J. Cent. South Univ. Technol. (2010) 17: 824-829

DOI: 10.1007/s11771-010-0562-9

Load distribution model and voltage static profile of Smart Grid

SUN Qiu-ye(孙秋野), LI Zhong-xu(李钟旭), YANG Jun(杨珺), LUO Yan-hong(罗艳红)

Institute of Electrical Automation, Northeastern University, Shenyang 110004, China

? Central South University Press and Springer-Verlag Berlin Heidelberg 2010

Abstract: Voltage profiles of feeders with the connection of distributed generations (DGs) were investigated. A unified typical load distribution model was established. Based on this model, exact expressions of feeder voltage profile with single and double DGs were derived and used to analyze the impact of DG’s location and capacity on the voltage profile quantitatively. Then, a general formula of the voltage profile was derived. The limitation of single DG and necessity of multiple DGs for voltage regulation were also discussed. Through the simulation, voltage profiles of feeders with single and double DGs were compared. The voltage excursion rate is 7.40% for only one DG, while 2.48% and 2.36% for double DGs. It is shown that the feeder voltage can be retained in a more appropriate range with multiple DGs than with only one DG. Distributing the total capacity of DGs is better than concentrating it at one point.

Key words: Smart Grid; distributed generation; typical load distribution model; voltage profile

1 Introduction

The existing power system is aging and under  stress, resulting in compromised reliability and power quality. There has been much recent discussion on what distribution systems can be and should look like in the future. Faced by challenges, the concept of “Smart Grid” or “IntelliGrid” is put forward by European and American with the same meaning. The Smart Grid should be meshed and intelligent [1], with some key characteristics such as being self-healing and tolerant of attack, providing high power quality, minimizing operation and maintenance expenses, and accommodating a wide variety of distributed generation and storage options [2-4]. There are some major technologies in the design of the Smart Grid, such as advanced metering infrastructure (AMI), distribution automation (DA), and distributed generation (DG). The functions of AMI, DA, and DG will be integrated as a whole [4-5].

DG plays a key role in the design and realization of the Smart Grid. The quality, reliability, and flexibility of the power systems can be enhanced by DG [6], which are required by the Smart Grid.

However, for the sake of the nonlinear feature of the power system [7-9], the connection of DGs may introduce complex technical challenges, involving voltage profile, system stability, power flow, and relay protection [10-14].

Many researches, up to date, have been devoted to optimal dispatch scheme for DG. For minimizing power loss, LE et al [15] presented an optimization methodology based on the sequential quadratic programming (SQP). The benefits of different DG dispatch schemes were quantified with several constraints by taking the cost of DG into consideration. In Ref.[16], a method for the placement of DG was presented, considering power flow continuation and determination of the most sensitive buses to voltage collapse.

As a key problem of power quality, the improvement of the voltage profile needs special attention. In Ref.[17], to improve the voltage profile, a solution methodology based on the voltage stability index was proposed for the optimal location of DGs. The optimal sizing problem of DGs was carried out using genetic algorithm for better voltage regulation. CARVALHO et al [18] investigated the impact of DG in different types of distribution systems on the voltage profile and short circuit analysis. Voltage profiles at different locations and capacities of DG were tested in three types of distribution systems: radial, loop and network.

However, there was no typical load distribution model established for quantitative research in Refs. [15-18]. In this work, a unified typical load distribution model was established. Based on this model, exact expressions of voltage profile of feeder with single and double DGs were derived. The limitation of single DG and necessity of multiple DGs for voltage regulation were also discussed. Through simulation, the impact of single and double DGs on voltage profiles was compared and analyzed.

2 Unified typical load distribution model

Due to its common existence in power grids, the waist-form load distribution model is selected to analyze the voltage profile of feeder with DGs connected, as shown in Fig.1, where L is the length of the feeder; points k₁, k₂…, k_n denote the locations of DGs; points A and B are the beginning and the end of the feeder, respectively; point C is called the waist point, which is equal to λL, λ[0, 1] in distance to point A; P₀+jQ₀ is the load at the transformer (beginning of the feeder); P₀ and Q₀ denote the real and reactive power loads, respectively; m and t are the slops of the load distribution curves, and concretely,

                            (1)

m₁ and t₁ denote the slopes of the real power load curve, m₂ and t₂ denote the slopes of the reactive power load curve; and d is the distance between the beginning of the feeder and point D.

