ratio between the rated power and the rated storage capacity.Therefore
and.If the ratio is notfixed,we can introduce another design variable for the rated power,and modify the investment cost so that it depends on both and.
The cost of each type of energy storage during time period, denoted by,includes the amortized investment cost and the operational/maintenance ,
(6)
In this equation,the operational/maintenance cost depends on the amount of charge and discharge,and anyfixed cost can be included as a constant term in this cost function.
Note that we made substitution(2),and therefore con-straints(1)and(3)are equivalent if only one element of each pair is non zero for all.Theorem1(in Section IV-A)guarantees that this condition is satisfied,and therefore the two constraints are indeed equivalent.
B.Generator Model
The generators are classified into traditional diel generators and renewable generators.For diel generators,the constraints include the generation capacity and generator ramp constraints. Let denote the t of all diel generators,and denote the generation of generator type during time period.We then have
(7)
(8)
where denotes the maximum generation capacity,and and denote ramp down and ramp up constraints, The cost of diel generators consists of the amor-tized investment cost and the operational/maintenance cost,de-noted by
(9) Usually a cond-order quadratic function or piece-wi linear function is ud for.Any environmental tax can also be included in this cost function.
We employ multiple types of renewable generators,including wind and solar,which are considered as non-dispatchable gener-ations.Let denote the renewable generation from type generator during time period,and denote the installed capacity.Then the generation can be written as
(10) where is a random variable denoting the renewable genera-tion per unit generation capacity.The cost for renewable energy during time period is then
怎么升级win10(11)
In addition to the generator types we discuss here,other types of ,hydro and nuclear generators can also be modeled similarly and included in the planning problem.
C.Load Balance Constraint
The total generation should equal the total demand in a power grid at all times.Let denote the energy shortage for an iso-lated grid,or the energy drawn from the main grid for a micro-grid.The total generation and discharge from the energy stor-ages should be equal to the total consumption and charge to the energy storages.We can then write the load balance constraint as follows:
(12) where denotes the demand from urs.Note that can be negative,which denotes energy injection to the main grid from a micro-grid,or dumped energy in an isolated grid.
IV.S TORAGE AND R ENEWABLE G ENERATION P LANNING
A.Optimal Planning Problem
The planning goal is tofind the optimal portfolio of different types of energy storage and generators,so that the total cost(in-cluding investment and operational/maintenance)is minimized, while most of the needs of the grid can be satisfied.Let de-note the planning horizon,and the objective function can then be written as
(13) Due to the intermittency of renewable energy sources,it is pos-sible that in extreme cas,the total local generation will not meet the total demand.We write the grid reliance constraint(for micro-grids)or the energy shortage constraint(for isolated grids)as
(14) where is a threshold which can be a function of current time and demand.There are also constraints on the minimum and maximum capacity for each type of storage and generator, which are denoted as
(15) We then formulate the optimization problem for energy plan-ning as
(16)
Variables such as the charging and discharging schedule of different energy storages,generation of diel generators,etc. are all optimization variables.Since they are not design param-eters of interest,we omit them in(16)for notational simplicity. One problem with this formulation is whether constraints(1) and(3)are equivalent.Bad on the problem tup,we have the following theorem.
Theorem1:In a cost minimization context,given an in-creasing positive operational cost function for charging and discharging,by making the substitution(2),we have that for ,only one of and can be non-zero for any given time period.
Proof:See Appendix A.
Additional costs and constraints can also be easily included in this formulation.For example,an envir
onmental tax for tradi-tional diel generators,and government incentives for renew-able generations can be included in the corresponding cost func-tions.The maximum allowed diel generation capacity speci-fied by certain energy policies can be included in the generator constraints.
