Energy Storage Benchmark Problems

Transcription

1 Energy Sorage Benchmark Problems Daniel F. Salas 1,3, Warren B. Powell 2,3 1 Deparmen of Chemical & Biological Engineering 2 Deparmen of Operaions Research & Financial Engineering 3 Princeon Laboraory for Energy Sysems Analysis Princeon Universiy, Princeon, NJ April 6, 2013 Conens 1 Purpose 1 2 A General Sorage Problem The Sae of The Sysem The Decisions The Consrains The Exogenous Informaion Process The Discree Uniform Disribuion The Discree Pseudonormal Disribuion The Transiion Funcion The Objecive Funcion Linear Programming 4 4 Dynamic Programming The Wind Process The Price Process The Demand Process The Markov Decision Process (MDP) The Benchmarks 6 1

2 5.1 The MATLAB Files The.x Files Purpose In his documen, we lay ou he model for a general energy sorage conrol problem and hen design a se of benchmark problems which can be solved exacly. The exac soluion o each one of hese problems is included in he accompanying.zip file. This is mean o be a collecion of problems ha anyone can use o asses he opimaliy of heir algorihm. We undersand ha real energy markes are much more complex han in his seing bu we are confiden ha he maerial presened in his documen serves as a generally realisic benchmark for energy sorage conrol algorihms. Furhermore, we wan o noe ha some of he model parameers were picked fairly arbirarily, hough we made sure hey led o ineresing models in he conex of energy sorage. Ulimaely, he purpose of his documen is o serve as a algorihmic benchmark and no an exac model of wind energy sorage. Furhermore, variables are presened as uniless wih he undersanding ha he appropriae unis of energy, power, price, ec. are implied. For example, one can hink of ime as hr, energy as MWh, power as MW, price as $, ec. or any oher uni sysem as long as all variables are of consisen dimensions. 2 A General Sorage Problem The nework consiss of a single energy sorage device which is conneced o a wind farm and o he elecriciy grid. Elecriciy may flow direcly from he wind farm o he sorage device or i may be used o saisfy he demand. Energy from sorage may be sold o he grid a any given ime, and elecriciy from he grid may be bough o replenish he energy in sorage or o saisfy he demand. We le T = {0,, 2,..., T, T } be a finie ime horizon. 2.1 The Sae of The Sysem The variable S = (R, E, D, P ) describes he sae of he sysem a ime is given by: R : The amoun of energy in he sorage device a ime. E : The ne amoun of wind energy available a ime. D : The aggregae energy demand a ime. P : The price of elecriciy a ime in he spo marke. 1

3 2.2 The Decisions A any poin in ime, he decision is given by he column vecor x = (x W D where x IJ, x RD, x GD, x W R, x GR, x RG ), is he amoun of energy ransferred from I o J a ime. The superscrip W sands for wind, D for demand, R for sorage and G for grid. 2.3 The Consrains We require ha all componens of x be nonnegaive for all. We le R max be he oal capaciy of he baery, η c and η d be he charging and discharging efficiencies, respecively, and γ c and γ d be he maximum charging and discharging raes, respecively. For energy sorage, a any ime we also require ha he oal amoun of energy sored in he device from he wind does no exceed he capaciy available, x W R + x GR R max R. (1) We also make he assumpion ha all demand a mus be saisfied a ime : x W D + η d x RD + x GD = D. (2) Addiionally, he amoun wihdrawn from he device a ime o saisfy demand plus any amoun of energy sold o he grid afer saisfying demand mus no exceed he amoun of energy ha is available in he device when we make he decision o sore or wihdraw: x RD + x RG R. (3) The oal amoun of energy charged o or wihdrawn from he device is also consrained by he maximum charging and discharging raes: x W R x RD + x GR γ c, (4) + x RG γ d. (5) Finally, flow conservaion requires ha: x W R + x W D E. (6) The feasible acion space, X, is he convex se defined by (1)-(6). We le X π (S ) be he decision funcion ha reurns x X, where π Π represens he ype of policy (which we deermine laer). 2.4 The Exogenous Informaion Process For he purpose of his model, W = (Ê, ˆP ). Noe ha he demand is assumed o be fixed (hough no necessarily consan). Ê = The change in he energy beween imes and. ˆP = The change in he price of elecriciy beween imes and. 2

