Optimal Control of Air Pollution

Transcription

1 Punjab University Journal of Mathematis (ISSN ) Vol. 49(1)(2017) pp Optimal Control of Air Pollution Y. O. Aderinto and O. M. Bamigbola Mathematis Department, University of Ilorin, Ilorin, Nigeria Corresponding Author: Reeived: 31 July, 2016 / Aepted: 29 November, 2016 / Published online: 15 February, 2017 Abstrat.In this artile, mathematial expressions that represent the dynamis of air pollution assoiated with eletri power generating industries ( in partiular, atmospheri Co 2 ) is proposed using an optimal ontrol theory approah. The model is haraterized via Pontyagins maximum/minimum priniples. Eletri power plants effiieny improvement was introdued through applying tehnologies. The optimality system is established in attempt to minimize both the ost of applying tehnology for effiieny improvement as well as the atmospheri emission while maximizing the eletri power generation output. AMS (MOS) Subjet Classifiation Codes: 49-XX; 34K35; 35H05 Key Words: Optimal ontrol, air pollution, fossil fuels, eletri power generation, emission free equilibrium. 1. INTRODUCTION Air pollution is a mixture of solid and gaseous partiles (biologial partiles or other harmful materials) suh as ar emissions, dust, fatories hemials, and so on in the air, whih may results in humans/living organisms diseases, damage or death. Eletri power generation is responsible for a large measure of air pollution whih is aused by burning onventional fuels like oal, oil or gas whih ours in thermal eletri power plants. Therefore, substantial investments should be targeted towards the development of power plants effiieny and improvement through variety of adjustments. Eletri power plants an be ategorized generally into two groups. Power plants are powered by fossil fuel and by non- fossils fuel. The fossil fuel power plants are those that use oal, natural gas, and oil as their soure of eletrial energy. The hemial energy stored in the fossil fuels is transformed into eletri energy in steam-eletri power plants by burning the fuel in the ombustion hamber. Heat energy is released and is used to produe steam in the boiler, whih is passed through the steam turbines and drives the eletri generator. The overall proess is assoiated with air and thermal pollutant problems. Air pollutants are emitted via the exhaust gasses and the thermal pollutant is assoiated with the vast amounts of heat losses in 139

2 140 Y. O. Aderinto and O. M. Bamigbola the ondenser ooling water whih aused severe problem in aquati life and great environmental threat. The non- fossil fuel power plants are those plants that use nulear energy, hydro energy and alternatives(renewable energy like solar, wind, geothermal e.t..) as their soure of eletrial energy. This is perfetly air pollutionfree but very sare resoures. However, eletri power plant pollutants an be redue through a variety of adjustment in the plants. Suh as fuel balaning, fuel swithing, implementation of improvement tehnologies to existing power plants. As well as using non- fossil fuel as energy soure or using energy soure that are renewable suh as solar, wind turbines e.t.., Bamigbola and Aderinto (2010), Charles (1976), Harry et al (1996), Olle (1970), Rayput (2003),Shammakh et al (2006). Many researhers have worked on the eletri power generation as well as on air pollution and the ontrol of air pollution with appliation to resoure management ontrol. Genhi, et al (2002) used optimization model to assess the redution of Co2 emissions. Hashim et al (2005) studied Co2 emission and eletri energy planning. Maro, et al (2007) worked on Co2 emission management in attempt to redue its greenhouse effet. Shammakh, et al (2006) looked at some strategies to ontrol air pollution optimally in the power generating system setor. Dymtro, et al (2007) used variational inequalities to model fuel supply in eletri power networks, to mention a few. However, the present effort is to apply the Pontraygins maximum/minimum Priniples of optimal ontrol theory to air pollution assoiated with eletri power generation. The main aim is to propose mathematial expressions in form of optimal ontrol model that desribes the way, effet, and relation of Co2 emission with respet to eletri power generating systems, and appliation of the effiieny tehnology towards removal / redution of Co2 emission during power generation via optimal ontrol approah. And for better understanding the model is haraterized and the optimality system is established in an attempt to redue the Co2 emission with respet to eletri power generation. 2. FORMULATION OF MODEL Let X ij (t) be the onentration of atmospheri Co 2 emission from eletriity generating plants. E 1j (t) be the eletri energy generated from the power plant 1 using fuel type j and E 2j (t) be the eletri energy generated from the power plant 2 using j soure of energy (j = 1, 2). Eletri power plants with fuel type 1 are those plants that use fossil fuels as their soure of energy and power plants with fuel type 2 are those plants that use non-fossil fuel and alternatives (renewable energy suh as solar, wind, geothermal e.t..) as their soure of energy. The model is given by ( X ij (t) = r j + r j X ij 1 X ij E 1j = q F E 1j l 1 E 1j + u 1 X ij + α ik E 1j E 2j = q NF E 2j l 2 E 2j + he 1j ) α ij E 1j + (β + u 2 )X ij X ij (0) = X 0, E 1j (0) = E 1j 0, E 2j (0) = E 1j 0; (1) X ij (t) 0, E 1j (t) 0, E 2j (t) 0, i = 1, 2; j = 1, 2.

