Optimal Economic Power Dispatch in the Presence of ...

Optimal Economic Power Dispatch in the Presence of Intermittent Renewable Energy Sources Salem Elsaiah, Student Member, IEEE, Mohammed Benidris, Student Member, IEEE, Joydeep Mitra, Senior Member, IEEE, and Niannian Cai, Student Member, IEEE Department of Electrical and Computer Engineering Michigan State University East Lansing, Michigan 48824, USA (elsaiahs, benidris, mitraj, and cainiann @msu.edu)

Abstract—This paper describes a method for solving the economic power dispatch problem in the presence of renewable energy sources. The method proposed in this paper uses linear programming because linear programming based formulations tend to be flexible, reliable, and faster than their nonlinear counterparts. A linearized network model in the form of DC power flow is utilized in this paper. Thermal limits of transmission lines and real power constraints have both been considered in the proposed model. In addition, piecewise linear models for generating units cost curves are developed during the realization of the presented work. A micro-power optimization model based on Homer software is used to account for the uncertainty in the output power of the intermittent renewable energy sources. The proposed linear programming based method is tested on the standard IEEE 30 bus system. Test results are reported, discussed, and thoroughly analyzed. Index Terms—Economic dispatch, renewable energy, wind turbine generator, linear programming.

I. I NTRODUCTION In recent years, the number of renewable energy sources installed on power systems has been rapidly increased. The vast majority of these renewable energy sources are installed at sub-transmission level and distribution system level. The most popular renewable energy sources are the solar panels and the wind turbine generators (WTG). Wind power generating system, which is addressed in this paper, usually consists of a wind turbine, a gearbox, and an induction generator. From operating perspective, wind turbine generator system can be operated either in stand-alone mode or in the gridconnected mode. According to their speed, however, wind turbine generating systems can be classified as variable-speed variable-frequency such as self-excited induction generator, constant-speed variable-frequency such as doubly fed or doubly output induction generator, and constant-speed constantfrequency, where the speed of the prime-mover is held constant by the continuous adjustment of the blade pitch of the turbine. Despite their numerous advantages, the inclusion of wind turbine generators in the steady-state analysis of power system such as static and dynamic economic dispatch, for instance, has also brought up a number of technological and operational issues that need to be addressed. The power dispatch problem was first formulated as a network constrained economic dispatch problem in 1962 by

Carpentier [1] and defined later as an optimal power flow problem by Dommel and Tinney [2]. The task of performing optimal economic power dispatch aims essentially at determining the optimal settings of the given power network by optimizing certain functions, e.g. minimization of total generation cost, total loss, or total emission, while satisfying a set of operational and technical constraints. Numerous control variables, which include generator’s real power, transformer’s tap changers and phase shifters, static and dynamic VAR compensators are also involved in the optimization procedures. A great variety of solution techniques have been proposed in the literature to solve the economic dispatch problem since its inception. Examples of these methods include, nonlinear programming [3]–[5], quadrature programming [6], [7], Newton’s method [8], [9], heuristic and swarm intelligence [10], [11], interior point method [12], [13], and linear programming (LP) based methods [14]–[17]. Amongst these methods, LP based methods are recognized as viable and promising tools in solving the constrained economic dispatch problem. The earliest versions of the LP based economic power dispatch were developed based on the pure DC power flow model (DCPF) [18]–[20]. Lately, the technical constraints in the LP models were linearized and treated by nonlinear power flows in order to impose them exactly. The inherent features of the LP based economic dispatch methods include reliability of optimization, flexibility of the solution, rapid convergence characteristics, and fast execution time. More prominently, the solution accuracy obtained by the LP based methods is almost achieved for most of practical optimization problems [21]. Several approaches have been recently proposed in the literature to solve the economic dispatch problem including renewable energy sources. A heuristic method based optimal dynamic power dispatch with renewable energy sources is presented in [11]. The problem of economic power dispatch in the presence of intermittent solar and wind energy sources is carried out using sequential quadratic programming in [22]. Sequential quadratic programming is able to handle the convex cost curves of generating units efficiently since the latter is inherently a quadratic function. Nonetheless, sequential quadratic programming requires considerable computational effort. More significantly, the standard simple form of the