Fig.1 Feeder of waist-form load with DGs

When D[0, λL], where D represents an arbitrary point along the feeder, the loads before and after point d are respectively:

                        (2)

     (3)

When D[λL, L], the loads before and after point d are respectively:

              (4)

    (5)

When the slope of the waist-form load distribution curve after point C equals that before point C, that is, m=t, the model (shown in Fig.1) turns into a unified load model, which can represent three models, i.e., uniform, increasing and decreasing models, as shown in Fig.2. It is a special case of the waist-form load model.

Fig.2 Unified model of uniform, increasing and decreasing loads with DGs

When m=t=0, the model turns into the uniform model; when m=t＞0, the model turns into the increasing model; and when m=t＜0, the model turns into the decreasing model.

3 Voltage profile with single DG

Single DG with a capacity of P_DG+jQ_DG is connected to the feeder at point k. The voltage profile should be discussed in the following four cases.

Case 1: D[0, λL] and D≤k (D between points A and C, before DG)

According to Eqs.(2)-(3), the loads before and after point D are:

                        (6)

     (7)

The voltage at point D is

U_D=U₀-?U₁-?U₂                             (8)

where U₀ is the voltage at the beginning of the feeder. Voltage drops ?U₁ and ?U₂ are caused by the equivalent loads after d and the loads before d, respectively.

  (9)

           (10)

where U_N denotes the rated voltage of the feeder; r and x denote the resistance and reactance per unit length, respectively; and τ is the integral varible.

According to Eqs.(8)-(10), the voltage at point D is

              (11)

where

                       (12)

Case 2: D[0, λL] and k＜D≤λL (D between points A and C, after DG)

The same method is applied to calculating U_D as in Case 1 and the result is

              (13)

Case 3: D[λL, L] and λL≤D≤k (D between points C and B, before DG)

             (14)

where

        (15)

Case 4: D[λL, L] and k＜D≤L (D between points C and B, after DG)

             (16)

4 Voltage profile with double DGs

4.1 Necessity of multiple DGs

A feeder with a uniform load (m=t=0 in Fig.2) is taken as an example to show the limitation of single DG and the necessity of multiple DGs.

Without DG, the voltage profile along the feeder is

                (17)

＜0                  (18)

It can be proved from Eq.(18) that the voltage decreases along the feeder and the minimum voltage appears at the end of the feeder, which may fall below lower limit U_min. So it is necessary to place a DG for voltage support to ensure the voltage within the admissible range [U_min, U_max].

When single DG is placed at point k, the voltage profile is

 (19)

The derivative of Eq.(19) is

  (20)

Let  in Eq.(20). A minimum value of voltage before DG can be obtained at point d^*=

 and the minimum voltage is

                      (21)

So, the minimum voltage of the feeder (U_{D, min}) may appear at end point L or at point d ^*:

U_{D, min}=min{U(d ^*), U(L)}                          (22)

where

       (23)

It can be observed from Eq.(20) that the voltage always decreases after DG, so the maximum voltage appears at point k (where DG is connected) regardless of the voltage at the beginning (U₀), that is,

U_{D, max}=U(k)                                (24)

where

    (25)

However, the situation may happen when the voltage at end U(L) is still under the lower limit if the capacity of DG is too small or DG is too far from the end, especially in a heavy loaded system, as shown in Fig.3.

Fig.3 Voltage profile with single DG of small capacity

When the capacity of DG is enlarged and the location is adjusted to support the voltage near the end, the voltage where DG is connected may be raised over the upper limit, as shown in Fig.4.

Fig.4 Voltage profile with single DG of large capacity

Whatever the location and capacity of DG are adjusted, if the feeder with single DG cannot satisfy the following boundary restriction:

                               (26)

it is necessary to place multiple DGs to ensure the voltage profile within the admissible range though more investment will be needed.