3 Optimal Control of Air Pollution 141 where r j is the emission rate from power plant using fuel type j, is the aring apaity of the air in terms of Co 2 from the power plant, α ik is the effiieny of power plant 1 if tehnology k is applied, β is emission rate of Co 2 through gross domesti produt, u 1 is the addition of power plant effiient tehnique towards the removal of Co 2 emission through power plant, k is the tehnology applied to inrease power plant effiieny, u 2 is the addition of effiieny tehnology towards removal of Co 2 due to gross domesti produt, q F is the generation rate for plant using fossil fuel as its soure of energy. q NF is the generation rate for plants using non-fuel as its soure of energy. l 1, l 2 are the power losses during generation for plants using fossil fuel and those that use non-fossil fuels as their soure of energy respetively. h is the rate at whih plant 1 hange from one type of fuel to another towards the redution of Co 2 emission. The model flow diagrams are shown in figures (1,2,3). Figure.1: Sheme of the model. Figure. 2: Power Plant Struture.

4 142 Y. O. Aderinto and O. M. Bamigbola Figure.3: Power Plant with Effiieny Improvement. 3. RESULTS: ANALYSIS OF THE MODEL The loal stability and steady state of model (1) is hallenging, however we are interested in the models ability to exhibit loally asymptotially stable and steady states. Table 1: Parameters value and desription (Maro, etal 2008, Shammakh etal,2006, and Sirikum, etal. 2007) Parameter Value Desription X ij (t) m/tons Conentration of Co 2 emission from eletri power generating plant. E 1j (t) 800 MW Power output from fossil fuel energy types E 2j (t) 500 MW Power output from non-fossil fuel energy r j 0.17 Emission rate of Co 2 from power plants 0.75 Carrying apaity of atmosphere in terms of Co 2 α ik 0.46 Effiieny of fossil fuel plant when tehnology is applied β 0.05 Emission rate of Co 2 from gross domesti produt u Effiieny tehnology applied to gross domesti produt q F 0.55 Generation rate of fossil fuel power plants q NF 0.60 Generation rate of non-fossil fuel power plants. h The rate of swithing to another type of fuel l Loss rate during generation with fossil fuel plant. l Loss rate during generation with non-fossil fuel plants. u Effiieny tehnology towards power generation with fossil fuel plant The emission free equilibrium and Stability. The emission free equilibrium (EFE) of model system (1) is denoted by E(0) is expressed as E(0) = (X ij (0), E 1j (0), E 2j (0)) = (r j, 0, 0) For all non-negative value of parameter, model system (1) has equilibrium E0 in the boundary of G and the endemi equilibrium point ours inside G. Theorem 3.1: The emission free equilibrium is loally asymptotially stable when the basi Emission is less than zero and unstable when its greater than zero (that is stable if the Eigen value λ i < 0, and unstable for λ i > 1.

