# Distributed Prescribed Finite Time Consensus Scheme for Economic Dispatch of Smart Grids with the Valve Point Effect

Authors: Yingjiang Zhou; Shigao Zhu; Qian Chen
Journal: Complexity
 (2020)

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## Abstract

The distributed prescribed finite time consensus schemes for economic dispatch (ED) of smart grids with and without the valve point effect are researched in this paper. First, the optimization problem is transformed into a consensus of multiagent system problem, where both with and without the valve point effect are considered. Second, for the directed balance network, a prescribed finite time method has been arranged to solve the ED problem with and without the valve point effect. Third, with considering the constraints of generation units, the prescribed finite time result is also achieved. Finally, from the simulations, the efficiency of the proposed algorithms is validated.


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## Body

## 1. Introduction

In the recent years, the research about the ED problem has attracted tremendous attentions, which is particularly important in the smart grid. The aim of the ED problem is to find the optimization of the minimum total generation cost. When there are certain practical constraints, the aim of the ED problem is to find the optimal outputs to minimum total generation cost while meeting the power demand. The traditional ways of solving the ED problem (e.g., genetic algorithm [1, 2], particle swarm optimization [3–6], and multiobjective collective decision optimization algorithm [7]) are centralized methods.Recently, many research results focus on the distributed approach to resolve the ED problem [8–17], where the generation units only get the neighbor’s information. The low communication cost, easy implementation and maintenance, and strong robustness against communication uncertainties are the benefits of distributed ED algorithms [8]. Many pioneering work about distributed ED were pointed in [8–10], but these results are not fully distributed. The fully distributed ED algorithms were first proposed in [11], where the distributed consensus methods in multiagent systems (MASs) have been used. When there exist unknown communication uncertainties, the ED problem is developed by using the adaptive consensus-based robust strategy [12]. For sparse communication networks and time delay, a distributed scheme is provided based on consensus strategy [13]. In order to reduce the amount of communication of the smart grids, the event-triggered control solution was devised to achieve the distributed reactive power sharing control [14–16]. In the smart grids, the second-order consensus methods have been used to solve the ED problem [17]. Considering the complex networks with the reaction diffusion terms and the probabilistic Boolean networks, the synchronization and stabilization methods were investigated in [18–20], and we will focus on the ED algorithms in these kinds of networks.The rate of convergence is a key factor of solving the ED problem [21–25], and fast convergence rate and strong robustness are the advantages of the finite time method [26, 27]. Considering the ED problem of generation, the distributed finite-step iterative strategy is arranged [21]. If the topology is jointly connected, distributed finite time ED in smart grid is derived [22]. In [15, 22], the homogeneous method is applied to solve the finite time ED problem, but it is hard to estimate the convergence time. A novel fully distributed finite time algorithm is devised to address the ED problem in the smart grids [23]. The convergence time in the abovementioned algorithms is connected with the initial condition, if the initial condition is large, the convergence time will be very long. The prescribed finite time method was proposed for the first time in [24], where the convergence time can be set by the designer. Very recently, a distributed prescribed finite time method for the ED problem was proposed [25].Another important topic is about the valve point effect [28–41]. Many meaningful works are focused on the valve point effect, such as evolutionary programming [28, 29], genetic algorithm [30–33], and particle swarm optimisation [34, 35]. The complicated, nonlinear, and nonconvex ED problem with the valve point effect has been settled by using the Maclaurin series-based Lagrangian algorithm [36]. In [37], the valve point loading effect and transmission losses are considered, where the ED problem has been resolved with the gradient and Newton methods. But, most algorithms are centralized, and relatively speaking, distributed methods are more consistent with the actual requirements. In [38], the ED problem with the valve point effect is resolved with a distributed pattern search scheme.The contributions of this paper can be listed in the following three aspects. (1) For the directed balance network, a prescribed finite time approach has been pointed out to solve the ED problem. (2) The prescribed finite time method was also derived for solving the ED problem with considering the valve point effect in this paper. (3) With considering the constraints of generation units, the prescribed finite time result is still correct.The remainder of this article is organized as follows. The graph theory and problem statement are introduced in Section2. Section 3 proposes distributed prescribed finite time consensus algorithms of solving the optimization without and with the valve point effect and the generation constraints of generation units. Three simulation examples are proposed to validate our results in Section 4. In Section 5, the conclusion has been presented.


## 2. Graph Theory and Problem Statement

In this section, the graph theory and the problem statement are introduced.


