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An economical generation direction for power system static voltage stability
Electric Power Systems Research 76 (2006) 1075–1083
An economical generation direction for power system static voltage stability Arthit Sode-Yome, Nadarajah Mithulananthan ∗ Energy Field of Study, Asian Institute of Technology, Pathumthani 12120, Thailand Received 7 July 2005; accepted 30 December 2005 Available online 3 March 2006
Abstract This paper proposes an economical generation direction for static voltage stability. The proposed generation direction minimizes the total operating cost at any loading level, up to the point of collapse. The proposed approach, named as EGD approach, is based on the economic load dispatch with load flow. Two alternative methods to identify the economic generation direction are proposed. Economical generation direction is used in the continuation power flow process to obtain the static voltage stability margin. Static voltage stability margin with economical generation direction is compared with the margins resulting from other existing generation directions in the modified IEEE 14-bus test system under various system operating conditions. Cost of providing loading margin as well as PV curves and losses under different generation directions resulting from various generation direction approaches are also studied and compared. The proposed approach provides an alternative way for utilities to obtain the lowest total operating cost with voltage stability constraint by using any existing commercial software. © 2006 Elsevier B.V. All rights reserved. Keywords: Economic load dispatch with load flow; Economical generation direction; Lowest operating cost; Voltage stability study
1. Introduction At present, electric power systems are complex and interconnected networks that consist of thousands of loads and hundreds of generators [1,2]. Most of those networks have reached a saturation point in terms of new installations such as generation and transmission facilities due to many reasons. Hence, power systems are operating closer to their stability limits and become vulnerable to various instability problems such as voltage instability when load increase or any contingency happened [3,4]. Voltage instability is the main reason for several partial/full power interruptions in the past several years [4,5]. Voltage instability due to the lack of the ability to foresee the impact of contingencies is one of the main reasons for the recent and worst North American power interruptions on August 14, 2003. In addition, with an open access in the deregulated market environment, poorly scheduling of generation due to inappropriate competitive bidding is one of many reasons for voltage instability problem. According to this, electric utilities have devoted
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a great deal of efforts in solving or at least minimizing voltage instability problem. According to practical experiences and theoretical research works, major contributory factors to voltage instability are power system configuration, including location of reactive power resources, load pattern, and generation pattern [3,4,6–8]. Power system configuration or topology can be modified to alleviate voltage instability by adding shunt capacitors and/or Flexible AC Transmission System (FACTS) controllers at the appropriate locations [7–9]. The load pattern is beyond the operator’s control; however, under exceptional circumstances such as at peak load with contingency conditions, load-shedding and price incentives can be used as ways to adjust the load pattern to avoid a possible voltage collapse. Generation pattern or generation direction (GD), on the other hand, is easier to control by system operators compared to load pattern. Customarily, in a typical static voltage stability study, the generation of each generator with adequate capacity is raised at a pre-defined rate (e.g. uniform rate or according to their spinning reserves) [6,8,9]. However, increasing generation at this rate may not result in minimum total operating cost at various loading levels up to the maximum loading point. Moreover, operating cost of the system is an important factor in the present
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utility operating environment and it needs to be minimized at all times [10]. Thus, from the operational point of view, it is economical to increase the generation according to the economic load dispatch, if the voltage stability is maintained. However, in stressed conditions, the operators have to maintain the security of the system with “adequate margin” and do whatever it takes to secure the system from a possible “disaster,” certainly, with the trade-off of high operating cost. In [8], generation direction of a pre-defined rate is used in static voltage stability study. In this work, voltage stability margin or loading margin (LM) with various shunt compensation devices is calculated and compared. If a fixed GD is used, neither the highest LM nor the minimum operating cost can be guaranteed. In [11], maximum loading margin (MLM) approach is proposed in finding the generation directions that maximize the loading margin of the system. Although this method can provide the maximum LM, it may not provide low operating cost to the system. In [12], voltage stability study with the objective to minimize the total operating cost with voltage stability constraint is proposed. In this study, optimization technique is used to find optimal dispatch of all participating generators. This method, however, involves in solving all necessary conditions and it may not be practical for utilities to use existing software for this study. Based on the above, attention is drawn in this paper to propose an economical generation direction that gives the lowest operating cost with the help of economic load dispatch with load flow (ELDL). Two alternative approaches to identify GD are proposed. The proposed technique can be applied with the help of any existing commercial software for load flow studies. The total operating cost and voltage stability margin resulting from the proposed GD are compared with those of various existing GDs in the modified IEEE 14-bus test system under various system conditions. Merits and demerits of all the methods are also discussed and compared. This paper is structured as follows: Section 2 briefly explains the concept of the static voltage stability. In Section 3, existing methods for determining generation directions for static voltage stability are summarized. The proposed GD based on economical load dispatch with load flow is presented in Section 4. Test systems along with analysis tools used throughout the paper are given in Section 5. Numerical results with discussion are presented in Section 6. Finally, a summary of the main conclusions and contributions of the paper is presented in Section 7.
In static voltage instability, slowly developing changes in the power system occur that eventually lead to a shortage of reactive power and declining voltage. This phenomenon can be seen from the plot of the power transferred versus the voltage at receiving end. The plots are popularly referred to as PV curve or “nose” curve. The maximum load that the system can support before reaching the collapse point is called loading margin of the system. As the power transfer increases, the voltage at the receiving end decreases. Eventually, the critical (nose) point, the point at which the system reactive power is out of usage, is reached where any further increase in active power transfer will lead to a very rapid decrease in voltage magnitude. Before reaching the critical point, a large voltage drop due to heavy reactive power losses can be observed. Voltage stability margin of the system can be enhanced by various methods. Introducing reactive power sources, i.e. shunt capacitors or FACTS device at the appropriate location is an effective way to increase loading margin. This method, however, requires some investment cost. Moreover, these devices may be problematic in some system due to many reasons such as adverse interaction between FACTS devices and existing facilities, inability to control reactive power and voltage at the connected bus of shunt capacitors, etc. [8]. Another effective method to increase the loading margin of the system is by reducing reactive power losses occurred in the transmission system. This can be done by dispatching generation from existing generators to redirect power from congested transmission lines to less congested ones. If the losses in transmission system are reduced, more reactive power can be transferred to the load, thus increasing LM of the system. However, if one wants to reduce the total operating cost the existing generation facilities should be increased in a direction based on economic load dispatch.
2. Static voltage stability
The generation direction or KG of each generator is very crucial to voltage stability. Customarily, the generation of each participating generator is raised at the same rate or a pre-defined rate and it is referenced as conventional approach in this paper. Existing methods to identify GDs in static voltage stability margin are summarized below.
Static voltage instability is mainly associated with reactive power imbalance. The loadability of a bus in a system depends on the reactive power support that the bus can receive from the system. Both reactive and real power losses increase rapidly as the system approaches the maximum loading point or voltage collapse point. In general reactive power losses are higher than the real power losses. Thus, reactive power support should be local. By providing reactive power locally, real power can be transferred with lower losses and hence the total operating cost can be reduced.
