Online Appendices for: Optimal Recharging Policies for Electric Vehicles

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1 Submitted to Trasportatio Sciece mauscript TS R2 Olie Appedices for: Optimal Rechargig Policies for Electric Vehicles Timoth M. Sweda Scheider, 30 S. Packerlad Dr., Gree Ba, WI 5433, Iria S. Doliskaa, Diego Klabja Departmet of Idustrial Egieerig ad Maagemet Scieces, Northwester Uiversit, 245 Sherida Rd., Evasto, IL 60208, Appedix A: Supplemetal proofs A.. Proof of Lemma Note that π P is feasible because r i = h j esures that the vehicle will reach ode m albeit with 0 charge remaiig, r m = r m r i h j esures that the vehicle departs ode m with the same charge level as it would have i π P, ad all other rechargig amouts are the same as i π P. We have [ ] l l Cπ P Cπ P = c r l, r j h j c r l, r j h j l= i i = c r i, r j h j c r i, r j h j c rm, r j h j c r m, r j h j, sice r l = r l ad l r j h j = l r j h j for all l {,..., i, m,..., } ad l l c r l, r j h j = c r l, r j h j = 0 for all l {i,..., m } due to r l = r l = 0 ad c0, = 0. We also have i i r j h j = r j h j = r j h j = 0

2 b defiitio of i ad m. It follows that Cπ P Cπ P = cr i, 0 cr i, 0 c rm, r i h j cr m, 0 = s γr i f r i s γr i f h j s γr m f r i h j r m f r i h j s γr m f r m r i h j = f r i f f r i f h j f r i h j h j r i h j 2 f r i f r i 3 = 0, where 2 is true sice both h j > 0 ad r i h j > 0, ad 3 is true sice f is covex ad icreasig. Because π P is a optimal polic, Cπ P = Cπ P ad π P is also a optimal polic. A.2. Proof of Lemma 2 Suppose r 0 ad r are the amouts to recharge at odes 0 ad, respectivel, specified b polic π P. Let π P = [r,..., r ] deote the rechargig polic for P that satisfies { r r l = 0 r h 0, l = r l, l S P \{}. This polic is feasible because r = r 0 r h 0 esures that the vehicle departs ode with the same charge level as i π P, ad all other rechargig amouts are the same as i π P with the exceptio of ode 0, which is ot part of P ad therefore has o correspodig rechargig amout i π P. The the total cost of π P exceeds that of π P b Cπ P Cπ P = cr 0, 0 cr, r 0 h 0 cr 0 r h 0, 0 = [ s γr 0 f r 0 ] [ si {r >0} γr f r 0 h 0 r f r 0 h 0 ] [ s γr 0 r h 0 f r 0 r h 0 ] = si {r >0} γh 0 f r 0 f r 0 h 0 γh 0 because f is icreasig. Sice π P is a optimal rechargig polic for P, it follows that as desired. Cπ P Cπ P γh 0 Cπ P γh 0, 2

3 A.3. Proof of Lemma 3 Note that π P is feasible sice r i = r i h m esures that the vehicle departs ode i with just eough charge to reach ode m ad r i b suppositio, ad r m = r m rm h m esures that the vehicle departs ode m with the same charge level as i π P. We defie the followig subpaths of P alog with correspodig rechargig policies: P =,..., m, π P = [r,..., r]; P 2 = m,...,, π P 2 = [rm,..., r]; P =,..., m, π P = [r,..., r m ]; P 2 = m,...,, π P 2 = [r m,..., r ]; where π P ad π P 2 are subpolicies of π P, ad π P ad π P 2 are subpolicies of π P. These policies are all feasible due to Propert, which esures that the rechargig policies ca be subdivided at a ode where the vehicle stops to recharge sice the charge level there will be 0. The we have Cπ P Cπ P = Cπ P Cπ P 2 Cπ P Cπ P 2 Cπ P Cπ P γh m b Lemma 2 = cr i, 0 cr i h m, 0 γh m = γh m f r i f r i h m γh m = 0 sice r i h m. Because π P is a optimal polic, Cπ P = Cπ P ad π P is also a optimal polic. A.4. Proof of Lemma 4 Let µ i deote the idex of the i th rechargig stop where ˆr µi > 0 for all i {,..., ˆ}. Due to the costructio of ˆπ P, it must be true that < ˆr µi ˆr µi for a i {,..., ˆ }. The we have ˆ ˆ ˆ < ˆr µi ˆr µi < 2 ˆr µi = 2, i= which implies ˆ < 2, ad 2 ˆ. I the extreme case where h i = αq 2 max ε ε > 0 for all i S P, the above boud is tight sice ˆ = αq 2 max ε < lim ε 0 αq = 2. 2 max ε We thus have 2 ˆ, which is also equal to 2/ if 2/ is a iteger, ad equalit holds whe Therefore, the lemma statemet is valid. ε = i= 2/ 2. 3