### 2.1. Graph Theory

We defineG=V,E,A as a weighted directed graph, where V=v1,…,vn represents the set of nodes, E⊆V×V represents the set of edges, vi,vj∈E represents that nodes i can receive information from node j, and A=aij∈Rn×n is the weighted adjacency matrix. If and only if there is a directed edge vi,vj in G, then the weight of the edge aij>0; otherwise, aij=0, (i≠j, i,j=1,…,n). Assume that aii=0 for all i∈1,…,n. The Laplacian matrix of a directed graph G is L=lij∈Rn×n, where lii=∑j=1naij and lij=−aij, i≠j, i,j=1,…,n.The direct path from the nodei1 to node im is i1,i2,i2,i3,…,im−1,im. If one node in a directed graph at least has a directed path to another node, the graph contains a directed spanning tree. If each node has a directed path to all other nodes in a directed graph, the directed graph is said to be strongly connected. If all the nodes satisfy dini=∑j=1Naij=douti=∑j=1Naji, i,j=1,…,n and the in-degree is equal to the out-degree, then the direct graph is said to be a balance graph.Define an undirected mirror graphG^=V,E^,A^, which has the same nodes V as G. The set of edges is E^⊆V×V, and the weighted symmetric adjacency matrix is A^=a^ij, where(1)a^ij=a^ji=aij+aji2≥0.


### 2.2. Problem Statement

Suppose the MASs consist ofn agents. Also, we can describe the dynamics of the agent as follows:(2)x.i=ui,where ui is the control inputs, i=1,…,n.Definition 1.
The MASs (2) is said to reach consensus in prescribed finite time, if for any preselected time T>0 such that limt⟶Txi=xj and for all t≥T, xi=xj, i,j∈1,…,n.Lemma 1.
(see [10]). For the irreducible Laplacian matrix L, the algebraic connectivity aL>0, where aL=minxTξ=0,x≠0xTL^x/xTΞx, with L^=ΞL+LTΞ/2, ξ=ξ1,ξ2,…,ξn>0, Ξ=diagξ1,…,ξn, ∑i=1nξi=1, and ξTL=0.
In this paper, the ED problem with and without the valve point effect is considered. Suppose there aren generating units. The cost function of each generator is as follows:(3)CiPi=aiPi2+biPi+ci+disineiPimin−Pi,where CiPi is the cost of the i-th generator, Pi is the real power generation of the i-th unit, Pimin is the lower bound of the generation capacity, and ai, bi, ci, di, and ei are the positive cost function coefficients. disineiPimin−Pi is the valve point effect. If di=0, it means that the valve point effect is not exist.
Our research objective is to minimize the total cost ofn power generation systems in the case of power demand and supply balance.
The optimization problem can be summarized as follows:(4)min∑i=1nCiPi.
Subjecting to the power balance constraint,(5)∑i=1nPi=PD,where PD is the total load of the power system.


## 2.1. Graph Theory

We defineG=V,E,A as a weighted directed graph, where V=v1,…,vn represents the set of nodes, E⊆V×V represents the set of edges, vi,vj∈E represents that nodes i can receive information from node j, and A=aij∈Rn×n is the weighted adjacency matrix. If and only if there is a directed edge vi,vj in G, then the weight of the edge aij>0; otherwise, aij=0, (i≠j, i,j=1,…,n). Assume that aii=0 for all i∈1,…,n. The Laplacian matrix of a directed graph G is L=lij∈Rn×n, where lii=∑j=1naij and lij=−aij, i≠j, i,j=1,…,n.The direct path from the nodei1 to node im is i1,i2,i2,i3,…,im−1,im. If one node in a directed graph at least has a directed path to another node, the graph contains a directed spanning tree. If each node has a directed path to all other nodes in a directed graph, the directed graph is said to be strongly connected. If all the nodes satisfy dini=∑j=1Naij=douti=∑j=1Naji, i,j=1,…,n and the in-degree is equal to the out-degree, then the direct graph is said to be a balance graph.Define an undirected mirror graphG^=V,E^,A^, which has the same nodes V as G. The set of edges is E^⊆V×V, and the weighted symmetric adjacency matrix is A^=a^ij, where(1)a^ij=a^ji=aij+aji2≥0.


## 2.2. Problem Statement

Suppose the MASs consist ofn agents. Also, we can describe the dynamics of the agent as follows:(2)x.i=ui,where ui is the control inputs, i=1,…,n.Definition 1.
The MASs (2) is said to reach consensus in prescribed finite time, if for any preselected time T>0 such that limt⟶Txi=xj and for all t≥T, xi=xj, i,j∈1,…,n.Lemma 1.
(see [10]). For the irreducible Laplacian matrix L, the algebraic connectivity aL>0, where aL=minxTξ=0,x≠0xTL^x/xTΞx, with L^=ΞL+LTΞ/2, ξ=ξ1,ξ2,…,ξn>0, Ξ=diagξ1,…,ξn, ∑i=1nξi=1, and ξTL=0.
In this paper, the ED problem with and without the valve point effect is considered. Suppose there aren generating units. The cost function of each generator is as follows:(3)CiPi=aiPi2+biPi+ci+disineiPimin−Pi,where CiPi is the cost of the i-th generator, Pi is the real power generation of the i-th unit, Pimin is the lower bound of the generation capacity, and ai, bi, ci, di, and ei are the positive cost function coefficients. disineiPimin−Pi is the valve point effect. If di=0, it means that the valve point effect is not exist.
Our research objective is to minimize the total cost ofn power generation systems in the case of power demand and supply balance.
The optimization problem can be summarized as follows:(4)min∑i=1nCiPi.
Subjecting to the power balance constraint,(5)∑i=1nPi=PD,where PD is the total load of the power system.


