1. Introduction
Congestion of transmission lines occurs when the networks fail to accommodate all the desired transactions due to the system operating limits such as branch power flow limits, voltage limits, etc. Congestion in one or more transmission lines leads to higher risk of electricity consumption, even unexpected widespread power blackouts [1]. Various congestion management approaches suitable for traditional power systems without intermittent energy, have been reported in recent literatures. However, the global rapid growth of wind power capacity increases the uncertainties in congestion management. Therefore, it is necessary to propose an efficient and reliable method to process congestion problem with random variables.
Optimal rescheduling of generators is generally adopted to manage the congestion for its low-cost and simplicity. Ashwani Kumar proposed a zonal sensitivity-based optimum real and reactive power generation rescheduling method for congestion management [2]. Another technique for optimum selection of generators to be rescheduled is demonstrated in [3], which is based on generator sensitivities to the power flow on congested lines. Congestion management techniques in different deregulated electricity markets are estimated in [4]. Besides, congestion management scheme based on optimal power flow (OPF) is an excellent alternative method in a power system with deterministic operational constraints.
An OPF-based congestion management approach proposed in [5] is based on the nodal pricing framework and the pool model. Another OPF-based scheme which aims to minimize both congestion and service costs is presented in [6]. Various kinds of traditional congestion management models are essentially the OPF problems associated with security constraints. As system with high level wind power integration has enhanced uncertainties, a noval probabilistic OPF-based congestion management approach is addressed in this paper.
Cumulants and Gram-Charlier expansion theory are combined to approximate the probabilistic distribution functions (PDFs) of transmission line flows in [7]. As the cumulant method can’t be used to process correlated input random variables, a two PEM [8] is applied. Literature [9] reveals that the 2m+1 PEM provides the best performance among the Hong’s PEMs. So this paper uses the 2m+1 PEM [9-10] to process the PPF in the congestion management. In recent years, compared with other heuristic mothods [11-14], ABC algorithm proposed by D. Karaboga [11] for fewer parameters, faster convergence and higher precision [12] has been widely used to solve complex nonlinear optimization problems, such as congestion management [13], OPF [14], etc.
Congestion management methods available in most literatures use simple power flow method without considering the uncertainties and correlations of the input variables. In this paper, the ABC algorithm combined with an extended 2m+1 PEM method is proposed to process the complex congestion problem with correlated random injections. Moreover, the impact of direct electricity purchase on the results of congestion is discussed.
This paper is organized as follows. The mathematical formulation of the congestion management model is presented in Section 2. Section 3 gives the PPF model coupled with correlated variables in detail. In Section 4, the ABC algorithm coupling with the PPF model to solve the complex congestion problem is proposed. Section 5 gives several case studies. Finally, Section 6 provides the relevant conclusions.
2. Mathematical Formulation
Congestion management is an optimization problem aims to minimize the congestion cost while satisfying the system and unit constraints. This section gives the congestion management model based on both pool and nodal pricing model in detail.
2.1 Objective functions
2.1.1 Cost of rescheduling in the pool model [15-16]
In the pool model, the ISO determines the market-clearing price Ci,M and the output PGi of generator i without considering system constraints aiming to minimize the cost of electricity purchase. The original total cost of electricity purchase can be calculated by:
where NG is the number of generators. When the congestion occurs, the cost of “constraint on” and “constraint off” generators are:
where and are the number of “constraint on” and “constraint off” generators, respectively; is the output of generator i after rescheduling; is the actual price bids submitted by generator i. Combining (1-2), the cost of rescheduling in the pool model is obtained:
In the above formula, the superscript ‘ − ’ denotes the expectation of the random variables.
2.1.2 Cost of breach of direct electricity purchase contract
Actually, the direct electricity purchase contract is a kind of bilateral transactions [16]. If there is no static and dynamic security violation, all the requested contracts or transactions should be satisfied. Otherwise, a breach of contract will occur. Mathematically, the breach cost of a direct electricity purchase contract at generator i can be calculated according to [16] as follows:
where Pi,dc is the trading power of the direct electricity purchase contract of generator i, P’i is the actual active power of generation i, and k is a penalty coefficient of breach of contract.