3. Generation direction Generation direction or “pattern” is the rate of change of generation of generators that have adequate reserve to serve a load increase from the base load. Let KGi be the direction of generator active power increase and PGi,o be the generation at the base case of generator i, then generator output for a load increase is given by (1): PGi = PGi,o (1 + KGi )
(1)
3.1. Conventional approach Conventionally, the generation of the system is increased by a fixed percentage as pre-specified in the planning stage. The real power output of generator i after the load increase can be
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written as: PGi = PGi (1 + KGi ) = PGi,o + PGi and PG =
PGi = PD + Ploss
(2)
(3)
NG
where PGi is the generation power of generator i, PGi,o the generation of generator i at base load, PG the total generator increase of all participating generators, PGi the increase of generation at generator i, PD the total load increase, Ploss the total loss increase and NG is the number of participating generators. 3.2. Cost participation factor (Cost PF) approach Cost participation factor is the easiest and simple way to determine generation direction to obtain economic load dispatch without considering losses in the system. Generator cost participation factors are calculated based on generators’ incremental cost [11,13] as follows: 1/C PGi = NG i PD j=1 (1/Cj )
(4)
|PGi |max the lower and upper power limits of generator i, |Ui |min , |Ui |max the lower and upper limits of voltage magnitude at bus i, Pij , Qij , Sij the real, reactive and apparent power in line ij and Sij,max is the MVA (Thermal) limit of line ij. Generation direction, in this approach, can be worked out by subtracting the new dispatch from the old dispatch for individual generators. This approach, however, requires all necessary conditions to be simultaneously solved for all state variables including the generation dispatch, hence computationally very demanding. 3.4. Maximum loading margin approach The MLM approach is a method to identify a vector of the GDs of generators that gives maximum LM by approximating the surface of the LM as a function of the generation direction [11]. If one can approximate the LM surface as a function of all generation direction variables (KGi ), optimization techniques can be used to provide the highest LM point. Mathematically, the method can be formulated as follows: Maximize
where Ci is the cost function of generator i and Ci is the second derivative of the cost function i.
LM = Bo +
3.3. Optimal power flow (OPF) approach
subject to
Optimal power flow can be formulated to include voltage stability criteria as follows [12]: Minimize C(PGi ) =
⎛ ⎝
n p=1
NG
⎞
(5)
NG
subject to PGi −(1+λ)PDi,o −
n
|Ui ||Uj |(Gij cos δij + Bij sin δij ) = 0
j=1
QGi −(1+λ)QDi,o −
n
(6) |Ui ||Uj |(Gij sin δij −Bij cos δij ) = 0
j=1
(7)
|PGi |min ≤ |PGi | ≤ |PGi |max
(8)
|Ui |min ≤ |Ui | ≤ |Ui |max Sij = Pij2 + Q2ij ≤ Sij,max
(9) (10)
where C is total operating cost of the system, aGi , bGi , cGi the cost coefficients of generator i, λ the load incremental parameter or loading factor (LF), PGi , QGi the real and reactive power generation at bus i, PDi,o , QDi,o the real and reactive power demand at bus i at base load, n the number of buses in the system, |PGi |min ,
p
Bi,p KGi ⎠
KGi = 1
(11)
(12)
i=1
0 ≤ KGi ≤ 1 2 aGi PGi + bGi PGi + cGi
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(13)
where KGi is generation direction of generator i, Bi,p the coefficients terms, Bo the constant term, p the power term and n is the order of the approximation of the polynomial equation. If generation is increased according to this direction, the system will have the maximum loading margin. Among the existing methods, only optimal power flow method provides lowest operating cost as the operation cost is minimized by using optimization technique. This method, however, involves in solving all necessary conditions and may not be practical for utilities that use existing commercial software for the studies. In the following section, economic generation direction is proposed for minimizing operating cost with voltage stability constraints based on ELDL. ELDL method can be applied in any existing commercial software tool for load flow studies. 4. The proposed approach Economical generation direction (EGD) can be calculated in two steps. In the first step, Step I, economic load dispatch is carried out with load flow to find the optimal generation. In the second step, Step II, the economical GD is computed based on the optimal generation found in Step I.
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4.1. Step I In the first step, ELDL is formulated to include voltage stability criteria as follows: Minimize C(PGi ) =
2 aGi PGi + bGi PGi + cGi
(14)
NG
subject to: PGi −(1+λ)PDi,o −
n
|Ui ||Uj |(Gij cos δij + Bij sin δij ) = 0
j=1
QGi −(1+λ)QDi,o −
n
(15) |Ui ||Uj |(Gij sin δij − Bij cos δij ) = 0
j=1
(16)
|PGi |min ≤ |PGi | ≤ |PGi |max
(17)
|Ui |min ≤ |Ui | ≤ |Ui |max Sij = Pij2 + Q2ij ≤ Sij,max
(18)
λ ≤ λmax
(20)
(19)
where λ is load incremental parameter or loading factor (LF), PGi and QGi the real and reactive power generation at bus i, PDi , and QDi ,o the real and reactive power demand, respectively, at bus i at base load, n the number of buses in the system, |PGi |min and |PGi |max the lower and upper power limits of voltage magnitute at bus i, Pij , Qij , and Sij the real, reactive and apparent power in line ij, Sij,max the MVA (Thermal) limit of line ij; and λmax is load incremental parameter at the voltage collapse point. The ELDL is an iterative approach that computes the optimal generation of all participating generators at specific loading level based on sensitivity method. It can be viewed as an extension to the cost participation factor method, where the operating cost of individual generator is considered without total power losses of the system. The procedure behind the ELDL approach can be summarized as shown in Fig. 1. The load is increased at each loading factor in the outer loop. In the inner loop, economic load dispatch and load flow calculation are repeatedly performed to obtain the optimal solution. Note that the maximum number of load flow iterations in this study is set to 200. The ELDL method provides optimal dispatching generation of each generator, based on optimality condition [13,14]. ELDL is used instead of optimization technique to make use of existing commercial software for the study. In optimization technique, the necessary conditions are required to be solved for all state variables. This may be a problem for practical sized power system and for utilities, who wish to use existing software for the studies. ELDL approach, on the other hand, can be applied with the help of the existing commercial software, since it is based on load flow and sensitivity method.
Fig. 1. Illustration of ELDL process for static voltage stability study.
Although ELDL approach is a practical solution for utilities, it faces difficulties in obtaining the solution when the system is close to the collapse point. At that point, sensitivity of the losses to the generation increase is high, which provides convergence problem. In order to use ELDL approach for voltage stability study, EGD based on dispatching generation is proposed and used in CPF process to avoid convergence difficulties in ELDL during the stressful condition. EGD calculation based on ELDL method is explained in Section 4.2. 4.2. Step II Two alternative methods are proposed for finding the EGD from optimal generator dispatches. The first method, named as “EGDI”, is based on dispatching generation dispatches at various loading levels. The second method, on the other hand, is based on the slope of generation increase. The second method is named as “EGDII”. 4.2.1. EGDI GD of generator i at each LF can be found from the ratio between the generation increase of the generator i and the total generation increase of all participating generators. This can be formulated by (21): KGi =
PGi PGi = PG NG PGi
(21)
where PGi is the increase of power generation at generator i and PG is the total generator increase of all participating generators.