Remark1:The problem formulation can be slightly mod-ified into a chance-constrained problem.Instead of the deter-ministic constraint(14),we can u the following probabilistic constraint:
(17) where is the maximal energy shortage probability al-lowed.Constraint(17)means that local generators and storages have a probability less than or equal to to be short of energy greater than.In this ca,using the results from[19],[20], the probabilistic constraint can be approximated by a t of de-terministic constraints,sampled from the probability distribu-tion of the random parameters from the probabilistic constraint. To be more specific,let each scenario be a random realization of load,renewable generation,and initial conditions of the en-ergy storages.The number of required scenarios
is determined by the number of design parameters and the prob-ability measure.Let denote the number of design param-eters.According to[19],if the number of scenarios is no less than,then the solution to the scenario approximation problem has a proba-bility at least to satisfy the original chanc
e constrained problem.The problem formulation and method of solving the problem are very similar to(16).We will point out the differ-ence in Remark2.For examples of using scenario approxima-tion to solve chance constrained optimization,plea refer to [21]and[22],where the authors employed this framework to solve the problem of optimizing distributed renewable energy source management.
B.Formulation of Connsus Problem
The renewable generation and ur loads in(16)are all random.In practice,historical data is ud in the problem formulation.With a large number of realizations of the random parameters from historic data,the problem becomes increas-ingly difficult to solve due to the increa of dimensionality. In the rest of this ction,we will reformulate the original
problem(16)as a connsus problem,which can be solved in a distributed manner.
We divide the entire planning horizon into sub planning horizons,which we call scenarios for simplicity.Let de-note the t of all sub horizons,and we have that. For convenience,we assume’s are arranged in the of time.Let denote the design parame-ters for the th scenario,denote the global design parameters,and denote the feasible t for the design variables of the th scenario,with.In prac-tice,the energy in storages at the beginning of each time period is not random,but rather depen
ds on the energy from the pre-vious time period.Assuming the energy stored at the beginning of a scenario should be equal to the energy stored at the end of the previous scenario,we need additional constraints to en-sure this condition is satisfied.Let denote the energy storage at the beginning of the th scenario,and denote the energy storage at the end of the th scenario.In we followed the approach in[15],and impod an additional assumption that the energy in each energy storage at the end of the optimization horizon should be equal to that at the beginning of the optimiza-tion ,.However,this assumption makes the solution suboptimal.In this work,we eliminate this assumption and add additional connsus constraints across sce-narios.
Let denote the boundary parameters for the th denote the global boundary parameters. Let denote the mapping for the indices of the boundary conditions for the th scenario.To be specific,denotes the global boundary parameters corresponding to. We also u the scalar function to denote element-wi index ,corresponds to.The con-straints can then be written as
.Using the notations,, and,we then formulate the original optimiza-tion problem(16)as follows:
(18) The global design parameters from solving(18)will satisfy that.Note that similar to(16),we omit some optimiza-tion variables that are not design variables for notational sim-plicity.
Remark2:If the probabilistic constraint is considered,and scenario approximation approach is ud,the formulation have to be slightly revid.According to[19],the random samples for each scenario has to be generated from independent identical distributions.Note that the starting energy stored in the storages also has to be drawn from certain probability distributions.The connsus formulation for the energy storage boundary condi-tions can then be removed.The number of generated scenarios has to be greater than or equal to the minimum number described in Remark1.
C.Distributed Optimization
The challenge in solving(18)is that as the number of sce-narios increas,the problem becomes increasingly difficult due to high time complexity.We propo to solve the problem in a distributed manner bad on the alternating direction method of multipliers(ADMM)[18],which mitigates the time complexity issue and makes the problem scalable.
To enforce the equality(connsus)constraint in(18),an additional quadratic term is added to the original Lagrangian, forming the augmented Lagrangian which can be written as
(19) where denote the dual variables,and is a pre-defined parameter which is the dual variable update step size.The quadratic term penalizes the difference between the local vari-ables and corres
ponding entries of the global variable, denoted by.
The ADMM algorithm iterates among the following steps, with subscript denoting the iteration number.
1)-Minimization Step:For each,the following local minimization problems are solved in parallel:
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(20) 2)-Minimization Step:
(21) To solve for the-minimization step,we consider and parately.Decompo,and we then rewrite(21) as
(22) Solving(22),we obtain that
(23)
(24) When the algorithm converges,the resulting global design variable has to satisfy the constraints of each sub-problem,
<,.Therefore we have that.
3)Dual-Variable Update:For each,the dual variables are updated in parallel:
(25) Since(20)and(25)can be parallelized,the problem is scal-able as the number of scenarios increas.The convergence of
