Management of several purifying plants in the same area: A multi-objective optimal control problem

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1 Managemen of several purifying plans in he same area: A muli-objecive opimal conrol problem L. J. Alvarez-Vázquez, N. García-Chan 2 A. Marínez, and M.E. Vázquez-Méndez 2 Deparameno Maemáica Aplicada II. ETSI Telecomunicación. Universidad de Vigo. 330 Vigo. Spain. lino@dma.uvigo.es, aurea@dma.uvigo.es 2 Deparameno Maemáica Aplicada. EPS. Universidad de Saniago de Composela Lugo. Spain. neog g@homail.com, erneso@usc.es Summary. In his paper we deal wih a parabolic muli-objecive opimal conrol problem relaed o he managemen of a wasewaer reamen sysem. The problem is sudied from a non-cooperaive poin of view (looking for a Nash equilibrium), and also from a cooperaive poin of view (looking for Pareo soluions beer han he Nash equilibrium). Numerical resuls for a real world siuaion in he esuary of Vigo (NW Spain) are presened. The muli-objecive opimal conrol problem We consider a shallow waer domain Ω locaed in an urban area wih a wasewaer reamen sysem consising of several purifying plans. We assume ha each of he plans is conrolled by a differen organizaion and we suppose ha each of hem has o ake care of some sensiive areas, in such a way ha a penaly is imposed on he plan if he waer polluion levels in one of is associaed zones is greaer han a hreshold level. In each plan here is a purificaion cos associaed o he purificaion process, and he problem consiss of finding he discharge sraegy in each plan minimizing coss (purificaion cos and penalies) a every plan. In his paper we assume N E purifying plans discharging wasewaer in poins P,...,P NE Ω, ake faecal coliform baceria (FC) as indicaor of he waer qualiy and denoe by m j () he mass flow rae of coliform discharged in P j (wih low and up bounds, respecively 0 < m j < m j ). If we define M j = {m j L (0,T) : m j m j () m j, a.e. in (0,T)} and M = N E M j, hen he problem can be formulaed (see []) as he following muli-objecive opimal conrol problem (P): Find he discharge sraegy m() = (m (),m 2 (),...,m NE ()) M which, for j =,...,N E, minimizes he funcionals

2 2 Alvarez-Vázquez e al. J j (m) = T 0 n j f j (m j ())d + i= 2ǫ j i A j i (0,T) ( ) 2 ρ(x,) σ j i dxd, () + where f j represens he purificaion cos a j plan, A j,...,aj n j Ω are he sensiive areas associaed o ha plan, σ j i is he FC hreshold in Aj i, ǫj i is a penaly parameer, (.) + denoes he posiive par funcion, and ρ(x,) is he FC concenraion given by: ρ + u ρ β ρ + κρ = N E m j ()δ(x P j ) in Ω (0,T), h (2) ρ(x,0) = ρ 0 (x) in Ω, ρ n = 0 on Ω (0,T). In his sysem δ(x P j ) denoes he Dirac measure a P j, n is he uni normal ouward vecor and h(x, ) (heigh of waer), u(x, ) (deph-averaged horizonal velociy of waer), ρ 0 (x) (iniial FC concenraion), β (viscosiy coefficien collecing urbulen and dispersion effecs) and κ (experimenal coefficien relaed o he loss rae of FC) are known daa. 2 A non-cooperaive sudy: Nash equilibria Firs we recall ha each plan is conrolled by a differen organizaion which looks for is own discharge sraegy (m j M j ) in order o minimize is own objecive funcional J j. So, we look for a whole discharge sraegy (vecor m M) acceped by all of he plan managers in he sense ha none can change is sraegy wihou increasing is cos funcional, if he ohers do no change heir sraegies. This vecor m M is known as a Nash equilibrium: Definiion. We say ha m = (m,...,m NE ) M is a Nash equilibrium of problem (P) if i verifies ha, for all j =,...,N E, J j (m,...,m j,...,m NE ) = min m j Mj J j (m,...,m j,m j,m j+,...,m NE ) (3) Nash equilibria can be characerized by using classical opimal conrol heory of parial differenial equaions: For each j =,...,N E we inroduce he j-h adjoin problem: q j β q j div(q j u) + κq j = n j i= ǫ j i χ A j(ρ σ j i ) + in Ω (0,T), i q j (x,t) = 0 in Ω, β q j n + q j u n = 0 on Ω (0,T), where χ A j denoes he characerisic funcion of he se A j i, i.e. χ i A j = i(x) only if x A j i. Then we have he following very useful resul (see [2]): (4)