4 To avoid violaing he nonanicipaiviy condiion, we assume ha any variable ha is indexed by is F -measurable. As a resul, W is defined o be he informaion ha becomes available beween imes and. A sample realizaion of W is denoed W n = W (ω n ) for sample pah ω n Ω The Discree Uniform Disribuion We le U(a, b) for a, b R be he uniform disribuion which defines he evoluion of a discree random variable X wih meshsize X. Then each elemen in X = {a, a+ X, a+2 X,..., b X, b} has he same probabiliy of occurring. The probabiliy mass funcion is given by: for all x X. u X (x) = The Discree Pseudonormal Disribuion X b a + X, Le X be a normally disribued random variable and le f X (x; µ X, σx 2 ) be he normal probabiliy densiy funcion wih mean µ X and variance σx 2. We define a discree pseudonormal probabiliy mass funcion for a discree random variable X wih suppor X = {a, a + X, a + 2 X,..., b X, b} as follows, where a, b R are given and X is he mesh size. For x i X we le: g X(x i ; µ, σ 2 ) = f X (x i ; µ X, σ 2 X ) X x j=0 f X(x j ; µ X, σ 2 X ) be he probabiliy mass funcion corresponding o he discree pseudonormal disribuion. We say ha X N (µ X, σ 2 X ) if X is disribued according o he discree pseudonormal disribuion. We recognize his is non-sandard noaion bu i simplifies he noaion in his documen. We include sample m-files creaep ricep robabiliy.m and creaew indp robabiliy.m for creaing he pseudonormal probabiliy densiy funcion and he cumulaive disribuion funcion. 2.5 The Transiion Funcion The ransiion funcion is given by S + = S M (S, x, W + ). The ransiion funcion for he energy in sorage is given by: R + = R + φ T x, where φ = (0, 1, 0, η c, η c, 1) is an incidence column vecor ha models he flow of energy from one node o anoher. The ransiion dynamics for he wind, price and demand processes are given in secions The Objecive Funcion The funcion C(S, x ) represens he conribuion from being in he sae S and making he decision x a ime. Assuming ha he demand a ime mus always be saisfied a ime, our conribuion a ime 3

5 is jus he oal amoun of money paid or colleced when we ransfer energy o and from he grid a ime : where c h = is a holding cos. C(S, x ) = P D P (x GR η d x RG + x GD ) c h (R + φ T x ), We consider he conrol problem of maximizing he oal un-discouned expeced conribuions over he finie ime horizon T : The objecive funcion is hen given by: F π 3 Linear Programming [ = max E C ( S, X π (S ) )]. (7) π Π T If he sae variable evolves deerminisically and he dynamics are known a priori, we can solve he conrol problem using a sandard bach linear program (LP): F = max C(S, x ), x 0,,x T such ha x X for each and subjec o ransiion dynamics expressed as a se of consrains linking all ime poins. This formulaion is mos useful when we can make exac predicions abou he wind, demand and price rends. This is hardly ever he case wih physical processes ha are inrinsically sochasic, bu deerminisic problems are useful o es he abiliy of he algorihm o learn he soluion in he presence of holding coss, when energy should be sored in he device as laes as possible in order o avoid incurring exra coss. These es problems also allow us o es he capabiliy of he algorihm o learn o sore energy in cases where he impac of soring is no fel unil hundreds of ime periods ino he fuure. We es differen ypes of deerminisic ransiions dynamics, as specified in able 1. The acual values for each of hese is given in he accompanying daase, as explained in secion 5. T 4 Dynamic Programming In order o solve he sochasic problem exacly, we assume he device is perfecly efficien. For his reason, we use a modified cos funcion in our simulaions: C(S, x ) = P D P (x GR ρx RG where ρ = 0.98 is a sof cos added o avoid degenerae soluions. 4.1 The Wind Process + x GD ) c h (R + k x ), The wind process E is modeled using a firs-order Markov chain: E + = E + Ê+ T \ {T }, 4