quadratic programming is not often used because convergence is not always guaranteed. The dynamic economic power dispatch including wind turbine generators is presented in [23]. The stochastic optimal power dispatch in the presence of wind turbine generators has also been proposed in [24]–[26]. This paper describes an LP based method for solving the economic power dispatch problem in the presence of renewable energy sources. A linearized network model in the form DCPF is utilized in this paper. Line flow capacities and real power constraints have both been considered in the proposed model. A micro-power optimization model based on Homer software is used to account for the uncertainty in the output power of the intermittent renewable energy sources. The proposed LP based method has been demonstrated on the standard IEEE 30 bus system. The rest of this work is organized as follows: In Section II the problem is formulated and the solution method is described. The implementation of the proposed method on the standard IEEE 30 bus system is presented in section III. Concluding remarks are provided in section IV.

ˆ= The augmented node susceptance matrix is calculated as B T ˆ ˆ A b A. Further, in accordance with the IEEE standard 15472003 for interconnecting distributed resources with electric power systems, it has been emphasized that renewable energy sources based distributed generators are not recommended to regulate bus voltages, thereby they should be modeled as PQ nodes not PV nodes. With that being said, and in order to accommodate the penetration of the wind turbine generators in the optimization problem, the right-hand side of the vector equation (2) is formulated as,

II. P ROBLEM F ORMULATION A. Objective Function and Constraints The objective function used in this paper is to minimize the total generation cost. This problem is posed as [27],

Here PD represents the total system demand, Pm represents the output power of the renewable energy source, and Nr represents the number of renewable energy sources.

Ng

The cost function of any conventional generating unit is usually expressed as a second-order quadratic polynomial as,

C = min

X

F (PGi )

(1)

0 PD = (1 − ηp ) × PD

When dealing with renewable energy sources, including wind turbine generators, it is customary to define their penetration level rather than using their rated output power. Therefore, we define the penetration level of the renewable energy sources (ηp ) as, Nr 1 X Pm (7) ηp = PD m=1

B. Linearization of Cost Functions

i=1

Subject to: 1) Vector of Power Balance Injections 0 ˆ + PG = PD Bθ

2 Fi (PGi ) = αi + βi PGi + γi PGi

(2)

2) Upper and Lower Real Power limits PGmin ≤ PG ≤ PGmax

(3)

3) Forward and Reverse Flow Capacities bAˆ θ ≤ F max f

−bAˆ θ ≤ Frmax θ is unrestricted

(5)

(8)

This paper adapted the linearization model developed in [18]. Now, let us consider the typical nonlinear cost curve of a generating unit i, which is depicted below in Fig. 1. Such cost curve can be approximated by a number of linear segments with different slopes. For instance, the cost function shown below in Fig.1 is approximated with three linear segments of Pi1 , Pi2 , and Pi3 , with slopes of µ1 , µ2 , and µ3 , respectively. The cost function is then approximated as,

(4)

4) Angle Constraints

(6)

Fi (PGi ) = Fi (PGmin ) + µ1 Pi1 + µ2 Pi2 + µ3 Pi3 i

(9)

The linear approximation of the cost function can be made by summing up all linear segments in the PGi values. That is,

Where,

PGi = PGmin + Pi1 + Pi2 + Pi3 i

(10)

PGmin ≤ PGi ≤ PGmax i i

(11)

with,

Ns

=

Number of buses

Nt

=

Number of transmission lines

Ng

=

Number of generator buses

PG

=

Vector of real power generation (Ns × 1)

PD ˆ B

=

Vector of active power demand (Ns × 1)

C. Modeling of Wind Turbine Generators

=

Augmented node susceptance matrix (Ns × Ns )

θ

=

Vector of bus voltage angles (Ns × 1)

F

=

Vector of flow capacities (Ns × 1)

A micro-power optimization model based on Homer software [28] is used to account for the uncertainty in the output power of the intermittent renewable energy sources. The typical relationship between the steady wind speed and the output power of the WTG is depicted below in Fig. 2. This curve can be divided into four regions as the following,

b = Aˆ =

Diagonal matrix of lines suseptances (Nt × Nt ) Element-node incidence matrix (Nt × Ns )

Where PGmin and PGmax are the upper and lower bounds i i of the real power of each generating unit.

