# Minimization of Fuel Costs and Gaseous Emissions of Electric Power Generation by Model Predictive Control

Authors: A. M. Elaiw; X. Xia; A. M. Shehata
Journal: Mathematical Problems in Engineering
 (2013)

---

## Abstract

The purpose of this paper is to present a model predictive control (MPC) approach for the periodic implementation of the optimal solutions of two optimal dynamic dispatch problems with emission and transmission line losses. The first problem is the dynamic economic emission dispatch (DEED) which is a multiobjective optimization problem which minimizes both fuel cost and pollutants emission simultaneously
under a set of constraints. The second one is the profit-based dynamic economic emission dispatch (PBDEED) which is also a multi-objective optimization problem which maximizes the profit and minimizes the emission simultaneously under a set of constraints. Both the demand and energy price are assumed to be periodic and the total transmission loss is assumed to be a quadratic function of the generator power outputs. We assume that there are certain disturbances or uncertainties in the execution of the optimal controller and in the forecasted demand. The convergence and robustness of the MPC algorithm are demonstrated through the application of MPC to the DEED and PBDEED problems with five-unit and six-unit test systems, respectively.


---

## Body

## 1. Introduction

Reducing fuel cost in electric-power-generating plants has received considerable attention in recent years. Dynamic economic dispatch (DED) is a real-time power system problem which is used to schedule the committed electric-power-generating units’ outputs so as to meet the load demand over a dispatch period at minimum operating cost while satisfying ramp rate constraints and other constraints (see, e.g., [1–11]). Since the ramp rate constraints couple the time intervals, the DED problem is a difficult optimization problem. Since the DED problem was introduced, several optimization techniques and procedures have been used for solving the DED problem with complex objective functions or constraints (see the review paper [1]). After deregulation of the electric power system, the objective of the dynamic dispatch problem has been changed from cost minimization to profit maximization and the problem is known as profit-based dynamic economic dispatch (PBDED) problem [11].The emission of gaseous pollutants including SO2, NOx, CO, and CO2 from fossil-fuel-fired thermal plants affects the human health directly or indirectly. Therefore, electric utilities and generation companies (GENCOs) are requested to reduce emission from their plants. As a result of public awareness of environmental protection, diverse compliance strategies have emerged. These strategies include installation of pollutant cleaning, switching to low-emission fuels, replacement of the aged fuel burners with cleaner ones, and emission dispatching [12]. The last strategy is usually preferred to the existing systems because it is easy to implement and requires less additional cost.The emission can be taken into the DED problem by minimizing both fuel cost and emission simultaneously under load demand constraint, ramp rate constraint, and other constraints, resulting in a multiobjective optimization problem. This problem is referred to as dynamic economic emission dispatch (DEED) [13–15]. The emission can also be considered into the PBDED problem by formulating the profit-based dynamic economic emission dispatch (PBDEED) problem which is also a multi-objective optimization problem which maximizes the profit and minimizes the emission simultaneously under ramp rate constraint and other constraints.Recently, many optimization techniques have been used to solve the DEED problem with complex objective functions or constraints (see, e.g., [13–18]). In [16], by assuming that the decision maker has goals for each of the two objective functions, the multi-objective optimization problem is transformed into a single-objective optimization by the goal-attainment method and is solved by particle swarm optimization (PSO) method. In [17], it was assumed that the decision maker has a fuzzy goal for each of the objective functions. The optimal noninferior generation schedule is determined by the evaluation-programming- (EP-) based fuzzy satisfying method. In [13], the multi-objective optimization problem is solved by nondominated sorting genetic algorithm-II. In [18], the multi-objective optimization problem is converted into a single-objective optimization and is solved by an improved pattern-search- (PS-) based algorithm. Pandit et al. [14] proposed an improved bacterial foraging algorithm for solving the DEED problem. Group search optimizer with multiple producers has been used to solve the DEED problem in [15].In [13–18], the main attention focuses on finding the optimal dispatch over a fixed time horizon and the periodic implementation of such an optimal dispatch solution has been though regarded as straightforward and thus left to the practitioners. This periodicity assumption comes from the fact that the demand is periodic due to cyclic consumption behavior and seasonal changes [2]. From a control theoretical point of view, both the DEED and PBDEED formulations provide only open-loop optimal solutions to the generation dispatch problem; that is, the optimal solutions are predetermined before actual execution, and there is no measurement on the system states which is fed back to the optimization model. Therefore, a closed-loop control by the MPC method is introduced in this paper so that the measurement of states can be fed back to the optimization model, and the optimal solution is updated according to the feedback information at each time step. The MPC method has been successfully applied in power systems in [2, 19–22]. MPC has many advantages including its facility of handling constraints, being able to use simple models, and its closed-loop stability and inherent robustness. Moreover, MPC solves the optimal control problem online for the current state of the plant which is a mathematical programming problem and is much simpler than determining the feedback solution by dynamic programming [2].MPC has been proposed for the periodic implementation of the optimal solutions of the DEED and PBDEED problems in [20]. However, the transmission line losses were not included in the DEED and PBDEED formulations. The system loss is a very essential factor to be considered in the power system analysis. Particularly, while the DEED and PBDEED are concerned, the system loss will impose a great impact on optimal pattern of the real power generations at different power plants. In the present paper we propose an MPC approach for the periodic implementation of the optimal solutions of the DEED and PBDEED problems taking into account the transmission losses. The transmission losses will be expressed as a quadratic function of the generator’s power outputs.The remainder of this paper is organized as follows. In Section2, we introduce the DEED and PBDEED problems under regulated and deregulated power systems. In Section 3, we outline an MPC approach for the DEED problem and summarize the main results obtained in [2]. The simulation results for the application of MPC to the DEED and PBDEED problems are given in Section 4. The last section is the conclusion. Throughout the paper, the following notations and definitions will be used. For a sampling period T, the dynamic dispatch problem is considered over dispatch intervals, [iT,(i+N)T)), where the optimization is considered, for i≥0 and NT is the dispatch period. For simplicity, we make the convention throughout the paper that [i,j) denotes the time interval [iT,jT). For any m≥0, k≥1 define Pm=(P11+m,P21+m,…,Pn1+m,P12+m,P22+m,…,Pn2+m,…,P1N+m,P2N+m,…,PnN+m), and Pk=(P1k,P2k,…,Pnk)′. Define U=(u11,u21,…,un1,u12,u22,…,un2,…,u1N-1,u2N-1,…,unN-1) and D=(D1,D2,…,DN)′. The total fuel cost and emission from all units and over the dispatch period [m,m+N) are denoted by C(Pm) and E(Pm), respectively. The demand Dt and the energy price SPt are assumed to be periodic with period N.


## 2. Optimal Dynamic Dispatch with Emission

In this section we formulate both DEED and PBDEED problems. Under regulated power systems utilities have the objective to minimize the fuel cost and are obliged to serve all customers and meet all demands. After deregulation and restructuring of the electric power systems, GENCO has the aim to maximize its own profit. In this case, GENCO is not obliged to meet all demands but may sell its energy at less than the system’s forecasted demand equilibrium if this will maximize its profit.


