The Hybrid Analysis Method Based on Loop Current and Nodal Voltage in Power Network

Wang Chengmin and Jiang Chuanwen
Department of Electrical Engineering
Shanghai Jiaotong University
Shanghai, Huashan Road, No. 1954
China

Abstract: -A novel hybrid analysis method for power flow calculation  based on  nodal voltage and loop current equations of electric power network is presentd. In this method the load is considered as voltage source. The impedance branch and ground branch were treated as the link and tree branch respectively in the  loop model . Moreover, the problems about the node and loop degradation are particularly expatiated as well as nodal type and load model treatment during the process of power flow calculation. It is showed that amount of foundational loops eaual to the quantity of total lines minus the number of degradation loops. Then, the relations combined the loop analysis with the voltage stability are revealed by analyzing the existence and multi-value of the power flow solutions. The case study is made at the IEEE-30 system and the result shows that the persented hybrid analysis method is effective for the power flow calculation.

Key-Words: - Nodal voltage; Loop current; Power flow calculation; Voltage stability; Hybrid analysis; Transmittion capacity;

1   Introduction
The conventional power flow arithmetic based on nodal analysis was applied widely to the electric power system due to its simplicity and effectiveness.
     In [1] the voltage correction power flow, which was applied to weakly meshed distribution and transmission networks, was presented. The weakly meshed network was converted into radial configuration by breaking all loops, then to achieve fast calculation of powers at branch ends and efficient calculation of node voltages in radial network. After that, the link flow was simulated by appropriate powers injected at related sides of the breakpoints, and the process was continued by efficient construction of loop impedance matrix and node voltage correction up-to convergence. The method generally was applied to the network reconstruction [2] due to the radial distribution network can be converged to weakly meshed configuration by closing all tie switches and the single loop optimization problem was performed in order to use the optimal flow pattern method [3]. Moreover, the branch-current-based state estimation method was proposed in [4]. The loop impedance method was applied to calculate the power flow of the distribution network with constant impedance load model. In this method the ground branch was ignored [5]~[8] and the good result in convergence was obtained. However all of these methods could only be applied to the distribution network with radial or weakly meshed construction 
A novel hybrid analysis method for power flow calculation based on nodal voltage and loop current equations of electric power network are presentd. In this method the load is considered as voltage source. The impedance branch and ground branch are treated as the link and tree branch respectively in the  loop model. Moreover, the problems about the node and loop degradation are particularly expatiated as well as nodal type and load model treatment during the process of power flow calculation. It is showed that amount of foundational loops eaual to the quantity of total lines minus the number of degradation loops. Then, the relations combined the loop analysis with the voltage stability are revealed by analyzing the existence and multi-value of the power flow solutions. The proposed method has been applied on IEEE-30 system to illustrate its effectiveness.

2   Foundational circuit model
To establish the foundational loop which amount is equal to the quantity of link branch is the key problem in the loop analysis. The -model circuit by a line composed of one impedance branch and two ground ones is showed in figure 1, which denotes a foundational loop while all impedance and ground branches are respectively considered as the links and tree branches where the amount of links and lines is equal to that of impedance branches.

Fig. 1 Foundational Loop Figure

where,  and  are injected power;  and  denote the nodal voltage; the loop current is ; subscript  denotes the nodal number; subscript  denotes the loop number. Therefore, loop current equation can be represented:
     	     (1)
It is obtained by expanding formula (1):
                  (2)
     The ground branch circuit is showed in figure 1 where the load is considered as voltage source, so the loop current passes two sub-branches, which are the capacitive branch and the load branch respectively. However, the voltage at node  of the equivalent voltage source of the load branch is:
                   (3)
It follows from (3) that:
         (4)
Above equation is represented as the plural form:
	 (5)
Furthermore, it is obtained:
      (6)
     As for node , there is the same form of nodal voltage function with the node . The equations (2) and (6) are the basic models of the power system.     

