ENEECO-02159; No of Pages 10 Energy Economics xxx (2011) xxx–xxx
Contents lists available at SciVerse ScienceDirect
Energy Economics j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / e n e c o
Optimum policy for integration of renewable energy sources into the power generation system M. Suruz Miah 1, N.U. Ahmed ⁎, 2, Monjur Chowdhury 3 School of Information Technology and Engineering, University of Ottawa, ON, Canada K1N 6N5
a r t i c l e
i n f o
Article history: Received 24 December 2010 Received in revised form 8 July 2011 Accepted 6 August 2011 Available online xxxx Keywords: Mathematical models State equations Renewable and conventional energy Pontryagin minimum principle Control (decision) policy Optimal controls
a b s t r a c t In this paper we propose a dynamic model representing the temporal evolution of the levels of power generation (installed capacity) from two competing sources. These are renewable and conventional (fossil) sources. The percentage penetration rates of renewable and nonrenewable sources are considered as control or decision variables. We introduce an objective functional based on energy demand, production of pollution associated with usage of fossil fuels, and the cost of their systematic substitution by renewable sources. Pontryagin minimum principle is used to determine optimal control policy through minimizing energy generation from fossil fuels while meeting the energy demand as closely as possible through gradual replacement of nonrenewable sources by renewable ones. For different choices of plan periods, optimal generation path along with the corresponding control policies are presented. These results demonstrate that modern control theory can be used effectively to formulate optimal socio-economic policies. © 2011 Elsevier B.V. All rights reserved.
1. Introduction In his presentation (Steinberger, 2009), Nobel laureate Jack Steinberger illustrated that in less than 90 years the population of the planet has grown 3 fold, from 2 billion to 6.6 billion. During this time, global energy use and Green House Gas (GHG) production have increased 8 fold. Since the beginning of the industrial age, atmospheric CO2 has risen from 280 ppm (parts per million) to 380 ppm. The temperature has risen about 0.8 °C. The sea level has risen 20 cm. According to his assumption, with the given population growth (1% per year), and per capita energy use growth (2.5% per year), the known low cost fossil fuel reserves would be depleted within 60 years. At the end of this time horizon, he predicts that the atmospheric CO2 level would be around 700 ppm, temperature rise would be 6 °C and sea level rise may reach 5 m. This has been validated by numerous scientists and researchers as seen in (Alberts et al., 2007; Edinger and Kaul, 2000; Ramakumar et al., 1992; Shi et al., 2008).
⁎ Corresponding author. E-mail address:
[email protected] (N.U. Ahmed). 1 Ph.D. Candidate, School of Information Technology and Engineering, University of Ottawa, ON, Canada. 2 Professor, School of Information Technology and Engineering, University of Ottawa, ON, Canada. 3 Postdoctoral Fellow, School of Information Technology and Engineering, University of Ottawa, ON, Canada.
In contrast, there are opinions among the scientific communities and policymakers that there is no cause for panic, given the present state of energy reserve and its utilization. They argue that there are still ample energy reserves under the earth, and their studies show a continuous discovery of more and more fossil fuels (Shafiee and Topal, 2009). It is to be recognized that the challenges of efficient implementation of renewable technologies are enormous especially the issues related to intermittency for both wind and solar energies (Macdonald, 2009). At the same time, any transition would cost trillions of dollars. For example, to maintain oil production through the year 2030, one needs 20 trillion while transition to a world powered by green energy for the same time requires at least 100 trillion as estimated by a group of Stanford scientists (Leonhard and Grobe, 2004). In spite of their disagreement to the urgency of implementation, scientists, policy makers and the world leadership agreed that transition to renewable energy sources is the path for continuous prosperity of human civilization(Alberts et al., 2007; Benson and Franklin, 2008; EREC, 2004; Moss and Kwoka, 2010; Youngquist, 2000). This can be supported by the fact that the rate of natural replenishment of fossil fuel based resources is much smaller than the rate of energy demand. Further, meeting the increased energy demand from fossil fuel would result in increased production of GHG. In view of these two factors, it is clear that there is a need for gradual and optimal transition toward renewable sources. In this respect, finding an optimal solution that takes care of multiple constraints remains appropriate for various reasons (Grafton and Silva-Echenique, 1997; Meadows and Meadows, 1973). This can
0140-9883/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.eneco.2011.08.002
Please cite this article as: Miah, M.S., et al., Optimum policy for integration of renewable energy sources into the power generation system, Energy Econ. (2011), doi:10.1016/j.eneco.2011.08.002
2
M.S. Miah et al. / Energy Economics xxx (2011) xxx–xxx
Nomenclature CO2 GHG R&D
Carbon dioxide Green House Gas Research and Development
List of symbols ψ Adjoint (costate variable) D(t) Total demand at time t H Hamiltonian function I ≡ [t0, tf] Plan period in years J Objective (cost) function ℓ Running cost integrand Φ Terminal cost P(t) Pollution level at time t generated by conventional sources U Set of decision (or control) policies Uad Set of admissible decision (or control) policies x1, x2 Installed capacity of renewable, conventional (fossil fuel) energy sources xdℓ ; xd2 Desired renewable, conventional (fossil) energy at the end of a plan period 〈(⋅)1, (⋅)2〉Scalar product of (⋅)1 and (⋅)2 (⋅) T Transposition of (⋅)