### 2.1. Dynamic Economic Emission Dispatch

It is well known that the fuel cost and the amount of emission conflict with each other. Minimization of fuel cost maximizes the amount of emission and vice versa. Therefore it is necessary to find out an operating point that strikes a balance between fuel cost and emission. This can be done by formulating the DEED problem which is a multi-objective optimization problem with two conflicting objectives, the fuel cost and emission.The following objectives and constraints are taken into account in the formulation of the DEED problem.Cost.The fuel cost curves are assumed to be a quadratic function of the generator’s active power output. Therefore, the total fuel cost over the dispatch period [0,N] is given by [15]
(1)C(P0)=∑t=1N‍∑i=1n‍Ci(Pit)=∑t=1N‍∑i=1n‍ai+biPit+ci(Pit)2.Emission.The amount of emission of SO2, CO2, and NOx can be expressed as a quadratic function of the generator’s active power output. The total pollutants emission over the dispatch period [0,N] is given by [23]
(2)E(P0)=∑t=1N‍∑i=1n‍Ei(Pit)=∑t=1N‍∑i=1n‍αi+βiPit+γi(Pit)2.Constraints. (i) Power balance constraint
(3)∑i=1n‍Pit=Dt+Losst,t=1,2,…,N,
where Losst is the transmission line loss at time t. The B-coefficient method is one of the most commonly used method by the power utility industry to calculate the network losses. In this method the network losses are expressed as a quadratic function of the generator’s power outputs that can be approximated in the following [13]:
(4)Losst=∑i=1n‍∑j=1n‍PitBijPjt,t=1,2,…,N.
(ii) Generation limits
(5)Pimin≤Pit≤Pimax,i=1,2,…,n,t=1,2,…,N.
(iii) Generating unit ramp rate limits
(6)-DRi·T≤Pit+1-Pit≤URi·T,i=1,2,…,n,t=1,2,…,N-1,(7)-DRi·T≤Pi1-PiN≤URi·T,i=1,2,…,n.The Optimization Problem. Aggregating the objectives and constraints, the DEED problem can be mathematically formulated as a nonlinear constrained multi-objective optimization problem which can be converted into a single-objective optimization using the weighting method as follows
(8)minP0H(P0)=αC(P0)+(1-α)E(P0),(9)subjecttoPit∈ΩDEED(P0),i=1,2,…,n,t=1,2,…,N,
where the feasible domain ΩDEED is defined to be the set of (Pit:i=1,2,…,n,t=1,2,…,N) satisfying constraints (3)–(7); α is a weighting factor, α∈[0,1]. It will be noted that, when α=1, the problem (8)-(9) determines the optimal amount of the generated power by minimizing the cost regardless of emission and the DEED problem leads to the DED problem [2]. If α=0 then, the DEED problem determines the optimal amount of the generated power by minimizing the emission regardless of cost and the DEED problem leads to the pure dynamic emission dispatch (PDED) [13].The constraints (3)–(6) are usually used in the conventional DEED problem [13, 16, 17]. Since the demand and constraints are periodic, one may obtain the solution of the conventional DEED problem (i.e., problem (8) under constraints (3)–(6)) over, for example, 24 hours (N=24 and T=1), and then, this solution is implemented not only for the first day but also for all the other week days. Sometimes such an optimal solution is not able to be practically implemented, or, in other words, the solution is not practically feasible. The ramp rate constraint may be violated when the generators are moved from the 24th hour of a day to the first hour of the next day. This problem can be resolved by including the ramp limit on the difference between Pi24 and Pi25=Pi1. This can be achieved by adding the constraint (7) to the conventional DEED problem (see [2, 20]).We note that the above DEED problem can be solved over the dispatch period[m,m+N] for any m≥0 and it can be formulated as
(10)minPmH(Pm)subjecttoPit∈ΩDEED(Pm),i=1,2,…,n,t=m+1,m+2,…,m+N.
We define the following set of parameters:
(11)ΓDEED={n,N,α,T,DRi,URi,Pimin,Pimax,ai,bi,ci,αi,βi,γi,Bij,i,j=1,…,n}.
Since the demandD is periodic and ΓDEED do not change over time, then Pm+1=Pm+N+1, and ΩDEED satisfies
(12)ΩDEED(Pm+1)=ΩDEED(Pm+2,…,Pm+N,Pm+N+1)=ΩDEED(Pm+2,…,Pm+N,Pm+1)=ΩDEED(Pm)
and then ΩDEED is shift invariant (see [21]). The shift-invariant property of ΩDEED is needed for the application of the MPC approach to the DEED problem.


### 2.2. Profit-Based Dynamic Economic Emission Dispatch

Now we introduce the PBDEED formulation with the aim to produce electricity with minimum operating cost and sell it with maximum profits and environmental protection by limiting the emission of greenhouse gases into the atmosphere.The total revenue of the GENCO over the dispatch period[0,N] is given by [11]
(13)RV(P0)=∑t=1N‍∑i=1n‍SPt·Pit,
where SPt is the forecasted energy price at time t.The total profit over the dispatch period[0,N] is given by [11, 20]
(14)PF(P0)=RV(P0)-C(P0).
Let us define a function G(·)=-PF(·) which measures the profit attained by the conversion of the energy available in fossil fuels into electric energy.The objective of the PBDEED is to simultaneously minimize the emission and maximize the profit satisfying a set of constraints. The PBDEED can be mathematically stated as follows:(15)minP0αG(P0)+(1-α)E(P0)subjecttoPit∈ΩPBDEED(P0),i=1,2,…,n,t=1,2,…,N,
where the feasible domain ΩPBDEED is defined to be the set of (Pit:i=1,2,…,n,t=1,2,…,N) satisfying the following constraints:
(16)∑i=1n‍Pit≤Dt+Losst,t=1,2,…,N,(17)Pimin≤Pit≤Pimax,i=1,2,…,n,t=1,2,…,N,(18)-DRi·T≤Pit+1-Pit≤URi·T,i=1,2,…,n,t=1,2,…,N-1,(19)-DRi·T≤Pi1-PiN≤URi·T,i=1,2,…,n.
It will be noted that, when α=1, the problem (15) determines the optimal amount of the generated power by maximizing the profit regardless of emission and the PBDEED problem leads to the PBDED problem [11]. If α=0, then the PBDEED problem determines the optimal amount of the generated power by minimizing the emission regardless of profit [24]. Of course this case is not useful for GENCOs.Constraint (16) means that under the deregulated environment GENCO is not obliged to meet all demands and energy lost due to transmission lines but may sell its energy at less than the system’s forecasted demand plus power loss. Similar to Section 2.1, one can show that the set ΩPBDEED is shift invariant.In the next section we propose an MPC approach for the periodic implementation of the optimal solutions of the DEED problem. MPC obtains a feedback control which is constructed by solving a finite horizon optimal control problem at each sampling instant using the current state of the plant as the initial state for the optimization and applying only “the first part” of the optimal control [25].


## 2.1. Dynamic Economic Emission Dispatch

It is well known that the fuel cost and the amount of emission conflict with each other. Minimization of fuel cost maximizes the amount of emission and vice versa. Therefore it is necessary to find out an operating point that strikes a balance between fuel cost and emission. This can be done by formulating the DEED problem which is a multi-objective optimization problem with two conflicting objectives, the fuel cost and emission.The following objectives and constraints are taken into account in the formulation of the DEED problem.Cost.The fuel cost curves are assumed to be a quadratic function of the generator’s active power output. Therefore, the total fuel cost over the dispatch period [0,N] is given by [15]
(1)C(P0)=∑t=1N‍∑i=1n‍Ci(Pit)=∑t=1N‍∑i=1n‍ai+biPit+ci(Pit)2.Emission.The amount of emission of SO2, CO2, and NOx can be expressed as a quadratic function of the generator’s active power output. The total pollutants emission over the dispatch period [0,N] is given by [23]
(2)E(P0)=∑t=1N‍∑i=1n‍Ei(Pit)=∑t=1N‍∑i=1n‍αi+βiPit+γi(Pit)2.Constraints. (i) Power balance constraint
(3)∑i=1n‍Pit=Dt+Losst,t=1,2,…,N,
where Losst is the transmission line loss at time t. The B-coefficient method is one of the most commonly used method by the power utility industry to calculate the network losses. In this method the network losses are expressed as a quadratic function of the generator’s power outputs that can be approximated in the following [13]:
(4)Losst=∑i=1n‍∑j=1n‍PitBijPjt,t=1,2,…,N.
(ii) Generation limits
(5)Pimin≤Pit≤Pimax,i=1,2,…,n,t=1,2,…,N.
(iii) Generating unit ramp rate limits
(6)-DRi·T≤Pit+1-Pit≤URi·T,i=1,2,…,n,t=1,2,…,N-1,(7)-DRi·T≤Pi1-PiN≤URi·T,i=1,2,…,n.The Optimization Problem. Aggregating the objectives and constraints, the DEED problem can be mathematically formulated as a nonlinear constrained multi-objective optimization problem which can be converted into a single-objective optimization using the weighting method as follows
(8)minP0H(P0)=αC(P0)+(1-α)E(P0),(9)subjecttoPit∈ΩDEED(P0),i=1,2,…,n,t=1,2,…,N,
where the feasible domain ΩDEED is defined to be the set of (Pit:i=1,2,…,n,t=1,2,…,N) satisfying constraints (3)–(7); α is a weighting factor, α∈[0,1]. It will be noted that, when α=1, the problem (8)-(9) determines the optimal amount of the generated power by minimizing the cost regardless of emission and the DEED problem leads to the DED problem [2]. If α=0 then, the DEED problem determines the optimal amount of the generated power by minimizing the emission regardless of cost and the DEED problem leads to the pure dynamic emission dispatch (PDED) [13].The constraints (3)–(6) are usually used in the conventional DEED problem [13, 16, 17]. Since the demand and constraints are periodic, one may obtain the solution of the conventional DEED problem (i.e., problem (8) under constraints (3)–(6)) over, for example, 24 hours (N=24 and T=1), and then, this solution is implemented not only for the first day but also for all the other week days. Sometimes such an optimal solution is not able to be practically implemented, or, in other words, the solution is not practically feasible. The ramp rate constraint may be violated when the generators are moved from the 24th hour of a day to the first hour of the next day. This problem can be resolved by including the ramp limit on the difference between Pi24 and Pi25=Pi1. This can be achieved by adding the constraint (7) to the conventional DEED problem (see [2, 20]).We note that the above DEED problem can be solved over the dispatch period[m,m+N] for any m≥0 and it can be formulated as
(10)minPmH(Pm)subjecttoPit∈ΩDEED(Pm),i=1,2,…,n,t=m+1,m+2,…,m+N.
We define the following set of parameters:
(11)ΓDEED={n,N,α,T,DRi,URi,Pimin,Pimax,ai,bi,ci,αi,βi,γi,Bij,i,j=1,…,n}.
Since the demandD is periodic and ΓDEED do not change over time, then Pm+1=Pm+N+1, and ΩDEED satisfies
(12)ΩDEED(Pm+1)=ΩDEED(Pm+2,…,Pm+N,Pm+N+1)=ΩDEED(Pm+2,…,Pm+N,Pm+1)=ΩDEED(Pm)
and then ΩDEED is shift invariant (see [21]). The shift-invariant property of ΩDEED is needed for the application of the MPC approach to the DEED problem.