2.1.3 Congestion cost in different modes
Combining (3-4), the objective function in the pool mode is:
where w is a penalty coefficient trying to perform all the direct electricity purchase contracts.
Though the formula (4) is still applicable in the nodal pricing model [5], the objective function is modified to:
where S is the set of the two endpoints of the branches; Pij is the active power flow from bus i to j; LMPi and LMPj is the nodal prices of bus i and j, which are two endpoints of a branch, respectively. All the nodal prices are calculated by minmizing the cost of generation.
2.2 Constrains and limits
2.2.1 Power flow equations
where Pi, Qi and Vi are the active power, reactive power and voltage amplitude of bus i, respectively; Gij and Bij are conductance and susceptance between bus i and j, respectively; Nbus is the number of buses; θij is the phase angle difference between bus i and j.
2.2.2 Power constraints of generators
where and are the minimum and maximum active power of generator i; and are the minimum and maximum reactive power of generator i; PGi and QGi are the active and reactive power of generator i.
2.2.3 Bus voltage limits
where Pro(·) denotes the probability of the event (·); and are the lower and upper bound of voltage amplitude of bus i; Vi is the voltage amplitude of bus i; αi is the confidence level of the bus i’s constraint.
2.2.4 Line power flow limits
where are the upper limit of transmission power flow of branch i;PFi is the power flow of branch i; βi is the confidence level of the branch i’s constraint; Nbranch is the number of the branches.
2.2.5 Direct electricity purchase contract
In a practical system, not all of the generators have direct electricity purchase contract with loads and vice-versa. Mathematically, if a generator at bus i have this contract the active power inequality constraint of (8) is modified into:
For the objective function has already considered the penalty term of this constraint violation, only constraint (8) need to be taken into account.
3. PPF Model Managing To Process Correlations
To perform the impact of the uncertainties of loads and wind farms (WFs), probabilistic models are established. Since the correlations among loads and WFs do affect the power flows [17], a modified 2m+1 PEM [10] capable of processing the correlations is introduced.
3.1 Probabilistic load model
Generally, load demand is supposed to follow a normal distribution [10, 17]. So the active and reactive power of load bus L can be expressed by:
where μPL and σPL are the mean and standard deviation of active load; μQL and σQL are the mean and standard deviation of reactive load; exp(·) represents the exponential function.
3.2 Probabilistic correlated WFs model
Different from the probabilistic load model, the output of WFs depends on the wind speed which follows the Weibull distribution. Thus, the joint PDF of the correlated WFs’ output at bus t can be obtained through the correlated wind speed using Monte Carlo method.
The PDF and cumulative distribution function (CDF) of the wind speed vj of the j-th WF at bus t are as follows [10]:
where kj is the shape factor and λj is the scale factor of vj. Weibull distributed wind speed can be transformed into normally distributed variable through Nataf transformation.
The correlated wind speed sampling matrix VNs×NWFs,t = [v1,v2,⋯,vNWFs,t] (Ns represents the sampling times) of the NWFs,t WFs at bus t can be generated as follows:
1. Generate a matrix RNs×NWFs,t =[r1, r2,⋯,rNWF,t] by the Matlab random number generator which represents NWFs,t independent standard normal distributed random variables. 2. For the given correlation coefficient matrix Cg of the wind speeds, the modified correlation coefficient matrix Cmd by the Nataf transformation [19] are obtained using [10, 20]:
where and are the elements at m-th row, n-th column of Cg and Cmd, respectively; γm and γn are the coefficients of variation of vm and vn. Then decompose Cmd by the Cholesky decomposition method [10, 21] into Cm=LLT, where L is an inferior triangular matrix.
3. Using the transformation Y=LR and the inverse Nataf transformation we can obtain the matrix VNs×NWFs,t = [v1,v2,⋯,vNWFs,t] with NWFs,t Weibull distributed variables with a correlation coefficient matrix Cd.
Next, the wind speed vector vj from the j-th column of V is used to determine the output column vector PWF,j [Ns×1] of WF j with nj WTGs at the bus t as follows:
where Pr is the rated power of a single WTG; vin, vr and vout are the cut-in, rated and cut-out wind speed, respectively; nj is the number of WTGs at the bus t.