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GD of generation i or KGi are found at various LFs. If these GDs are converged to some values at high LF, EGD can be found from these converged values. EGD is approximated from the trend of the GDs at various LFs based on this technique. 4.2.2. EGDII The second approach to identify EGD is based on the slope of the generation increase. If the graph of the generation increase of generation i with respect to LF is plotted and it can be represented by a linear equation with the slope KGi,un . The equation of generation increase with respect to LF or λ for generator i can be represented by: PGi = λ × KGi,un
(22)
where KGi,un is the slope of plot between generation and LF for generator i. The total generation increase of all participating generators is, therefore, PG = (23) PGi = λ × KGi,un NG
NG
Thus, EGD or KGi of the generator i can be obtained by: PGi KGi,un = NG PGi NG KGi,un
KGi =
(24)
Eq. (24), EGDII, is as same as Eq. (21), known as EGDI, thus resulting in the same EGD. EGD based on EGDI is obtained from the final converged values of GDs while EGD based on EGDII is obtained from the slope of the generation increase. Compared to EGDI, EGDII may be of interest to utilities because it needs only two dispatching points to find EGD. The EGD based on the proposed method is used in CPF or direct method to obtain LM in the stressful condition, since ELDL approach itself may not provide the solutions during stressful condition.
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5. System and analysis tools A single line diagram of the modified IEEE 14-bus test system is depicted in Fig. 2, which consists of five synchronous machines, including two synchronous compensators used only for reactive power support and three generators located at buses 1, 2 and 6. The modification is only at generator 3 located at bus 6, changing from a synchronous condenser to a generator. There are 20 branches and 14 buses with 11 loads totaling 259 MW and 81.4 Mvar. Results presented in the paper were produced with the help of UWPFLOW [15] and tools developed in MATLAB [16]. UWPLOW is commercial grade software that has been designed to calculate load flow and various static voltage stability parameters including the maximum loading margin of the power system. In ELDL approach, a program developed in MATLAB is used to manipulate UWPFLOW for load flow calculation in iterative process. Optimization toolbox in MATLAB is used to provide the solution for ELD in ELDL process [16]. To validate the proposed approach with other existing methods, voltage stability study is done with the help of UWPFLOW. In this study, in order to obtain the PV curves hence the loading margin of the system for different cases, all the loads were represented as constant PQ and increased simultaneously, i.e. by keeping constant power factor, as follows [17]: PD = Po (1 + λ) QD = Qo (1 + λ)
(25)
where Po and Qo correspond to the base loading conditions and λ is the loading factor.
Fig. 2. Single line diagram of the modified IEEE 14-bus test system.
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6. Numerical results The EGD approach is validated by comparing the results with those of three existing methods, namely, conventional, Cost PF and MLM methods. The results are based on the modified IEEE 14-bus test system for various system operating conditions, including base case and N − 1 contingencies. 6.1. Base case system 6.1.1. Cost functions, the starting point and cost participation factors Quadratic cost functions are used for all three generators throughout the study. The cost coefficients, minimum and maximum generation limits, and Cost PF of each generator computed using (4) are shown in Table 1 [10,14]. Note that, at the starting point, generator 2 has the lowest incremental cost, whereas generator 3 gives the highest incremental cost. The cost functions of all generators are almost linear, as the coefficients of quadratic term are close to zero. In order to compare loading margins resulting from the different generation directions, initial points should be set at the same values for all cases. The ELDL is used to produce the starting point, since it gives the lowest operating cost while considering the network losses. The optimal generation set points of generators 1–3 are 150, 77.94, and 40 MW, respectively, at the base load of 259 MW. 6.1.2. MLM method [11] The MLM method is based on the loading margins of the system at various possible generation directions in the “generation direction space”. The plot of LM in generation direction spaces of generators 2 and 3 is illustrated in Fig. 3. Note that the generation direction of generator 1 is 1 minus the generation directions of generators 2 and 3 combined, i.e. 1 − (KG2 + KG3 ). From Fig. 3, it is obvious that the maximum loading margin of the system occurs at generation directions 0.7, 0.3 and 0 for generators 2, 3 and 1, respectively. As expected, increasing generation at generators 2 and 3 can reduce the losses and increase the LM; generator 2 has more influence in the system losses than generator 3, since it is situated closer to most of the load. If the generators are increased according to this direction, the system will have the maximum LM.
Fig. 3. Approximate LM surface at all possible GDs.
ELDL is used to find the optimal generation at each loading level based on the methodology presented in Fig. 1. Fig. 4 shows generation of each participating generator at various loading factors based on ELDL method. From Fig. 4, the curves of generation are linear for all participating generators in a given case. It is noticed from Fig. 4 that the solution can be obtained up to 0.8 LF. ELDL method faces convergence problem, at LF higher than 0.8 p.u., due to high sensitivity of losses with respect to the generation increase. 6.1.3.1. EGDI. The generation direction based on EGDI can be found from Eq. (21) for various LFs. Fig. 5 shows GDs for all participating generators at various LFs. It can be observed from the figure that GDs of generators 1–3 are converged to 0, 0.67 and 0.33, respectively, at higher LFs. These converged GDs are EGD, which is used for LF higher than 0.8 p.u. in the stressful condition based on CPF process. 6.1.3.2. EGDII. An alternative way to find the EGD is to use the slope of the generation increase, which is shown in Fig. 4. From Fig. 4, the slopes of generation at generators 1–3 are 0, 1.9074,
6.1.3. Economical generation direction The proposed EGD is found based on the methodology proposed in Section 4. First, ELDL method is used to find generators output as explained in Section 4.1. Next, EGD is computed from the generators output as explained in Section 4.2. Table 1 Cost coefficients, limits, and Cost PF of all generators G
a
b
C
Pmin
Pmax
Cost PF
1 2 3
0.001562 0.001940 0.004820
7.92 7.85 7.97
561 310 78
150 40 40
500 500 500
0.469655 0.378145 0.152200 Fig. 4. ELDL based generation outputs.
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Fig. 6. PV curves with different generation directions.
Fig. 5. Economical GDs of generators.