6 such ha E min E E max, and where Ê is eiher pseudonormally or uniformly disribued (see Table 2). In he case where Ê N (µ E, σe 2 ), is suppor is { 3, 3 + E, E,..., 0,..., 3 E, 3}. 4.2 The Price Process We es wo differen sochasic processes for P : Sinusoidal: P + = µ P + + ˆP 0,+ T \ {T }, where µ P = sin ( ) 5π 2T and ˆP0, N (µ P, σp 2 ). 1s-order Markov chain: P + = P + ˆP 0,+ + 1 {u+ p} ˆP 1,+ T \ {T }, such ha P min P P max, and where ˆP 0, is eiher pseudonormally or uniformly disribued as indicaed in Table 2. In he case where ˆP 0, N (µ P, σp 2 ), is suppor is { 8, 8 + P, P,..., 0,..., 8 P, 8}. We le u U(0, 1), and we le p = for problems where jumps may occur and p = 0 oherwise, and ˆP 1, N (0, 50 2 ) wih suppor { 40, 40 + P, P,..., 0,..., 40 P, 40}. 4.3 The Demand Process The demand is assumed o be deerminisic and given by D = max [ 0, 3 4 sin ( )] 2π T, where is he floor funcion. 4.4 The Markov Decision Process (MDP) The opimal soluion o sochasic problems can be found for problems which have denumerable and relaively small sae, decision and oucome spaces, S, X and W, respecively. In hese cases, Bellman s opimaliy equaion can be wrien as: S V ( (S ) = max C(S, x ) + x X s =1 P (s S, x )V + (s ) ) for T, (8) where P (s S, x ) is he ime-dependen condiional ransiion probabiliy of going from sae S o sae s given he decision x, and where we assume ha VT + = 0. Afer solving (8), we can simulae he model as a MDP by sepping forward in ime following he opimal policy, π, defined by he opimal value funcions (V ) T. For a given sample pah ω Ω, we can simulae he MDP by solving: S ( (ω) X π (S (ω)) = arg max C(S (ω), x ) + P (s S (ω), x )V+ (s S (ω), x ) ) for T, x X where S +1 (ω) = S M (S (ω), X π (S (ω)), W +1 (ω)). s =1 5

7 Label Price, P Wind Energy, E Demand, D F 1000 D1 Sinusoidal Consan Sinusoidal 99.99% D2 Sinusoidal Sep Sep 99.92% D3 Sinusoidal Sep Sinusoidal 99.97% D4 Sinusoidal Sinusoidal Sep 99.98% D5 Consan Consan Sinusoidal 99.97% D6 Consan Sep Sep 99.93% D7 Consan Sep Sinusoidal 99.98% D8 Consan Sinusoidal Sep 99.99% D9 Flucuaing Flucuaing Sinusoidal 99.97% D10 Flucuaing Flucuaing Consan 99.96% Table 1: Deerminisic es problems. Resource, R Wind, E Price, P Label Levels R Levels E Ê Levels Process ˆP0, S U( 1, 1) 7 Sinusoidal N (0, 25 2 ) S N (0, ) 7 Sinusoidal N (0, 25 2 ) S N (0, ) 7 Sinusoidal N (0, 25 2 ) S N (0, ) 7 Sinusoidal N (0, 25 2 ) S U( 1, 1) 41 1s-order + jump N (0, ) S U( 1, 1) 41 1s-order + jump N (0, ) S U( 1, 1) 41 1s-order + jump N (0, ) S U( 1, 1) 41 1s-order + jump N (0, ) S N (0, ) 41 1s-order + jump N (0, ) S N (0, ) 41 1s-order + jump N (0, ) S N (0, ) 41 1s-order + jump N (0, ) S N (0, ) 41 1s-order + jump N (0, ) S N (0, ) 41 1s-order + jump N (0, ) S N (0, ) 41 1s-order + jump N (0, ) S N (0, ) 41 1s-order + jump N (0, ) S N (0, ) 41 1s-order N (0, ) S N (0, ) 41 1s-order N (0, ) S N (0, ) 41 1s-order N (0, ) S N (0, ) 41 1s-order N (0, ) S N (0, ) 41 1s-order N (0, ) S N (0, ) 41 1s-order N (0, ) Table 2: Sochasic es problems. Since he ransiion from S o s is sochasic, a saisical esimae of (7) may be found by simulaing K sample pahs ω 1,..., ω K Ω : 5 The Benchmarks F = 1 K K ) C (S (ω k ), X π (S (ω k )). k=1 T The se of deerminisic benchmark problems in given in able 1. Tha of sochasic benchmark problems is given in Table 2. Each problem is given a label. Some parameers are fixed for all problem insances where hey are used. These are given in ables 3 and The MATLAB Files Deerminisic problems: The daa for each problem insance is in a srucure named label. Each srucure consains six componens: 6