## 2.2. Profit-Based Dynamic Economic Emission Dispatch

Now we introduce the PBDEED formulation with the aim to produce electricity with minimum operating cost and sell it with maximum profits and environmental protection by limiting the emission of greenhouse gases into the atmosphere.The total revenue of the GENCO over the dispatch period[0,N] is given by [11]
(13)RV(P0)=∑t=1N‍∑i=1n‍SPt·Pit,
where SPt is the forecasted energy price at time t.The total profit over the dispatch period[0,N] is given by [11, 20]
(14)PF(P0)=RV(P0)-C(P0).
Let us define a function G(·)=-PF(·) which measures the profit attained by the conversion of the energy available in fossil fuels into electric energy.The objective of the PBDEED is to simultaneously minimize the emission and maximize the profit satisfying a set of constraints. The PBDEED can be mathematically stated as follows:(15)minP0αG(P0)+(1-α)E(P0)subjecttoPit∈ΩPBDEED(P0),i=1,2,…,n,t=1,2,…,N,
where the feasible domain ΩPBDEED is defined to be the set of (Pit:i=1,2,…,n,t=1,2,…,N) satisfying the following constraints:
(16)∑i=1n‍Pit≤Dt+Losst,t=1,2,…,N,(17)Pimin≤Pit≤Pimax,i=1,2,…,n,t=1,2,…,N,(18)-DRi·T≤Pit+1-Pit≤URi·T,i=1,2,…,n,t=1,2,…,N-1,(19)-DRi·T≤Pi1-PiN≤URi·T,i=1,2,…,n.
It will be noted that, when α=1, the problem (15) determines the optimal amount of the generated power by maximizing the profit regardless of emission and the PBDEED problem leads to the PBDED problem [11]. If α=0, then the PBDEED problem determines the optimal amount of the generated power by minimizing the emission regardless of profit [24]. Of course this case is not useful for GENCOs.Constraint (16) means that under the deregulated environment GENCO is not obliged to meet all demands and energy lost due to transmission lines but may sell its energy at less than the system’s forecasted demand plus power loss. Similar to Section 2.1, one can show that the set ΩPBDEED is shift invariant.In the next section we propose an MPC approach for the periodic implementation of the optimal solutions of the DEED problem. MPC obtains a feedback control which is constructed by solving a finite horizon optimal control problem at each sampling instant using the current state of the plant as the initial state for the optimization and applying only “the first part” of the optimal control [25].


## 3. MPC Approach to DEED

In this section, we introduce an MPC approach proposed in [2, 20] for the optimal dynamic dispatch problem with emission and transmission line losses. We first show the application of MPC to the DEED problem and then show that this MPC version can also be applied to the PBDEED problem. We introduce the control variables uit as [8, 9]
(20)uit=Pit+1-PitT,i=1,2,…,n,t=1,2,…,N-1,
where uit is the ramping action of unit i at time t. This equation actually defines coordinate transformation between the variables {Pit:i=1,2,…,n,t=1,2,…,N} and the variables {uit:i=1,2,…,n,t=1,2,…,N-1}. It is obvious that the inverse coordinate transformation is given by
(21)Pit=Pi1+∑j=1t-1‍Tuij,t=2,3,…,N.
The optimal solution of the DEED problem is implemented repeatedly at instants which equal to multiples of N. To introduce the MPC approach, let us consider the DEED problem starting at an arbitrary instant t=m and over a dispatch interval [m,m+N). Then the optimization variables are changed into {Pim+1,Pim+2,…,Pim+N,i=1,2,…,n}. By the transformation defined in (21), the optimization variables {Pim+1,Pim+2,…,Pim+N,i=1,2,…,n} are transformed into {Pim+1,uim+1,…,uim+N-1,i=1,2,…,n}.In the MPC method, a finite-horizon optimal control problem is repeatedly solved and the input is applied to the system based on the obtained optimal open-loop control. In our case, the horizon is chosen to beN. Instead of solving the DEED problem with nN number of variables {Pim+1,uim+1,…,uim+N-1,i=1,2,…,n}, the MPC algorithm solves the following problem which has only n(N-1) number of variables {uim+1,…,uim+N-1,i=1,2,…,n}.Problem
M
P
C
D
E
E
D
P
m
+
1
(
u
,
[
m
,
m
+
N
)
)
. Given ΓDEED, D and Pm+1, let
(22)Pi1:=Pim+1,uij:=uim+j,Dt:=Dm+t,i=1,2,…,n,t=1,2,…,N,j=1,2,…,N-1,
and solves the following minimization problem:
(23)minU∑t=1N‍∑i=1n‍αCi(Pi1+∑s=1t-1‍Tuis)+(1-α)Ei(Pi1+∑s=1t-1‍Tuis)subjecttouit∈ΩD(P1,U),i=1,2,…,n,t=1,2,…,N-1,
where the feasible domain ΩD(P1,U) is defined to be the set of {Pi1,uit:i=1,2,…,n,t=1,2,…,N-1} satisfying
(24)∑i=1n‍(Pi1+∑s=1t-1‍Tuis)=Dt+∑i=1n‍∑j=1n‍(Pi1+∑s=1t-1‍Tuis)Bij(Pj1+∑s=1t-1‍Tujs),t=1,2,…,N,Pimin≤Pi1+∑s=1t-1‍Tuis≤Pimax,i=1,2,…,n,t=1,2,…,N,-DRi≤uit≤URi,i=1,2,…,n,t=1,2,…,N-1.
The notation MPCDEEDPm+1  (u,[m,m+N)) denotes that the optimization problem is solved over the interval [m,m+N) with variables uit and for known inputs Pim+1,  i=1,2,…,n,t=m+1,…,m+N-1.In order to make the MPCDEED problem solvable, the following hypothesis is needed as in [2, 5, 8].Feasibility Hypothesis. After the change of variables in (22) over any dispatch interval [m,m+N) with m≥0, the set ΩD(P1,U) is not empty. This hypothesis ensures the solvability of the problem MPCDEEDPm+1(u,[m,m+N)). Denote the optimal solution of MPCDEED for a given initial generation P-m+1 by u¯m(P-m+1)={u¯im+j(P-m+1),i=1,2,…,n,j=1,2,…,N-1}. In the MPC method, the optimal solution u¯m is applied only in the first sampling period [m,m+1); that is, u¯im+1(P-m+1) is applied to the state P-im+1. Since u¯m(P-m+1) depends on the current state P-m+1, in this way a feedback control can be obtained. We define the MPC feedback controller by vim:=u¯im+1.The closed-loop solution P-im+2 given by P-im+2=P-im+1+Tvim(P-m+1) is actually executed over the time period [m+1,m+2). The idea of the MPC can be formulated into the following MPC algorithm.MPC Algorithm Initialization. Input the initial status P-1≜P1=(P11,P21,…,Pn1) and let m=0.(1)
Compute the open-loop optimal solutionu¯m to the problem MPCDEEDP-m+1(u,[m,m+N)).(2)
The (closed-loop) MPC controllervim is applied to the plant in the sampling interval [m,m+1) to obtain the closed-loop MPC solution
(25)P-im+2=P-im+1+Tvim(P-m+1)
over the period [m+1,m+2).(3)
Letm:=m+1 and go to step (1).Theorem 1 (see [2]).
Suppose the Feasibility Hypothesis holds,P* is the globally optimal solution of the DEED problem, and the initial power output P1 at time t=1 satisfies Pi1∈ΩDEED, then  the MPC algorithm converges to P*.This theorem tells that the solutions of the MPC algorithm converge to the optimal solution of the DEED problem. Now we consider the inherent robustness properties of the MPC algorithm. The uncertainties in energy demand, price, and reserve demand for the PBDED problem are discussed by fuzzy optimization in [11]. However, no theoretical result is given. First, we suppose that the disturbances happen in the execution of the controller. That is, the disturbance happens only in step (2) of the MPC algorithm so that when the control vim is applied to the plant in the sampling interval [m,m+1), the system actually executes
(26)P-im+2=P-im+1+Tvim(P-m+1)+Twim+1
over the period [m+1,m+2), where wim+1 is the disturbance. We assume that the disturbances satisfy the following bound:
(27)∥wim+1∥<e,e>0,i=1,2,…,n,m≥0.Theorem 2 (see [2]).
Suppose the Feasibility Hypothesis holds,P* is the globally optimal solution of the DEED problem, the norm of the gradient of the function H of DEED problem has the upper bound L on ΩDEED, ϵ is a small enough positive constant, c is a positive constant which is less than ϵ, (26) is executed in step (2) of the MPC algorithm instead of (25), and the constant disturbance wik satisfies (27) where e is small enough so that e<min{c/L,(ϵ-c)/L}, then there exists an integer N0 such that for any k>N0, the optimal MPC solution P-k+1 of the kth loop in the MPC algorithm belongs to the domain Ω¯:={P:||P-P*||<c}.Theorem2 shows that the MPC algorithm is robust against certain disturbances in the execution of the optimal controller. It may happen that there is disturbance or uncertainty in the forecasted demand; that is, the demand Dk is disturbed so that the actual demand is D~k. It has been shown in [2] that the demand disturbances or uncertainties can also be written in the form of (26) and the result of Theorem 2 is applicable when ||D~k-Dk|| is small enough.Theorems1 and 2 are based on the assumption that the objective function of the DEED problem is strictly convex and differentiable over the set ΩDEED which is bounded. Since both the fuel cost and emission functions are assumed to be quadratic, then all the objective functions of the DEED and PBDEED problems are strictly convex and differentiable over their feasible constraint sets. Also since the demand and energy price and all constraints are assumed to be periodic, then all feasible constraint sets, ΩDEED and ΩPBDEED, are shift invariant. Therefore, Theorems 1 and 2 are valid for both the DEED and PBDEED problems.