sources and x2(t) denotes the installed capacity of conventional (fossil) power sources. The rate of change of installed capacity from renewable power (clean power) sources is denoted by x˙ 1 . Similarly, x˙ 2 denotes the rate of change of installed capacity of conventional power sources. The parameter α1 is the percentage growth rate of capacity generated from clean sources and α2 is the percentage growth rate of capacity generated from fossil fuels. These factors depend on the availability of both renewable and nonrenewable sources and their exploitation rates. The parameter β1 represents the impact of the level of conventional power generation on the growth rate of clean power generation. Similarly, the coefficient β2 represents the impact of the clean power generation level on the production rate of conventional power. Note that the interaction is not necessarily symmetric. In the model, the state vector (x1, x2) is the main indicator of the path that the system takes. The larger the coefficient α1, the greater is the penetration of renewable sources. On the other hand, the larger the coefficient ha2, the greater is the dependence on polluting sources. In view of this it is natural to consider these parameters as control variables and since they can be varied with time as required, we can choose them as functions of time. Therefore, we set α1 ≡ u1(t) and α2 ≡ u2(t) for t ∈ I. It is natural to restrict the control variables to the closed bounded interval [− 1, 1]. The upper bound (lower bound) is assumed to be one (minus one), meaning that the percentage growth rate (retirement rate) is not allowed to exceed hundred percent. This implies that − 1 ≤ u1(t) ≤ 1 and − 1 ≤ u2(t) ≤ 1. Thus the range of the control vector u ≡ (u1, u2) is the set n o 2 2 U≡ u = ðu1 ; u2 Þ∈R : −1≤u1 ; u2 ≤1 ⊂R :
be achieved by developing an optimal policy based on modern Systems and Optimal Control Theory (Ahmed, 1988; Costello, June, 2007; Pontryagin et al., 1965; Teo et al., 1991; Tora and El-Halwagi, 2009). Our record indicates that not much work has been done in the area of optimum energy management and generation planning involving transition from fossil to renewable sources. Therefore, the motivation of this paper is to provide with a simple but effective tool (methodology) to support the social scientists and policy makers. The rest of the paper is organized as follows: In Section 2, we have developed a mathematical model based on Lotka–Volterra prey–predator system [(Moss and Kwoka, 2010), p. 24–25, 267–268], leading to the problem formulation. A brief description of the general Pontryagin minimum principle is provided in Section 3. In Section 4, we apply the minimum principle to our Energy Problems. Numerical algorithm, based on the minimum principle, which is used for the solution of the problems considered in this paper, is included in Section 5. In Section 6, numerical results and graphs are presented with brief explanations. We end the paper with concluding remarks in Section 7. 2. Mathematical model and problem formulation We were motivated by the Lotka–Volterra model that describes the dynamics of competing (or/and cooperating) populations in any given habitat. In fact we feel that similar model can be used for social systems in any environment of competing or cooperating agents. Inspired by this, we propose a pair of first-order, non-linear, differential equations to describe the dynamics of two competing energy sources: conventional and renewable. The model can be described as follows: x˙ 1 =
dx1 = α1 x1 −β1 x1 x2 dt
x˙ 2 =
dx2 = α2 x2 −β2 x1 x2 ; dt
ð1Þ
over any time period I ≡ [t0, tf] called the plan period. For any time t ∈ I, x1(t) denotes the level of installed capacity of renewable power
Therefore, from now on we will consider the system (1) as a controlled system giving dx1 = f1 ðx; uÞ = u1 x1 −β1 x1 x2 dt dx2 = f2 ðx; uÞ = u2 x2 −β2 x1 x2 : x˙ 2 = dt
x˙ 1 =
ð2Þ
A compact form of equation (2) can be written as x˙ = f ðx; uÞ; T
where x˙ = ½x˙1 x˙2 and f(x, u) = [f1(x, u)f2(x, u)] T. Our objective is gradual replacement of polluting sources by renewable sources. This requires investment of financial resources, technological development (R&D) as well as social acceptance. Given this, the planner must now set desirable goals including the associated cost of implementation. This is realized by the following objective functional also called cost functional, (Ahmed and Ahsan, 1984) o 2 2 1 tf 1 tn d d 2 2 dt + ∫t0f q1 u1 + q2 u2 dt w1 x1 ðt Þ−x1 ðt Þ + w2 x2 ðt Þ−x2 ðt Þ ∫ 2 t0 2 1 d 2 d 2 ν1 x1 tf −x1 + ν2 x2 tf −x2 + : 2
J ð uÞ =
ð3Þ This functional is given by the sum of three terms. The first term, known as the running cost, is the mismatch between actual power generation and the level of demand at any time during the plan period, the second is the cost related to control efforts, and the third, known as the terminal cost, represents the gap between the target at the end of the plan period and the actual generation level reached. The functions x1d(t) and x2d(t) appearing in the running cost are the desired levels of generation at time t from renewable and polluting sources respectively. These are determined by demand and possibly other socio-economic factors. The second term representing the control cost is determined by the level of investment toward integration of
Please cite this article as: Miah, M.S., et al., Optimum policy for integration of renewable energy sources into the power generation system, Energy Econ. (2011), doi:10.1016/j.eneco.2011.08.002
M.S. Miah et al. / Energy Economics xxx (2011) xxx–xxx