## 3. Main Theoretical Results

### 3.1. Optimization by Consensus without the Valve Point Effect

In this section, the distributed control method is devised to solve the optimization problem without the valve point effect.Definition 2.
The incremental cost of each generatori without valve point effect is defined as ICi=∂CiPi/∂Pi=riPi+bi, i=1,2,…,n, where ri=2ai.
The Lagrange multiplier algorithm is applied to solve the optimization problem.(6)LPi,λ=∑i=1nCiPi+λPD−∑i=1nPi=∑i=1naiPi2+biPi+ci+λPD−∑i=1nPi,where λ is the Lagrange multiplier.
The minimum value of (6) can be obtained by differentiating the abovementioned equation.(7)∂L∂Pi=∂CiPi∂Pi−λ=riPi+bi−λ=0.
So,∑i=1nPi=PD. We have(8)λ=riPi+bi.
We defineyit=λi=riPi+bi. Then, a distributed prescribed finite time protocol is designed for the optimization problem.(9)y.it=crikt∑j=1na^ijyjt−yit=−crikt∑j=1nL^ijyjt.
With initial conditionsyi0=riPi0+bi, the following equation is satisfied:(10)∑i=1nPi0=PD.
kt=k1+k2/T−tsign1+signT−t, k1>0, k2>0, and c≥1/aΞ−1L^>0 are constants. T>0 is a prespecifiable convergence time.
From [10], condition (10) is satisfied.Theorem 1.
If the topological graph is a strongly connected direct balance graph, the distributed prescribed finite time protocol (9) is designed to solve the prescribed finite time optimization problem without the valve point effect (4) via the initial conditions in (10).Proof.
First, we will prove the balance between demand and supply of powers.
Letξm=1/rrm, and y∗t=∑m=1nξmymt=1/r∑m=1nbm/rm+Pmt be the weighted average value of all the incremental costs, where r=∑m=1n1/rm, m=1,2,…,n. We know L^ is zero row-sum and symmetric, so(11)y.∗t=−cktr∑m=1n∑j=1nL^mjyjt=0=1r∑m=1nP.mt.
This means(12)∑m=1nPmt=∑m=1nPm0=PD.
From (12), we know the total output power of all the generators is a constant value, which means the balance between the demand and supply of powers is always true.
Then, we will prove the prescribed finite time consensus for the incremental cost of each generator.
We defineδi=yi−y∗ as the error states between the average value and the i-th generator, δ=δ1,…,δnT. It is easy to see that ∑k=1nξk=1 and ∑k=1nξkδk=0. So, we can get the error system:(13)δ.it=crikt∑j=1na^ijδjt−δit=−crikt∑j=1nL^ijδjt.
We choose the following Lyapunov function:(14)Vt=∑i=1nδiTδiri.
By using Lemma1, the derivative of Vt can be described as(15)V.t=2∑i=1nδiTδ.iri=−2ckt∑i=1n∑j=1nδiTL^ijδj=−2cktδTL^δ≤−cktaΞ−1L^δTΞδ≤−cktaΞ−1L^Vt.
Askt=k1+k2/T−tsign1+signT−t, we will prove the result in two intervals 0,T and T,∞.
Fort∈0,T, kt=k1+k2/T−t. We have(16)V.t≤−ck1+k2T−taΞ−1L^V=−ck1aΞ−1L^V−ck2T−taΞ−1L^V.
We definez=T−t−k2/2, t∈0,T. Then, z.=k2/2T−t−k2/2−1, z./z=k2/2T−t. As c≥1/aΞ−1L^>0, we have(17)V.t≤−k1Vt−k2T−tVt=−k1Vt−2z.zVt.
Then,(18)z2V.t≤−k1z2Vt−2zz.Vt.
So,(19)dz2Vtdt=z2V.t+2zz.Vt≤−k1z2Vt.
Then,(20)z2Vt≤exp−k1tz02V0,(21)Vt≤z−2exp−k1tz02V0=T−tT2exp−k1tV0.
From (21), we know limt⟶T−Vt=0.
Fort∈T,∞, kt=k1. We have(22)V.t≤−ck1aΞ−1L^V<0.
It meansV will not rise anymore. So, ∀t≥T, Vt=VT≡0.
The prescribed finite time optimization problem without the valve point effect has been proved. The proof is completed.Remark 1: the prescribed finite time fully distributed method is designed for solving the ED problem in Theorem1. The result here is a fully distributed result, and it is different from the centralized algorithms.Remark 2: note that the graph in Theorem1 is a direct strongly connected balance graph, and it is an improvement over an undirected graph. The result here can also be used for any undirected connected graph. In the future work, we will focus on other direct graphs.Remark 3: the prescribed finite time result has two advantages compared with the finite time result. Firstly, the convergence time can be preassigned as needed by the designer, as we know that the convergence time of the finite time results is connected with the initial condition. Secondly, the controller in the prescribed finite time result is continuous, while the controller in the finite time result is discontinuous. The continuous controller can make the system state change smoothly.