Applying V and (16), the output sampling matrix PWFs [Ns×1] of bus t with NWFs,t WFs is calculated by:
Using PWFs, the mean μWFs,t and standard deviation σWFs,t of the WFs’ output at bus t can be obtained:
where PWFs(i) is the i-th element of PWFs. The z-th (z>2) order standardized central moments of the bus t with WFs can be calculated as:
So far, probabilistic models of loads and WFs have been established. No matter the random injections are continuous or discrete, the traditional 2m+1 PEM can be just applied to uncorrelated ones. Thus, Part C introduces a modified 2m+1 PEM capable of dealing with correlated random injections at each bus.
3.3 2m+1 PEM for correlated random variables
As mentioned above, independent input random variables are required in the 2m+1 PEM proposed in [9, 10]. For this purpose, the orthogonal transformation [10] based on Cholesky decomposition method [21] is used. A detailed description of the orthogonal transformation is given in [10], which can convert a set of correlated input variables into an uncorrelated one. Based on the principle of the 2m+1 PEM [10], processing the correlations of the input random variables is to process the correlations of their standardized central moments. The 2m+1 PEM for correlated input random variable is as follows:
Step 1. According to the correlation coefficient matrix of the input random variables p=[p1, p2,…, pm]T, obtain the variance-covariance matrix Cp. Then get the matrix B by the Cholesky decomposition method using Cp=LLT and B=L-1.
Step 2. Transform the correlated input variables p into a new set of independent variables q=[q1, q2,…, qm]T whose first four central moments satisfy:
where μp and μq are the mean vectors of p and q;σp and σq are the standard deviation vectors of p and q; Im is the m-dimensional identity matrix; λql,j (j=3,4) are the coefficients of skewness and kurtosis of ql; bli is the element at the l-th row, i-th column of B.
Step 3. Calculate the new transformed pairs (ql,k, ωl,k) of independent q defining the new 2m+1 PEM using follows:
where ξl,k can be calculated by:
Step 4. Construct the new 2m+1 points in the form (μq1,⋯,ql,k,⋯,μqm), k=1,2 and (μq1,⋯,μql,⋯,μqm). Let q2m+1,k, k=1, 2, 3, be a m×m matrix each row of which is one point of the 2m+1 points with l from 1 to m. Then transform the 2m+1 points to the original space using p2m+1,k=B−1q2m+1,k.
Step 5. Calculate the deterministic power flow for each row of p2m+1,k for 2m+1 times using:
The solution vectors si(l, k) is obtained.
Step 6. Estimate the the j-th raw moment using:
And the PDF and CDF of the output random variable si can be calculated through Gram-Charlier expansion [7, 16]:
where φ(·) and Φ(·) are the PDF and CDF of standard normal distribution, respectively; ci are constant coefficients of which detailed calculation can refer to [7, 16]; μsi and σsi are the mean and standard deviation of si.
4. Solution Method
A PPF model capable of managing correlated random injections has been described in the previous section to simulate the uncertainties and correlations in a congestion management problem. This section will give a detailed introduction of the ABC algorithm coupling with the PPF model to solve the complex congestion model.
4.1 Overview of the ABC algorithm
Initially, the ABC algorithm is proposed for optimizing numerical unconstrained problems, which is a swarm-based meta-heuristic algorithm [22]. Then it is modified in [23] to handle constrained optimization problems.
In the ABC algorithm, every food source represents a possible solution of an optimization problem, and nectar amount of a food source represents the fitness of the corresponding food source. The process of artificial bees’ searching for the best food source is the optimization process. The colony of artificial bees includes three groups of bees: employed bees, onlookers and scout bees. The search of food source implemented by the artificial bees can be summarized as following: 1. Employed bees find the food source within the neighborhood of the previous food source in their memory and record the nectar amount of the new food source. 2. According to the information offered by the employed bees, onlookers judge the merits of the food source and select a food source probabilistically. 3. Employed bee at abandoned inferior food source becomes a scout bee and starts search a new food source randomly.
The main steps of the ABC algorithm are as follows:
Step 1. Initialize the randomly distributed food-source positions Xi=[xi,1, xi,2, …, xi,d] (solutions population) according to the upper and lower limits of the decision variables, where i=1,2,…,N (N represents the number of employed bees, onlooker bees and food sources), xi,j(j=1,2,…,d) represents the j-th decision variable of the solution Xi and d is the number of decision variables.