and 0.95195 p.u., respectively. These slopes are used to find EGD based on Eq. (24), which are 0, 0.67, and 0.33 for generators 1–3, respectively. Table 2 shows EGDs obtained from EGDI and EGDII. GDs obtained from these two methods are identical. EGDI is based on dispatching generation and it needs many dispatching points. EGDII, on the other hand, requires the slopes of the generation and it can be used when only two dispatching points are known. EGDII may be of interest to utilities since the computation effort is less. If the generations are increased at these directions, the system will have the lowest operating cost. EGDI and EGDII methods provide the same GD, thus in the following subsections only EGDI method is used to compare with the existing GD methods. EGDI is onwards named as EGD. 6.1.4. PV curves and loading margins In order to compare the static voltage stability margins of EGD with that of other GDs, the thermal and voltage limits are neglected. However, the limits and constraints can be easily incorporated in the proposed approach compared to the other existing generation direction methods. The generation directions used for conventional method is 0.85, 0.15 and 0 [8], while, for Cost PF method, they are assigned at 0.469655, 0.378145 and 0.152200 for generators 1–3, respectively, as given in Table 1. For MLM method, 0, 0.7 and 0.3 are used for generation direction of generators 1–3, respectively [11]. EGD based on the proposed methodology is 0, 0.67 and 0.33 for generators 1–3, respectively, as shown in Table 2. The PV curves of the test system with various generation direction methods, namely, conventional, Cost PF, EGD and MLM methods are depicted in Table 2 Economical GD of EGDI and EGDII methods Generator i
1 2 3
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Fig. 6. From Fig. 6, as expected, the MLM method offers the highest loading margin followed by EGD, Cost PF, and conventional methods. The maximum LMs based on conventional, Cost PF, EGD and MLM methods are given in Table 3. From Fig. 6 and Table 3, the MLM approach yields the highest LM since it attempts to maximize LM of the system. The conventional method, on the other hand, provides the lowest LM since the GD is fixed. Cost PF approach gives higher LM than the conventional approach. The MLM and EGD methods give almost the same LM and a better voltage profile compared to other generation directions as shown in Fig. 6. Total real power losses of the system based on each generation direction method are compared in Fig. 7. From the figure, it can be seen that EGD method offers lowest losses and incremental losses followed by MLM, Cost PF, and the conventional methods, respectively. MLM method, on the other hand, results in the maximum loading margin with small difference in the losses compared to the EGD approach. The conventional and Cost PF generation direction approaches lead to high losses since they do not consider the network losses, hence, making the loading margin less. The conventional method results in more losses than Cost PF approach. 6.1.5. Total operating cost and generation dispatch Total operating cost with various GDs at different loading levels up to the collapse point are depicted in Fig. 8. From Fig. 8, the EGD approach gives the lowest total operating cost at any loading levels. The MLM method gives almost a similar operating cost as the EGD method, but still yielding the highest loading margin. Table 3 Loading margins for different generation directions
EGDI
EGDII
KGi
Slope of PGi (KGi,un )
KGi
0 0.67 0.33
0 1.9074 0.95195
0 0.67 0.33
Method
GD
LM
Conventional Cost PF EGD MLM
0.85, 0.15, 0 0.469655, 0.378145, 0.1522 0, 0.67, 0.33 0, 0.7, 0.3
0.8473 0.9870 1.0927 1.0931
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Fig. 9. Cost difference between conventional and EGD approaches.
Fig. 7. Total real power losses with GDs.
6.2. Contingency cases
Fig. 8. Total operating cost at various loading factors for different generation directions.
Fig. 9 shows the difference in cost between the conventional and EGD methods. The total operating cost resulting from the EGD method is much lower than that of conventional method, especially in high loading condition. The improvement in cost is an exponential function of LF. Information of cost at the collapse points (CP), respective loading margins and the ratio of Cost/LM are given in Table 4. From the table, the EGD gives the lowest ratio of Cost/LM while MLM is the second best. EGD and MLM method offer a close value of Cost/LM. Table 4 LM, cost at LM and cost of all methods Method
LM [p.u.]
Cost at CP [R/h]
Cost/LM [R/h]/p.u.
Conventional Cost PF EGD MLM
0.8473 0.9870 1.0927 1.0931
5613.3 5858.0 6082.5 6089.7
6624.9 5935.2 5566.5 5571.0
In case of contingencies, the system becomes more stressful and results in lower loading margin [8,18]. For simplicity, only line contingencies are considered in this study. The contingency ranking of the system can be carried out by the CPF method, which is based on the actual loading margin resulting from a contingency. The first three worst contingencies are the outages of lines 1–2, 2–3 and 1–5, respectively. Loading margins, cost at LM and Cost/LM for the line outage cases are given in Table 5 for all the GD methods. From Table 5, it is obvious that EGD and MLM methods provide the highest LM compared to other methods for all contingency cases. Cost/LM using EGD approach is lowest for all the contingency cases and it is close to that of MLM method. This can be better viewed in Fig. 10, where the values of Cost/LM are plotted for various methods for all contingency cases. The MLM generation direction should give a higher loading margin among all the generation directions as the objective of the method is maximizing the loading margin. However, due to the step size considered and the approximation in the continuation Table 5 LM, cost at LM, Cost/LM of all methods for various line outages Contingency
Method
LM [p.u.]
Cost at LM [R/h]
Cost/LM [R/h]/p.u.
1–2
Conventional Cost PF EGD MLM
0.17806 0.27535 0.43479 0.43323
3802.5 3998.1 4300.9 4294.2
21355 14520 9892 9912
2–3
Conventional Cost PF EGD MLM
0.33038 0.36775 0.40846 0.40495
4343.8 4400.9 4480.4 4473.6
13148 11967 10969 11047
1–5
Conventional Cost PF EGD MLM
0.50415 0.62186 0.74672 0.73990
4752.2 4960.1 5194.4 5181.0
9426 7976 6956 7002
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ties, who wish to use existing commercial software for obtaining the right generation direction. In practical system, the lowest operating cost should be obtained if the voltage stability criterion is not violated. The information on appropriate generation directions can be made available on-line to assist system operators. References
Fig. 10. Cost/LM of all the methods with various contingency cases.
power flow process the margins obtained with MLM generation direction is slightly less than that obtained with EGD. Based on the results, EGD approach provides the lowest operating cost with voltage stability constraint for all the cases in the modified IEEE 14-bus test system. From the operation viewpoint, one would suggest to operate the system according to the EGD method in non-stressed conditions, if the solution is attainable, and according to MLM method in stressed conditions. This will certainly give low operating cost and maximum loading margin in the lower and higher loading conditions, respectively. In transition from one loading point to the other, operator can re-dispatch the generation to fit the appropriate approach based on the margin requirement and total cost trade-off. 7. Conclusion In this paper, an economical generation direction is proposed for static voltage stability. The proposed generation direction is compared with three other generation directions, namely conventional, cost participation factor and MLM, on loading margin, total losses and total operating cost in the modified IEEE 14-bus test system for various system conditions. The EGD gives the lowest total operating cost at any loading level for different contingency conditions. The proposed generation direction can be obtained with the help of any existing commercial software with more realistic limits and constraints. This is useful for utili-
[1] B. Gao, G.K. Morison, P. Kundur, Towards the development of a systematic approach for voltage stability assessment of large-scale power systems, IEEE Trans. Power Syst. 11 (August (3)) (1996) 1314–1324. [2] T. Nagao, K. Tanaka, K. Takenaka, Development of static and simulation programs for voltage stability studies of bulk power system, IEEE Trans. Power Syst. 12 (February (1)) (1997) 273–281. [3] IEEE/PES Power System Stability Subcommittee, Voltage Stability Assessment: Concepts, Practices and Tools, Special Publication, Final Draft, August 2003. [4] P. Kundur, Power System Stability and Control, McGraw Hill, 1994. [5] Blackout of 2003: Description and Responses, available: http://www. pserc.wisc.edu/. [6] B.H. Lee, K.Y. Lee, A study on voltage collapse mechanism in electric power systems, IEEE Trans. Power Syst. 6 (August) (1991) 966–974. [7] C.A. Canizares, A.C.Z. De Souza, V.H. Quintana, Comparison of performance indices for detection of proximity to voltage collapse, IEEE Trans. Power Syst. 11 (August (3)) (1996) 1441–1447. [8] A. Sode-Yome, N. Mithulananthan, Comparison of shunt capacitor, SVC and STATCOM in static voltage stability margin enhancement, Int. J. Electr. Eng. Edu. UMIST 41 (July (3)) (2004). [9] B.H. Lee, K.Y. Lee, Dynamic and static voltage stability enhancement of power systems, IEEE Trans. Power Syst. 8 (1) (1993) 231–238. [10] K.Y. Lee, A. Sode-Yome, J.H. Park, Adaptive Hopfield neural networks for economic load dispatch, IEEE Trans. Power Syst. 13 (May (2)) (1998). [11] A. Sode-Yome, N. Mithulananthan, Maximizing static voltage stability margin in power systems using a new generation pattern, Aust. J. Electr. Electron. Eng. (AJEEE) 2 (3) (2005) 255–261. [12] W.D. Rosehart, C.A. Canizares, V.H. Quintana, Effect of detailed power system models in traditional and voltage-stability-constrained optimal power-flow problems, IEEE Trans. Power Syst. 18 (February (1)) (2003) 27–35. [13] K.Y. Lee, F.M. Nuroglu, A. Sode-Yome, Real power optimization with load flow using adaptive Hopfield neural networks, J. Eng. Intell. Syst. 8 (March (1)) (2000) 53–58. [14] A.J. Wood, B.F. Wollenberg, Power Generation Operation and Control, John Wiley & Sons, 1984. [15] C.A. Ca˜nizares, et al., UWPFLOW: Continuation and Direct Methods to Locate Fold Bifurcations in AC/DC/FACTS Power Systems, University of Waterloo, September 2004, available at http://www.power. uwaterloo.ca. [16] Matlab Functions, available at: http://www.mathworks.com, June 2004. [17] N. Amjady, M. Esmaili, Voltage security assessment and vulnerable bus ranking of power systems, Electr. Power Syst. Res. 64 (2003) 227–237. [18] Z. Fend, V. Ajjarpu, D.J. Maratukulam, A practical minimum load shedding strategy to mitigate voltage collapse, IEEE Trans. Power Syst. 13 (November (4)) (1998) 12855–12891.