### 3.2. Valve Point Effect

In this section, the valve point effect is introduced. In the interval0,π, the piecewise linearization is employed. The effect of the valve point effect on the cost function is the type of the sine-wave function, as in Figure 1(a). The active real power generated can obtained from the derivative of the sine-wave function, as in Figure 1(b). As the function in Figure 1(b) is periodic, the piecewise linearization is introduced to approximate the sine-wave function, as in Figure 1(c) [42].Figure 1
Piecewise linearization of the sine-wave part.
(a)(b)(c)Taking the derivative of formula (3),(23)dCiPidPi=2aiPi+bi+dieicosmodeiPimin−Pi,π,where mod means the MOD function.The piecewise linearization with different slopes is introduced:(24)dieicosmodeiPimin−Pi,π=giPi−qiπei+ki,where gi and ki are constants and qi=Pi−Pimin/π/ei is an integer number of intervals. From (23) and (24),(25)dCiPidPi=2aiPi+bi+giPi−qiπei+ki=2ai+giPi−giqiπei+bi+ki.


### 3.3. Optimization by Consensus with the Valve Point Effect

Next, the distributed control approach is devised of solving the optimization problem with the valve point effect.Definition 3.
We define the incremental cost of each generatori as ICi=∂CiPi/∂Pi=r¯iPi+ηi, where r¯i=2ai+gi and ηi=−giqiπ/ei+bi+ki, i=1,2,…,n.
The Lagrange multiplier algorithm is used to solve the optimization problem with the valve point effect.(26)LPi,λ=∑i=1nCiPi+λPD−∑i=1nPi=∑i=1naiPi2+biPi+ci+disineiPimin−Pi+λPD−∑i=1nPi.
The minimum value of (26) can be obtained by differentiating equation (26).(27)∂L∂Pi=∂CiPi∂Pi−λ=r¯iPi+ηi−λ=0.
So,∑i=1nPi=PD. We have(28)λ=r¯iPi+ηi.
We definexit=λi=r¯iPi+ηi, i=1,2,…,n. Then, a distributed prescribed finite time protocol is designed for the optimization problem.(29)x.it=crikt∑j=1na^ijxjt−xit=−crikt∑j=1nL^ijxjt.
With initial conditionsxi0=r¯iPi0+ηi, the following equation is satisfied:(30)∑i=1nPi0=PD.
kt=k1+k2/T−tsign1+signT−t, c≥1/aΞ−1L^>0, k1>0, k2>0 are constants. T>0 is a prespecifiable convergence time.Theorem 2.
If the topological graph is a strongly connected direct balance graph, the distributed prescribed finite time protocol (29) is designed to solve the prescribed finite time optimization problem with the valve point effect (4) via the initial conditions in (30).
The proof here is similar to the proof in Theorem1. We omitted the proof here.


### 3.4. Optimization by Consensus with the Power Generation Constraints

Next, considering the power generation constraints of the generation-demand constraint, we need to further revise the distributed algorithms (9)-(10) to solve the ED problem.The following three steps are derived to solve the constraints problem:Step 1: by using the algorithm (9)-(10) in Theorem 1, we can get the optimal incremental cost λ∗ and the optimal power generation value Pi∗.Step 2: from Step 1, we getPi∗. We check to see whether P∗ is in the interval Pi,min,Pi,max. If Pi∗>Pi,max, let Pi∗=Pi,max. If Pi∗<Pi,min, let Pi∗=Pi,min.We defineΩp as the generation units whose optimal values of power generation are Pi∗=Pi,max or Pi∗=Pi,min. Two auxiliary variables x^i, y^i are introduced, and the initialize condition are(31)x^i=λ∗−bi2ai−Pi∗,i∈Ωp,0,i∉Ωp,y^i=12ai,i∉Ωp,0,i∈Ωp.The distributed average algorithms are introduced:(32)x^.it=crikt∑j=1na^ijx^jt−x^it,(33)y^.it=crikt∑i=1na^ijy^jt−y^it.Corollary 1.
If the topological graph is a strongly connected direct balance graph, the distributed prescribed finite time algorithms (32) and (33) can solve the optimization problem with the power generation constraints in the preselected finite time T1, i.e.,(34)limt⟶T1x^∗t=∑i=1nξix^i0=∑i∈Ωpξiλ∗−bi2ai−Pi∗,(35)limt⟶T1y^∗t=∑i=1nξiy^i0=∑i∈Ωpξi2ai.Proof.
When we setai=0.5, it is easy to get the results (34) and (35), and the proof is similar to the proof in Theorem 1.
From (34) and (35), each generation unit obtains the average values of x^i∗ and y^i∗. We can get the new incremental cost λ∗∗ as follows:(36)λ∗∗=λ∗+x^i∗y^i∗.
The new optimal valuePi∗∗ can be obtained by(37)P∗∗=λ∗∗−bi2aii∉ΩpPi,minorPi,maxi∈Ωp.Step 3: check to see weatherP∗∗ is in the interval Pi,min,Pi,max. If P∗∗ is not in the interval, set λ∗=λ∗∗ and repeat Step 2. Otherwise, P∗∗ is the final value.