Step 2. Compute the nectar amount of the food source Xi using their fitness values:
where f is the objective function value at solution Xi.
Step 3. Determine neighborhood positions for the employed bees according to the exiting food-source positions using:
where xi,j is the j-th parameter of solution Xi that was selected to be modified; U is a random number between [-1,1]; k≠i and k ∈ {1,2,…,N} ; i ∈ {1,2,…,N} ; j ∈ {1,2,…,d}. Then record the fitness values of the new neighborhood positions using (27).
Step 4. If the fitness value of a new neighborhood position is larger than the old one, replace the old one with the new one; otherwise, keep the old one.
Step 5. Calculate the selection probability Probi of the solution Xi for the outlook bees applying:
Step 6. Select the onlooker bee depending on the probability value. For the selected onlooker bee Xi, a new neighborhood position is created using (28); else go to Step 8. Then record the fitness values of the new neighborhood positions using (27).
Step 7. Follow the Step 4.
Step 8. Find the abandoned food sources for scout bees. If a food source is still not updated by a predetermined number of trials known as ‘limit’ value Limmax, then that food source is abandoned and the corresponding employed bee becomes a scout. Otherwise, no abandoned food sources exist and go to Step 10.
Step 9. For an abandoned food source Xi, update it with a completely new food source Vi through:
where ui is a random number between [0,1]; and are the maximum and minimum parameter of Xi, respectively.
Step 10. Storage the global best solution obtained so far.
Step 11. If the current iteration number (Iter) is larger than the maximum iteration number of the search process (Itermax), stop and output the results. Otherwise, go back to Step 3.
There are three control parameters need to be set: 1) food-source size N, representing the number of employed bees or onlooker bees; 2) ‘limit’ value, which is the number of trials determining a food-source position abandoned or not (at least 0.5 × N × d suggested in [11]); 3) Itermax, that is the maximum iteration number.
4.2 Congestion management strategy
After alleviating congestion in transmission grids, the congestion may occur again due to the uncertainties and correlations of loads and wind power. Moreover, direct electricity purchase contracts affect the rescheduling of generators obviously. In case of congestion again, all the generators participating in congestion management must be rescheduling properly. This paper proposes a congestion management approach using PPF considering direct electricity purchase. If congestion cannot be removed just by rescheduling, a breach of contract is done between contracted parties based on its liquidated damage.
This paper uses the following methods to handle the constraints (7-10). Mathematically, equality constraint (7) is solved during the determined power flow calculation. Inequality constraint (8) can be handled as follows:
Reactive power constraints of generators are processed by the similar method. Besides, if reactive power generation of any PV bus gets violated, the PV bus is treated as PQ bus.
The chance constraints (9) and (10) are handled as follows:
4.2.1 Calculate the penalty terms:
where Pen_V and Pen_P are the penalty terms of bus voltage limits and line power flow limits; ΔVi and ΔPi are computed by:
where and
4.2.2 Modify the original objective function f into:
where wV and wP are the penalty factors.
4.3 ABC algorithm coupling with PPF model
ABC algorithm coupling with PPF model is proposed to solve the congestion management problem with correlated random injections. The flowchart of the proposed ABC algorithm is illustrated in Fig. 1. Further studies in [22-28] has proved that the ABC algorithm has a better performance in results and solutions compared with other popular population-based and heuristic optimization algorithms.
Fig. 1.Flowchart for the ABC algorithm coupling with PPF
5. System Studies
The proposed congestion management approach has been illustrated on the modified IEEE 30-bus test system [29-30] and IEEE 57-bus test system [31] in the pool model in 5.1 and 5.2. The performance of the ABC algorithm is compared with that of evolutionary algorithm (EA) [32] and particle swarm optimization (PSO) [32] in 5.1. Similar study of IEEE 30-bus is conducted in nodal pricing model in 5.3. All the studies are implemented in MATLAB 2012a.