An economical generation direction for power system static voltage stability Arthit Sode-Yome, Nadarajah Mithulananthan ∗ Energy Field of Study, Asian Institute of Technology, Pathumthani 12120, Thailand Received 7 July 2005; accepted 30 December 2005 Available online 3 March 2006
Abstract This paper proposes an economical generation direction for static voltage stability. The proposed generation direction minimizes the total operating cost at any loading level, up to the point of collapse. The proposed approach, named as EGD approach, is based on the economic load dispatch with load flow. Two alternative methods to identify the economic generation direction are proposed. Economical generation direction is used in the continuation power flow process to obtain the static voltage stability margin. Static voltage stability margin with economical generation direction is compared with the margins resulting from other existing generation directions in the modified IEEE 14-bus test system under various system operating conditions. Cost of providing loading margin as well as PV curves and losses under different generation directions resulting from various generation direction approaches are also studied and compared. The proposed approach provides an alternative way for utilities to obtain the lowest total operating cost with voltage stability constraint by using any existing commercial software. © 2006 Elsevier B.V. All rights reserved. Keywords: Economic load dispatch with load flow; Economical generation direction; Lowest operating cost; Voltage stability study
1. Introduction At present, electric power systems are complex and interconnected networks that consist of thousands of loads and hundreds of generators [1,2]. Most of those networks have reached a saturation point in terms of new installations such as generation and transmission facilities due to many reasons. Hence, power systems are operating closer to their stability limits and become vulnerable to various instability problems such as voltage instability when load increase or any contingency happened [3,4]. Voltage instability is the main reason for several partial/full power interruptions in the past several years [4,5]. Voltage instability due to the lack of the ability to foresee the impact of contingencies is one of the main reasons for the recent and worst North American power interruptions on August 14, 2003. In addition, with an open access in the deregulated market environment, poorly scheduling of generation due to inappropriate competitive bidding is one of many reasons for voltage instability problem. According to this, electric utilities have devoted
∗
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0378-7796/$ – see front matter © 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.epsr.2005.12.024
a great deal of efforts in solving or at least minimizing voltage instability problem. According to practical experiences and theoretical research works, major contributory factors to voltage instability are power system configuration, including location of reactive power resources, load pattern, and generation pattern [3,4,6–8]. Power system configuration or topology can be modified to alleviate voltage instability by adding shunt capacitors and/or Flexible AC Transmission System (FACTS) controllers at the appropriate locations [7–9]. The load pattern is beyond the operator’s control; however, under exceptional circumstances such as at peak load with contingency conditions, load-shedding and price incentives can be used as ways to adjust the load pattern to avoid a possible voltage collapse. Generation pattern or generation direction (GD), on the other hand, is easier to control by system operators compared to load pattern. Customarily, in a typical static voltage stability study, the generation of each generator with adequate capacity is raised at a pre-defined rate (e.g. uniform rate or according to their spinning reserves) [6,8,9]. However, increasing generation at this rate may not result in minimum total operating cost at various loading levels up to the maximum loading point. Moreover, operating cost of the system is an important factor in the present
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utility operating environment and it needs to be minimized at all times [10]. Thus, from the operational point of view, it is economical to increase the generation according to the economic load dispatch, if the voltage stability is maintained. However, in stressed conditions, the operators have to maintain the security of the system with “adequate margin” and do whatever it takes to secure the system from a possible “disaster,” certainly, with the trade-off of high operating cost. In [8], generation direction of a pre-defined rate is used in static voltage stability study. In this work, voltage stability margin or loading margin (LM) with various shunt compensation devices is calculated and compared. If a fixed GD is used, neither the highest LM nor the minimum operating cost can be guaranteed. In [11], maximum loading margin (MLM) approach is proposed in finding the generation directions that maximize the loading margin of the system. Although this method can provide the maximum LM, it may not provide low operating cost to the system. In [12], voltage stability study with the objective to minimize the total operating cost with voltage stability constraint is proposed. In this study, optimization technique is used to find optimal dispatch of all participating generators. This method, however, involves in solving all necessary conditions and it may not be practical for utilities to use existing software for this study. Based on the above, attention is drawn in this paper to propose an economical generation direction that gives the lowest operating cost with the help of economic load dispatch with load flow (ELDL). Two alternative approaches to identify GD are proposed. The proposed technique can be applied with the help of any existing commercial software for load flow studies. The total operating cost and voltage stability margin resulting from the proposed GD are compared with those of various existing GDs in the modified IEEE 14-bus test system under various system conditions. Merits and demerits of all the methods are also discussed and compared. This paper is structured as follows: Section 2 briefly explains the concept of the static voltage stability. In Section 3, existing methods for determining generation directions for static voltage stability are summarized. The proposed GD based on economical load dispatch with load flow is presented in Section 4. Test systems along with analysis tools used throughout the paper are given in Section 5. Numerical results with discussion are presented in Section 6. Finally, a summary of the main conclusions and contributions of the paper is presented in Section 7.
In static voltage instability, slowly developing changes in the power system occur that eventually lead to a shortage of reactive power and declining voltage. This phenomenon can be seen from the plot of the power transferred versus the voltage at receiving end. The plots are popularly referred to as PV curve or “nose” curve. The maximum load that the system can support before reaching the collapse point is called loading margin of the system. As the power transfer increases, the voltage at the receiving end decreases. Eventually, the critical (nose) point, the point at which the system reactive power is out of usage, is reached where any further increase in active power transfer will lead to a very rapid decrease in voltage magnitude. Before reaching the critical point, a large voltage drop due to heavy reactive power losses can be observed. Voltage stability margin of the system can be enhanced by various methods. Introducing reactive power sources, i.e. shunt capacitors or FACTS device at the appropriate location is an effective way to increase loading margin. This method, however, requires some investment cost. Moreover, these devices may be problematic in some system due to many reasons such as adverse interaction between FACTS devices and existing facilities, inability to control reactive power and voltage at the connected bus of shunt capacitors, etc. [8]. Another effective method to increase the loading margin of the system is by reducing reactive power losses occurred in the transmission system. This can be done by dispatching generation from existing generators to redirect power from congested transmission lines to less congested ones. If the losses in transmission system are reduced, more reactive power can be transferred to the load, thus increasing LM of the system. However, if one wants to reduce the total operating cost the existing generation facilities should be increased in a direction based on economic load dispatch.