renewable sources and replacement of conventional sources. Considering the terminal cost, the quantities xd1 and xd2 represent the desired target of production level of renewable and conventional sources respectively at the end of the plan period. The parameters, {w1, w2 ; ν1, ν2 ; q1, q2}, are the weights assigned to specific terms of the cost functional J(⋅). The parameters, w1 and w2, represent the relative importance of the difference between the desired and actual energy produced from the renewable and the conventional sources, respectively. The parameters ν1 and ν2 represent the relative importance given to the mismatch between the desired goal and what is actually achieved at the end of the plan period I. The weights, q1 and q2, represent the relative cost of control efforts in terms of capital investment, maintenance, and cost of replacement of existing infrastructure by new machines and machineries. From now on, we will denote the terminal cost by 1 d 2 d 2 ν1 x1 tf −x1 + ν2 x2 tf −x2 = Φ x tf 2
ð4Þ
2 2 1 d d w1 x1 ðt Þ−x1 ðt Þ + w2 x2 ðt Þ−x2 ðt Þ 2 +
ð5Þ
o 1n 2 2 q u + q2 u2 : 2 1 1
Therefore, the cost functional (equation (3)) can now be written as t J ðuÞ = ∫t f ℓðt; xðt Þ; uðt ÞÞdt + Φ x tf ; 0
3. Pontryagin minimum principle Consider the n-dimensional (n = 2 in our case) system x˙ = f ðt; x; uÞ; t ∈ I; xð0Þ = x0 ; where x(t) represents the state of the system at time t and u(t) represents the control. The cost functional is formulated as equation (6). Let U be any closed bounded subset of R m representing the control constraint, that is, the controls are allowed to take values only from the set U. The problem is: find a control policy u o(t), t ∈ I, that minimizes the functional J(u) as defined above. That is J(u o) ≤ J(u) for all controls u taking values from the set U. In order to solve this problem, Pontryagin introduces the Hamiltonian function H(·) which is given by the following expression: H ðt; x; ψ; uÞ ≡ h f ðt; x; uÞ; ψi + ℓðt; x; uÞ: Then he introduces a pair of canonical differential equations in terms of the Hamiltonian giving
and the running cost by ℓðt; xðt Þ; uðt ÞÞ =
3
ð6Þ
t
where Φ(x(tf)) and ∫tf0 ℓðt; xðt Þ; uðt ÞÞ represent terminal cost and running cost, respectively. The problem is to determine the control policy u(t), t ∈ I, that minimizes the cost functional J(u) subject to the control constraint u(t) ∈ Uad and the dynamic constraints (equation (2)). 2.1. Optimum energy policy Despite significant advantages, renewable energy sources suffer from several drawbacks such as discontinuity of power generation due to its dependence on the climate. Nevertheless, due to the expansion of global economy and the increased energy demand, the fossil fuel resources (non-renewable energy) are becoming scarce and inefficient (for example crude oil and Alberta tar sand) and hence increasing carbon emissions to the atmosphere. Obviously this changes the climate, as such, reducing the availability of renewable energy resources. As a result, complex design principles and planning based on modern control and optimization methods are required to meet the growing energy demand due to evolution of civilization while minimizing the environmental pollution. There are two strategies to resolve the above problem (Banos et al., 2011). First, energy saving programs (focused on energy demand reduction and energy production efficiency in industrial (Lee and Chen, 2009) and domestic fields (Martiskainen and Coburn, 2010)) can be applied for reducing energy consumption thereby reducing dependence on fossil fuel resources. Second, increasing the use of renewable energy sources. However, this is a constraint on the evolution of civilization as mentioned. The current work solves this energy problem using the celebrated Pontryagin minimum principle (Ahmed, 1988) while minimizing the pollution due to evolution of civilization. We present this principle in its abstract form in the following section and then apply this result to our specific problem in the subsequent sections. There we present all the necessary equations for determination of optimal energy policies.
x˙ = Hψ = f ðt; x; uÞ; xð0Þ = x0
ð7Þ
T T ψ˙ = −Hx = −fx ðt; x; uÞψ−ℓx ðt; x; uÞ; ψ tf = Φx x tf ;
ð8Þ
where fx is the Jacobian matrix of the vector f and fxT is its transposition. The first equation is the original system equation. The second is known as the adjoint or costate equation. In classical calculus of variations,ψ is also known as the Lagrange multiplier. Now we can state the celebrated Pontryagin minimum principle. Let Uad denote the class of (measurable) functions defined on the time interval I and taking values from the set U. 3.1. Necessary conditions of optimality Let u o ∈ Uad, and x o ∈ C(I, R n) be the solution of equation (7) corresponding to the (control) policy u o. For the control policy u o to be optimal, it is necessary that there exists a co-state ψ o such that the following inequality and equations hold: o o o o o H t; x ðt Þ; ψ ðt Þ; u ðt Þ ≤ H t; x ðt Þ; ψ ðt Þ; u for all u ∈ U; t ∈ I
ð9Þ
o o o o o o o x˙ = Hψ t; x ðt Þ; ψ ðt Þ; u ðt Þ = f t; x ðt Þ; u ðt Þ ; x ð0Þ = x0
ð10Þ
o o o o o T o o ψ˙ = −Hx t; x ðt Þ; ψ ðt Þ; u ðt Þ = −fx t; x ðt Þ; s ðt Þ ψ ðt Þ o o −ℓx t; x ðt Þ; u ðt Þ ; o
ψ
o tf = Φx x tf :
ð11Þ
Any reader who is interested in the proof of the above result may consult any of the following books, Pontryagin (Benson and Franklin, 2008), Ahmed (Moss and Kwoka, 2010), Teo (Meadows and Meadows, 1973). According to the above result, an optimal control must satisfy all the necessary conditions (9)–(11). Controls that satisfy these necessary conditions are generally called extremal controls. There may be multiple controls satisfying the necessary conditions. At least one of these extremal controls is optimal. In the following section we apply this result and present the necessary conditions of optimality for our specific problems as stated in Section 2.
