dynamic economic dispatch using model predictive control algorithm

Dynamic Economic Dispatch Using Model Predictive Control Algorithm

Thy Selaroth Department Of Electrical and Energy Engineering

Institute of Technology of Cambodia [email protected]

Abstract— This paper presents potential benefits of applying model predictive control (MPC) Algorithm and Quadratic program to solving the dynamic economic dispatch problem in electric power systems. The main objective of the economic load dispatch problem is to determine the optimal schedule of output powers of all generating units so as to meet the required load demand at minimum operating cost while satisfying system equality and inequality constraints. We take Dynamic Economic Dispatch in 24 hour to implement. Simulation is implemented in a power system comprising six generators to illustrate potential benefits from this look-ahead dispatch of both intermittent and more conventional power plants. For the result of this method, we compare with the Dynamic Economic Load Dispatch of Thermal Power System Using Genetic Algorithm by W. M. Mansour , M. M. Salama, S. M. Abdelmaksoud, H. A. Henry. The Computational results manifest that the method has a lot of excellent performances, and it is superior to other methods in many respects.

I. INTRODUCTION In power markets there is an increasing need for improving the representation of high-voltage transmission networks in order to better support market design alternatives, price formation mechanisms, and for general operation and planning decisions. In most cases, this process involves the definition of more complex mathematical models [1]. With the development of modern power systems, economic load dispatch (ELD) problem has received an increasing attention. The primary objective of ELD problem is to minimize the total generation cost of units while satisfying all units and system equality and inequality constraints [2]. It involves meeting the load demand at minimum total fuel cost while satisfying various unit and system constraints. The economic dispatch (ED) model is a optimization power demand , generation limit constraints, ramp rate limits ,and system loss. The ED for power systems can be divided into traditional static ED and DED. The static ED seeks to achieve an optimal objective for the power system at a specific time, but will not take into account the intrinsic link between the systems at different time moments. The DED takes into account of the coupling effect of system at different time

moments, such as the limit on the generator ramping rate. As a result, its computation process is more complex than that of a static optimal dispatch [3]. For this paper we use the model predictive control algorithm to do the step of Dynamic Economic Dispatch. Each step, we use the quadratic program to the optimization problem. MPC has been proposed for the periodic implementations of the optimal solutions of the DED problem in .The system loss is a very essential factor to be considered in the power system analysis. In the present paper we present the MPC method for the periodic implementations of the optimal solutions of the DED problem taking into account the transmission losses. Model predictive control is a receding horizon optimization based control technique. The basic concept of MPC is that at each control step, a finite-horizon optimal control problem is solved but only the first step of control sequences gets implemented. The state space trajectory over the prediction horizon is described by a predictive model, with the initial state being the measured state of the actual system. After the implementation of the first step, the system control waits until the next step. With the new measurement, the optimal control routine is re-run. As a result of this online optimization, the MPC approach has been successfully applied to many real-world process control problems [4].

II. ECONOMIC DISPATCH PROBLEM In the practical case the fuel cost of generator can be represented as the quadratic form. We will be considering the operation of generating i units [5].

2i i i i i iC a P b P c (1)

The general formula of total power loss in transmission line is expressed by (from kron’s loss formula)

0 00

1 1 1

g g gn n n

loss i ij j i ii j i

P PB P B P B

(2)

The total demand power Pd:

1

gn

i lossi

P Pd P

(3)

The power output of generator should not exceed its rating

nor should be below its rating.

(min) (max)i i iP P P (4)

where Pd is the real power load; Pi is the real power output at generator bus i; Bij, B0j, B00 are the B-coefficients of the transmission loss formula;

(min)iP is the minimal real power

output at generator i; (max)iP is the maximal real power output

at generator i; For this optimization we also us the Lagrange multiplier

1 1

( , , ) ( ) ( ) ( )gh

mm

j j jj j

L x f x jh x g x

(5)

According to the optimization theory, the Kuhn-tucker (KT)

for the optimum point * * *( , , )x are :

* * *( , , ) 0i

Lx

x

i=1,2……N (6)

*( ) 0jh x j=1,2….. hm (7)

*( ) 0jg x j=1,2….. gm (8)

* * *( ) 0, 0j j jg x j=1,2….. gm (9)

By using the Lagrange multiplier ,the Kuhn-tucker (KT) , and adding additional terms to include the equality constraints (1) to (9) we obtain :

( ) ( )0

( )( )

(1 ) 2

2( )

gnk k

i i ij ii jk

i ki ii

B b B P

Pa B

(10)

We sum all of power and using Taylor’s series we obtain

( )0

( )( ) 2

1 1

(1 ) 2

( )2( )

g

g g

nk

i i ii i i ij in ni jki

ki i i ii

a B B b a B PP

a B

(11)

( )

1

gnk

loss ii

P Pd P P

(11)

( )( )

( )

1

( )g

kk

nki

i

P

dP

d

(12)

( 1) ( ) ( )k k k (13)

By using this one and consider the

(min) (max)i i iP P P ,

By (10) to (13) we can find the power of all generator.