2. Static voltage stability
The generation direction or KG of each generator is very crucial to voltage stability. Customarily, the generation of each participating generator is raised at the same rate or a pre-defined rate and it is referenced as conventional approach in this paper. Existing methods to identify GDs in static voltage stability margin are summarized below.
Static voltage instability is mainly associated with reactive power imbalance. The loadability of a bus in a system depends on the reactive power support that the bus can receive from the system. Both reactive and real power losses increase rapidly as the system approaches the maximum loading point or voltage collapse point. In general reactive power losses are higher than the real power losses. Thus, reactive power support should be local. By providing reactive power locally, real power can be transferred with lower losses and hence the total operating cost can be reduced.
3. Generation direction Generation direction or “pattern” is the rate of change of generation of generators that have adequate reserve to serve a load increase from the base load. Let KGi be the direction of generator active power increase and PGi,o be the generation at the base case of generator i, then generator output for a load increase is given by (1): PGi = PGi,o (1 + KGi )
(1)
3.1. Conventional approach Conventionally, the generation of the system is increased by a fixed percentage as pre-specified in the planning stage. The real power output of generator i after the load increase can be
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written as: PGi = PGi (1 + KGi ) = PGi,o + PGi and PG =
PGi = PD + Ploss
(2)
(3)
NG
where PGi is the generation power of generator i, PGi,o the generation of generator i at base load, PG the total generator increase of all participating generators, PGi the increase of generation at generator i, PD the total load increase, Ploss the total loss increase and NG is the number of participating generators. 3.2. Cost participation factor (Cost PF) approach Cost participation factor is the easiest and simple way to determine generation direction to obtain economic load dispatch without considering losses in the system. Generator cost participation factors are calculated based on generators’ incremental cost [11,13] as follows: 1/C PGi = NG i PD j=1 (1/Cj )
(4)
|PGi |max the lower and upper power limits of generator i, |Ui |min , |Ui |max the lower and upper limits of voltage magnitude at bus i, Pij , Qij , Sij the real, reactive and apparent power in line ij and Sij,max is the MVA (Thermal) limit of line ij. Generation direction, in this approach, can be worked out by subtracting the new dispatch from the old dispatch for individual generators. This approach, however, requires all necessary conditions to be simultaneously solved for all state variables including the generation dispatch, hence computationally very demanding. 3.4. Maximum loading margin approach The MLM approach is a method to identify a vector of the GDs of generators that gives maximum LM by approximating the surface of the LM as a function of the generation direction [11]. If one can approximate the LM surface as a function of all generation direction variables (KGi ), optimization techniques can be used to provide the highest LM point. Mathematically, the method can be formulated as follows: Maximize
where Ci is the cost function of generator i and Ci is the second derivative of the cost function i.
LM = Bo +
3.3. Optimal power flow (OPF) approach
subject to
Optimal power flow can be formulated to include voltage stability criteria as follows [12]: Minimize C(PGi ) =
⎛ ⎝
n p=1
NG
⎞
(5)
NG
subject to PGi −(1+λ)PDi,o −
n
|Ui ||Uj |(Gij cos δij + Bij sin δij ) = 0
j=1
QGi −(1+λ)QDi,o −
n
(6) |Ui ||Uj |(Gij sin δij −Bij cos δij ) = 0
j=1
(7)
|PGi |min ≤ |PGi | ≤ |PGi |max
(8)
|Ui |min ≤ |Ui | ≤ |Ui |max Sij = Pij2 + Q2ij ≤ Sij,max
(9) (10)
where C is total operating cost of the system, aGi , bGi , cGi the cost coefficients of generator i, λ the load incremental parameter or loading factor (LF), PGi , QGi the real and reactive power generation at bus i, PDi,o , QDi,o the real and reactive power demand at bus i at base load, n the number of buses in the system, |PGi |min ,
p
Bi,p KGi ⎠
KGi = 1
(11)
(12)
i=1
0 ≤ KGi ≤ 1 2 aGi PGi + bGi PGi + cGi
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(13)
where KGi is generation direction of generator i, Bi,p the coefficients terms, Bo the constant term, p the power term and n is the order of the approximation of the polynomial equation. If generation is increased according to this direction, the system will have the maximum loading margin. Among the existing methods, only optimal power flow method provides lowest operating cost as the operation cost is minimized by using optimization technique. This method, however, involves in solving all necessary conditions and may not be practical for utilities that use existing commercial software for the studies. In the following section, economic generation direction is proposed for minimizing operating cost with voltage stability constraints based on ELDL. ELDL method can be applied in any existing commercial software tool for load flow studies. 4. The proposed approach Economical generation direction (EGD) can be calculated in two steps. In the first step, Step I, economic load dispatch is carried out with load flow to find the optimal generation. In the second step, Step II, the economical GD is computed based on the optimal generation found in Step I.
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4.1. Step I In the first step, ELDL is formulated to include voltage stability criteria as follows: Minimize C(PGi ) =
2 aGi PGi + bGi PGi + cGi
(14)
NG
subject to: PGi −(1+λ)PDi,o −
n
|Ui ||Uj |(Gij cos δij + Bij sin δij ) = 0
j=1
QGi −(1+λ)QDi,o −
n
(15) |Ui ||Uj |(Gij sin δij − Bij cos δij ) = 0
j=1
(16)
|PGi |min ≤ |PGi | ≤ |PGi |max
(17)
|Ui |min ≤ |Ui | ≤ |Ui |max Sij = Pij2 + Q2ij ≤ Sij,max
(18)
λ ≤ λmax
(20)
(19)
where λ is load incremental parameter or loading factor (LF), PGi and QGi the real and reactive power generation at bus i, PDi , and QDi ,o the real and reactive power demand, respectively, at bus i at base load, n the number of buses in the system, |PGi |min and |PGi |max the lower and upper power limits of voltage magnitute at bus i, Pij , Qij , and Sij the real, reactive and apparent power in line ij, Sij,max the MVA (Thermal) limit of line ij; and λmax is load incremental parameter at the voltage collapse point. The ELDL is an iterative approach that computes the optimal generation of all participating generators at specific loading level based on sensitivity method. It can be viewed as an extension to the cost participation factor method, where the operating cost of individual generator is considered without total power losses of the system. The procedure behind the ELDL approach can be summarized as shown in Fig. 1. The load is increased at each loading factor in the outer loop. In the inner loop, economic load dispatch and load flow calculation are repeatedly performed to obtain the optimal solution. Note that the maximum number of load flow iterations in this study is set to 200. The ELDL method provides optimal dispatching generation of each generator, based on optimality condition [13,14]. ELDL is used instead of optimization technique to make use of existing commercial software for the study. In optimization technique, the necessary conditions are required to be solved for all state variables. This may be a problem for practical sized power system and for utilities, who wish to use existing software for the studies. ELDL approach, on the other hand, can be applied with the help of the existing commercial software, since it is based on load flow and sensitivity method.