The average wind speeds used in this paper are based on the hourly data from 1996 − 2006 measurements, which are obtained from automated stations at reporting airports [30]. This data is plugged into the Homer software to generate hourly wind speed profile based on Weibull probability distribution. The 8760 hours data used in this paper are acquired from St Paul Island Airport reporting station at the state of Alaska. The wind speed data of the 8760 hours is depicted in Fig. 3

Fi

Pi3

Wind Speed (m/s)

Pi2 Pi1 Pi Pi(min)

Fig. 1: Cost curve linearization

15 10 5 0

Rated Output Speed Cut-out Speed Rated Output Power

Power Output (KW)

Pr

Wind Resource

20

Jan

Feb

Mar

Apr

May Jun Jul Month

Aug Sep

Oct

Nov Dec

Fig. 3: Average wind speeds of the hourly data from 1996 − 2006 measured at St Paul Island Airport The wind speed profile is generated for the entire year, however, due to space constraints, we only show the wind profile for the month of January in which the speed reaches its peak value of 25 m/sec at about the hour 14.

Cut-in Speed Vr

Vi

Jan

Vo

Steady Wind Speed (m/s)

24

20

Fig. 2: Steady wind speed versus output power of a typical wind turbine generator  Pw ,    Pr , P (V ) =  0,   0,

Vr > V > Vi Vo > V ≥ Vr V ≤ Vi V ≥ Vo

12

(12) 8 0

As can be seen from Fig. 2, the region between Vi and Vr can almost be represented by a third-order polynomial. The typical formula that is used to calculate the output power of the wind turbine while in this region can be expressed as [29], Pw =

1 · Cp · ρ · A · V 3 2

16

(13)

Where Cp is the coefficient of performance of the wind machine, ρ is the air density in kilograms per cubic meter (kg/m3 ), A is the swept rotor area in square meters (m2 ), and V is the wind speed in meters per second (m/sec). In the presented work, a more simplified formula is used [29]. This formula gives a rough estimate of the output power from the wind turbine, which can be expressed as,

6

12

18

24

Fig. 4: A Typical steady wind speed for 24 hours in the month of January As has been mentioned before, the Weibull probability distribution is used in the Homer software. The probability density function (frequency versus wind speed) is given below in Fig. 5. The best-fit Weibull is found to be at 17.51 m/sec, which is corresponding to a shape parameter of 2.

(14)

III. D EMONSTRATION AND D ISCUSSION The results obtained by the proposed linear programming method are presented and discussed in this section. For the sake of validation, the proposed linear programming method has been compared with a method based on nonlinear AC power flow (ACPF). Then, several case scenarios are conducted in this section. Test results are reported, analyzed, and thoroughly discussed.

Where K is a constant which is chosen to be equal to 0.2 [29], Pa is the mean average power output in watts over one year, and Vm is the mean annual wind speed in (m/sec).

A. Case Scenario I The proposed linear programming method has been implemented on the IEEE 30 bus system [32]. This system consists

Pa = K · A · Vm3

TABLE II: Total Generation Cost and Total Generation of the IEEE 30 Bus System–Case Scenario I

Scaled data PDF

7

6

Method Cost ($/hr) Generation (MW)

Frequency (%)

5

4

Based DCPF 572.95 189.2

Based ACPF 578.29 192.011

3

2

1

0

0

10

20 Wind speed data

30 40 Value (m/s) Best-fit Weibull (k=1.98, c=17.50 m/s)