## 3.1. Optimization by Consensus without the Valve Point Effect

In this section, the distributed control method is devised to solve the optimization problem without the valve point effect.Definition 2.
The incremental cost of each generatori without valve point effect is defined as ICi=∂CiPi/∂Pi=riPi+bi, i=1,2,…,n, where ri=2ai.
The Lagrange multiplier algorithm is applied to solve the optimization problem.(6)LPi,λ=∑i=1nCiPi+λPD−∑i=1nPi=∑i=1naiPi2+biPi+ci+λPD−∑i=1nPi,where λ is the Lagrange multiplier.
The minimum value of (6) can be obtained by differentiating the abovementioned equation.(7)∂L∂Pi=∂CiPi∂Pi−λ=riPi+bi−λ=0.
So,∑i=1nPi=PD. We have(8)λ=riPi+bi.
We defineyit=λi=riPi+bi. Then, a distributed prescribed finite time protocol is designed for the optimization problem.(9)y.it=crikt∑j=1na^ijyjt−yit=−crikt∑j=1nL^ijyjt.
With initial conditionsyi0=riPi0+bi, the following equation is satisfied:(10)∑i=1nPi0=PD.
kt=k1+k2/T−tsign1+signT−t, k1>0, k2>0, and c≥1/aΞ−1L^>0 are constants. T>0 is a prespecifiable convergence time.
From [10], condition (10) is satisfied.Theorem 1.
If the topological graph is a strongly connected direct balance graph, the distributed prescribed finite time protocol (9) is designed to solve the prescribed finite time optimization problem without the valve point effect (4) via the initial conditions in (10).Proof.
First, we will prove the balance between demand and supply of powers.
Letξm=1/rrm, and y∗t=∑m=1nξmymt=1/r∑m=1nbm/rm+Pmt be the weighted average value of all the incremental costs, where r=∑m=1n1/rm, m=1,2,…,n. We know L^ is zero row-sum and symmetric, so(11)y.∗t=−cktr∑m=1n∑j=1nL^mjyjt=0=1r∑m=1nP.mt.
This means(12)∑m=1nPmt=∑m=1nPm0=PD.
From (12), we know the total output power of all the generators is a constant value, which means the balance between the demand and supply of powers is always true.
Then, we will prove the prescribed finite time consensus for the incremental cost of each generator.
We defineδi=yi−y∗ as the error states between the average value and the i-th generator, δ=δ1,…,δnT. It is easy to see that ∑k=1nξk=1 and ∑k=1nξkδk=0. So, we can get the error system:(13)δ.it=crikt∑j=1na^ijδjt−δit=−crikt∑j=1nL^ijδjt.
We choose the following Lyapunov function:(14)Vt=∑i=1nδiTδiri.
By using Lemma1, the derivative of Vt can be described as(15)V.t=2∑i=1nδiTδ.iri=−2ckt∑i=1n∑j=1nδiTL^ijδj=−2cktδTL^δ≤−cktaΞ−1L^δTΞδ≤−cktaΞ−1L^Vt.
Askt=k1+k2/T−tsign1+signT−t, we will prove the result in two intervals 0,T and T,∞.
Fort∈0,T, kt=k1+k2/T−t. We have(16)V.t≤−ck1+k2T−taΞ−1L^V=−ck1aΞ−1L^V−ck2T−taΞ−1L^V.
We definez=T−t−k2/2, t∈0,T. Then, z.=k2/2T−t−k2/2−1, z./z=k2/2T−t. As c≥1/aΞ−1L^>0, we have(17)V.t≤−k1Vt−k2T−tVt=−k1Vt−2z.zVt.
Then,(18)z2V.t≤−k1z2Vt−2zz.Vt.
So,(19)dz2Vtdt=z2V.t+2zz.Vt≤−k1z2Vt.
Then,(20)z2Vt≤exp−k1tz02V0,(21)Vt≤z−2exp−k1tz02V0=T−tT2exp−k1tV0.
From (21), we know limt⟶T−Vt=0.
Fort∈T,∞, kt=k1. We have(22)V.t≤−ck1aΞ−1L^V<0.
It meansV will not rise anymore. So, ∀t≥T, Vt=VT≡0.
The prescribed finite time optimization problem without the valve point effect has been proved. The proof is completed.Remark 1: the prescribed finite time fully distributed method is designed for solving the ED problem in Theorem1. The result here is a fully distributed result, and it is different from the centralized algorithms.Remark 2: note that the graph in Theorem1 is a direct strongly connected balance graph, and it is an improvement over an undirected graph. The result here can also be used for any undirected connected graph. In the future work, we will focus on other direct graphs.Remark 3: the prescribed finite time result has two advantages compared with the finite time result. Firstly, the convergence time can be preassigned as needed by the designer, as we know that the convergence time of the finite time results is connected with the initial condition. Secondly, the controller in the prescribed finite time result is continuous, while the controller in the finite time result is discontinuous. The continuous controller can make the system state change smoothly.