III. MODEL PREDICTIVE CONTROL Model predictive control is a receding horizon optimization based control technique. The basic concept of MPC is that at each control step, a finite-horizon optimal control problem is solved but only the first step of control sequences gets implemented. The state space trajectory over the prediction horizon is described by a predictive model, with the initial state being the measured state of the actual system. After the implementation of the first step, the system control waits until the next step. With the new measurement, the optimal control routine is re-run. As a result of this online optimization, the MPC approach has been successfully applied to many real-world process control problems [6]. The MPC problem at control step i is

0 1 2

1

0

min , { , , ,..... }

. . ( ), 0,1, 2.... 1

( ) 0, 0,1, 2.... 1

( )

nU

k k k

k k

J U u u u u

s t x f x u k N

g x u k N

x Z k

(14)

Where N is the prediction horizon, the optimal solution to the above problem is denoted by * * * * *

0 1 2{ , , ,..... }nU u u u u .

Only *0u after this iteration of optimization is implemented.

The process repeats in time. We can apply MPC to dynamic economic dispatch by algorithm below [7].

1. input the initial status 1 1 1 1 11 2 3{ , , ,..... }nP P P P P

2. Compute the open loop optimal solution of DED

* * * * *0 1 2{ , , ,..... }nU u u u u

And 2 1 *i i iP P Tu

3. Compute the Close loop optimal solution of (MPC)

*( 1) *( 1) *( 1) *( 1) *( 1)1 2 3{ , , ..... }m m m m m

nU u u u u

1 *( 1)m m mi i iP P Tu

4. Let m=m+1 and go to step 1 By this algorithm we can optimize cost by the dynamic way. For making optimization we use the quadratic programming.

IV. QUADRATIC PROGRAMMING QP is an effective optimization method to find the global

solution if the objective functions is quadratic form and the constraints are linear. It can be applied to optimization problems having non quadratic objective and nonlinear constraints by the approximating the objective to quadratic function and constraints as linear [8].

The classic objective function of a QP problem is as follows [9]

1min( )

2T

xX QX CX (15)

Subject to the linear inequality and equality and bound

constrains

eq eq

Ax b

A x b

lb X ub

(16)

where C is an n - dimensional row vector describing the coefficients of the linear terms in the objective function; Q is an ( n × n ) symmetrical matrix describing the coefficients of the quadratic terms or hessian matrix and T in (1) denote the transposed vector [5]. As in linear programming, the decision variables are denoted by the n - dimensional column vector x, and the constraints are defined by an ( m × n ) matrix (A) and an m - dimensional column vector B of right - hand – side coefficients. For the real power ED problem, we know that a feasible solution exists and that the constraint region is bounded. When the objective function F (x) is strictly convex for all feasible points, the problem has a unique local minimum, which is also the global minimum. A sufficient condition to guarantee strict convexity is for Q to be positive definite. This is generally true for most of economic dispatch problems. To map the ED to QP, the objective function variables are given by the power generation output vector as follow:

Take 1 *( 1)m m m

i i iP P Tu (17)

We optimization every time T =1 Hour So that (17) will become

1 *( 1)m m mi i iP P u (18)

Operation cost of generating i units is

2 21 1 1(2 )i i i i i i i i i i i iC a u P a b u b P c a P (19)

We will optimize iu by quadratic programming

The general formula of total power loss in transmission line is expressed by (from kron’s loss formula)

0 00

1 1 1

g g gn n n

loss i ij j i ii j i

P PB P B P B

(20)

We suppose that ix P and by (16)

We can get

11 1

1 2 3

1

.....

([1,1,1...,1] [ , , ..., ] : :

.....

N

eq N

N NN

B B

A P P P P

B B

00 00 0001 02 03 0

1 2

[ , , ..., ] [ , ,..., ]NN

B B BB B B B

P P P (21)

2eq lossb Pd P (22)

By (18) So that form (21) will become

11 1( 1) ( 1)

1 1

1

.....

[ ,..., ] : :

.....

Nm m m m

eq N N

N NN

B B

A P u P u

B B

01 02 03 0[1,1,1...,1] [ , , ..., ]NB B B B

00 00 00( 1) ( 1) ( 1)

1 1 2 2

[ , ,..., ]m m m m m m

N N

B B B

P u P u P u

(23)

( 1)1 1

11 1 ( 1)( 1) ( 1) 2 2

1 1

1 ( 1)

...

[ ,..., ] : ::

...

m m

N m mm m m m

loss N N

N NN m mN N

P uB B

P uP P u P u

B BP u

( 1)1 1

( 1)2 2

01 02 03 0 00

( 1)

[ , , ..., ]:

m m

m m

N

m mN N

P u

P uB B B B B

P u

(24)

To find Hessian matrix Q we take

2 2

21 1

2 2

21

...