5 Optimal Control of Air Pollution 143 Proof Let (X ij (0), E 1j (0), E 2j (0)) = (r j, 0, 0) = E(0) J(X ij, E 1j, E 2j ) = r j 2r jx ij + β + u 2 α ik 0 u 1 q F l 1 + α ik 0 0 h q NF l 2 J(E(0)) = r j 2r 2 j + β + u 2 α ik 0 u 1 q F l 1 + α ik 0 0 h q NF l 2 where J is the Jaobian matrix of a funtion. Finding the eigen-value of the Jaobian matrix at equilibrium point E ( 0) we have det[j(e(0)) λi]. Thus we obtain ( = r j 2r2 j + β + u 2 [( ) r j 2r2 j + β + u 2 λ 1 (q F l 1 + α ik ) λ 2 0 h (q NF l 2 ) λ 3 ) λ 1 ] [(q F l 1 + α ik ) λ 2 ] [(q NF l 2 ) λ 3 ] = 0 λ 1 = r j 2r2 j + β + u 2, λ 2 = (q F l 1 + α ik ), λ 3 = (q NF l 2 ) Hene, stability at this point depends ritially on the following onditions: u 2 < 2r2 j (r j + β), or u 2 < 2r 2 j (r j + β), or < 1 u 2 [2r 2 j (r j + β)] l 1 < q F + α ik and l 2 < q NF The eigen - values of det[j(e 0 ) λi], have negative real part whenever the above three onditions hold. This implies that E 0 is loally asymptotially stable. After substituting the value of the parameters in Table 1, we obtained 3 eigen-values of J in whih none of them is negative; λ 1 = , λ 2 = , λ 3 = Hene, E 0 is not asymptotially stable. For the prove of similar result see Bhunu, et al (2008). Lemma 3.1 Let f : [0, ] R be a bounded funtion and twie differentiable with bounded seond derivative. Let t n and f(t n ) onverge to f or f for n, then lim f(tn) = 0 where f = lim inf t f(t), f = lim sup t f(t) and f is a real valued funtion, Burghes and Graham (2002). Theorem 3.2: The emission free equilibrium is globally asymptotially stable for E 0 < 0. Proof: Let λ i hoose a sequene t n suh that X ij (t n ) X ij, Using d dt X ij in (1) and Lemma 3.1, we have d dt X ij(t n ) 0 0 (β + u 2 )X ij α ik E 1j + r j X ij ( ) 1 X ij

6 144 Y. O. Aderinto and O. M. Bamigbola = X ij (β + u 2 + r j ) ± (β + u 2 + r j ) 2 4(r j 1 α ik E 1j ) 2r j 1 (2) Similarly, hoose a sequene r n suh that E 1j (r n ) E 1j, d dt E 1j(r n ) 0 and using d dt E 1j and Lemma (q F l 1 + α ik )E 1j + u 1 X ij = E 1j u 1 X ij (q F l 1 + α ik ) Also, hoosing sequene s n suh that E 2j (s n ) E 1j, d dt E 1j(s n ) 0 and using equation (1) and Lemma 3.1 we have, 0 (q NF l 2 )E 2j + he 1j (3) substituting (3) and (4) into (2), we have X ij (β + u 2 + r j ) ± = E 2j he 1j (q NF l 2 ) (β + u 2 + r j ) 2 4(r j 1 α ik ) 2r j 1 ( u1 Xij ) (q F l 1 +α ik ) (4) = (β + u 2 + r j ) ± (β+u2 +r j ) 2 (q F l 1 +α ik )+4r j α ik u 1 X ij (q F l 1+α ik ) 2r j 1 = (β + u 2 + r j ) 2 ± (β+u 2 +r j ) 2 (q F l 1 +α ik )+4r j α ik u 1 X ij (q F l 1 +α ik ) 4r 2 j 2 = 23 (β + u 2 + r j ) 2 (q F l 1 + α ik ) + 4r j α ik u 1 X ij 4r 2 j (q F l 1 + α ik ) E 0 = E + P X ij (5) Q This implies Xij 0 whenever E 0 < 0. Substituting the data in table 1, we have the following results. E 0 = > 0 meaning that Xij does not less or equal to zero. Hene, without loss of generality, E 0 is not globally asymptotially stable. 4. CONTROL FORMULATION We onsider a ontrol problem given by tf J(u 1, u 2 ) = (AX ij + Bu Cu 2 2)dt (6) t 0 Subjet to the state equation (1), where A, B and C are adjusted weights to reflet the relative importane of the variables X ij, u 1, and u 2. Variables u 1 and u 2, are the ontrol variables suh as (i) introdution of power plant effiieny through a variety of adjustment in the plants, (ii) Restore Plants to design ondition (iii) retrofit improvement, e.t..