3   The nodal voltage and loop current analysis
It is obtained from equation (2):
	            (7)
     The classical nodal voltage equations represented as the form in rectangular coordinates can be obtained by substituting formula (7) into equation (6). For the same reason, when the conductance of the ground branch is ignored as  and , , , the formulas can be obtained from equation (6) as follows:
                   (8)
   (9)
where,  is the loop current vector. Accordingly, loop current equations are:
  (10)
     Thus it can be observed that loop current equations are linear after the corresponding nodal voltages are determined. However, there usually are two values of the nodal voltage. The value which is close to the initial ooperation situation is adopted in the process of calculation.

4   Loop degradation and foundational loop
In the process of actual calculation with loop current method, the amount of foundational loop is always less than that of the lines because of the existence of loop degradation phenomenon. The loop degradation phenomenon will happen if the ground branches are neglected as  and the load at node  is zero as  because of the decrease of the amount of tree branches, thus the corresponding link should be complemented from the impedance branches in order to form enough foundational loop. The loop degradation is firstly reflected as node degradation, so corresponding nodal voltage cannot be determined through equation (8) and (9) and it is obtained:
		(11)
     The corresponding nodal voltage equations from equation (6) are replaced by nodal injected current formula from equation (11). Although the variables and amount of total equations do not change, but the corresponding nodal voltage cannot be obtained from equation (11), which is not related to nodal voltage, so the loop current equations including corresponding nodal voltage in (2) are required to exchange with the equations (11). For example, three nodes system is shown in the figure 2.

Fig. 2 Three Nodes system

     In figure 2, there is not a ground branch at node 2 and the injected power is zero. The nodal voltage equations at node 1 and 3 are represented as below:
                (12)
              (13)
For node 2, the formulas are given by:
                     (14)
     The loop current equations are represented as follows:
                  (15)
                 (16)
     In the two loops which are connected with node 2 there is only one is independent because of the degradation at node 2. That is to say, if the amount of degradation node is , the amount of foundational loop is . The network equations are composed of the nodal voltage equations (12), (13), (16) and loop current equations (15), (14), on the assumption that loop 2 is dependent by exchanging the equation (16) with (14).

5   Newton method

5.1	Correction equation
Above discursion has shown that network equations contain four parts: non-degenerated nodal voltage equations, degenerated loop current equations, non-degenerated loop current equations and degenerated nodal voltage equations. By linearizing the network equations and taking into account , , , , the formulas are obtained as:
        (17)
where, , , and  are the increment vectors for injected nodal power, loop voltage, nodal voltage and loop current respectively. The subscript  represents non-degenerated node or loop and the subscript  represents degenerated node or loop. The elements of the Jacob matrix are given as follows:
(18)
 and  have the same configuration as below:
   (19)
     The matrix (19) contains  rows where the left  rows denotes  and the right  rows denotes ;  is corresponding branch-node relative matrix;  and  are negative corresponding branch-node relative matrix;  is corresponding node-branch relative matrix and  is negative corresponding node-branch relative matrix.

(20)
     The above is diagonal block matrix where upper  rows denote  and latter  rows denote .

5.2	Predigestion of Newton correction equation
In the process of actual calculation, it is difficult to compute the voltage at degenerated node and the current in degenerated loop. Therefore it is necessary to eliminate these variables to predigest the calculation. In equation (17), supposing , then:
		(21)
Then, supposing  we obtain:
	(22)
Substituting equation (21) into (22), we obtain:

(23)
     The successful predigestion is achieved by substituting the equations (21) and (23) into other sub-equations of (17) as:
                     (24)
where:
                              (25)

(26)
      (27)
	(28)

6   Existence and multi-value of the power flow solution
From formula (9), it is obvious that power flow solutions are existent when the formulas are satisfied as follow:
  	(29)
viz.:
	(30)
     It means that solutions are existent when the square of nodal injected current amplitude locates in the circle, which is centered at  and   the radius is .
     From formula (6), it is obvious that only one solution of the node voltage is actual when the equal sign of formula (29) is available, otherwise two solutions which include one higher voltage solution and one lower one are existent when the sign of inequality from (29) is available. The amount of power flow solutions is determined by the solution amount of the of node voltage equations which equal to .  denotes the amount of non-degenerated nodes or loops. Therefore, there are some errors in [9] and the number of power flow solutions cannot be .
     The transmission capacity reaches the upper limit in the electric power network when the equal sign of formula (29) is available. In other words the saddle bifurcation point is reached on the PV curve, which means thet operation point of the power system is very close to the edge of voltage stability. Therefore, the formula (30) also provides the region of static voltage stability in the space of state variables.