## 4. Simulation Results

In this section we present two test systems consisting of five units and six units. The first system is used to show that the MPC algorithm converges to the optimal solutions of the DEED problem. The inherent robustness properties of the model predictive control (IRP-MPC) algorithm are also shown. In the second test system, we show the converge and robustness properties of the proposed MPC algorithm to the PBDEED problem. The load demand and energy price are assumed to be periodic over a dispatch period of one day and the sampling period is chosen to be one hour. The optimization problem is solved by sequential quadratic programming. All computations were carried out by the MATLAB program. In particular, the optimal control sequence is computed by the fmincon code of the MATLAB Optimization Toolbox.


### 4.1. Five-Unit Test System

This example presents an application of MPC to the DEED problem consisting of five units. The technical data of the units are taken from [16]. First, we present the optimal solutions of the DEED problems for different values of the weighting factor α and then show that the solutions of the MPC converge to the optimal solutions of the DEED problem. The optimal solutions of the DEED problem with different values of α are given in Tables 1 and 2. Table 1 shows the hourly power schedule and transmission loss obtained from the DEED problem with α=1 and α=0.7. Table 2 shows the hourly power schedule and transmission loss obtained from the DEED problem with α=0.5 and α=0. The optimal cost and emission obtained from the DEED problem with different values of α are given in Table 3. It is seen from Table 3 that the cost is $40,121 under DEED (α=1), but it increases to $40,851 under DEED (α=0) and emission obtained from DEED (α=1) is 20,363 lb but decreases to 16,546 lb under DEED (α=0). Figure 1 shows the effect of the transmission loss on the outputs of the power units. We can see that neglecting transmission loss would often introduce inaccuracy into the resulting dispatch.Table 1
Hourly generation (MW) schedule obtained from DEED (α=1,α=0.7) for the 5-unit system.
H
DEED (α=1)
DEED (α=0.7)
P
1
t
P
2
t
P
3
t
P
4
t
P
5
t
Losst
P
1
t
P
2
t
P
3
t
P
4
t
P
5
t
Losst
1
15.3041
72.5541
61.5777
118.5306
145.6491
3.6156
35.3428
63.4632
109.2221
114.8838
90.5390
3.4508
2
16.4010
75.4029
68.8115
127.0094
151.4306
4.0554
37.5960
67.1865
115.4316
122.8094
95.8659
3.8894
3
18.1616
79.9660
80.4027
140.5893
160.6954
4.8151
41.2062
73.1509
125.3825
135.5049
104.4026
4.6471
4
20.5883
86.2528
96.3755
159.2916
173.4643
5.9725
46.1827
81.3656
139.0964
152.9899
116.1669
5.8014
5
21.8270
89.4580
104.5228
168.8264
179.9781
6.6123
48.7217
85.5538
146.0921
161.9040
122.1678
6.4395
6
24.0431
95.1914
119.0980
185.8755
191.6321
7.8400
53.2657
93.0427
158.6080
177.8437
132.9037
7.6638
7
24.8524
97.2620
124.3507
192.0138
195.8299
8.3088
54.9040
95.7422
163.1213
183.5888
136.7751
8.1313
8
26.0898
100.4824
132.5307
201.5877
202.3761
9.0666
57.4540
99.9456
170.1488
192.5335
142.8049
8.8869
9
27.6901
104.6328
143.0725
213.9197
210.7768
10.0918
61.2354
106.1787
175.0000
205.7698
151.7344
9.9184
10
28.3138
106.2316
147.1587
218.7101
214.0919
10.5060
62.9302
108.9742
175.0000
211.6974
155.7389
10.3407
11
29.0370
108.0749
151.8361
224.1873
217.8548
10.9901
64.8728
112.1722
175.0000
218.4748
160.3159
10.8356
12
29.9330
110.3833
157.7003
231.0449
222.5497
11.6112
67.3023
116.1719
175.0000
226.9531
166.0457
11.4730
13
28.3139
106.2315
147.1587
218.7100
214.0919
10.5060
62.9301
108.9741
175.0000
211.6974
155.7390
10.3407
14
27.6893
104.6333
143.0726
213.9197
210.7770
10.0918
61.2356
106.1785
175.0000
205.7699
151.7344
9.9184
15
26.0894
100.4711
132.5392
201.5873
202.3796
9.0665
57.4541
99.9456
170.1488
192.5334
142.8050
8.8869
16
22.8134
91.9940
110.9206
176.3178
185.0932
7.1390
50.7195
88.8473
151.5954
168.9141
126.8884
6.9647
17
21.8267
89.4583
104.5227
168.8266
179.9780
6.6123
48.7218
85.5537
146.0920
161.9040
122.1679
6.4395
18
24.0532
95.2077
119.0874
185.8684
191.6233
7.8400
53.2655
93.0428
158.6080
177.8437
132.9038
7.6638
19
26.0883
100.4706
132.5394
201.5881
202.3802
9.0665
57.4540
99.9458
170.1488
192.5334
142.8049
8.8869
20
28.3130
106.2317
147.1590
218.7103
214.0920
10.5060
62.9305
108.9740
175.0000
211.6973
155.7388
10.3407
21
27.2534
103.4720
140.1400
210.4958
208.4400
9.8012
60.0237
104.1856
175.0000
201.5410
148.8725
9.6229
22
23.9089
94.8507
118.2197
184.8516
190.9324
7.7633
52.9926
92.5929
157.8562
176.8867
132.2589
7.5872
23
20.4569
85.9088
95.5033
158.2704
172.7666
5.9060
45.9111
80.9171
138.3475
152.0352
115.5243
5.7351
24
17.6333
78.5962
76.9229
136.5133
157.9141
4.5799
40.1227
71.3606
122.3952
131.6944
101.8397
4.4126Table 2
Hourly generation (MW) schedule obtained from DEED (α=0.5,α=0) for the 5-unit system.
H
DEED (α=0.5)
DEED (α=0)
P
1
t
P
2
t
P
3
t
P
4
t
P
5
t
Losst
P
1
t
P
2
t
P
3
t
P
4
t
P
5
t
Losst
1
43.1673
60.7831
115.0952
113.0510
81.3489
3.4456
54.6445
57.2094
119.8650
110.1053
71.6251
3.4492
2
45.8553
64.6456
121.0726
120.7786
86.5315
3.8834
57.9634
61.1854
125.5602
117.5630
76.6146
3.8865
3
50.1626
70.8326
130.6514
133.1568
94.8363
4.6397
63.2806
67.5546
134.6871