7 Optimal Control of Air Pollution 145 The objetive funtion (6) expresses our goal to minimize both the ost of retrofit for swithing from one type of fuel to another and ost of applying tehnology for effiieny improvement, as well the atmospheri Co2 emissions while maximizing eletri power output. Therefore, we seek an optimal ontrol pair (u 1, u 2) suh that J(u 1, u 2) = min {J(u 1, u 2 ) (u 1, u 2 ) U} subjet to the system of ordinary differential equation (1), where u = (u, u) u i is measurable, a i u i b i, t [t 0, t f ], i = 1, 2 is the ontrol set. Many researhers have used a ontrol theoretial approah to formulate and study problems. Adams et al (2004) used optimal ontrol approahes on dynami multidrug therapies for HIV. Sirikum (2007) worked on power generation expansion planning with emission ontrols. Fister et al (1998) use ontrol strategies to optimize hemotherapy in an HIV model. However, the present work is based on the use of optimal ontrol approah applied to eletri power generation with Co 2 emission. 5. THE OPTIMALITY SYSTEM AND APPLICATIONS We established the existene of an optimal ontrol pair for the optimality system using results from literatures. That is by showing that the right hand sides of equation (1) are bounded by the state and ontrol variables and that the integrand of the objetive funtion (6) is onave in u and is bounded. These bounds give the ompatness to establish the existene of the optimal ontrols. Theorem 5.1: Consider the optimal ontrols and the state system (1), establish that there exists an optimality system for the optimal ontrol pair. Proof: By the appliation of Pontryagin Maximum / Minimum Priniple, we define the Lagrangian funtion to be: L(X ij (, E 1j, E 2j, u 1 (, u 2, λ 1, λ) 2, λ 3 ) = AX ij + Bu Cu ) 2 2 +λ 1 r j + r j X ij 1 Xij α ik E 1j + (β + u 2 )X ij +λ 2 (q F E 1j l 1 E 1j + u 1 X ij + α ik E 1j ) + λ 3 (q NF E 2j l 2 E 2j + he 2j ) ω 11 (u 1 a 1 ) ω 12 (b 1 u 2 ) ω 21 (u 2 a 2 ) ω 22 (b 2 u 2 ) where λ i are Langrange Multipliers and W ij 0 are penalty Multipliers satisfying ω 11 (t)(u 1 (t) a 1 ) = ω 12 (t)(b 1 u 2 (t)) = 0 at u 1 = u 1 and ω 21 (t)(u 2 (t) a 2 ) = ω 22 (t)(b 2 u 2 (t)) = 0. By Differentiating the Lagrangian with respet to state variables X ij, E 1j and E 2j respetively, the following equations were obtained for the adjoint variables λ i. λ 1 = L, λ 2 = L, λ 3 = X ij E 1j L E 2j The adjoint variables λ 1, λ 2, λ 3 satisfy the following ordinary differential equations from (7). ( λ 1 = {A + λ 1 r j (1 2X ) } ij ) + β + u 2 + λ 2 u 1 λ 2 = {λ 1 ( α ik ) + λ 2 (q F l 1 + α ik ) + λ 3 l 2 } (7)