5.1 Modified IEEE 30-bus case study
The modified IEEE 30-bus test system consists of 6 generators (Table 1), 41 branches, 20 load buses whose data are from Table 2-3 of [30], and 2 WFs located at bus 28 as shown in Fig. 2. The active power consumption of each load is considered to be normally distributed with means equal to the values provided in Table 3 of [30] with the value at bus 5 zeroed out, and standard deviations of 10% with respect to such mean values. For simplicity, the power factor of each load is kept constant. Each WF contains five 3MW WTGs and the wind speed is assumed to follow the Weibull distribution with scale and shape parameters 9, 2.205, respectively. Both WFs are correlated with a correlation coefficient 0.9 and a power factor 0.9 lag. Besides, the involved parameters are set as follows: wV=106, wP=106, Itermax=200 and Limmax=100.
Fig.2.Modified IEEE 30-bus test system
Table 1.Bidding function: fB(P)= a P + b $/MWh
Table 2.Probable congested line details of 30-bus system with different rL (Line limit 32MVA)
Table 3.Comparison results of different algorithms
Firstly, the original outputs of the generators are calculated by minimizing the electricity purchase cost without constraints (9)-(11). Here, the output of the WFs and the loads are assumed to be independent and the correlation coefficients among load buses are rL. Table 2 gives the details of the line congestion in condition of the original outputs with rL=0, 0.2, 0.4, 0.5, 0.6, 0.8, 0.9, respectively. From Table 2, with rL increased from 0 to 0.9, the standard deviation of the power flow through the congested line 6-8 grows from 3.7584MVA to 3.9340 MVA, which means a 4.67% increase, while the expected power flow has no significant increase. Also, if the confidence level of the chance constraints (9)-(10) is 0.95, branch 6-8 is the only congested line.
Fig. 3 illustrates the evolution of the congestion probability of branch 6-8 with rL=0.5. As the line limit increase from 32MW to 40MW, the congestion probability decrease from 0.7662 to 0.0854. In order to obtain a wide range of feasible region for congestion management, the line limit of branch 6-8 is raised to 40MW. Table 4 gives the simulated results of the congestion management with rL = 0.5 under different confidence levels. From the table, the cost of congestion management increases obviously with the confidence level raises, especially when αi = βi>0.95. What’s more, when αi = βi = 0.97, Pen_P>0 which means the line congestion can’t be eliminate just by reschedule the generators. To avoid the violations without load shedding, the line connected with or near the WFs must be expanded.
Fig. 3.Line limit (line 8-9) effect on the congestion probability
Table 4.Confidence level effect on results of congestion management (rL=0.5)
According to the previous study, the parameters of the system are set as follows: limit of line 6-8 is 40MW, rL=0.5 and αi = βi = 0.95. Different combinations of market structures comprising pool model and mix of pool plus direct electricity purchase contracts taken for study are: P: pool model without direct electricity purchase contracts independent; P1: pool model with one direct electricity purchase contract between buses 2–8; P2: pool model with two direct electricity purchase contracts between buses 2–8 and 27–7; P3: pool model with two direct electricity purchase contracts between buses 2–8, 12; P4: pool model with three direct electricity purchase contracts between buses 2–8, 12 and 27-7; P5: pool model with four direct electricity purchase contract between buses 2–8, 12 and buses 27–7, 24.
Here, we assume that each direct electricity purchase contract is the total active load consumption of the load bus, k=8 and w=10.
Table 3 gives the comparison results of the performance of ABC algorithm, EA [32] and PSO [32] for market structure P using the same population and iteration. From the table, the ABC algorithm has a better performance in the optimization results and can be used to solve the congestion problem properly.
The detailed simulated results and rescheduling of generators are present in Table 5 and Fig. 4. The table and the figure demonstrate that direct electricity purchase contracts decrease the feasible region of generation rescheduling. Comparing P1-P2 and P3-P4, the added contract between buses 27-7 have no effect on the existing contracts. However, some added contract may affect the rescheduling results significantly based on the comparison of P-P1, P1-P3 and P4-P5. Especially, in model P5, G27 needs to cut the contract by 0.92MW to make the system relieve the probable congestions. This can guide the generators to sign rational direct electricity purchase contracts.