4.2 Voltage profile with double DGs

In the waist-form load distribution model, two DGs with capacities of and  are placed at points k₁ and k₂ (k₁＜k₂), respectively. According to the relations of the waist point (point C in Fig.1) and DGs’ locations, the voltage profile should be discussed in the following six cases.

Case 1: D[0, λL] and 0≤D≤k₁

The loads before and after point D are:

                        (27)

      (28)

The voltage drops are

         (29)

      (30)

According to Eqs.(8) and (29)-(30), the voltage at point D is

(31)

Case 2: D[0, λL] and k₁＜D≤k₂

(32)

Case 3: D[0, λL] and k₁＜k₂＜D≤λL

(33)

Case 4: D[λL, L] and λL≤D≤k₁

(34)

Case 5: D[λL, L] and k₁＜D≤k₂

(35)

Case 6: D[λL, L] and k₂＜D≤L

(36)

A general formula can be concluded.

n DGs are located at points k₁, k₂,  k_t,  k_n within [0, L]. Let k₀=0, k_n+1=L, without loss of generality, suppose k₁, k₂,  k_t[0, λL], k_t+1,  k_n[λL, L].

When D[k_i-1, k_i] (i=1, 2,t+1) and D≤λL,

           (37)

When D[k_i-1, k_i] (i=t+1, t+2,n+1) and D＞λL,

           (38)

where which is the convergence

condition for classical power flow calculation and can reflect the impact of the capacity of DG on the voltage profile.

5 Simulation

A 10 kV feeder with a uniform load (m=0 in Fig.2) is designed with the following parameters L=10 km, r= 0.20 mΩ/m, x=0.45 mΩ/m, P₀=0.6 MW, Q₀=0.4 MVar, U₀=10.5 kV, and power factor of DG cos θ=0.9.

Single DG with capacity of P_DG=4 MW and cos θ= 0.9 is placed at point k=8 km. Double DGs with the same capacity P_DG=2 MW and cos θ=0.9 are placed at points k₁=6 km, k₂=8 km, respectively.

It can be observed from Fig.5 that the voltage is pulled up largely，especially at the connected points with DG.

Fig.5 Voltage profile with single and double DGs

For this single DG, the voltage at the connected point of DG (k=8 km) is 10.74 kV. The voltage excursion rate (η_VER) at the connection point of DG is

×100% =

×100%=7.40%        (39)

η_VER is raised over the upper limit 7.00%, which is the admissible range in 10 kV distribution network.

For double DGs, η_VER at the connected points of DGs (k₁=6 km, k₂=8 km) is:

×100% =

×100%=2.48%       (40)

×100% =

×100%=2.36%       (41)

which are approximately equal and both in the admissible range.

This shows that voltage profile with double DGs changes more smoothly and stably, especially at the connected points of DG.

6 Conclusions

(1) Voltage profile with DGs is an important reference to the planning of futuristic Smart Grid. The waist-form load distribution model, which can reflect the feature of load distribution on a feeder, is established. Quantitatively and succinctly, expressions of voltage profile with single and multiple DGs can be obtained based on this model.

(2) Multiple DGs have better effect on improving voltage profile than the single DG. Voltage profile is smoother and more stable with multiple DGs. The application of multiple DGs is advisable though more investment will be needed.

(3) Distributing the total capacity of DG is better for voltage profile than concentrating it at one point. The total capacity of single and double DGs is the same. However, smoother voltage profile appears when the total capacity is dispersed.

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Foundation item: Projects(60904101, 60972164) supported by the National Natural Science Foundation of China; Project(N090404009) supported by the Fundamental Research Funds for the Central Universities; Project(20090461187) supported by China Postdoctoral Science Foundation

Received date: 2009-09-23; Accepted date: 2010-03-16

Corresponding author: SUN Qiu-ye, PhD; Tel: +86-13998811006; E-mail: sunqiuye@mail.neu.edu.cn

(Edited by CHEN Wei-ping)