129.5096
84.6101
4.6420
4
56.0966
79.3553
143.8531
150.2058
106.2813
5.7922
70.6064
76.3274
147.2663
145.9645
95.6288
5.7935
5
59.1235
83.7007
150.5876
158.8981
112.1194
6.4293
74.3486
80.7966
153.6820
154.3540
101.2488
6.4301
6
64.5383
91.4710
162.6369
174.4415
122.5642
7.6518
75.0000
89.8771
166.7190
171.3855
112.6714
7.6531
7
66.4908
94.2716
166.9820
180.0439
126.3304
8.1187
75.0000
93.1922
171.4836
177.6064
116.8388
8.1210
8
69.5320
98.6463
173.7475
188.7748
132.1728
8.8734
75.0000
99.2991
175.0000
189.0543
124.5301
8.8835
9
74.4893
105.7396
175.0000
202.9461
141.7335
9.9084
75.0000
108.2654
175.0000
205.8482
135.8169
9.9305
10
75.0000
108.9451
175.0000
209.3405
146.0470
10.3326
75.0000
111.7567
175.0000
212.3860
140.2141
10.3568
11
75.0000
112.7466
175.0000
216.9235
151.1615
10.8316
75.0000
115.7496
175.0000
219.8622
145.2453
10.8572
12
75.0000
117.5046
175.0000
226.4089
157.5615
11.4750
75.0000
120.7459
175.0000
229.2153
151.5412
11.5024
13
75.0000
108.9451
175.0000
209.3406
146.0470
10.3326
75.0000
111.7567
175.0000
212.3860
140.2141
10.3568
14
74.4892
105.7396
175.0000
202.9462
141.7335
9.9084
75.0000
108.2654
175.0000
205.8482
135.8169
9.9305
15
69.5320
98.6463
173.7475
188.7748
132.1728
8.8734
75.0000
99.2991
175.0000
189.0543
124.5300
8.8835
16
61.5044
87.1178
155.8856
165.7339
116.7120
6.9538
75.0000
84.7250
159.3200
161.7208
106.1884
6.9543
17
59.1234
83.7006
150.5877
158.8982
112.1195
6.4293
74.3485
80.7966
153.6820
154.3540
101.2488
6.4301
18
64.5383
91.4709
162.6369
174.4415
122.5642
7.6518
75.0000
89.8771
166.7190
171.3855
112.6714
7.6531
19
69.5321
98.6462
173.7475
188.7748
132.1728
8.8734
75.0000
99.2991
175.0000
189.0543
124.5300
8.8835
20
75.0000
108.9451
175.0000
209.3406
146.0469
10.3326
75.0000
111.7567
175.0000
212.3860
140.2141
10.3568
21
73.0654
103.6961
175.0000
198.8682
138.9840
9.6138
75.0000
105.7732
175.0000
201.1807
132.6788
9.6326
22
64.2130
91.0043
161.9132
173.5081
121.9368
7.5754
75.0000
89.3248
165.9258
170.3495
111.9764
7.5765
23
55.7726
78.8901
143.1321
149.2750
105.6562
5.7260
70.2064
75.8485
146.5793
145.0661
95.0270
5.7273
24
48.8696
68.9755
127.7758
129.4415
92.3432
4.4056
61.6847
65.6428
131.9472
125.9237
82.2098
4.4082Table 3
The optimal cost and emission obtained from DEED with different value ofα for the  5-unit system.
Value  ofα
Cost  ($)
Emission  (lb)
α
=
1
40,121
20,363
α
=
0.9
40,215
18,100
α
=
0.8
40,343
17,340
α
=
0.7
40,465
16,966
α
=
0.6
40,578
16,752
α
=
0.5
40,670
16,638
α
=
0.4
40,731
16,587
α
=
0.3
40,775
16,563
α
=
0.2
40,808
16,552
α
=
0.1
40,832
16,547
α
=
0
40,851
16,546Figure 1
The generation output of unit 5 under DEED (α=0.5) from the five-unit system.Turning now to our goal of this section, we apply MPC strategy. The initialPi1 is chosen such that ∑i=1n‍Pi1=D1+∑i=1n‍∑j=1n‍Pi1BijPj1. The proposed MPC strategy is implemented over 48 hours. Figures 2, 3, 4, 5, and 6 show that the MPC closed-loop solutions asymptotically approach the optimal solutions of the DEED problem for different values of α.Figure 2
Convergence of the closed-loop MPC solutions to those of DEED (α=1) for the five-unit system.Figure 3
Convergence of the closed-loop MPC solutions to those of DEED (α=0.7) for the five-unit system.Figure 4
Convergence of the closed-loop MPC solutions to those of DEED (α=0.5) for the five-unit system.Figure 5
Convergence of the closed-loop MPC solutions to those of DEED (α=0.2) for the five-unit system.Figure 6
Convergence of the closed-loop MPC solutions to those of DEED (α=0) for the five-unit system.Now we show the IRP-MPC algorithm against two sources of disturbances.(
1
) Disturbance arises from the execution of the optimal controller. In this case, we execute (26) with disturbance wim generated by
(28)wim=-εi+2εir(m),
where the parameters r(m)’s are uniformly distributed random numbers on [0,1]. Denote ε=(ε1,ε2,…,ε5). We choose two different bounds of disturbances:IRP-MPC-(I):ε=(3,3,2,3,2),IRP-MPC-(II):ε=(6,6,4,6,4).(
2
) Disturbance arises from the forecasted demand. In this case, the actual demand D~t can be defined as
(29)D~t={Dt,ift=1,Dt+q(1-2r(t))Dtift=2,3,…,N,
where q(1-2r) is the relative change between the actual D~t demand and the forecasted demand Dt.Here, we assume the demand is disturbed with 5% and 10% of the nominal one; then we have the following cases:IRP-MPC-(III):q=0.05,IRP-MPC-(IV):q=0.1.In these cases the initialPi1,  i=1,2…,5 for the MPC are chosen as the optimal solution of the DEED problem at t=1; that is, Pi1=P-i1, and the weighting factor α is chosen α=0.5. From Figures 7, 8, 9, and 10, we can see that, although the disturbance increased, the MPC still keeps the disturbed system in the neighborhood of the optimal solution of the DEED problem. From these cases we observe that the size of this neighborhood depends on the bound of the disturbance.Figure 7
The generation output of the five-unit system under DEED (α=0.5) and IRP-MPC-(I).Figure 8
The generation output of the five-unit system under DEED (α=0.5) and IRP-MPC-(II).Figure 9
The generation output of the five-unit system under DEED (α=0.5) and IRP-MPC-(III).Figure 10
The generation output of the five-unit system under DEED (α=0.5) and IRP-MPC-(IV).