Fig. 4.Active power rescheduled of generators for modified 30-bus system
Table 5.Simulated cases for modified 30-bus system (total load 189.2MW)
5.2 Modified IEEE 57-bus case study
The modified IEEE 57-bus test system [31] consists of 7 generators, 80 branches and 42 load buses. WFs are located at bus 7 and 46 and each of them has 2 WFs with a correlation coefficient 0.9 and a power factor 0.9 lag. The two WF buses are correlated with a correlation coefficient 0.8 but they are both independent with other load buses. The characteristics of loads are set as the 30-bus system. Besides, we set the limit of branch 8-9 215MVA, rL=0.5, αi=βi=0.95, wV=109, wP=109, Itermax=200, Limmax=100, k=8 and w=103. Table 6 gives the detailed data of generators for the modified 57-bus system.
Table 6.Bidding function: fB(P)= a P + b $/MWh
Different combinations of market structures comprising pool model and mix of pool plus direct electricity purchase contracts are as follows: C: pool model without any direct electricity purchase contracts. C1: pool model with a direct electricity purchase contract between buses 6-8. C2: pool model with two direct electricity purchase contracts between buses 6-8 and 8-12 (320MW). C3: pool model with two direct electricity purchase contracts between buses 6-8, 13. C4: pool model with three direct electricity purchase contracts between buses 6-8, 13 and 8-12 (320MW). C5: pool model with three direct electricity purchase contracts between buses 6-8, 13 and 8-12 (350MW).
Here, all direct electricity purchase contracts are the total active load consumption of the lo ad bus except those with a brackets mark.
Table 7 and Fig. 5 depict the simulated results for modified IEEE 57-bus system in different market structures. Original outputs of generators are optimization results aiming to minimize the electricity purchase cost without constraints. From Table 7, congestion cost increases as the number of direct electricity purchase contracts grows overall. Some direct electricity purchase contracts added to the system have no effects on the original system (comparing C1-C2) but some may have (comparing C3-C4). In C5, in order to remove the congestion the contract between buses 8-12 is decreased by 8.35MW. As a result, signing direct electricity purchase contracts rationally can help relief congestion somehow. Cumulative probability distribution (CPD) of branch 8-9 before and after rescheduling for market structure C is presented in Fig. 6. The value of the power flow range decrease obviously through the rescheduling.
Fig. 5.Active power rescheduled of generators for modified 57-bus system
Fig. 6.CPD of power flow in line 8-9 (model C)
Table 7.Simulated cases for congested line 8-9 in modified 57-bus system (total load 1250.8MW)
5.3 Nodal pricing model case study of IEEE 30-bus
All the relative parameters in this part are set as the same as those in 5.1. Similar to the previous case studies, different combinations of market structures comprising nodal pricing model and mix of nodal pricing plus direct electricity purchase contracts are assumed, namely N, N1, N2, N3, N4, N5, respectively.
Table 8 shows the simulated optimization results for modified IEEE 30-bus system in nodal pricing model under different direct electricity purchase contracts. Based on the table, similar concolusions as those in 5.1 can be obtained. However, as the settlement of congestion cost in the nodal pricing model is quite different from that in the pool model, the results in Table 8 are quite different from those in Table 5. This indicates that suitable scheduling approaches should be adopted in different market modes. Also, Table 8 demonstrates that the proposed congestion management model capable of processing uncertainties and correlations can be adoptd in the nodal pricing model.
Table 8.Simulated cases in nodal pricing for modified IEEE 30-bus system (total load 189.2MW)
6. Conclusion
In this paper, a new congestion management approach based on probabilistic power flow has been presented to process uncertainties and correlations in congestion problem. The probabilistic power flow model has been formed based on the combined method of 2m+1 PEM and Cholesky decomposition. An optimal probabilistic power flow model minimizing the congestion cost considering market structures with pool, nodal pricing and direct electricity purchase contracts has been studied. The simulated results on the modified IEEE 30-bus system and IEEE 57-bus system reveal the following conclusions.
1) The correlations between buses with WFs or loads have significant effect on the congestion probability but little effect on the expected power flow. 2) Higher confidence level leads to more congestion cost. Dispatchers can select appropriate confidence level according to the demand of system operation. 3) The congestion management approach based on probabilistic power flow provides a way to balance both congestion cost and congestion probability. 4) Signing direct electricity purchase contracts rationally can help relief the congestion without burdening generation rescheduling. 5) The proposed congestion management model can be used in both pool and nodal pricing model.
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