Please cite this article as: Miah, M.S., et al., Optimum policy for integration of renewable energy sources into the power generation system, Energy Econ. (2011), doi:10.1016/j.eneco.2011.08.002
4
M.S. Miah et al. / Energy Economics xxx (2011) xxx–xxx
4. Minimum principle applied to energy problems
and
The current work applies the Pontryagin minimum principle to three different problems:
H ðt; x; ψ; uÞ = f ðx; uÞψ + F ðx2 Þ;
1. Meyer problem: reaching desired target at the end of the plan period, 2. Bolza problem: reaching desired generation level at the end of the plan period while minimizing the accompanied pollution over the period, 3. Lagrange problem: Balancing power generation from the two sources to follow the demand and the desired program of gradual replacement of conventional sources by renewable ones while taking into account investment cost (cost of implementation).
respectively. Hence, the adjoint equation can be written as
4.1. Meyer problem The objective here is to reach a pre-specified power generation level from the two sources (x1, x2) at the end of the plan period (at t = tf). This is achieved by using the Meyer problem which is defined by the terminal cost Φ(x(tf)) as in equation (4) Ahmed (Moss and Kwoka, 2010). The goal is to find a control policy u ∈ Uad that minimizes the terminal cost Φ(x(tf)). The running cost of equation (3) is eliminated by setting ℓð·Þ = 0. As a result, the objective function J(u) and the corresponding Hamiltonian H(·) for this problem are given by J ðuÞ≡ Φ x tf
ð12Þ
and T
H ðt; x; ψ; uÞ = f ðx; uÞψ ; respectively. Hence, the adjoint equation can be written as T ψ˙ = −Hx = −fx ðx; uÞψ:
T
T ψ˙ = −Hx = −fx ðx; uÞψ−Fx ðx2 Þ:
The expression for the gradient Hu remains the same as in the previous case. 4.3. Lagrange problem In the preceding problems, it was tacitly assumed that the cost of implementation is insignificant compared to the environmental cost. Here we include all the relevant costs (except the terminal cost, Φ(x(tf))=0). In this case, the cost function J(u) in equation (6) is given by t
J ðuÞ = ∫t0f ℓðt; xðt Þ; uðt ÞÞdt;
ð15Þ
where ℓðt; xðt Þ; uðt ÞÞ is defined in equation (5). As stated in the definition of the Lagrange problem, the objective here is to introduce a control policy so as to balance the power generation from the two sources in order to meet the specified demand and follow the desired program of gradual replacement of conventional sources by renewable ones while taking into account investment cost (cost of implementation). In order to solve this problem, it is necessary for the planner to specify the functions, x1d(t), x2d(t), and D(t). The planner can do so on the basis of projection of future energy demand which is clearly tied with the population growth and the growth of industrial demand for power. It is natural that the combined generation from the two main sources should satisfy the demand D(t) leading to the equation d
d
Dðt Þ = x1 ðt Þ + x2 ðt Þ; t ∈ I:
ð16Þ
The expression for the gradient Hu is given by Hu =
∂H ψ1 x1 = ψ2 x2 ∂u
For simplicity, we assume that the desired level of generation from conventional sources follows an exponential law given by d
d
−γt
with ψ = [ψ1 ψ2] T.
x2 ðt Þ = x2 ð0Þe
4.2. Bolza problem
where γ(≥ 0) must be equal or less than the depletion rate. Note that the choice of γ determines the retirement rate (program) for fossil fuels. Similarly, we assume that the demand D(t) also follows an exponential law given by
The purpose of this problem is to find a control policy in order to reach, at the end of the plan period, a desired level of power generation from each of the two sources while keeping the environmental pollution as low as possible. This is modeled by the Bolza problem which incorporates the terminal cost Φ(x(tf)) as well as the cumulative pollution cost. The production rate of pollution is proportional to the power generation from fossil fuels. As such, the pollution level tP(t) at any time t is given by a nonnegative and nondecreasing function F(⋅) of the generation level x2(t), that is, P(t) = F(x2(t)) and hence the cumulative emission over the plan period is given by t
t
∫tf0 P ðt Þdt = ∫t0f F ðx2 ðt ÞÞdt:
ð13Þ
;
λt
Dðt Þ = Dð0Þe ;
ð18Þ
where λ ≥ 0. It is clear that λ will depend on the growth rate of population and industrial activities. However, γ should be chosen by the decision makers according to social needs. The Hamiltonian in this case is given by Hðt; x; ψ; uÞ = f1 ψ1 + f2 ψ2 + ℓðt; x; uÞ = ðu1 x1 −β1 x1 x2 Þψ1 + ðu2 x2 −β2 x1 x2 Þψ2 o 1 1n d 2 d 2 2 2 w1 x1 −x1 + w2 x2 −x2 q u + q2 u2 : ð19Þ + + 2 2 1 1
In our numerical study, we choose F(x2) = ν3|x2| q for some q ≥ 1 and ν3 N 0. For this problem, the necessary conditions given in the preceding section are slightly modified by replacing ℓ by F. The objective function and the Hamiltonian in this case are given by
d ψ˙ 1 = −Hx1 = β1 ψ1 x2 + β2 ψ2 x2 −uψ1 −w1 x1 −x1
t t J ðuÞ≡∫t0f P ðt Þdt + Φ x tf = ∫tf0 F ðx2 ðt ÞÞdt + Φ x tf ;
d ψ˙ 2 = −Hx2 = β1 ψ1 x1 + β2 ψ2 x1 −vψ2 −w2 x2 −x2 :