7   Other problems
The load model is considered as constant power in the process of discursion above. When the constant impendence load model is adopted, the formula (5) becomes:
        (31)
where,  is the equivalent load admittance at node , furthermore we can obtain:
     (32)
     One of the two solutions is zero and it can be obtained from equation (32). It is obvious that loop impedance model proposed in [5] is a particular case of the loop current model. The only difference between them consists in the choices of the foundational circuit.
     If the nodal type is considered as PV node rather than a PQ node, the formula used to replace the corresponding reactive balance equation in formula (6) is given as follows:
                                (33)
Similarly, the nodal equation is not required for slack bus because the corresponding nodal voltage is known.

8   Case study
The case study is made at IEEE-30 system to test the loop analysis method. Supposing the , we can obtain the independent loop current correction formula from (24):
                  (34)
The Newton method is used to solve loop current equations. The node data is shown in Table 1.

Table 1 Node Data
Node
No.	Voltage	Phase Angle	Active Power
of Generator	Reactive Power
of Generator	Active Power
of  Load	Reactive Power
of Load	Node
Type	Parallel
Capacitor	
1	1.0088	12.5615	0	0	0.106	0.019	2	0	
2	1.0338	2.75009	0.3586	-0.116837	0	0	1	0	
3	1.03142	4.70425	0	0	0.024	0.012	2	0	
4	1.02646	5.63532	0	0	0.076	0.016	2	0	
5	1.0058	9.01951	-0.6964	0.0272589	0	0	1	0	
6	1.02178	-6.51598	0	0	0.001	0.0005	2	0	
7	1.0075	-8.07121	0	0	0.228	0.109	2	0	
8	1.023	-6.51522	0.05	-0.00924631	0	0	1	0	
9	1.05929	-8.18472	0	0	0.001	0.0005	2	0	
10	1.05461	-10.0328	0	0	0.058	0.02	2	0.19	
11	1.0913	-6.33596	0.1793	0.160241	0	0	1	0	
12	1.06136	-9.26996	0	0	0.112	0.0075	2	0	
13	1.0883	-8.09557	0.1691	0.19387	0	0	1	0	
14	1.04737	-10.1593	0	0	0.062	0.016	2	0	
15	1.04351	-10.2722	0	0	0.082	0.025	2	0	
16	1.05134	-9.86612	0	0	0.035	0.018	2	0	
17	1.04834	-10.1886	0	0	0.09	0.058	2	0	
18	1.03537	-10.8732	0	0	0.032	0.009	2	0	
19	1.03366	-11.041	0	0	0.095	0.034	2	0	
20	1.03812	-10.8459	0	0	0.022	0.007	2	0	
21	1.04236	-10.4932	0	0	0.175	0.112	2	0	
22	1.0429	-10.4875	0	0	0.001	0.0005	2	0	
23	1.03476	-10.7099	0	0	0.032	0.016	2	0	
24	1.03145	-10.9531	0	0	0.087	0.067	2	0.04	
25	1.03107	-10.847	0	0	0.001	0.0005	2	0	
26	1.01361	-11.2541	0	0	0.035	0.023	2	0	
27	1.03957	-10.5167	0	0	0.001	0.0005	2	0	
28	1.01798	-6.93227	0	0	0.001	0.0005	2	0	
29	1.02007	-11.7076	0	0	0.024	0.009	2	0	
30	1.05	0	1.38498	-0.0320089	0	0	3	0	

The calculating results are showed in table 2 where the second and third row are the final loop current values. The fourth and fifth row in table 2 are the loop current errors of three iterations.