8 Parameer Value R max R min 0.00 R η c 0.90 η d 0.90 γ c 0.10 γ d 0.10 T Table 3: Lis of parameers for deerminisic es problems. Parameer Value R max R min 0.00 R P max P min P 1.00 E max 7.00 E min 1.00 η c 1.00 η d 1.00 γ c 5.00 γ d 5.00 T K 256 Table 4: Lis of parameers for sochasic es problems. label.c A scalar C, where C = F represens he opimal soluion. label.r A (T + 1) 1 vecor R, where R represens he energy in sorage a ime. label.e A (T + 1) 1 vecor E, where E represens he energy available from wind a ime. label.p A (T + 1) 1 vecor P, where P represens he elecriciy price a ime. label.d A (T + 1) 1 vecor D, where D represens he energy demand a ime. label.x A 6 (T + 1) marix x, where x : represens he opimal decision vecor a ime. Sochasic problems: The daa for each problem insance is in a srucure named label, conained in label.ma. consains eigh componens: Each srucure label.c A (T + 1) K marix C, where C k represens he conribuion earned a ime for sample pah k, i.e. C k = C ( S (ω k ), X π (S (ω k )) ). label.r A (T + 1) K marix R, where R k represens he energy in sorage a ime for sample pah k. label.e A (T + 1) K marix E, where E k represens he energy available from wind a ime for sample pah k. 7

9 label.eha A (T + 1) K marix Ê, where Êk represens he change in energy available from wind beween imes 1 and for sample pah k. label.p A (T + 1) K marix P, where P ij represens he elecriciy price a ime i for sample pah j. label.pha A (T + 1) K marix ˆP, where ˆP k represens he change in elecriciy price beween imes 1 and for sample pah k. label.d A (T + 1) 1 vecor D, where D represens he energy demand a ime. label.x A 6 (T + 1) K ensor x, where x :k represens he opimal decision vecor a ime for sample pah k, X π (S (ω k )). 5.2 The.x Files Deerminisic problems: The daa for each problem insance is in a folder named label. In he folder, here are six.x files, one for each of he following componens: C.x A scalar C, where C = F represens he opimal soluion. R.x A (T + 1) 1 vecor R, where R represens he energy in sorage a ime. e.x A (T + 1) 1 vecor E, where E represens he energy available from wind a ime. p.x A (T + 1) 1 vecor P, where P represens he elecriciy price a ime. D.x A (T + 1) 1 vecor D, where D represens he energy demand a ime. x.x A 6 (T + 1) marix x, where x : (he h column) represens he opimal decision vecor a ime. Sochasic problems: The daa for each problem insance is in a folder named label. In he folder, here are 263.x files, one for each of he following componens: C.x A (T + 1) K marix C, where C k represens he conribuion earned a ime for sample pah k, i.e. C k = C ( S (ω k ), X π (S (ω k )) ). R.x A (T + 1) K marix R, where R k represens he energy in sorage a ime for sample pah k. e.x A (T + 1) K marix E, where E k represens he energy available from wind a ime for sample pah k. eha.x A (T + 1) K marix Ê, where Êk represens he change in energy available from wind beween imes 1 and for sample pah k. p.x A (T + 1) K marix P, where P ij represens he elecriciy price a ime i for sample pah j. 8

10 pha.x A (T + 1) K marix ˆP, where ˆP k represens he change in elecriciy price beween imes 1 and for sample pah k. D.x A (T + 1) 1 vecor D, where D represens he energy demand a ime. xk.x A 6 (T + 1) marix x k, where x k : represens he opimal decision vecor a ime for sample pah k, X π (S (ω k )). 9