4 A.5. Proof of Lemma 5 Note that π P,q is feasible because the vehicle ca reach ode m without rechargig, r m = r i esures that the vehicle departs ode m with the same charge level as it would have i π P,q, ad all other rechargig amouts are the same as i π P,q. We have [ ] l l Cπ P,q Cπ P,q = c r l, q r j h j c r l, q r j h j l= i = c r i, q h j c r i, q h j i i = s γr i f q h j r i f q h j s γr i f q h j r i f q h j i f q h j r i h j f q h j 0 b covexit of f h j sice i < m, r i > 0, ad f is icreasig. Because π P,q is also a optimal polic. is a optimal polic, Cπ P,q = Cπ P,q ad π P,q A.6. Proof of Theorem Suppose for some cotiuous rechargig polic ρ P, = [λ,..., λ ], [r,..., r ] that i l= r l = P λ i = λ i for all i {,..., }. Note that this coditio is the cotiuous aalog of Propert ad there exists a optimal rechargig polic that satisfies it. Also suppose that r i < r j for some i, j {,..., }. Let ρ P, = [λ,..., λ ], [r,..., r ] be a polic with r i ε, l = i r l = r j ε, l = j r l, l {,..., }\{i, j} for some ε 0, r j r i ad i r j = P λ i = λ i for all i {,..., }. That is, the locatios of rechargig stops i through j or j through i are adjusted such that i l= r l = P λ i for all i {,..., }. Note that this polic is feasible because the total amout recharged alog the path is still ad the vehicle ol recharges whe its charge level is 0. The we have Dρ P, Dρ P, = cr l, 0 cr l, 0 l= = cr i ε, 0 cr i, 0 cr j ε, 0 cr j, 0 = [ s γr i ε f r i ε ] [ s γr i f r i ] [ s γr j ε f r j ε ] [ s γr j f r j ] = f r i ε f r i 4

5 0 f r j f r j ε b covexit of f sice r i ε < r j. No such improvemet exists whe r =... = r, i which case we must have r l = / ad λ l = l / for all l {,..., } i order to satisf the coditio i l= r l = P λ i = λ i for all i {,..., }. Therefore, the cotiuous rechargig polic ρ P, is optimal. A.7. Proof of Theorem 2 For a, let ρ P, = [λ l = l / : l =,..., ], [r l = / : l =,..., ]. B Theorem, this polic is optimal for a give. The total cost of the polic ρ P, is thus Dρ P, = s k. 4 We cosider the followig two cases that describe the possible relatios of the rechargig amout at each stop, which is the same for all stops, to : Case : < i.e., the vehicle alwas overcharges wheever it stops ad Case 2: i.e., the vehicle ever overcharges whe it stops. The for a give case, icreasig the value of b oe i.e., addig a extra rechargig stop decreases the amout recharged at each stop to. This results i a trasitio from Case to Case 2 if < αq max or o chage otherwise. We defie the possible trasitios as follows: Trasitio : Case to Case, Trasitio 2: Case to Case 2, ad Trasitio 3: Case 2 to Case 2. The differece i cost whe icreasig the umber of stops b oe is k, Dρ P, Dρ P, = s k, αq < max 0, > for feasible i.e., /. The subcases 5a, 5b, ad 5c correspod to the cost differeces of Trasitios, 2, ad 3, respectivel. As icreases, ote that Trasitio caot occur after Trasitio 2, which i tur caot occur after Trasitio 3. We ca therefore order the trasitios ad compare them sequetiall. Comparig Trasitios ad 2 first, we fid that s k < s k sice / <, ad comparig Trasitios 2 ad 3 reveals that s k s because / 0. The fuctio Dρ P, is therefore covex ad also piecewise liear with respect to ad has up to three segmets. We are iterested i the value of that miimizes Dρ P,, or the smallest feasible iteger value of such that the cost differece is o-egative. There are three differet cases to cosider. 5 5a 5b 5c