### 4.2. Six-Unit Test System

This system consists of six units for the application of MPC to the PBDEED problem. The technical data of the units are taken from [20]. The transmission loss formula coefficients are given in the appendix. The initial Pi1 is chosen such that ∑i=1n‍Pi1≤D1+∑i=1n‍∑j=1n‍Pi1BijPj1. The result of the optimization is dependent on the energy price data. Indeed, minor changes in the energy price may give a significant change in the power generation of the units as well as the profit. We consider different energy price profiles which are given in Table 4.Table 4
The results of PBDEED (α=1) for the three price profiles.
H
Price-I
P
L
t
Losst
Price-II
P
L
t
Losst
Price-III
P
L
t
Losst
($)
(MW)
(MW)
($)
(MW)
(MW)
($)
(MW)
(MW)
1
10.31
561.02
4.25
11.46
740.1
7.5
13.75
955
11.76
2
10.80
621.59
5.39
12.00
873.8
10.0
14.40
942
11.51
3
11.56
745.91
7.50
12.84
935
11.3
15.41
935
11.40
4
9.49
482.50
3.01
10.54
615.2
5.3
12.65
930
11.21
5
10.89
625.04
5.46
12.10
866.8
9.8
14.52
935
11.37
6
11.86
835.62
9.34
13.18
963
11.9
15.82
963
12.00
7
12.42
989
12.45
13.80
989
12.5
16.56
989
12.58
8
11.21
803.18
8.62
12.46
1008
12.9
14.95
1023
13.28
9
13.25
1124.56
15.45
14.72
1126
15.7
17.66
1126
15.78
10
13.45
1150
16.17
14.94
1150
16.2
17.93
1150
16.36
11
13.37
1201
17.41
14.86
1201
17.5
17.83
1201
17.60
12
13.91
1235
18.29
15.46
1235
18.4
18.55
1235
18.50
13
11.74
805.70
8.75
13.04
1190
17.1
15.65
1190
17.23
14
12.42
996.35
12.61
13.80
1251
18.7
16.56
1251
18.82
15
10.98
645.56
5.86
12.20
932.2
11.2
14.64
1263
19.04
16
10.19
545.29
3.98
11.32
726.9
7.2
13.58
1250
18.66
17
11.25
698.05
6.76
12.50
999.9
12.7
15.00
1221
17.98
18
12.26
895.99
10.14
13.62
1202
17.2
16.34
1202
17.56
19
9.79
541.62
3.94
10.88
697.4
6.8
13.06
1159
16.37
20
11.03
655.09
6.03
12.26
943.6
11.5
14.71
1092
14.87
21
11.20
686.22
6.56
12.44
1002.2
12.7
14.93
1023
13.28
22
12.22
929.72
11.23
13.58
984
12.4
16.30
984
12.47
23
11.43
732.97
7.41
12.70
975
12.1
15.24
975
12.24
24
10.28
556.53
4.17
11.42
730.59
7.4
13.70
960
11.87
Profit=831.24 ($)
Profit=29080 ($)
Profit=94922.18 ($)The effect of the energy price over the time horizon on the total power which produced by the committed units for the PBDEED problem atα=1 is shown in Table 4. Now we define PLt=∑i=1n‍Pit-Losst. We can see from Table 4 that, for price-I and price-II, PLt<Dt in some t and PLt=Dt in some other t which are shown by bold face. It means that GENCO chooses to supply power less than the demand plus power loss in some hours of the day and equal to the demand plus power loss in other hours of the day when this satisfies maximum profit. For price-III, the optimal solution of the PBDEED satisfies PLt=Dt for all t. In this case, GENCO will supply power to satisfy the demand plus power loss over the whole day since this will maximize its profit. We also note that the profit increases according to the increase of the energy prices. According to the PBDEED formulation, GENCO will sell all produced power and the consumer is the one who will bear the energy loss.The effect of the weighting factorα on the total amount of power and the total profit in case of price-II for the PBDEED problem is given in Table 5. It can be seen that, as the weighting factor α is increased (i.e., the importance of the emission is decreased), both the profit and the total power are increased.Table 5
The results of PBDEED for price-II with different values ofα.
H
α
=
0.2
α
=
0.5
α
=
0.7
α
=
1
P
L
t
Loss
t
P
L
t
Losst
P
L
t
Losst
P
L
t
Losst
(MW)
(MW)
(MW)
(MW)
(MW)
(MW)
(MW)
(MW)
1
385.3
1.5
516.3
3.2
613
4.9
740.1
7.5
2
400.5
1.6
589
4.1
717.4
6.5
873.9
10.1
3
438.6
1.9
746
6.4
913.1
9.9
935
11.3
4
378.5
1.5
438.8
2.2
512.9
3.3
615.2
5.3
5
403.7
1.6
606
4.4
738.2
6.8
866.8
9.8
6
463.2
2.1
821.4
7.6
963
10.9
963
11.9
7
516.5
2.7
958.6
10
989
11.4
989
12.5
8
415.6
1.7
742.6
6.2
827.1
8.3
1008
12.9
9
610.2
3.7
1081.2
12.7
1126
14.4
1126
15.7
10
633.2
4.0
1150
14.2
1150
15.0
1150
16.2
11
624.8
3.9
1173.5
14.8
1201
16.2
1201
17.5
12
687.5
4.7
1235
16.3
1235
17.1
1235
18.4
13
451.7
2.0
790.4
7.1
973.7
11.1
1190
17.1
14
516.5
2.7
958.6
10
1187.7
15.9
1251
18.7
15
406.9
1.7
624.2
4.6
763.7
7.2
932.2
11.2
16
382.7
1.5
500.3
3.0
592
4.6
726.9
7.2
17
417.4
1.7
678.6
5.4
827.4
8.4
999.7
12.7
18
501.0
2.5
913.7
9.2
1075.7
13.1
1202
17.2
19
378.5
1.5
465.6
2.5
567.9
4.2
697.4
6.8
20
408.9
1.7
635.1
4.8
777.6
7.5
943.6
11.5
21
414.8
1.7
667.7
5.2
819.2
8.2
1002.2
12.7
22
497.6
2.5
910
9.1
984
11.3
984
12.4
23
429.5
1.8
714.9
5.9
879.2
9.3
975
12.2
24
384.5
1.5
511.7
3.1
605.4
4.8
0.730.6
7.4
Profit($)
3,225
23,505
27,596
29,080Figures11, 12, 13, and 14 show that the MPC closed-loop solutions asymptotically approach the optimal solutions of the PBDEED problem for price-II and for different values of α. To show the IRP-MPC algorithm, we choose different ranges of disturbances that arise from the execution of the optimal controller. We have two cases:IRP-MPC-(I):ε=(8,5,4,3,4,3),IRP-MPC-(II):ε=(32,20,16,12,16,12).Figure 11
Convergence of the MPC solution of the six-unit system to those of PBDEED (α=1) and price-II.Figure 12
Convergence of the MPC solution of the six-unit system to those of PBDEED (α=0.7) and price-II.Figure 13
Convergence of the MPC solution of the six-unit system to those of PBDEED (α=0.5) and price-II.Figure 14
Convergence of the MPC solution of the six-unit system to those of PBDEED (α=0.2) and price-II.From Figures15 and 16 we can see that, although the disturbance increased, the IRP-MPC-(II) still keep the disturbed system in the neighborhood of the optimal solution of the PBDEED problem. From these cases we observe that the size of this neighborhood depends on the bound of the disturbances.Figure 15
The generation output of the six-unit system under PBDEED (α=0.5), price-II, and IRP-MPC-(I).Figure 16
The generation output of the six-unit system under PBDEED (α=0.5), price-II, and IRP-MPC-(II).Beside the convergence and robustness of the MPC, it has the following advantages [20].(
1
)
Reduced Dimensions. For the five-unit system, the DEED problem must solve an optimization problem with 5×24=120 variables. However, in each iteration step of the MPC algorithm, the algorithm solves an optimization problem with 5×23=115 number of variables which reduces 5 (6 for the six-unit system) dimensions in the optimization problem and makes the computation easier.(
2
)
Easy Implementation. Because of the MPC convergence, restarting the MPC algorithm from any time will give rise to the same convergence, which further implies that the MPC algorithm can be executed at any sampling time point. Thus the MPC algorithm is more favorable for practical applications than other open-loop algorithms [2].