ð14Þ
ð17Þ
The co-state (or the adjoint) equations are given by
Please cite this article as: Miah, M.S., et al., Optimum policy for integration of renewable energy sources into the power generation system, Energy Econ. (2011), doi:10.1016/j.eneco.2011.08.002
M.S. Miah et al. / Energy Economics xxx (2011) xxx–xxx
By substituting equations (16)−(18) in equation (19) we obtain the following expression for Hamiltonian: h i2 2 H ðt; x; ψ; uÞ = w1 x1 ðt Þ− Dðt Þ−e−γt xd2 ð0Þ + w2 x2 ðt Þ−e−γt xd2 ð0Þ + q1 u21 + q2 u22 + ðu1 x1 −β1 x1 x2 Þψ1 + ðu2 x2 −β2 x1 x2 Þψ2 :
The associated adjoint equations are given by ψ˙1 = −Hx1 ψ2 = −Hx2
The state equations remain the same as in equation (2). 5. Numerical algorithm Here, we describe the basic algorithm based on the minimum principle presented above. This basic algorithm is based on gradient technique as presented in [(Moss and Kwoka, 2010), p. 301–304] which is implemented on MATLAB (Wang, 2009). The key steps of the proposed technique are described below. Let us consider u n(t), t ∈ I, to be the decision policies at the n th iteration. Step 1 Subdivide the interval I ≡ [t0, tf] into N equal subintervals and assume a piecewise-constant control n
n
ð20Þ
where ukn ∈ U for each k = 0, 1, 2, ⋯, N − 1. Step 2 Apply the assumed control u (n) ≡ u n(t), t ∈ I to integrate the state equations from t0 to tf with initial conditions x(t0) = x0 and store the state trajectory x (n). Step 3 Use u (n) and x (n) to integrate the costate equations backward in time starting from the costate ψ (n)(tf) at the terminal time tf. The terminal costate is given by ψ (n)(tf) = Φx(x n(tf)) where Φ(·) defines the terminal cost. Step 4 Using the triple {u n, x n, ψ n} compute the gradient gn ðt Þ =
n ∂H n n n n n x ðt Þ; ψ ðt Þ; u ðt Þ ≡ Hu x ðt Þ; ψ ðt Þ; u ðt Þ ; ∂un
ð21Þ
for t ∈ I and store this vector. Step 5 Construct the control for the next iteration as n + 1
u
n
ðt Þ = u ðt Þ−εgn ðt Þ; t ∈ I
ð22Þ
by choosing 0 b ε b 1 appropriately such that u n + 1 satisfies the constraint: u n + 1(t) ∈ U for t ∈ [t0, tf] and compute n n n n + 1 n n H x ðt Þ; ψ ðt Þ; u ðt Þ = H x ðt Þ; ψ ðt Þ; u ðt Þ 0 0 0 2 −ε∥Hu x ðt Þ; ψ ðt Þ; u ðt Þ ∥ + oðε Þ:
Step 6 Stop if n n n ∫ ∥Hu x ðt Þ; ψ ðt Þ; u I
+ 1
ðt Þ∥ dt≤δ 2
problems are solved by setting the initial renewable power to x1(0) = 0.2(× P) MW and the conventional power x2(0) = 0.8(× P) MW where P is any positive number suitable for the particular planning region. In other words we are solving the normalized problem by taking P = 1. Thus all the numerical results that follow are to be interpreted as normalized quantities. 6.1. Meyer problem
−γt = β1 ψ1 x2 + β2 ψ2 x2 −u1 ψ1 −2w1 x1 ðt Þ−Dðt Þ + e x2 ð0Þ ˙ −γt = β1 ψ1 x1 + β2 ψ2 x1 −u2 ψ2 −2w2 x2 ðt Þ−e x2 ð0Þ :
u ðt Þ = uk ; t ∈½tk ; tk + 1;
5
ð23Þ
(where δ N 0 is a specified tolerance) and compute the final results and exit; if not, use the new control u n + 1 given by equation (22) and go back to step 2 thereby closing the loop. 6. Numerical results In this section, we present simulation results for all the three optimization problems as stated in Subsections 4.1 (Meyer problem), 4.2 (Bolza problem), and 4.3 (Lagrange problem) respectively. These
The purpose of this experiment is to reach desired levels (mix) of power generation from the renewable and conventional sources. The parameters of the cost function defined in equation (12) are set to ν1 = 20, ν2 = 5. The desired levels are xd1 = 0:8, and xd2 = 0:2, which are simply the reversal of the initial state. We consider this for three different plan periods I = [0, 15] years, I = [0, 20] years, and I = [0, 30] years. Fig. 1(a) shows the generation levels of renewable and conventional power over the plan period of 15 years. The corresponding control policies are shown in Fig. 1(b). It is interesting to observe that at the end of the plan period, we can reach the desired levels of (power) generation exactly as specified. This means that the terminal cost (or mismatch) is zero as shown in Fig. 1(d). Fig. 1(c) gives the phase-portrait illustrating the inverse relationship between x1 and x2. For the plan periods of 20 and 30 years, the decision policies shown in Figs. 2(b) and 3(b) reflect the fact that they could still satisfactorily (terminal cost of almost zero (see Figs. 2(d)–3(d))) reach the desired levels of generation as recorded in Figs. 2(a) and 3(a). We have also computed the yearly average control efforts (equivalent to yearly investment) which are 0.021, 0.014, and 0.008 for 15, 20, and 30 years, respectively. It is clear from this that the average yearly investment decreases with the increase of the length of the plan period. Thus planners may choose the plan period that suits the available capital for the project. 