6 If the expressio i 5b is egative, the ol the third segmet of the fuctio has a o-egative slope. It is also implied that is o-iteger i this case because otherwise we would have = 0 ad cosequetl s < 0, which is ot possible sice the stoppig time is assumed to be o-egative. That segmet icludes values of that are greater tha, ad the smallest iteger value of such that >, where is o-iteger, is. This result correspods to the case 4a, ad because, where is the miimum possible umber of stops, it is feasible. If the expressio i 5a is egative ad the oe i 5b is ot, the both the secod ad third segmets of the fuctio have o-egative slopes. Thus, the optimal umber of stops is icluded i the secod segmet if feasible, which cosists of values satisfig <. The ol iteger value of that satisfies the iequalities is, but it caot be less tha the miimum possible umber of stops,. This result correspods to the case 4b. If the expressio i 5a is o-egative, the all three segmets of the fuctio have o-egative slopes. Therefore, the fuctio is miimized whe is miimized, ad the smallest feasible iteger value of is. This is also the umber of stops made whe is ifeasible for i the previous case, ad the result correspods to the case 4c. Therefore, the stated expressio for argmi { Dρ P, } holds. A.8. Proof of Theorem 3 Suppose for some equidistat rechargig polic σ P, = [µ,..., µ ], [r,..., r ] satisfig Propert that r i h < r j for some i, j {,..., }. Let σ P, = [µ,..., µ ], [r,..., r ] be the polic i which r i h, l = i r l = r j h, l = j r l, l {,..., }\{i, j} ad correspodig [µ,..., µ ] are defied based o Propert. Note that this polic is feasible because the total amout recharged alog the path is still h ad the vehicle ol recharges whe its charge level is 0. The we have Eσ P, Eσ P, = cr l, 0 cr l, 0 l= = cr i h, 0 cr i, 0 cr j h, 0 cr j, 0 = [ s γr i h f r i h ] [ s γr i f r i ] [ s γr j h f r j h ] [ s γr j f r j ] = f r i h f r i 0 f r j f r j h b covexit of f sice r i h < r j. No such improvemet exists whe max i,j {,...,} {r i r j } h, ad the total cost caot be further reduced b violatig Propert. Therefore, the equidistat rechargig polic σ P, is optimal. 6

7 A.9. Proof of Theorem 4 Let σ P, = [µ,..., µ ], [r,..., r] with r =... = r = h ad r =... = r = h. B Theorem 3, this polic is optimal for a give. The total cost of polic σ P, is thus Eσ P, = s k r i. 6 i= If we relate the two possible values for the amout to recharge at each stop to, we see that there are three possible cases to cosider: Case : < h h i.e., the vehicle alwas overcharges wheever it stops, Case 2: h < h i.e., the vehicle overcharges at the first stops but ot at the remaiig stops, ad Case 3: h h i.e., the vehicle ever overcharges whe it stops. The for a give case, icreasig the value of b oe i.e., addig a extra rechargig stop decreases the amouts recharged to h ad h for the correspodig stops. This could result i a ew relatioship of the rechargig amouts to ad trasitio us to a differet case. For example, we trasitio from Case to Case 2 whe < h h ad h < h. The possible trasitios are the followig: Trasitio : Case to Case, Trasitio 2: Case to Case 2, Trasitio 3: Case 2 to Case 2, Trasitio 4: Case to Case 3, Trasitio 5: Case 2 to Case 3, ad Trasitio 6: Case 3 to Case 3. Note that is the maximum possibl fractioal umber of stops such that the amout recharged at each stop is greater tha, ad is the maximum feasible umber of stops such that ot all of the amouts recharged at each stop are less tha. If >, the there is o feasible value for such that h αq max < h. Thus, we ol eed to cosider Trasitios, 4, ad 6. Alterativel, if, the all trasitios except Trasitio 4 are possible. We ow compute the costs for each of the six possible trasitios. The differece i cost whe icreasig the umber of stops b oe is E σp, E σ P, = 7

8 k, < k h, = < ad k h, < ad s k h, = < ad > k h, = < ad 0, for feasible i.e., /. The six subcases i 7 correspod to the cost differeces of Trasitios through 6, i order. We first cosider the case where > 7a 7b 7c 7d 7e 7f. As icreases, ote that Trasitio caot occur after Trasitio 4, which i tur caot occur after Trasitio 6. We ca therefore order the trasitios ad compare them sequetiall. Begiig with Trasitios ad 4, we compare 7a ad 7d ad fid that s k s k h h = s k h sice because > case. h < h ad s k h h 0. Comparig Trasitios 4 ad 6, we ote that < s k h h = s ad h >. Thus, E is covex ad piecewise liear with respect to i this We ext cosider the case where. As i the previous case, the trasitios ca be ordered ad compared sequetiall. We first compare 7a ad 7b Trasitios ad 2 ad see that s k k h because s k 0 ad h 0. We ext compare Trasitios 2 ad 3 ad fid that h s k h = s k h. Comparig 7c ad 7e, or Trasitios 3 ad 5, gives s k h < s k h, ad the expressio i 7e is o greater tha s, the cost of Trasitio 6. Therefore, i each of the cases > ad, the fuctio E is covex ad also piecewise liear with respect to. We are iterested i the value of that miimizes E, or the smallest feasible iteger of such that the cost differece is o-egative. There are several cases to cosider. 8