## 3.2. Valve Point Effect

In this section, the valve point effect is introduced. In the interval0,π, the piecewise linearization is employed. The effect of the valve point effect on the cost function is the type of the sine-wave function, as in Figure 1(a). The active real power generated can obtained from the derivative of the sine-wave function, as in Figure 1(b). As the function in Figure 1(b) is periodic, the piecewise linearization is introduced to approximate the sine-wave function, as in Figure 1(c) [42].Figure 1
Piecewise linearization of the sine-wave part.
(a)(b)(c)Taking the derivative of formula (3),(23)dCiPidPi=2aiPi+bi+dieicosmodeiPimin−Pi,π,where mod means the MOD function.The piecewise linearization with different slopes is introduced:(24)dieicosmodeiPimin−Pi,π=giPi−qiπei+ki,where gi and ki are constants and qi=Pi−Pimin/π/ei is an integer number of intervals. From (23) and (24),(25)dCiPidPi=2aiPi+bi+giPi−qiπei+ki=2ai+giPi−giqiπei+bi+ki.


## 3.3. Optimization by Consensus with the Valve Point Effect

Next, the distributed control approach is devised of solving the optimization problem with the valve point effect.Definition 3.
We define the incremental cost of each generatori as ICi=∂CiPi/∂Pi=r¯iPi+ηi, where r¯i=2ai+gi and ηi=−giqiπ/ei+bi+ki, i=1,2,…,n.
The Lagrange multiplier algorithm is used to solve the optimization problem with the valve point effect.(26)LPi,λ=∑i=1nCiPi+λPD−∑i=1nPi=∑i=1naiPi2+biPi+ci+disineiPimin−Pi+λPD−∑i=1nPi.
The minimum value of (26) can be obtained by differentiating equation (26).(27)∂L∂Pi=∂CiPi∂Pi−λ=r¯iPi+ηi−λ=0.
So,∑i=1nPi=PD. We have(28)λ=r¯iPi+ηi.
We definexit=λi=r¯iPi+ηi, i=1,2,…,n. Then, a distributed prescribed finite time protocol is designed for the optimization problem.(29)x.it=crikt∑j=1na^ijxjt−xit=−crikt∑j=1nL^ijxjt.
With initial conditionsxi0=r¯iPi0+ηi, the following equation is satisfied:(30)∑i=1nPi0=PD.
kt=k1+k2/T−tsign1+signT−t, c≥1/aΞ−1L^>0, k1>0, k2>0 are constants. T>0 is a prespecifiable convergence time.Theorem 2.
If the topological graph is a strongly connected direct balance graph, the distributed prescribed finite time protocol (29) is designed to solve the prescribed finite time optimization problem with the valve point effect (4) via the initial conditions in (30).
The proof here is similar to the proof in Theorem1. We omitted the proof here.