: :

...

N

N N

f f

x x x

Q

f f

x x x

(25)

By (19) & (25) Hessian matrix Q is equal

1 .... 0

: :

0 ... N

a

Q

a

(26)

To find C the coefficients of the linear objective function

1 1 1

2 2 2

(2 )

(2 )

:

(2 )N N N

P a b

P a bf

P a b

(27)

We use the flowing Matlab code formulated to do the

optimization [10]

x=quadprog (H, f, A, b, Aeq, beq, lb, ub)

% solves the the quadratic programming problem:

min 0.5*x'*H*x + f'*x

% while satisfying the constraints

A*x ≤ b

Aeq*x = beq

lb <= x <= ub

V. APPLY FLOW CHART For this section we will apply the combine Algorithm of

MPC and Quadratic programming

VI. SIMULATION RESULTS To verify the effectiveness of the proposed algorithm, a six unit thermal power generating plant was tested. The proposed algorithm has been implemented in MATLAB language. The proposed algorithm is applied to 26 buses, 6 generating units with generator constraints, ramp rate limits and transmission losses [21]. The results obtained from the proposed method will be compared with the outcomes obtained from the Genetic Algorithm and PSO method in terms of the solution quality and computation efficiency. The fuel cost data and ramp rate limits of the six thermal generating units were given in Table I. The load demand for 24 hours is given in Table II. B-loss coefficients of six units system is given in Equation

(17). Table III gives the optimal scheduling of all generating units, power loss and total fuel cost for 24 hours by using PSO technique. Table IV gives the optimal scheduling of all generating units, power loss and total fuel cost for 24 hours by using Genetic Algorithm and Table V gives the optimal scheduling of all generating units, power loss and total fuel cost for 24 hours by using the proposed method.

B-loss coefficients of six units system is given in Equation

TABLE III GIVES THE OPTIMAL SCHEDULING OF ALL GENERATING UNITS, POWER LOSS AND TOTAL FUEL COST FOR 24

HOURS BY USING PSO

TABLE IV GIVES THE OPTIMAL SCHEDULING OF ALL GENERATING UNITS, POWER LOSS AND TOTAL FUEL COST FOR 24

HOURS BY USING GENETIC ALGORITHM

TABLE V GIVES THE OPTIMAL SCHEDULING OF ALL GENERATING UNITS, POWER LOSS AND TOTAL FUEL COST FOR 24 HOURS BY USING THE PROPOSED METHOD

Figure 1. Fuel cost of unit 1 versus 24 hr by the three used method






Figure 7. Total fuel cost of 6 unit versus 24 hr by the three used method

VII. CONCLUSIONS

In this paper, Dynamic Economic Dispatch Using Model Predictive Control Algorithm is used to solve the DED problem. The proposed algorithm has been successfully implemented for solving the DED problem of a power system consists of 6 units with different constraints such as real power balance, generator power limits and ramp rate limits. For the results, some of the generator (1; 4; 6) has the lower cost than other method while generators (2; 3; 5) are higher. Even though some are lower and some are high cost, the total cost of propose method is lower that other methods. it is clear From the that the total fuel cost obtained by Dynamic Economic Dispatch Using Model Predictive Control Algorithm is comparatively less compared to other methods. Simulation results demonstrate that the proposed method is powerful and practical tool for obtaining minimum of total fuel cost.

REFERENCES [1] , F. Benhamida , S. Souag , F. Z. Gharbi, R. Belhachem ,A.

Graa “Constrained Dynamic Economical Dispatch”. [2]. W. M. Mansour, M. M. Salama, S. M. Abdelmaksoud,

H. A. Henry, “Dynamic Economic Load Dispatch of Thermal Power System Using Genetic Algorithm”.

[3]. A. M. Elaiw , X. Xia and A. M. Shehata “Application ` Of Model Predictive Control To Dynamic Economic

Emission Dispatch With Transmission Losses” [4]. Hadi Saadat “Power System Analysis” [5]. Le Xie, Student Member, IEEE, and Marija D. Ili´c,

Fellow, IEEE “Model Predictive Economic /Environmental Dispatch of Power Systems with Intermittent Resources

[6]. A.M.Elaiw, X.Xia, and A.M.Shehata” An application of model predictive control to the dynamic economic emission with transmission loss

[7]. R.Belhachem, F.Benhamida, S.souag, I.Ziane ,and Y.salhi “Dynamic Economic Load Dispatch Using Quadratic Programming and Gams

[8]. Jizhong Zhu “Optimization of power system Operation” [9]. matlab code optimization http://www.mathworks.com

/help/optim/ug/quadprog.html

dynamic economic dispatch using model predictive control algorithm

Documents