9 If the expressio i 7d is egative whe > or the expressio i 7e is egative whe, the ol the last segmet of the fuctio has a o-egative slope. That segmet icludes values of that are at least, ad the smallest iteger value of such that is. This result correspods to cases 5a ad 6a, ad because, where is the miimum /h /h possible umber of stops, it is feasible. If the expressio i 7a is egative ad the oe i 7d is ot whe, or if the expressio i 7c is egative ad the oe i 7e is ot whe, the if is a feasible umber of stops, the last two segmets of the fuctio have o-egative slopes. Thus, the optimal umber of stops is icluded i the secod-to-last segmet, which cosists of values satisfig = <. The ol iteger value of that satisfies the expressio is, where must be o-iteger. This result correspods to cases 5b ad 6b. If the expressio i 7b is egative ad the oe i 7c is ot whe, the the third, fourth, ad fifth segmets of the fuctio have o-egative slopes. The optimal umber of stops is therefore icluded i the third segmet if feasible whe, which cosists of values satisfig <. The smallest iteger value of satisfig the expressio is, but it must be feasible ad caot be less tha /h. This result correspods to case 6c. If the expressio i 7a is egative ad the oe i 7b is ot whe, ad if is a feasible umber of stops, the the secod through fifth segmets of the fuctio have o-egative slopes. Thus, the optimal umber of stops is icluded i the secod segmet whe, which cosists of values satisfig = <. The ol iteger value of that satisfies the expressio is, where must be o-iteger. This result correspods to case 6d. If the expressio i 7a is o-egative, the all segmets of the fuctio have o-egative slopes. Therefore, the fuctio is miimized whe is miimized, ad the smallest feasible iteger value of is. This is also the umber of stops that is made if the value i a of the previous cases is /h ifeasible, ad the result correspods to cases 5c ad 6e. Therefore, the stated expressios for argmi { Eσ P, } hold. A.0. Proof of Theorem 5 Let K deote the overchargig cost icurred b the polic ˆπ P, where K = f h i i= sice the polic ol overcharges wheever h i >. The the total cost of the polic ˆπ P satisfies Cˆπ P = ˆs 2 K s K b Lemma 4. It follows that the ratio of the cost of the heuristic polic to the cost of a optimal polic must satisf 2 2 Cˆπ P CπP s K s s K f i= r i s K 9

10 because a feasible rechargig polic must icur overchargig costs of at least K. Furthermore, sice 2 s > s, it follows that 2 2 as desired. A.. Proof of Theorem 6 Cˆπ P Cπ P < B extesio of Lemma 3, it ca be see that the polic π P = 2 α, 8 miimizes the umber of rechargig stops alog P. For α =, the lemma states that the polic π P is optimal, ad its correspodig cost would be s, where is the umber of stops i the polic. Therefore, must be the miimum feasible umber of stops. For the case where α <, however, the polic ma ot be optimal due to the possibilit of overchargig costs. At most, the amout overcharged alog the etire path is α. If we let deote the umber of stops i the optimal polic π P, the we have C π P CπP s k α s k. For s/k <, the cost of a optimal polic is bouded below b s, which comes from the cotiuous result give i Theorem 2. It follows that C π P CπP k α s k sice k α s /αqmax αqmax = α. / s/k Whe s/k, the cost of a optimal polic is bouded below b s k from Theorem 2, ad therefore C π P CπP k α k sice s k k α k s k α sice s/k α, qmax = αqmax 0 α qmax, > 0 /α, 0 αqmax, > 0. Thus, the stated boud for C π P /Cπ P is valid. 0

11 A.2. Proof of Theorem 7 For i, if the optimal polic recharges more tha j l=i h l q i at ode i, the the total cost ca be reduced b rechargig less at i ad more at j. For ii, if the optimal polic recharges less tha q i at ode i, the the total cost ca be reduced b rechargig more at i ad less at j. For iii, if c i r i r i, q i > c j r i r i r i, q i r i j h l=i l, the the total cost ca be reduced b rechargig less at i ad more at j, or if c i r i r i, q i c j r i r i r i, q i r i j h l=i l, the the total cost ca be reduced b rechargig more at i ad less at j. Therefore, r i must satisf the stated properties. Appedix B: Supplemetal figure of heuristic performace for trips loger tha 300 miles Figure euristic rechargig polic performace for trips loger tha 300 miles; mea Cost Ratios with 0th- 90th iterpercetile rages are show