3 Managemen of several purifying plans 3 Theorem. A vecor m = (m,...,m NE ) in(m) is a Nash equilibrium of he problem (P) if and only if i verifies he opimaliy sysem given by: Sae sysem (2), Adjoin sysems (4), for j =,...,N E. f j(m (5) j ) + h(p j,) q j(p j,) = 0 in (0,T), for j =,...,N E. Then, o obain a Nash equilibrium we inroduce a ime discreizaion: we ake N N, = T N, and n = n, for n = 0,...,N. We define M = NE [m j,m j] N, and consider he discree conrol m = (m ( ),...,m ( N ),...,m NE ( ),...,m NE ( N )) M. The opimaliy sysem (5) is now approximaed by: Find m M verifying F(m ) = 0, () where he funcion F : M R N NE R N NE is given by: Algorihm. (Compuaion of F(m ) ) Iniial inpus: Polygonal approximaion Ω h of Ω, admissible riangulaion τ h of Ω h, and m M. - Sep.: Numerical resoluion of he sae sysem: Taking m M as daa, we solve sysem (2) by using a characerisic- Galerkin mehod (see [3]) and obain, for n = 0,...,N, funcions ρ n h (x) verifying ρ n h (x) ρ(x,n ) in Ω h. - Sep.2: Numerical resoluion of he adjoin sysems: Taking approximaions ρ n h (x) as daa, we solve sysems (4) by using he previous characerisic-galerkin mehod and obain, for n = N,...,0 and j =,...,N E, funcions qjh n (x) verifying qn jh (x) q j(x, n ) in Ω h. - Sep.3: Time discreizaion of he opimaliy condiion: We compue F(m ) = ((f j(m j ( n )) + h(p j, n ) qn jh(p j )) N n=) NE Finally, a discree approximaion of a Nash equilibrium is obained from solving problem () by any sandard numerical mehod for nonlinear sysems. 3 A cooperaive sudy: Pareo soluions Once we have already obained a Nash equilibrium, we wonder if i is an opimal soluion. Tha is, he Nash equilibrium is a discharge sraegy (m) acceped by all plan managers because if one of hem (j plan) changes is paricular sraegy (m j ), hen is paricular cos funcional (J j ) necessarily increases. Bu now he quesion is: If all plan managers are ready o cooperae, can we obain a beer sraegy which brings off a simulaneously decrease of all cos funcionals? According o his we inroduce he concep of Pareo soluion:

4 4 Alvarez-Vázquez e al. Definiion 2. We say ha m = (m,...,m NE ) M is a Pareo soluion of problem (P) if here does no exis any m M such ha J j (m ) J j (m), for all j =,2,...,N E, and for a leas one j {,2,...,N E }, J j (m ) < J j (m). If m M is a Pareo soluion, he objecive vecor (J (m),...,j NE (m)) is called Pareo-opimal and he se of Pareo-opimal objecive vecors is called Pareo-opimal fronier. Fig. shows he geomerical inerpreaion for wo plans. An admissible se and is image are illusraed. The fa line is he Pareo-opimal fronier and, for a non Pareo soluion m M, dashed lines bound objecive vecors corresponding o sraegies m M beer han m. Sraegies m M wih image on he arch bounded by dashed lines are Pareo soluions beer han m. Pareo-opimal fronier Fig.. Geomerical inerpreaion of Pareo soluions and Pareo-opimal fronier Pareo soluions can be characerized by means of he weighing mehod. For each vecor λ = (λ,λ 2,...,λ NE ) R NE such ha λ i 0, for all i =,...,N E, and N E i= λ i =, we inroduce he weighing problem: N E minimize J(m) = λ j J j (m) subjec o m M. (7) We can prove he following very useful resul (see []): Theorem 2. Le f j C ([m j,m j ]) be sricly convex in [m j,m j ], for all j =,...,N E. For each vecor λ = (λ,λ 2,...,λ NE ) R NE, λ 0 and NE k= λ k =, he weighing problem (7) has only one soluion. Moreover, m M is a Pareo soluion of problem (P) if and only if here exiss λ = (λ,λ 2,...,λ NE ) R NE, λ 0 and N E k= λ k = such ha m is a soluion of (7). From his resul, Pareo soluions can be obained by solving (7) for every weigh vecor λ. From a compuaional viewpoin, i is divided in wo sages: Sage. We mus fix he number imax + of Pareo soluions we are ineresed in, and we have o choose heir corresponding weighs {λ 0,λ,...,λ imax }.