6.2. Bolza problem Here we emphasize the cost associated with the pollution level (cleanup cost, health cost etc.) which is proportional to the level of power generated by use of fossil fuels as formulated in Section 4.2. The terminal cost functional Φ(x(tf)) remains exactly the same as in the Meyer problem and the parameters determining the pollution cost F(⋅) in (13) are chosen as q = 2 and ν 3 = 3 to perform the computation. As before, we consider three different plan periods I = [0, 15] years, I = [0, 20] years, and I = [0, 30] years by setting the parameters for Φ(x(tf)) as ν1 = 20, ν2 = 5 in equation (14). The results are shown in Figs. 4–6. Examining Figs. 4(a)–6(a) it is observed that the desired level of power generation from renewable sources specified by x1(tf) = 0.8 can be met at the end of each of the plan periods (15 years, 20 years, and 30 years) using the control efforts shown in Figs. 4(b)–6(b) as expected. It is interesting to note that the target level of conventional power generation specified by x2 (tf) = 0.2 is not met. In fact the generation level from the conventional sources reaches below the target level (doing better than expected). This is natural since it reduces the cost associated with pollution. In a trade off situation involving the sum of several objective functionals, the optimal policy always tries to reduce the one that (damages) costs the most. However, demand for pollution reduction at the exponential rate as suggested by the expression (13) naturally competes with the terminal cost. As compared to the Meyer problem, the Bolza problem ensures that the optimal policy reduces the power generation from the fossil fuel while it increases that from the renewable sources. This is clear from Figs. 4(c)−6(c) As expected, the cumulative costs, as recorded in Figs. 4(d)−6(d) (1.6397, 1.7017, and 1.8172), increase with the increase of the length of the plan period. It is important to note that these costs are given by the sum of the terminal costs and the running
Please cite this article as: Miah, M.S., et al., Optimum policy for integration of renewable energy sources into the power generation system, Energy Econ. (2011), doi:10.1016/j.eneco.2011.08.002
6
M.S. Miah et al. / Energy Economics xxx (2011) xxx–xxx
a
b x1 x2
0.9
1
u1 u2
0.8
0.8
0.6
Decesion policies
Renewable(x1), Fossil (x2) power [X 100 MW]
1
0.7 0.6 0.5 0.4 0.3
0.4 0.2 0 −0.2 −0.4
0.2
−0.6
0.1
−0.8
0 0
5
10
−1
15
0
5
Time (years)
c
10
Target Actual
0.9
9
0.8
8
0.7
Cost J (K)
Conventional (fossil fuel) power (x2) [X 100 MW]
15
d
1
0.6 0.5 0.4
7 6 5 4
0.3
3
0.2
2
0.1
1
0
10
Time (years)
0
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
0
1
Final cost J(K)=0
0
500
Renewable power (x1) [X 100 MW]
1000
1500
Number of Iteration
Fig. 1. Meyer problem: (a) Optimal trajectories {x1, x2} for the plan period of 15 years, (b) optimal control policies {u1, u2} (c) phase portrait {x1, x2}, and (d) cost vs iteration.
b
1
x1 x2
0.9
1 0.6
0.8 0.7 0.6 0.5 0.4 0.3
0.4 0.2 0 −0.2 −0.4
0.2
−0.6
0.1
−0.8
0
u1 u2
0.8
Decesion policies
Renewable(x1), Fossil (x2) power [X 100 MW]
a
0
2
4
6
8
10
12
14
16
18
−1 0
20
2
4
6
c
12
14
16
18
20
10
Target Actual
0.9
9
0.8
8
0.7
Cost J (K)
Conventional (fossil fuel) power (x2) [X 100 MW]
10
d
1
0.6 0.5 0.4
7 6 5 4
0.3
3
0.2
2
0.1
1
0
8
Time (years)
Time (years)
0
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
Renewable power (x1) [X 100 MW]
1
0
Final cost J(K)=0.0001
0
500
1000
1500
Number of Iteration
Fig. 2. Meyer problem: (a) Optimal trajectories {x1, x2} for the plan period of 20 years, (b) optimal control policies {u1, u2} (c) phase portrait {x1, x2}, and (d) cost vs iteration.
Please cite this article as: Miah, M.S., et al., Optimum policy for integration of renewable energy sources into the power generation system, Energy Econ. (2011), doi:10.1016/j.eneco.2011.08.002
M.S. Miah et al. / Energy Economics xxx (2011) xxx–xxx
b
1 x1 x2
0.9
1
u1 u2
0.8
0.8
0.6
Decesion policies
Renewable(x1), Fossil (x2) power [X 100 MW]
a
7
0.7 0.6 0.5 0.4 0.3
0.4 0.2 0 −0.2 −0.4
0.2
−0.6
0.1
−0.8
0 0
5
10
15
20
25
−1
30
0
5
10
20
25
30
d
1 Target Actual
0.9
10
0.8 8
0.7
Cost J (K)
Conventional (fossil fuel) power (x2) [X 100 MW]
c
15
Time (years)
Time (years)
0.6 0.5 0.4
6 4
0.3
Final cost J(K)=0.0003
0.2
2
0.1 0 0
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
0
1
0
500
Renewable power (x1) [X 100 MW]
1000
1500
Number of Iteration
Fig. 3. Meyer problem: (a) Optimal trajectories {x1, x2} for the plan period of 30 years, (b) optimal control policies {u1, u2}, (c) phase portrait {x1, x2}, and (d) cost vs iteration.
b
1
x1 x2
0.9
1 0.6
0.8 0.7 0.6 0.5 0.4 0.3
0.4 0.2 0 −0.2 −0.4
0.2
−0.6
0.1
−0.8
0
u1 u2
0.8
Decesion policies
Renewable(x1), Fossil (x2) power [X 100 MW]
a
0
5
10
−1
15
0
Time (years)
d
1
Target Actual
0.9
10
15
Time (years) 35 30
0.8
25
0.7
Cost J (K)
Conventional (fossil fuel) power (x2) [X 100 MW]
c
5
0.6 0.5 0.4 0.3
20 15 Final cost J(K)=1.6397
10
0.2 5
0.1 0
0
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
Renewable power (x1) [X 100 MW]
1
0
0
500
1000
1500
Number of Iteration
Fig. 4. Bolza problem: (a) Optimal trajectories {x1, x2} for the plan period of 15 years, (b) optimal control policies {u1, u2}, (c) phase portrait {x1, x2}, and (d) cost vs iteration.