## 3.4. Optimization by Consensus with the Power Generation Constraints

Next, considering the power generation constraints of the generation-demand constraint, we need to further revise the distributed algorithms (9)-(10) to solve the ED problem.The following three steps are derived to solve the constraints problem:Step 1: by using the algorithm (9)-(10) in Theorem 1, we can get the optimal incremental cost λ∗ and the optimal power generation value Pi∗.Step 2: from Step 1, we getPi∗. We check to see whether P∗ is in the interval Pi,min,Pi,max. If Pi∗>Pi,max, let Pi∗=Pi,max. If Pi∗<Pi,min, let Pi∗=Pi,min.We defineΩp as the generation units whose optimal values of power generation are Pi∗=Pi,max or Pi∗=Pi,min. Two auxiliary variables x^i, y^i are introduced, and the initialize condition are(31)x^i=λ∗−bi2ai−Pi∗,i∈Ωp,0,i∉Ωp,y^i=12ai,i∉Ωp,0,i∈Ωp.The distributed average algorithms are introduced:(32)x^.it=crikt∑j=1na^ijx^jt−x^it,(33)y^.it=crikt∑i=1na^ijy^jt−y^it.Corollary 1.
If the topological graph is a strongly connected direct balance graph, the distributed prescribed finite time algorithms (32) and (33) can solve the optimization problem with the power generation constraints in the preselected finite time T1, i.e.,(34)limt⟶T1x^∗t=∑i=1nξix^i0=∑i∈Ωpξiλ∗−bi2ai−Pi∗,(35)limt⟶T1y^∗t=∑i=1nξiy^i0=∑i∈Ωpξi2ai.Proof.
When we setai=0.5, it is easy to get the results (34) and (35), and the proof is similar to the proof in Theorem 1.
From (34) and (35), each generation unit obtains the average values of x^i∗ and y^i∗. We can get the new incremental cost λ∗∗ as follows:(36)λ∗∗=λ∗+x^i∗y^i∗.
The new optimal valuePi∗∗ can be obtained by(37)P∗∗=λ∗∗−bi2aii∉ΩpPi,minorPi,maxi∈Ωp.Step 3: check to see weatherP∗∗ is in the interval Pi,min,Pi,max. If P∗∗ is not in the interval, set λ∗=λ∗∗ and repeat Step 2. Otherwise, P∗∗ is the final value.


## 4. Numerical Examples

In this section, in order to verify the effectiveness of prescribed finite time algorithm for the ED problem of smart grids with the valve point effect, three numerical examples are listed, prescribed finite time optimization by consensus, prescribed finite time optimization by consensus with power generation constraints of generation units, and prescribed finite time optimization by consensus with the valve point effect. In this paper, a simulation model with 5 generators is selected, and the communication topology of the generator model can be seen in Figure2.Figure 2
The topology graph of the generator model.The topology is balance directed, the adjacent matrix of the graph can be written asA=0200640303030000000245020, and the Laplacian matrix of the MASs is L=8−200−6−410−30−30−33000002−2−4−50−211. The mirror diagram can be further obtained as A^=0300530304030000000254020, and the Laplacian matrix of mirror diagram is L^=8−300−5−310−30−40−33000002−2−5−40−211.Forkt=k1+k2/T−tsign1+signT−t, select k1=1, k2=2, c=2 in equation (9). Set a1=0.096, a2=0.072, a3=0.105, a4=0.082, a5=0.103, b1=1.22, b2=1.41, b3=1.53, b4=1.02, b5=1.50, c1=51, c2=31, c3=78, c4=42, and c5=81. The original values of λi0 are λ10=12.74, λ20=13.61, λ30=23.53, λ40=17.14, and λ50=22.01.


### 4.1. Prescribed Finite Time Optimization by Consensus

In this section, distributed ED algorithm (9) in a directed topology is used to solve the optimization problem without considering power generation constraints.The convergence time can be set as 1 s. Figure3 shows the incremental cost of the generator λi, and the stable consensus λ∗=17.38 after 1s. We can see that the generator power demand is about 447.34MW from Figure 4, which satisfies the balance condition of power demand. It can be seen from Figure 5 that the optimal values of the generator power output are P1=84.16MW, P2=110.89MW, P3=75.47MW, P4=99.74MW, and P5=77.08MW.Figure 3
The incremental cost of generatorλi.Figure 4
Total power demand of the generator.Figure 5
The output power of generatorPi.


### 4.2. Prescribed Finite Time Optimization by Consensus with Power Generation Constraints

Power generation constraints of generator units are taken into account in this simulation. The maximum values of each generator constraints areP1H=200, P2H=190, P3H=180, P4H=120, and P5H=180. The minimum power of each generators are P1L=60, P2L=40, P3L=50, P4L=30, and P5l=20. The initial condition of the two auxiliary variables are x^10=0, x^20=0, x^30=0, x^40=0, x^50=10, y^10=5.21, y^20=6.94, y^30=4.76, y^40=0.10, and y^50=4.65. The convergence time is selected as 5 s. The incremental cost of the generator is λ∗=17.03, as in Figure 6. According to (33) and (34), x^i∗=1.74 in Figure 7 and y^i∗=4.35 in Figure 8 reach consensus in 5 s. From (37), we can get the final incremental cost λ∗=17.43. It can be seen from Figure 9 that the total generator demand is 437.67MW, which satisfies the balance condition of power demand. From Figure 10, we found that each Pi will reach a value in prescribed finite time, that is, P1∗∗=82.35MW,P2∗∗=108.48MW,P3∗∗=73.81MW,P4∗∗=97.63MW,andP5∗∗=75.40MW.Figure 6
The incremental cost of generatorλi with power generation constraints.Figure 7
The incremental cost of generatorxi with power generation constraints.Figure 8
The incremental cost of generatoryi with power generation constraints.Figure 9
Total power demand of generator with power generation constraints.Figure 10
The output power of generatorPi with power generation constraints.