Fig. 1. Illustration of ELDL process for static voltage stability study.
Although ELDL approach is a practical solution for utilities, it faces difficulties in obtaining the solution when the system is close to the collapse point. At that point, sensitivity of the losses to the generation increase is high, which provides convergence problem. In order to use ELDL approach for voltage stability study, EGD based on dispatching generation is proposed and used in CPF process to avoid convergence difficulties in ELDL during the stressful condition. EGD calculation based on ELDL method is explained in Section 4.2. 4.2. Step II Two alternative methods are proposed for finding the EGD from optimal generator dispatches. The first method, named as “EGDI”, is based on dispatching generation dispatches at various loading levels. The second method, on the other hand, is based on the slope of generation increase. The second method is named as “EGDII”. 4.2.1. EGDI GD of generator i at each LF can be found from the ratio between the generation increase of the generator i and the total generation increase of all participating generators. This can be formulated by (21): KGi =
PGi PGi = PG NG PGi
(21)
where PGi is the increase of power generation at generator i and PG is the total generator increase of all participating generators.
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GD of generation i or KGi are found at various LFs. If these GDs are converged to some values at high LF, EGD can be found from these converged values. EGD is approximated from the trend of the GDs at various LFs based on this technique. 4.2.2. EGDII The second approach to identify EGD is based on the slope of the generation increase. If the graph of the generation increase of generation i with respect to LF is plotted and it can be represented by a linear equation with the slope KGi,un . The equation of generation increase with respect to LF or λ for generator i can be represented by: PGi = λ × KGi,un
(22)
where KGi,un is the slope of plot between generation and LF for generator i. The total generation increase of all participating generators is, therefore, PG = (23) PGi = λ × KGi,un NG
NG
Thus, EGD or KGi of the generator i can be obtained by: PGi KGi,un = NG PGi NG KGi,un
KGi =
(24)
Eq. (24), EGDII, is as same as Eq. (21), known as EGDI, thus resulting in the same EGD. EGD based on EGDI is obtained from the final converged values of GDs while EGD based on EGDII is obtained from the slope of the generation increase. Compared to EGDI, EGDII may be of interest to utilities because it needs only two dispatching points to find EGD. The EGD based on the proposed method is used in CPF or direct method to obtain LM in the stressful condition, since ELDL approach itself may not provide the solutions during stressful condition.
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5. System and analysis tools A single line diagram of the modified IEEE 14-bus test system is depicted in Fig. 2, which consists of five synchronous machines, including two synchronous compensators used only for reactive power support and three generators located at buses 1, 2 and 6. The modification is only at generator 3 located at bus 6, changing from a synchronous condenser to a generator. There are 20 branches and 14 buses with 11 loads totaling 259 MW and 81.4 Mvar. Results presented in the paper were produced with the help of UWPFLOW [15] and tools developed in MATLAB [16]. UWPLOW is commercial grade software that has been designed to calculate load flow and various static voltage stability parameters including the maximum loading margin of the power system. In ELDL approach, a program developed in MATLAB is used to manipulate UWPFLOW for load flow calculation in iterative process. Optimization toolbox in MATLAB is used to provide the solution for ELD in ELDL process [16]. To validate the proposed approach with other existing methods, voltage stability study is done with the help of UWPFLOW. In this study, in order to obtain the PV curves hence the loading margin of the system for different cases, all the loads were represented as constant PQ and increased simultaneously, i.e. by keeping constant power factor, as follows [17]: PD = Po (1 + λ) QD = Qo (1 + λ)
(25)
where Po and Qo correspond to the base loading conditions and λ is the loading factor.
Fig. 2. Single line diagram of the modified IEEE 14-bus test system.
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6. Numerical results The EGD approach is validated by comparing the results with those of three existing methods, namely, conventional, Cost PF and MLM methods. The results are based on the modified IEEE 14-bus test system for various system operating conditions, including base case and N − 1 contingencies. 6.1. Base case system 6.1.1. Cost functions, the starting point and cost participation factors Quadratic cost functions are used for all three generators throughout the study. The cost coefficients, minimum and maximum generation limits, and Cost PF of each generator computed using (4) are shown in Table 1 [10,14]. Note that, at the starting point, generator 2 has the lowest incremental cost, whereas generator 3 gives the highest incremental cost. The cost functions of all generators are almost linear, as the coefficients of quadratic term are close to zero. In order to compare loading margins resulting from the different generation directions, initial points should be set at the same values for all cases. The ELDL is used to produce the starting point, since it gives the lowest operating cost while considering the network losses. The optimal generation set points of generators 1–3 are 150, 77.94, and 40 MW, respectively, at the base load of 259 MW. 6.1.2. MLM method [11] The MLM method is based on the loading margins of the system at various possible generation directions in the “generation direction space”. The plot of LM in generation direction spaces of generators 2 and 3 is illustrated in Fig. 3. Note that the generation direction of generator 1 is 1 minus the generation directions of generators 2 and 3 combined, i.e. 1 − (KG2 + KG3 ). From Fig. 3, it is obvious that the maximum loading margin of the system occurs at generation directions 0.7, 0.3 and 0 for generators 2, 3 and 1, respectively. As expected, increasing generation at generators 2 and 3 can reduce the losses and increase the LM; generator 2 has more influence in the system losses than generator 3, since it is situated closer to most of the load. If the generators are increased according to this direction, the system will have the maximum LM.
Fig. 3. Approximate LM surface at all possible GDs.
ELDL is used to find the optimal generation at each loading level based on the methodology presented in Fig. 1. Fig. 4 shows generation of each participating generator at various loading factors based on ELDL method. From Fig. 4, the curves of generation are linear for all participating generators in a given case. It is noticed from Fig. 4 that the solution can be obtained up to 0.8 LF. ELDL method faces convergence problem, at LF higher than 0.8 p.u., due to high sensitivity of losses with respect to the generation increase. 6.1.3.1. EGDI. The generation direction based on EGDI can be found from Eq. (21) for various LFs. Fig. 5 shows GDs for all participating generators at various LFs. It can be observed from the figure that GDs of generators 1–3 are converged to 0, 0.67 and 0.33, respectively, at higher LFs. These converged GDs are EGD, which is used for LF higher than 0.8 p.u. in the stressful condition based on CPF process. 6.1.3.2. EGDII. An alternative way to find the EGD is to use the slope of the generation increase, which is shown in Fig. 4. From Fig. 4, the slopes of generation at generators 1–3 are 0, 1.9074,
6.1.3. Economical generation direction The proposed EGD is found based on the methodology proposed in Section 4. First, ELDL method is used to find generators output as explained in Section 4.1. Next, EGD is computed from the generators output as explained in Section 4.2. Table 1 Cost coefficients, limits, and Cost PF of all generators G
a
b
C
Pmin
Pmax
Cost PF
1 2 3
0.001562 0.001940 0.004820
7.92 7.85 7.97
561 310 78
150 40 40
500 500 500
0.469655 0.378145 0.152200 Fig. 4. ELDL based generation outputs.