Please cite this article as: Miah, M.S., et al., Optimum policy for integration of renewable energy sources into the power generation system, Energy Econ. (2011), doi:10.1016/j.eneco.2011.08.002
8
M.S. Miah et al. / Energy Economics xxx (2011) xxx–xxx
b
1 x1 x2
0.9
1
u u21
0.8
0.8
0.6
Decesion policies
Renewable(x1), Fossil (x2) power [X 100 MW]
a
0.7 0.6 0.5 0.4 0.3 0.2 0.1
0.4 0.2 0 −0.2 −0.4 −0.6 −0.8
0
0
2
4
6
8
−1 0
10 12 14 16 18 20
2
4
c
Target Actual
10 12 14 16 18 20
40
0.8
35
0.7
30
Cost J(K)
Conventional (fossil fuel) power (x2) [X 100 MW]
8
d
1 0.9
0.6 0.5 0.4
25 20
0.3
15
0.2
10
0.1
5
0 0
6
Time (years)
Time (years)
Final cost J(K)=1.7017
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
0 0
1
500
Renewable power (x1) [X 100 MW]
1000
1500
Number of Iteration
Fig. 5. Bolza problem: (a) Optimal trajectories {x1, x2} for the plan period of 20 years, (b) optimal control policies {u1, u2}, (c) phase portrait {x1, x2}, and (d) cost vs iteration.
b
1 x1 x2
0.9
1 0.6
0.8 0.7 0.6 0.5 0.4 0.3
0.4 0.2 0
0.2 0.1 0 0
5
10
15
20
25
0
30
Time (years)
c
5
10
15
20
25
30
Time (years)
d
1 Target Actual
0.9
60 50
0.8 0.7
Cost J (K)
Conventional (fossil fuel) power (x2) [X 100 MW]
u1 u2
0.8
Decesion policies
Renewable(x 1), Fossil (x2) power [X 100 MW]
a
0.6 0.5 0.4
40 30 20
0.3 0.2
Final cost J(K)=1.8172
10
0.1 0
0
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
Renewable power (x 1) [X 100 MW]
1
0 0
500
1000
1500
Number of Iteration
Fig. 6. Bolza problem: (a) Optimal trajectories {x1, x2} for the plan period of 30 years, (b) optimal control policies {u1, u2}, (c) phase portrait {x1, x2}, and (d) cost vs iteration.
Please cite this article as: Miah, M.S., et al., Optimum policy for integration of renewable energy sources into the power generation system, Energy Econ. (2011), doi:10.1016/j.eneco.2011.08.002
1.9
costs (due to pollution determined by equation (13)) as opposed to the terminal cost (Meyer problem) only. In this case also, we have computed the yearly relative control efforts for the three plan periods (15, 20, and 30 years) and they are 0.091, 0.070, 0.048, respectively.
1.8 1.7 1.6 1.5
6.3. Lagrange problem
1.4
The objective functional given by equation (15) takes into account the total demand equation (18) which is driven by population and industrial growth, and the cost of controls measured in terms of capital investment for renewable sources. For numerical experiment, we choose λ = 0.01, γ = 0.1, w1 = 1.5, w2 = 1.2, q1 = 1.1, and q2 = 0.75 for the cost function defined by equation (15) where l is given by the expression (5). The results are shown in Fig. 7. The desired levels of renewable power and conventional power are generated using equations (16), (17), and (18). Fig. 7(a) shows the actual power generated from renewable sources (x1) and fossil fuel (x2) over time using the control policies shown in Fig. (7a). Note that the power generation from renewable sources (x1) increases drastically in order keep pace with the increasing demand due to population and industrial growth. However, it reduces power generation from conventional sources (x2) in order to follow the exponential path specified by the expression (17). Recall that the total (power) demand at any time t is given by D(t) = x1d(t) + x2d(t) while the total (power) supply is given by the sum of the power actually generated by the two sources which is denoted by S(t) = x1(t) + x2(t).This is shown in Fig. 8. It is clear that initially the total power S(t) actually generated dropped and then it went back to follow the demand as expected. It is clear from Fig. 7(c) that at the end of the plan period power generation from renewable sources has increased while that from fossil fuel has decreased giving an overall increase of power generation to meet the increased demand. The reason for increase of power generation at the initial stages from fossil fuel, as seen in Fig. 7(a), is to support the
1.2 1.1 1 0.9
20
0.6
0.7
0.4
0.6 0.5 0.4 0.3 0.2
25
30
1 u1 u2
0.2 0
0.1 0
5
10
15
20
25
30
0
5
10
15
20
25
30
Time (years)
Time (years)
d
1.8
35
Desired Actual
1.6
30
1.4
Cost J (K)
Conventional (fossil fuel) power (x2) [X 100 MW]
15
0.8
0.8
0
c
10
Based on the numerical results, we now show the impact of evolution of civilization on the proposed control policy to meet energy demand in any plan period. We take into account the increasing worldwide demand for energy in the objective functional equation (3). This allows us to determine the investment cost to meet the desired energy demand by simply updating the weight parameters {w1, w2, ν1, ν2, ν3, λ, γ} of equation (3) based on economic factors and social goals appropriate for the period. It is important to articulate the fact that the proposed control policy resolves not only the problem of substituting the non-renewable energy sources by the renewable ones, but also resolves the problem of evolution of civilization as
x1 x2
0.9
5
7. Impact of evolution of civilization on proposed power generation policy
b
1
0
increasing demand by compensating insufficient generation from renewable sources.
Decesion policies
Renewable(x 1), Fossil (x2) power [X 100 MW]
a
1.3
1.2 1 0.8 0.6
25 20 15 Final cost J(K)=0.3277
10
0.4 5
0.2 0.5
1
1.5
2
2.5
Renewable power (x 1) [X 100 MW]
0
0
500
1000
1500
Number of Iteration
Fig. 7. Lagrange problem: (a) Optimal trajectories {x1, x2} for the plan period of 30 years, (b) optimal control policies {u1, u2}, (c) phase portrait {x1, x2}, and (d) cost vs iteration.