8 146 Y. O. Aderinto and O. M. Bamigbola λ 3 = {λ 3 (q NF l 2 )} λ 1 (t 1 ) = λ 2 (t 1 ) = λ 3 (t 1 ) = 0 (8) The optimality of the ontrol variables u 1 and u 2 requires that we find the derivatives of the Langrangian with respet to the ontrols u 1 and u 2 respetively. Thus, we have, Solving for the ontrols, we obtain L u 1 = 2Bu 1 + λ 2 X ij ω 11 + ω 22 = 0, at u 1 L u 2 = 2Cu 2 + λ 1 X ij ω 21 + ω 12 = 0, at u 2 u 1 = 1 2B (λ 2X ij ω 11 + ω 12 ) (9) u 2 = 1 2C (λ 1X ij ω 21 + ω 22 ) (10) To determine an expliit expansion for the optimal ontrol without ω 11, ω 12, ω 21 and ω 22 we onsider the following ases. (i) On {t a 1 = u 1(t)}, sine u 1(t) b 1, ω 11 = 0, then a 1 = u 1 = 1 2B (λ 2X ij + ω 12 ). Solving for ω 12 gives 2Ba 1 λ 2 X ij = ω Ba 1 λ 2 X ij and a 1 1 2B λ 2X ij (ii) On {t a 1 < u 1(t) < b 1 }. By definition of penalty multipliers ω 12 = ω 11 = 0, we have u 1(t) = 1 2B (λ 2X ij ). (iii) On {t b 1 = u 1(t)}, sine u 1(t) a 1, ω 12 = 0, we have b 1 = u 1(t) = 1 2B (λ 2X ij ω 11 ) 0 ω 11 = 2Bb 1 λ 2 X ij and b 1 1 2B λ 2X ij Hene, we have 1 2B (λ 2X ij ) if a 1 < 1 2B (λ 2X ij ) < b 1 u 1 = a 1 if 1 2B (λ 2X ij ) a 1 b 1 if 1 2B (λ 2X ij ) b 1 In ompat form, we have u 1 = min { { } } 1 max a 1, 2B (λ 2X ij ), b 1 Using similar arguments, we obtain the following expression for the seond ( optimal ontrol funtion. (i) On {t a 2 = u 2(t)}, ω 21 = 0, then a 2 = u 2(t) = 1 2C λ1 Xij + ω ) 22. 2Ca 2 λ 1 X ij = ω Ca 2 λ 1 Xij and a 2 1 2C λ 1Xij ( ) (ii) On {t a 2 < u 2(t) < b 2 }, ω 21 = ω 22 = 0 and u 2(t) = 1 2C λ1 Xij. (iii) On {t b 2 = u 2(t)}, sine u 2(t) a 2, ω 22 = 0, we have b 2 = u 2(t) = 1 ( λ1 X ) ij ω 11 2C (11)

9 Optimal Control of Air Pollution ω 21 = λ 1 Xij 2Cb 2 sine ω 0, 1 = 1, 2 and b 2 ( ) ( ) 1 2C λ 1Xij 1 Hene, u 2C λ1 Xij ( if a2 ) < 1 2C λ1 Xij < b2 2 = a 2 if 1 2C ( λ1 Xij ) a2 b 2 if 1 2C λ1 Xij b2 Combining these three ases, we have { { u 1 ( 2 = min max a 2, λ1 X ) } } ij, b 2 2C The optimality system omprises of the state system with the adjoint system (8) with initial and transversality onditions together with the haraterization of the optimal ontrol pair (11) and (12). The optimality system an then be solved via an iterative numerial method. The state equation is solved using Runge Kutta forward order sheme and the adjoint system is solved bakward in time. The ontrols are updated at the end of eah iteration using (11) and (12). (12) 6. CONCLUSION The mathematial model for better understanding of Co2 emission with appliation to eletriity power generating system model was formulated. We further haraterize the model using the existing results. Finally, the optimality system for the model was established. 7. ACKNOWLEDGMENTS The authors thank the referees for their valuable suggestions whih led to the improvement of this paper. Author s Contributions: The first author gave the idea of the main results. Both authors ontributed to the writing of this paper. Both authors read and approved the final manusript. REFERENCES [1] Dynami Multidrug therapies for HIV: Optimal and STI ontrol Approah, Mathematial Biosienes and Engineering 1, No. 3 (2004) [2] On the Charaterization of Optimal Control Model of Eletri Power Generating Systems, Third International Conferene on Siene and Development Studies, Book of Proeedings 3, No. 3 (2009) [3] Tuberulosis Transmission Model with Chemoprophylaxis and Treament, Bulletin of Mathematil Biology 70, (2008) [4] An Introdution to ontrol theory, inluding optimal ontrol, The Open University, Milton Keynes (1989) Canadian energy Outlook, G.a/es/ener2002 [5] Power System Analysis,, John Willey, New York, [6] Modelling of eletri power hain networks with fuel supply via variational inequalities, International Journal of Emerging Eletri Power Systems 8, No. 5 (2007) [7] Optimizing hemotherapy in an HIV Model, Eletroni Journal of Differential Equations, 32, (1996) [8] Assessment of CO 2 emission redution potential by using an optimization model for Regional energy supply system, Sixth International Conferene on Greenhouse gas ontrol tehnology (2002) 1-4. [9] Control of Eletri Power Generating Plants, Control Handbook, GE power system engineering, Sherretady, 1996 NY.

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