Table 2 Calculating Results
Line
No.					
1	1.28129	0.0980171	0	5.96745e-016	
2	0.673963	-0.0107051	-1.04219e-006	2.42779e-006	
3	0.418896	0.0058823	6.66134e-016	4.19109e-015	
4	0.649085	-0.0212779	1.04219e-006	-2.42779e-006	
5	0.644449	-0.0749591	2.22045e-016	-2.22045e-016	
6	0.553552	0.00067679	6.43929e-015	2.43139e-014	
7	0.606277	-0.0476339	5.66214e-015	2.01505e-014	
8	-0.0539811	-0.035317	-8.88178e-016	-1.72085e-015	
9	0.259948	-0.0965038	-7.21645e-015	-2.63123e-014	
10	0.0732552	0.217304	-8.21565e-015	-2.35367e-014	
11	0.287415	0.0331736	-5.9952e-015	-2.48135e-014	
12	-0.168863	0.0290358	9.21485e-015	2.28151e-014	
13	-0.10606	0.311694	-4.44089e-016	-1.66533e-016	
14	0.310048	-0.209491	-2.9976e-015	1.9984e-015	
15	0.386187	0.0313277	-4.44089e-016	-4.57967e-015	
16	-0.0858271	0.330815	0	-3.60822e-016	
17	0.081494	-0.0490809	-2.22045e-016	-1.11022e-016	
18	0.198265	-0.147111	0	-2.22045e-016	
19	0.0271356	-0.0186264	1.11022e-016	-1.11022e-016	
20	0.0899105	-0.0709868	0	0	
21	0.0091738	-0.0329653	2.88658e-015	-2.16493e-015	
22	0.0613379	-0.0454322	-3.33067e-016	-2.22045e-016	
23	0.0662545	-0.039776	-1.11022e-016	-5.55112e-017	
24	0.0380843	-0.022511	0	5.55112e-017	
25	-0.0436356	0.0363441	0	2.77556e-016	
26	0.0627732	-0.0490409	2.88658e-015	-1.88738e-015	
27	0.197454	-0.202942	2.9976e-015	-2.10942e-015	
28	0.108961	-0.106438	2.88658e-015	-1.9984e-015	
29	0.0599046	-0.0470274	-1.11022e-016	1.11022e-016	
30	0.0877386	-0.0814183	-4.44089e-016	-1.11022e-016	
31	0.0863906	-0.0776176	-4.44089e-016	-1.05471e-015	
32	0.0612657	-0.057184	-2.22045e-016	-7.77156e-016	
33	0.0709164	-0.0808088	6.66134e-016	9.99201e-016	
34	0.0281134	-0.0356871	-1.11022e-016	5.55112e-017	
35	-0.0418848	0.0381991	-3.33067e-016	-5.55112e-017	
36	-0.24915	0.196219	9.99201e-016	1.69309e-015	
37	0.0568599	-0.0367281	4.44089e-016	2.77556e-016	
38	0.0659498	-0.0396224	1.11022e-016	1.11022e-016	
39	0.0355825	-0.0192153	-3.33067e-016	-2.22045e-016	
40	0.0688237	-0.0871818	2.77556e-015	1.94289e-016	
41	0.274196	-0.149114	-5.55112e-015	-2.33147e-014	

The maximum error showed in table 3 is smaller form the first iteration because the initial value is close to the power flow solution. It is proved that loop current method is an effective way for power network analysis.

Table 3 Iteration Precision
Iteration	1	2	3	
Maximum
Error	9.32447e-005	5.23659e-006	4.89406e-007	

8   Conclusion
In the method presented the loads is treated as voltage source and loop current is introduced as a state variable. The main conclusions are represented as below:
     1) The loop current method can improve the convergence of power flow calculation.
     2) The amount of degenerative loop is obtained. Furthermore, the amount of foundational loop and its expression are presented.
     3) The existence condition of multi-solution for power flow is proved. The solution amount of power flow is presented.

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