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Fig. 6. PV curves with different generation directions.
Fig. 5. Economical GDs of generators.
and 0.95195 p.u., respectively. These slopes are used to find EGD based on Eq. (24), which are 0, 0.67, and 0.33 for generators 1–3, respectively. Table 2 shows EGDs obtained from EGDI and EGDII. GDs obtained from these two methods are identical. EGDI is based on dispatching generation and it needs many dispatching points. EGDII, on the other hand, requires the slopes of the generation and it can be used when only two dispatching points are known. EGDII may be of interest to utilities since the computation effort is less. If the generations are increased at these directions, the system will have the lowest operating cost. EGDI and EGDII methods provide the same GD, thus in the following subsections only EGDI method is used to compare with the existing GD methods. EGDI is onwards named as EGD. 6.1.4. PV curves and loading margins In order to compare the static voltage stability margins of EGD with that of other GDs, the thermal and voltage limits are neglected. However, the limits and constraints can be easily incorporated in the proposed approach compared to the other existing generation direction methods. The generation directions used for conventional method is 0.85, 0.15 and 0 [8], while, for Cost PF method, they are assigned at 0.469655, 0.378145 and 0.152200 for generators 1–3, respectively, as given in Table 1. For MLM method, 0, 0.7 and 0.3 are used for generation direction of generators 1–3, respectively [11]. EGD based on the proposed methodology is 0, 0.67 and 0.33 for generators 1–3, respectively, as shown in Table 2. The PV curves of the test system with various generation direction methods, namely, conventional, Cost PF, EGD and MLM methods are depicted in Table 2 Economical GD of EGDI and EGDII methods Generator i
1 2 3
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Fig. 6. From Fig. 6, as expected, the MLM method offers the highest loading margin followed by EGD, Cost PF, and conventional methods. The maximum LMs based on conventional, Cost PF, EGD and MLM methods are given in Table 3. From Fig. 6 and Table 3, the MLM approach yields the highest LM since it attempts to maximize LM of the system. The conventional method, on the other hand, provides the lowest LM since the GD is fixed. Cost PF approach gives higher LM than the conventional approach. The MLM and EGD methods give almost the same LM and a better voltage profile compared to other generation directions as shown in Fig. 6. Total real power losses of the system based on each generation direction method are compared in Fig. 7. From the figure, it can be seen that EGD method offers lowest losses and incremental losses followed by MLM, Cost PF, and the conventional methods, respectively. MLM method, on the other hand, results in the maximum loading margin with small difference in the losses compared to the EGD approach. The conventional and Cost PF generation direction approaches lead to high losses since they do not consider the network losses, hence, making the loading margin less. The conventional method results in more losses than Cost PF approach. 6.1.5. Total operating cost and generation dispatch Total operating cost with various GDs at different loading levels up to the collapse point are depicted in Fig. 8. From Fig. 8, the EGD approach gives the lowest total operating cost at any loading levels. The MLM method gives almost a similar operating cost as the EGD method, but still yielding the highest loading margin. Table 3 Loading margins for different generation directions
EGDI
EGDII
KGi
Slope of PGi (KGi,un )
KGi
0 0.67 0.33
0 1.9074 0.95195
0 0.67 0.33
Method
GD
LM
Conventional Cost PF EGD MLM
0.85, 0.15, 0 0.469655, 0.378145, 0.1522 0, 0.67, 0.33 0, 0.7, 0.3
0.8473 0.9870 1.0927 1.0931
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Fig. 9. Cost difference between conventional and EGD approaches.
Fig. 7. Total real power losses with GDs.
6.2. Contingency cases
Fig. 8. Total operating cost at various loading factors for different generation directions.
Fig. 9 shows the difference in cost between the conventional and EGD methods. The total operating cost resulting from the EGD method is much lower than that of conventional method, especially in high loading condition. The improvement in cost is an exponential function of LF. Information of cost at the collapse points (CP), respective loading margins and the ratio of Cost/LM are given in Table 4. From the table, the EGD gives the lowest ratio of Cost/LM while MLM is the second best. EGD and MLM method offer a close value of Cost/LM. Table 4 LM, cost at LM and cost of all methods Method
LM [p.u.]
Cost at CP [R/h]
Cost/LM [R/h]/p.u.
Conventional Cost PF EGD MLM
0.8473 0.9870 1.0927 1.0931
5613.3 5858.0 6082.5 6089.7
6624.9 5935.2 5566.5 5571.0
In case of contingencies, the system becomes more stressful and results in lower loading margin [8,18]. For simplicity, only line contingencies are considered in this study. The contingency ranking of the system can be carried out by the CPF method, which is based on the actual loading margin resulting from a contingency. The first three worst contingencies are the outages of lines 1–2, 2–3 and 1–5, respectively. Loading margins, cost at LM and Cost/LM for the line outage cases are given in Table 5 for all the GD methods. From Table 5, it is obvious that EGD and MLM methods provide the highest LM compared to other methods for all contingency cases. Cost/LM using EGD approach is lowest for all the contingency cases and it is close to that of MLM method. This can be better viewed in Fig. 10, where the values of Cost/LM are plotted for various methods for all contingency cases. The MLM generation direction should give a higher loading margin among all the generation directions as the objective of the method is maximizing the loading margin. However, due to the step size considered and the approximation in the continuation Table 5 LM, cost at LM, Cost/LM of all methods for various line outages Contingency
Method
LM [p.u.]
Cost at LM [R/h]
Cost/LM [R/h]/p.u.
1–2
Conventional Cost PF EGD MLM
0.17806 0.27535 0.43479 0.43323
3802.5 3998.1 4300.9 4294.2
21355 14520 9892 9912
2–3
Conventional Cost PF EGD MLM
0.33038 0.36775 0.40846 0.40495
4343.8 4400.9 4480.4 4473.6
13148 11967 10969 11047
1–5
Conventional Cost PF EGD MLM
0.50415 0.62186 0.74672 0.73990
4752.2 4960.1 5194.4 5181.0
9426 7976 6956 7002
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ties, who wish to use existing commercial software for obtaining the right generation direction. In practical system, the lowest operating cost should be obtained if the voltage stability criterion is not violated. The information on appropriate generation directions can be made available on-line to assist system operators. References
Fig. 10. Cost/LM of all the methods with various contingency cases.
power flow process the margins obtained with MLM generation direction is slightly less than that obtained with EGD. Based on the results, EGD approach provides the lowest operating cost with voltage stability constraint for all the cases in the modified IEEE 14-bus test system. From the operation viewpoint, one would suggest to operate the system according to the EGD method in non-stressed conditions, if the solution is attainable, and according to MLM method in stressed conditions. This will certainly give low operating cost and maximum loading margin in the lower and higher loading conditions, respectively. In transition from one loading point to the other, operator can re-dispatch the generation to fit the appropriate approach based on the margin requirement and total cost trade-off. 7. Conclusion In this paper, an economical generation direction is proposed for static voltage stability. The proposed generation direction is compared with three other generation directions, namely conventional, cost participation factor and MLM, on loading margin, total losses and total operating cost in the modified IEEE 14-bus test system for various system conditions. The EGD gives the lowest total operating cost at any loading level for different contingency conditions. The proposed generation direction can be obtained with the help of any existing commercial software with more realistic limits and constraints. This is useful for utili-
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