50

60

Fig. 5: Representation of frequency versus steady wind speed using probability density function of 30 buses, 41 transmission lines, and 2 shunt capacitors placed at buses 5 and 24, respectively. This system has six generating units placed at buses 1, 2, 13, 22, 23, and 27, respectively. The IEEE 30 bus system has also 4 tap-changing transformers placed between lines 4 − 12, 6 − 9, 6 − 10, and 27 − 28. The total real and reactive power peak loads on this system are 189.2 MW and 107.2 MVar, respectively. This standard system has been used to validate the accuracy of the proposed linear programming based method. In this case study, no wind turbine generators are presented. The results of the optimal power dispatch of each generating unit obtained by the proposed method are presented in Table I. The results obtained by the proposed LP method are also compared with those obtained by nonlinear programming based approach using Matpower [31]. As can be seen from Table I, the results obtained by the proposed method corresponding closely to those obtained by the nonlinear programming based method while requiring less computational effort. The total generation cost and total generation are respectively given below in Table II. As can be seen from Table II, the results of the total generation cost and total generation obtained by the proposed method are very close to those obtained by the by the nonlinear programming based approach. The deviations in the results obtained by the DCPF and linear programming and those obtained using ACPF and nonlinear programming of the total generation cost and total generation are only 0.92% and 1.46%, respectively. TABLE I: Results of the Optimal Power Dispatch of the IEEE 30 Bus System–Case Scenario I Output Power in MW Generating Unit # 1 Generating Unit # 2 Generating Unit # 13 Generating Unit # 22 Generating Unit # 23 Generating Unit # 27

Based DCPF 45 47.884 15 21 16 44.316

Based ACPF 39.92 53.03 14.96 22.46 16.06 45.58

B. Case Scenario II The proposed method has been implemented on the IEEE 30 bus system in the presence of the wind turbine generators. The

wind turbine generators used in this case scenario are assumed to be Enercon, type E33 with the following specifications: diameter of 33.4 m and tower height of 49 m [28]. From the probability density function shown in Fig. 5, the mean annual wind speed is estimated and the mean average power output in watts over one year is calculated using (14). In this case scenario, we assume that the penetration level of the wind turbine generators is 5%. Further, we assume that the total system demand is increased by 10% to simulate heavily loading conditions. The results for the optimal power dispatch of this case scenario are depicted below in Table III. The results of the total generation cost and total generation obtained by the proposed method are also presented in Table IV. As can be seen from Table III and Table IV, the total generation is reduced from 645.85 $/hr to 606.15 $/hr, when the wind turbine generators are included in the dispatch problem even in the case that the total load has been increased by 10%. The total generation is also reduced from 208.12 MW to 197.79 MW when the wind turbine generators are accounted for in the optimization problem. TABLE III: Results of the Optimal Power Dispatch of the IEEE 30 Bus System–Case Scenario II Output Power in MW Generating Unit # 1 Generating Unit # 2 Generating Unit # 13 Generating Unit # 22 Generating Unit # 23 Generating Unit # 27

Without WTG 45 68.95 15 21 16 42.171

With WTG 45 60.90 15 21 16 39.89

TABLE IV: Total Generation Cost and Total Generation of the IEEE 30 Bus System–Case Scenario II Method Cost ($/hr) Generation (MW)

Without WTG 645.85 208.12

With WTG 606.15 197.79

C. Case Scenario III In this case scenario the penetration level of the wind turbine generators is assumed to be 7.5%. Further, we assume that the total system demand is increased by 15% to simulate stressed and heavily loading conditions. The results of the optimal power dispatch for this case scenario are depicted below in Table V. The results of the total generation cost and total generation obtained by the proposed method are also presented in Table VI. As can be seen from Table V and Table VI, the total generation and the total generation cost have both been

reduced when the wind turbine generators are included in the optimal power dispatch problem even in the case that the total load has been increased. TABLE V: Results of the Optimal Power Dispatch of the IEEE 30 Bus System–Case Scenario III Output Power in MW Generating Unit # 1 Generating Unit # 2 Generating Unit # 13 Generating Unit # 22 Generating Unit # 23 Generating Unit # 27

Without WTG 45 79.48 15 21 16 41.09

With WTG 45 66.86 15 21 16 37.52

TABLE VI: Total Generation Cost and Total Generation of the IEEE 30 Bus System–Case Scenario III Method Cost ($/hr) Generation (MW)

Without WTG 682.30 217.58

With WTG 620.03 201.38

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