5 Managemen of several purifying plans 5 In his paper we use an algorihm generaing he family of weigh vecors by spliing he inerval [0,] in a regular way, as given by Caballero e al. [4]. Sage 2. For each i = 0,,...,imax, we have o solve he problem (7) aking λ = λ i. In order o do i, we recall he ime discreizaion inroduced in secion 2, and approach he problem (7) by he discree problem: where minimize J (m ) subjec o m M, (8) N E N J (m ) = λ j (f j (m j ( n )) + n= n j i=n j + 2ǫ i A i (ρ n h(x) σ i ) 2 + dx), and, for n =,...,N, ρ n h (x) is he approximaion of ρ(x,n ) obained as described in Sep. of algorihm. The gradien of J a m can be also approximaed by a discreizaion of adjoin sysems (4). To be exac, N E J (m ) ((f j(m j ( n )) + λ k h(p j, n ) qn kh(p j )) N n=) NE, k= where, for n = N,..., and k =,...,N E, qkh n (x) is he approximaion of q k (x, n ) obained as described in Sep.2 of algorihm. The discree problem (8) can now be solved by any mehod for convex differeniable opimizaion. 4 Numerical resuls Problem (P) has been solved in a realisic siuaion posed in he ría of Vigo (NW Spain). We have considered wo sewage purifying plans, and wo sensiive areas, each one associaed o is corresponding plan. For he numerical simulaion we considered a complee idal cycle (T = 2.4 hours), chose N =, supposed ρ 0 = 0, and used he heigh/velociy obained by solving he shallow waer equaions on his domain. Relaed o purificaion characerisics we have assumed ha area associaed o plan is more sensiive han area associaed o plan 2 (σ < σ 2 ), we have aken he same purificaion cos funcion for boh plans (f = f 2 ) and also same penaly parameers (ǫ = ǫ 2 ). Firs we have looked for a Nash equilibrium in his siuaion and he resul can be seen in fig. 2-a. Nex, we have looked for Pareo soluions: fig. 3 shows he Pareo-opimal fronier. Cos for plan is represened in he abscissa axis and cos for plan 2 is represened in he ordinae axis. An empy circle represens he cos associaed o he Nash equilibrium given in fig. 2-a. As we can see, he Nash equilibrium is no a Pareo soluion, and discharge sraegies wih cos inside he dashed lines are beer ha he discharge sraegy given by he Nash equilibrium. Plan managers have o negoiae o choose one of hem (for insance, a reasonable opion is ha giving a similar improvemen -

6 Alvarez-Vázquez e al. 0 mass flow rae of FC m () 2 0 mass flow rae of FC m () 2 m () m () ime ime Figure 2-a: Nash equilibrium Figure 2-b: Pareo soluion Fig. 2. Opimal discharge sraegies J (m) J (m) Fig. 3. Pareo-opimal fronier in cos reducion - for boh plans). Tha discharge sraegy, wih cos poined ou in fig. 3, is represened in fig. 2-b. Acknowledgemen: Work parially suppored by MEC of Spain (Projec MTM ), and CONACyT of Mexico (code 572). References. L.J. Alvarez-Vázquez, N. García-Chan, A. Marínez, and M.E. Vázquez- Méndez, Comp. Op. Appl. DOI: 0.007/s (in press) 2. N. García-Chan, R. Muñoz-Sola, and M.E. Vázquez-Méndez, ESAIM-Conrol Opim. Calc. Var. DOI: 0.05/cocv: (in press) 3. L.J. Alvarez-Vázquez, A. Marínez, C. Rodríguez, C. and M.E. Vázquez- Méndez, Appl. Mah. Model. 25, (200) 4. R. Caballero, L. Rey, F. Ruiz and M. González, Muliple crieria decision making Lecure Noes in Econ. and Mah. Sysems 448, (Springer, 997)