## 4.1. Five-Unit Test System

This example presents an application of MPC to the DEED problem consisting of five units. The technical data of the units are taken from [16]. First, we present the optimal solutions of the DEED problems for different values of the weighting factor α and then show that the solutions of the MPC converge to the optimal solutions of the DEED problem. The optimal solutions of the DEED problem with different values of α are given in Tables 1 and 2. Table 1 shows the hourly power schedule and transmission loss obtained from the DEED problem with α=1 and α=0.7. Table 2 shows the hourly power schedule and transmission loss obtained from the DEED problem with α=0.5 and α=0. The optimal cost and emission obtained from the DEED problem with different values of α are given in Table 3. It is seen from Table 3 that the cost is $40,121 under DEED (α=1), but it increases to $40,851 under DEED (α=0) and emission obtained from DEED (α=1) is 20,363 lb but decreases to 16,546 lb under DEED (α=0). Figure 1 shows the effect of the transmission loss on the outputs of the power units. We can see that neglecting transmission loss would often introduce inaccuracy into the resulting dispatch.Table 1
Hourly generation (MW) schedule obtained from DEED (α=1,α=0.7) for the 5-unit system.
H
DEED (α=1)
DEED (α=0.7)
P
1
t
P
2
t
P
3
t
P
4
t
P
5
t
Losst
P
1
t
P
2
t
P
3
t
P
4
t
P
5
t
Losst
1
15.3041
72.5541
61.5777
118.5306
145.6491
3.6156
35.3428
63.4632
109.2221
114.8838
90.5390
3.4508
2
16.4010
75.4029
68.8115
127.0094
151.4306
4.0554
37.5960
67.1865
115.4316
122.8094
95.8659
3.8894
3
18.1616
79.9660
80.4027
140.5893
160.6954
4.8151
41.2062
73.1509
125.3825
135.5049
104.4026
4.6471
4
20.5883
86.2528
96.3755
159.2916
173.4643
5.9725
46.1827
81.3656
139.0964
152.9899
116.1669
5.8014
5
21.8270
89.4580
104.5228
168.8264
179.9781
6.6123
48.7217
85.5538
146.0921
161.9040
122.1678
6.4395
6
24.0431
95.1914
119.0980
185.8755
191.6321
7.8400
53.2657
93.0427
158.6080
177.8437
132.9037
7.6638
7
24.8524
97.2620
124.3507
192.0138
195.8299
8.3088
54.9040
95.7422
163.1213
183.5888
136.7751
8.1313
8
26.0898
100.4824
132.5307
201.5877
202.3761
9.0666
57.4540
99.9456
170.1488
192.5335
142.8049
8.8869
9
27.6901
104.6328
143.0725
213.9197
210.7768
10.0918
61.2354
106.1787
175.0000
205.7698
151.7344
9.9184
10
28.3138
106.2316
147.1587
218.7101
214.0919
10.5060
62.9302
108.9742
175.0000
211.6974
155.7389
10.3407
11
29.0370
108.0749
151.8361
224.1873
217.8548
10.9901
64.8728
112.1722
175.0000
218.4748
160.3159
10.8356
12
29.9330
110.3833
157.7003
231.0449
222.5497
11.6112
67.3023
116.1719
175.0000
226.9531
166.0457
11.4730
13
28.3139
106.2315
147.1587
218.7100
214.0919
10.5060
62.9301
108.9741
175.0000
211.6974
155.7390
10.3407
14
27.6893
104.6333
143.0726
213.9197
210.7770
10.0918
61.2356
106.1785
175.0000
205.7699
151.7344
9.9184
15
26.0894
100.4711
132.5392
201.5873
202.3796
9.0665
57.4541
99.9456
170.1488
192.5334
142.8050
8.8869
16
22.8134
91.9940
110.9206
176.3178
185.0932
7.1390
50.7195
88.8473
151.5954
168.9141
126.8884
6.9647
17
21.8267
89.4583
104.5227
168.8266
179.9780
6.6123
48.7218
85.5537
146.0920
161.9040
122.1679
6.4395
18
24.0532
95.2077
119.0874
185.8684
191.6233
7.8400
53.2655
93.0428
158.6080
177.8437
132.9038
7.6638
19
26.0883
100.4706
132.5394
201.5881
202.3802
9.0665
57.4540
99.9458
170.1488
192.5334
142.8049
8.8869
20
28.3130
106.2317
147.1590
218.7103
214.0920
10.5060
62.9305
108.9740
175.0000
211.6973
155.7388
10.3407
21
27.2534
103.4720
140.1400
210.4958
208.4400
9.8012
60.0237
104.1856
175.0000
201.5410
148.8725
9.6229
22
23.9089
94.8507
118.2197
184.8516
190.9324
7.7633
52.9926
92.5929
157.8562
176.8867
132.2589
7.5872
23
20.4569
85.9088
95.5033
158.2704
172.7666
5.9060
45.9111
80.9171
138.3475
152.0352
115.5243
5.7351
24
17.6333
78.5962
76.9229
136.5133
157.9141
4.5799
40.1227
71.3606
122.3952
131.6944
101.8397
4.4126Table 2
Hourly generation (MW) schedule obtained from DEED (α=0.5,α=0) for the 5-unit system.
H
DEED (α=0.5)
DEED (α=0)
P
1
t
P
2
t
P
3
t
P
4
t
P
5
t
Losst
P
1
t
P
2
t
P
3
t
P
4
t
P
5
t
Losst
1
43.1673
60.7831
115.0952
113.0510
81.3489
3.4456
54.6445
57.2094
119.8650
110.1053
71.6251
3.4492
2
45.8553
64.6456
121.0726
120.7786
86.5315
3.8834
57.9634
61.1854
125.5602
117.5630
76.6146
3.8865
3
50.1626
70.8326
130.6514
133.1568
94.8363
4.6397
63.2806
67.5546
134.6871
129.5096
84.6101
4.6420
4
56.0966
79.3553
143.8531
150.2058
106.2813
5.7922
70.6064
76.3274
147.2663
145.9645
95.6288
5.7935
5
59.1235
83.7007
150.5876
158.8981
112.1194
6.4293
74.3486
80.7966
153.6820
154.3540
101.2488
6.4301
6
64.5383
91.4710
162.6369
174.4415
122.5642
7.6518
75.0000
89.8771
166.7190
171.3855
112.6714
7.6531
7
66.4908
94.2716
166.9820
180.0439
126.3304
8.1187
75.0000
93.1922
171.4836
177.6064
116.8388
8.1210
8
69.5320
98.6463
173.7475
188.7748
132.1728
8.8734
75.0000
99.2991
175.0000
189.0543
124.5301
8.8835
9
74.4893
105.7396
175.0000
202.9461
141.7335
9.9084
75.0000
108.2654
175.0000
205.8482
135.8169
9.9305
10
75.0000
108.9451
175.0000
209.3405
146.0470
10.3326
75.0000
111.7567
175.0000
212.3860
140.2141
10.3568
11
75.0000
112.7466
175.0000
216.9235
151.1615
10.8316
75.0000
115.7496
175.0000
219.8622
145.2453
10.8572
12
75.0000
117.5046
175.0000
226.4089
157.5615
11.4750
75.0000
120.7459
175.0000
229.2153
151.5412
11.5024
13
75.0000
108.9451
175.0000
209.3406
146.0470
10.3326
75.0000
111.7567
175.0000
212.3860
140.2141
10.3568
14
74.4892
105.7396
175.0000
202.9462
141.7335
9.9084
75.0000
108.2654
175.0000
205.8482
135.8169
9.9305
15
69.5320
98.6463
173.7475
188.7748
132.1728
8.8734
75.0000
99.2991
175.0000
189.0543
124.5300
8.8835
16
61.5044
87.1178
155.8856
165.7339
116.7120
6.9538
75.0000
84.7250
159.3200
161.7208
106.1884
6.9543
17
59.1234
83.7006
150.5877
158.8982
112.1195
6.4293
74.3485
80.7966
153.6820
154.3540
101.2488
6.4301
18
64.5383
91.4709
162.6369
174.4415
122.5642
7.6518
75.0000
89.8771
166.7190
171.3855
112.6714
7.6531
19
69.5321
98.6462
173.7475
188.7748
132.1728
8.8734
75.0000
99.2991
175.0000
189.0543
124.5300
8.8835
20
75.0000
108.9451
175.0000
209.3406
146.0469
10.3326
75.0000
111.7567
175.0000
212.3860
140.2141
10.3568
21
73.0654
103.6961
175.0000
198.8682
138.9840
9.6138
75.0000
105.7732
175.0000
201.1807
132.6788
9.6326
22
64.2130
91.0043
161.9132
173.5081
121.9368
7.5754
75.0000
89.3248
165.9258
170.3495
111.9764
7.5765
23
55.7726
78.8901
143.1321
149.2750
105.6562
5.7260
70.2064
75.8485
146.5793
145.0661
95.0270
5.7273
24
48.8696
68.9755
127.7758
129.4415
92.3432
4.4056
61.6847
65.6428
131.9472
125.9237
82.2098
4.4082Table 3
The optimal cost and emission obtained from DEED with different value ofα for the  5-unit system.
Value  ofα
Cost  ($)
Emission  (lb)
α
=
1
40,121
20,363
α
=
0.9
40,215
18,100
α
=
0.8
40,343
17,340
α
=
0.7
40,465
16,966
α
=
0.6
40,578
16,752
α
=
0.5
40,670
16,638
α
=
0.4
40,731
16,587
α
=
0.3
40,775
16,563
α
=
0.2
40,808
16,552
α
=
0.1
40,832
16,547
α
=
0
40,851
16,546Figure 1
The generation output of unit 5 under DEED (α=0.5) from the five-unit system.Turning now to our goal of this section, we apply MPC strategy. The initialPi1 is chosen such that ∑i=1n‍Pi1=D1+∑i=1n‍∑j=1n‍Pi1BijPj1. The proposed MPC strategy is implemented over 48 hours. Figures 2, 3, 4, 5, and 6 show that the MPC closed-loop solutions asymptotically approach the optimal solutions of the DEED problem for different values of α.Figure 2
Convergence of the closed-loop MPC solutions to those of DEED (α=1) for the five-unit system.Figure 3
Convergence of the closed-loop MPC solutions to those of DEED (α=0.7) for the five-unit system.Figure 4
Convergence of the closed-loop MPC solutions to those of DEED (α=0.5) for the five-unit system.Figure 5
Convergence of the closed-loop MPC solutions to those of DEED (α=0.2) for the five-unit system.Figure 6
Convergence of the closed-loop MPC solutions to those of DEED (α=0) for the five-unit system.Now we show the IRP-MPC algorithm against two sources of disturbances.(
1
) Disturbance arises from the execution of the optimal controller. In this case, we execute (26) with disturbance wim generated by
(28)wim=-εi+2εir(m),
where the parameters r(m)’s are uniformly distributed random numbers on [0,1]. Denote ε=(ε1,ε2,…,ε5). We choose two different bounds of disturbances:IRP-MPC-(I):ε=(3,3,2,3,2),IRP-MPC-(II):ε=(6,6,4,6,4).(
2
) Disturbance arises from the forecasted demand. In this case, the actual demand D~t can be defined as
(29)D~t={Dt,ift=1,Dt+q(1-2r(t))Dtift=2,3,…,N,
where q(1-2r) is the relative change between the actual D~t demand and the forecasted demand Dt.Here, we assume the demand is disturbed with 5% and 10% of the nominal one; then we have the following cases:IRP-MPC-(III):q=0.05,IRP-MPC-(IV):q=0.1.In these cases the initialPi1,  i=1,2…,5 for the MPC are chosen as the optimal solution of the DEED problem at t=1; that is, Pi1=P-i1, and the weighting factor α is chosen α=0.5. From Figures 7, 8, 9, and 10, we can see that, although the disturbance increased, the MPC still keeps the disturbed system in the neighborhood of the optimal solution of the DEED problem. From these cases we observe that the size of this neighborhood depends on the bound of the disturbance.Figure 7
The generation output of the five-unit system under DEED (α=0.5) and IRP-MPC-(I).Figure 8
The generation output of the five-unit system under DEED (α=0.5) and IRP-MPC-(II).Figure 9
The generation output of the five-unit system under DEED (α=0.5) and IRP-MPC-(III).Figure 10
The generation output of the five-unit system under DEED (α=0.5) and IRP-MPC-(IV).