### 4.3. Prescribed Finite Time Consensus with the Valve Point Effect

According to (23), d1=0.45, d2=0.6, d3=0.32, d4=0.26, d5=0.33, e1=0.041, e2=0.036, e3=0.028, e4=0.052, and e5=0.031 are selected in this paper. In line with formula (28), we can set the convergence time as 1s. The incremental cost of the generator with the valve point effect is λ∗=17.24, as shown in Figure 11. The total power requirement is 474.79MW, as in Figure 12, which meets the balance of power demand and supply. From Figure 13, it is found that all Pi can reach consensus in prescribed finite time, where the steady states are P1=91.22MW,P2=124.10MW,P3=77.28MW,P4=103.14MW,andP5=79.05MW.Figure 11
The incremental cost of generatorλi with the valve point effect.Figure 12
Total power demand of generator with the valve point effect.Figure 13
The output power of generatorPi with the valve point effect.


## 4.1. Prescribed Finite Time Optimization by Consensus

In this section, distributed ED algorithm (9) in a directed topology is used to solve the optimization problem without considering power generation constraints.The convergence time can be set as 1 s. Figure3 shows the incremental cost of the generator λi, and the stable consensus λ∗=17.38 after 1s. We can see that the generator power demand is about 447.34MW from Figure 4, which satisfies the balance condition of power demand. It can be seen from Figure 5 that the optimal values of the generator power output are P1=84.16MW, P2=110.89MW, P3=75.47MW, P4=99.74MW, and P5=77.08MW.Figure 3
The incremental cost of generatorλi.Figure 4
Total power demand of the generator.Figure 5
The output power of generatorPi.


## 4.2. Prescribed Finite Time Optimization by Consensus with Power Generation Constraints

Power generation constraints of generator units are taken into account in this simulation. The maximum values of each generator constraints areP1H=200, P2H=190, P3H=180, P4H=120, and P5H=180. The minimum power of each generators are P1L=60, P2L=40, P3L=50, P4L=30, and P5l=20. The initial condition of the two auxiliary variables are x^10=0, x^20=0, x^30=0, x^40=0, x^50=10, y^10=5.21, y^20=6.94, y^30=4.76, y^40=0.10, and y^50=4.65. The convergence time is selected as 5 s. The incremental cost of the generator is λ∗=17.03, as in Figure 6. According to (33) and (34), x^i∗=1.74 in Figure 7 and y^i∗=4.35 in Figure 8 reach consensus in 5 s. From (37), we can get the final incremental cost λ∗=17.43. It can be seen from Figure 9 that the total generator demand is 437.67MW, which satisfies the balance condition of power demand. From Figure 10, we found that each Pi will reach a value in prescribed finite time, that is, P1∗∗=82.35MW,P2∗∗=108.48MW,P3∗∗=73.81MW,P4∗∗=97.63MW,andP5∗∗=75.40MW.Figure 6
The incremental cost of generatorλi with power generation constraints.Figure 7
The incremental cost of generatorxi with power generation constraints.Figure 8
The incremental cost of generatoryi with power generation constraints.Figure 9
Total power demand of generator with power generation constraints.Figure 10
The output power of generatorPi with power generation constraints.


## 4.3. Prescribed Finite Time Consensus with the Valve Point Effect

According to (23), d1=0.45, d2=0.6, d3=0.32, d4=0.26, d5=0.33, e1=0.041, e2=0.036, e3=0.028, e4=0.052, and e5=0.031 are selected in this paper. In line with formula (28), we can set the convergence time as 1s. The incremental cost of the generator with the valve point effect is λ∗=17.24, as shown in Figure 11. The total power requirement is 474.79MW, as in Figure 12, which meets the balance of power demand and supply. From Figure 13, it is found that all Pi can reach consensus in prescribed finite time, where the steady states are P1=91.22MW,P2=124.10MW,P3=77.28MW,P4=103.14MW,andP5=79.05MW.Figure 11
The incremental cost of generatorλi with the valve point effect.Figure 12
Total power demand of generator with the valve point effect.Figure 13
The output power of generatorPi with the valve point effect.


## 5. Conclusions

This paper has solved the distributed prescribed finite time optimization ED problem with and without the valve point effect. The relationship between the consensus and optimization problem of ED has been derived, and we can easily transform the optimization problem into a consensus problem. For both an undirected network and balance directed network, the prescribed finite time consensus schemes for ED of smart grids have been investigated. We found that if there exist constraints of generation units, the prescribed finite time optimization ED problem can also be solved.



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