10
M.S. Miah et al. / Energy Economics xxx (2011) xxx–xxx
reflected in equation (14) (see Figs. 4(c)−6(c),). Moreover, renewable energy sources, such as solar, nuclear, plasma, to name a few, will definitely dominate in meeting the total energy demand while keeping the investment cost and environmental pollution low. These are illustrated in Section 4.3 with some numerical results shown in Figs. 7(c) and 8. 8. Concluding remarks In this paper we have presented a dynamic model for planning and management of energy production from renewable and fossil fuel based sources. This is given by a pair of nonlinear differential equations representing the dynamics of the two competing sources. We have proposed different cost functionals reflecting the social objectives demanding reduction of power generation from polluting sources while increasing generation from renewable ones. It is important to mention that the optimal policies are dependent on several constraints such as resources, time horizon (plan period) and social concerns (determining the weights in the objective functional). We have used modern control theory, in particular, Pontryagin minimum principle, to determine the optimal policies. Given the constraints as described, the results presented here show that it is possible to formulate an optimal policy to approach the desired target. We want to emphasize however that here we have presented a methodology whereby the planners can make optimum decisions taking into consideration all socio-economic factors. The numerical results are based on arbitrary parameters and they are presented for illustration only. Acknowledgments The second author would like to thank the National Science and Engineering Research Council of Canada for financial support under grant no. A7109. The first and the third authors would like to thank Mr. Bruno Randimbiarison, and Dr. Sk.Mizanur Rahman for their valuable assistance. Authors would like to thank the reviewers for their invaluable comments and suggestions for improving the quality of this manuscript. References Ahmed, N.U., 1988. Elements of Finite Dimensional System and Control Theory, Pitman Monographs and Surveys in Pure and Applied Mathematics, vol. 37. Longman Scientific and Technical with John Wiley, London, New York.
Ahmed, N.U., Ahsan, Q., 1984. A Dynamic Model for Generation Expansion Planning. University of Ottawa, Ontario, K1N 6N5 (Canada). (December 21). Alberts, B., Yongxiang, L., Alper Co-Chairs, H., 2007. IAC Report, “Lighting the way toward a sustainable energy future ”. Inter Academy Council, pp. 91–92 (118–119, October). Banos, R., Manzano-Agugliaro, F., Montoya, F.G., Gil, C., Alcayde, A., Gomez, J., 2011. Optimization methods applied to renewable and sustainable energy: a review. Renewable and Sustainable Energy Reviews 15, 1753–1766. Benson, S.M., Franklin, M., 2008. Sustainability and energy conversions. MRS Bulletin vol. 33 (April). Costello, K., June 2007. Diversity of generation technologies: implications for decisionmaking and public policy. The Electricity Journal 20 (5), 10–21. Edinger, R., Kaul, S., 2000. Humankinds detour toward sustainability: past, present, and future of renewable energies and electric power generation. Renewable and Sustainable Energy Reviews-4. EREC, 2004. Integration of renewable energy sources targets and benefits of large-scale deployment of renewable energy sources. Workshop Report, Renewable Energy Market Development Status and Prospects (April–May). Grafton, R.Q., Silva-Echenique, J., 1997. How to manage nature? Strategies, predator– prey models, and chaos. Marine Resource Economics vol. 12, 127–143. Lee, T.-Y., Chen, C.-L., 2009. Wind-photovoltaic capacity coordination for a time-of- use rate industrial user. IE Transactions on Renewable Power Generation 3 (2), 152–167. Leonhard, W., Grobe, M., 2004. Sustainable electrical energy supply with wind, biomass and pumped hydro storage: a realistic long-term strategy or Utopia? Presentation at the General Meeting of the IEEE Power Engineering Society. (June). Macdonald, G., 2009. Coal world: will energy transition deliver us a world of green energy, or return us to a world of coal? www.gregor.us 2009(posted by AGORACOM on, December 3 rd). Martiskainen, M., Coburn, J., 2010. The role of information and communication technologies (ICTs) in household energy consumption/prospects for the UK. Energy Efficiency. Meadows, D.L., Meadows, Donella H., 1973. Toward Global Equilibrium: Collected Papers, 02142. Wright-Allen Press, Inc, Cambridge, Massachusetts. Moss, D.L., Kwoka, J.E., 2010. Competition policy and the transition to a low-carbon, efficient electricity industry. The Electricity Journal vol. 23 (Issue 7), 6–15 (August– September). Pontryagin, L.S., Boltyanskii, V.G., Gamkrelidze, R.V., Mishchenko, E.F., 1965. The Mathematical Theory of Optimal Processes. John Wiley, New York. Ramakumar, R., Abouzahr, I., Ashenayi, K., 1992. A knowledge-based approach to the design of integrated renewable systems. IEEE Transactions on Energy Conversion vol. 7 (No.4) (December). Shafiee, S., Topal, E., 2009. When will fossil fuel reserves be diminished? Energy Policy 37, 181–189. Shi, Y., Liu, Y., Pang, N., 2008. Optimization of electric power source structure under the constraint of CO2 emission. International Conference on Information Management and Industrial Engineering. Steinberger, C.J., 2009. What future for energy and climate? Presentation in Bad Kissingen. (July). Teo, K.L., Goh, C.J., Wong, K.H., 1991. A Unified Computational Approach to Optimal Control Problems, Pitman Monographs and Surveys in Pure and Applied Mathematics, vol. 55. Longman Scientific and Technical with John Wiley, New York. Tora, E.A., El-Halwagi, M.M., 2009. Optimal design and integration of solar systems and fossil fuels for sustainable and stable power outlet. Clean Techn Environ Policy vol. 11, 401–407. Wang, X., 2009. Solving Optimal Control Problems with MATLAB-Indirect Methods. (November). Youngquist, W., 2000. Alternative Energy Sources. http://www.oilcrisis.com/youngquist/ altenergy.htm (October).
Please cite this article as: Miah, M.S., et al., Optimum policy for integration of renewable energy sources into the power generation system, Energy Econ. (2011), doi:10.1016/j.eneco.2011.08.002