## 4.2. Six-Unit Test System

This system consists of six units for the application of MPC to the PBDEED problem. The technical data of the units are taken from [20]. The transmission loss formula coefficients are given in the appendix. The initial Pi1 is chosen such that ∑i=1n‍Pi1≤D1+∑i=1n‍∑j=1n‍Pi1BijPj1. The result of the optimization is dependent on the energy price data. Indeed, minor changes in the energy price may give a significant change in the power generation of the units as well as the profit. We consider different energy price profiles which are given in Table 4.Table 4
The results of PBDEED (α=1) for the three price profiles.
H
Price-I
P
L
t
Losst
Price-II
P
L
t
Losst
Price-III
P
L
t
Losst
($)
(MW)
(MW)
($)
(MW)
(MW)
($)
(MW)
(MW)
1
10.31
561.02
4.25
11.46
740.1
7.5
13.75
955
11.76
2
10.80
621.59
5.39
12.00
873.8
10.0
14.40
942
11.51
3
11.56
745.91
7.50
12.84
935
11.3
15.41
935
11.40
4
9.49
482.50
3.01
10.54
615.2
5.3
12.65
930
11.21
5
10.89
625.04
5.46
12.10
866.8
9.8
14.52
935
11.37
6
11.86
835.62
9.34
13.18
963
11.9
15.82
963
12.00
7
12.42
989
12.45
13.80
989
12.5
16.56
989
12.58
8
11.21
803.18
8.62
12.46
1008
12.9
14.95
1023
13.28
9
13.25
1124.56
15.45
14.72
1126
15.7
17.66
1126
15.78
10
13.45
1150
16.17
14.94
1150
16.2
17.93
1150
16.36
11
13.37
1201
17.41
14.86
1201
17.5
17.83
1201
17.60
12
13.91
1235
18.29
15.46
1235
18.4
18.55
1235
18.50
13
11.74
805.70
8.75
13.04
1190
17.1
15.65
1190
17.23
14
12.42
996.35
12.61
13.80
1251
18.7
16.56
1251
18.82
15
10.98
645.56
5.86
12.20
932.2
11.2
14.64
1263
19.04
16
10.19
545.29
3.98
11.32
726.9
7.2
13.58
1250
18.66
17
11.25
698.05
6.76
12.50
999.9
12.7
15.00
1221
17.98
18
12.26
895.99
10.14
13.62
1202
17.2
16.34
1202
17.56
19
9.79
541.62
3.94
10.88
697.4
6.8
13.06
1159
16.37
20
11.03
655.09
6.03
12.26
943.6
11.5
14.71
1092
14.87
21
11.20
686.22
6.56
12.44
1002.2
12.7
14.93
1023
13.28
22
12.22
929.72
11.23
13.58
984
12.4
16.30
984
12.47
23
11.43
732.97
7.41
12.70
975
12.1
15.24
975
12.24
24
10.28
556.53
4.17
11.42
730.59
7.4
13.70
960
11.87
Profit=831.24 ($)
Profit=29080 ($)
Profit=94922.18 ($)The effect of the energy price over the time horizon on the total power which produced by the committed units for the PBDEED problem atα=1 is shown in Table 4. Now we define PLt=∑i=1n‍Pit-Losst. We can see from Table 4 that, for price-I and price-II, PLt<Dt in some t and PLt=Dt in some other t which are shown by bold face. It means that GENCO chooses to supply power less than the demand plus power loss in some hours of the day and equal to the demand plus power loss in other hours of the day when this satisfies maximum profit. For price-III, the optimal solution of the PBDEED satisfies PLt=Dt for all t. In this case, GENCO will supply power to satisfy the demand plus power loss over the whole day since this will maximize its profit. We also note that the profit increases according to the increase of the energy prices. According to the PBDEED formulation, GENCO will sell all produced power and the consumer is the one who will bear the energy loss.The effect of the weighting factorα on the total amount of power and the total profit in case of price-II for the PBDEED problem is given in Table 5. It can be seen that, as the weighting factor α is increased (i.e., the importance of the emission is decreased), both the profit and the total power are increased.Table 5
The results of PBDEED for price-II with different values ofα.
H
α
=
0.2
α
=
0.5
α
=
0.7
α
=
1
P
L
t
Loss
t
P
L
t
Losst
P
L
t
Losst
P
L
t
Losst
(MW)
(MW)
(MW)
(MW)
(MW)
(MW)
(MW)
(MW)
1
385.3
1.5
516.3
3.2
613
4.9
740.1
7.5
2
400.5
1.6
589
4.1
717.4
6.5
873.9
10.1
3
438.6
1.9
746
6.4
913.1
9.9
935
11.3
4
378.5
1.5
438.8
2.2
512.9
3.3
615.2
5.3
5
403.7
1.6
606
4.4
738.2
6.8
866.8
9.8
6
463.2
2.1
821.4
7.6
963
10.9
963
11.9
7
516.5
2.7
958.6
10
989
11.4
989
12.5
8
415.6
1.7
742.6
6.2
827.1
8.3
1008
12.9
9
610.2
3.7
1081.2
12.7
1126
14.4
1126
15.7
10
633.2
4.0
1150
14.2
1150
15.0
1150
16.2
11
624.8
3.9
1173.5
14.8
1201
16.2
1201
17.5
12
687.5
4.7
1235
16.3
1235
17.1
1235
18.4
13
451.7
2.0
790.4
7.1
973.7
11.1
1190
17.1
14
516.5
2.7
958.6
10
1187.7
15.9
1251
18.7
15
406.9
1.7
624.2
4.6
763.7
7.2
932.2
11.2
16
382.7
1.5
500.3
3.0
592
4.6
726.9
7.2
17
417.4
1.7
678.6
5.4
827.4
8.4
999.7
12.7
18
501.0
2.5
913.7
9.2
1075.7
13.1
1202
17.2
19
378.5
1.5
465.6
2.5
567.9
4.2
697.4
6.8
20
408.9
1.7
635.1
4.8
777.6
7.5
943.6
11.5
21
414.8
1.7
667.7
5.2
819.2
8.2
1002.2
12.7
22
497.6
2.5
910
9.1
984
11.3
984
12.4
23
429.5
1.8
714.9
5.9
879.2
9.3
975
12.2
24
384.5
1.5
511.7
3.1
605.4
4.8
0.730.6
7.4
Profit($)
3,225
23,505
27,596
29,080Figures11, 12, 13, and 14 show that the MPC closed-loop solutions asymptotically approach the optimal solutions of the PBDEED problem for price-II and for different values of α. To show the IRP-MPC algorithm, we choose different ranges of disturbances that arise from the execution of the optimal controller. We have two cases:IRP-MPC-(I):ε=(8,5,4,3,4,3),IRP-MPC-(II):ε=(32,20,16,12,16,12).Figure 11
Convergence of the MPC solution of the six-unit system to those of PBDEED (α=1) and price-II.Figure 12
Convergence of the MPC solution of the six-unit system to those of PBDEED (α=0.7) and price-II.Figure 13
Convergence of the MPC solution of the six-unit system to those of PBDEED (α=0.5) and price-II.Figure 14
Convergence of the MPC solution of the six-unit system to those of PBDEED (α=0.2) and price-II.From Figures15 and 16 we can see that, although the disturbance increased, the IRP-MPC-(II) still keep the disturbed system in the neighborhood of the optimal solution of the PBDEED problem. From these cases we observe that the size of this neighborhood depends on the bound of the disturbances.Figure 15
The generation output of the six-unit system under PBDEED (α=0.5), price-II, and IRP-MPC-(I).Figure 16
The generation output of the six-unit system under PBDEED (α=0.5), price-II, and IRP-MPC-(II).Beside the convergence and robustness of the MPC, it has the following advantages [20].(
1
)
Reduced Dimensions. For the five-unit system, the DEED problem must solve an optimization problem with 5×24=120 variables. However, in each iteration step of the MPC algorithm, the algorithm solves an optimization problem with 5×23=115 number of variables which reduces 5 (6 for the six-unit system) dimensions in the optimization problem and makes the computation easier.(
2
)
Easy Implementation. Because of the MPC convergence, restarting the MPC algorithm from any time will give rise to the same convergence, which further implies that the MPC algorithm can be executed at any sampling time point. Thus the MPC algorithm is more favorable for practical applications than other open-loop algorithms [2].


## 5. Conclusions

In this paper, we have proposed a model predictive control approach for the periodic implementation of the optimal solutions of two optimal dynamic dispatch problems with emission and transmission losses. The first problem is the dynamic economic emission dispatch (DEED), and the second is the profit-based dynamic economic emission dispatch (PBDEED). Both the demand and energy price are assumed to be periodic. We have assumed that there are certain disturbances or uncertainties in the execution of the optimal controller and in the forecasted demand. The convergence and robustness of the MPC algorithm are demonstrated through the application of MPC to the DEED problem with a five-unit system and to the PBDEED problem with a six-unit system.



---

*Source: 906958-2013-04-15.xml*