CN107194510B - Power distribution network reactive power optimization method based on simulated annealing chicken swarm algorithm - Google Patents
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Abstract
The invention belongs to the technical field of power flow calculation of power systems, and particularly relates to a reactive power optimization method for a power distribution network with distributed power supplies based on a simulated annealing chicken flock algorithm, which comprises the following steps: establishing an interval reactive power optimization model which comprises the constraints that the power grid loss is minimized as a target function, an interval power flow equation is an equality constraint and the output and reactive power compensation capacity of the distributed power supply are inequality constraints; solving the interval power flow equation by adopting an interval Newton method, giving initial voltage, expanding a monotonicity-containing interval of variable derivatives, obtaining an interval Newton operator, and then performing iteration to obtain a voltage and phase angle interval solution of the interval power flow equation; and solving the interval reactive power optimization model by using a simulated annealing chicken flock algorithm. The method has the advantages that the power distribution network reactive power optimization considering the uncertainty of the output of the distributed power supply is better optimized, the convergence rate is optimistic, the obtained target function is a more reasonable interval, the deterministic optimization result is included, and the method has good engineering application prospects.
Description
Technical Field
The invention belongs to the technical field of power flow calculation of power systems, and particularly relates to a reactive power optimization method for a power distribution network with distributed power supplies based on a simulated annealing chicken flock algorithm.
Background
The reactive power optimization of the power system refers to that under the condition that the distribution of active load, active power supply and active power flow of the system is determined, the values of certain control variables are determined through optimization calculation, and one or more performance indexes of the system are optimized on the premise that constraint conditions are met. Mathematically, reactive power optimization is a typical nonlinear programming problem and has the characteristics of nonlinearity, discontinuity, more uncertain factors and the like. The reactive power optimization of the power system is an effective means for improving the safety and the economy of the power system, and is an important measure for improving the voltage quality. The uncertainty of the distributed power supply and the load may cause that the reactive power optimization cannot obtain an optimal solution, so that the influence of the uncertainty on the reactive power optimization needs to be researched.
At present, the main method applied to reactive power optimization of a power system is (1) a linear programming method, and a mathematical model of the method is simple and intuitive, clear in physical concept and high in calculation speed; (2) the nonlinear programming method has the advantages that a mathematical model is accurate, calculation accuracy is high, calculation amount is large, memory is occupied, inequality constraint processing difficulty is high, and the nonlinear programming method is difficult to apply to a large-scale system; (3) the mixed integer programming method comprises the steps of firstly determining integer variables, and then coordinating with a linear programming method to process continuous variables, wherein the model is accurate, but the overall optimality is weakened by a two-step optimization method, oscillation divergence is easy to occur in the solving process, the calculating process is very complex, and the calculated amount is large; (4) the artificial intelligence algorithm can process various constraints and solve various objective functions, but the situation of local optimum is easy to occur, and the global optimum solution cannot be obtained.
Disclosure of Invention
Aiming at the problems, the invention provides a reactive power optimization method for a power distribution network with distributed power supplies based on a simulated annealing chicken flock algorithm, which comprises the following steps:
step 1, establishing an interval reactive power optimization model which comprises the constraints that the power grid loss is minimized as a target function, an interval power flow equation is an equality constraint and the output and reactive power compensation capacity of a distributed power supply are inequality constraints;
step 2, solving the interval power flow equation by adopting an interval Newton method, giving an initial voltage, expanding a monotonicity-containing interval of a variable derivative to obtain an interval Newton operator, and then performing iteration to obtain a voltage and phase angle interval solution of the interval power flow equation;
and 3, solving the interval reactive power optimization model by using a simulated annealing chicken flock-based algorithm.
The objective function is:
in the formula: n is the number of the nodes,
voltage interval values of the nodes i and j are respectively; g ij ，B ij Respectively is a conductance value and a susceptance value between the nodes i and j;
the phase angle difference interval value of the nodes i and j is obtained;
is the expected value of the voltage at node i; u shape imin 、U imax Respectively an upper limit value and a lower limit value of the voltage at the node i;
considering the reactive interval value at the reactive out-of-limit node j; q jmax Considering the maximum allowable reactive value at the reactive out-of-limit node j; mu.s 1 ,μ 2 Is an out-of-limit penalty factor.
The equation constraints are:
in the formula: delta P i Is the value of the active power change, Δ Q, at node i i Is the value of the reactive power change at node i,
the output interval of the active power and the reactive power of the generator set at the node i is set;
the output interval of the active power and the reactive power of the distributed power supply at the node i is set;
the fluctuation interval of the active load and the reactive load at the node i is set;
voltage interval values of the nodes i and j are respectively;
is the phase angle difference interval value of the node i, j.
The inequality constraints are:
in the formula: p Di 、Q Di Are respectively distributedActive power output and reactive power output of the power supply; p Dimin 、P Dimax Respectively are upper and lower limit values of active power output of the distributed power supply; q Dimin 、Q Dimax Respectively are upper and lower limit values of reactive power output of the distributed power supply; q Ci Is a reactive compensation capacity; q Cimin 、Q Cimax The reactive compensation capacity is an upper limit value and a lower limit value; t is a unit of k Is a tap gear of the on-load tap changing transformer; t is a unit of kmin 、T kmax The upper and lower limit values of the tap position of the on-load tap changing transformer; u shape i Is the voltage at node i; u shape imin 、U imax The upper and lower limit values of the voltage at the node i.
The step 2 comprises the following steps:
expression (2) is: h (x) is 0(4),
meanwhile, the mark value of the middle point value of the interval is mid, wherein, the mid represents the number of the interval,
given an initial voltage, the phase angle interval is:
in the formula:
in the voltage amplitude interval, U min 、U max Upper and lower voltage amplitude limits;
is a voltage phase angle interval, theta min 、θ max Upper and lower voltage phase angle limits; let the intermediate variable y ═ mid (X), denote the midpoint value of the interval variable X;
the inclusive monotonicity interval extension of the derivative of the voltage phase angle θ with respect to the voltage magnitude U in equation set (5) is:
this results in the interval newton operator N:
N＝y-[H'(X)] -1 h(y) (7)
iteration is carried out by using interval Newton operators:
X (k+1) ＝X (k) ∩N (k) ，(k＝1，2，3...)(8)
and obtaining the voltage and phase angle interval solution of the interval power flow equation (2).
The step 3 comprises the following steps:
step 301: the chicken flock individuals are initialized, and the chicken flock individuals are initialized,
in the formula:
voltage regulation interval for the voltage of the generator terminal;
compensating a capacity interval for reactive compensation equipment;
is a tap gear range of the on-load tap changing transformer; and determining the number N of the chicken flocks; the number of cocks, hens, chicks and mama hens is NR, NH, NC and NM respectively; maximum number of iterations k max Initializing to enable k to be 0;
step 302: if the k% G is 1, wherein G is a reconstruction grade system parameter, the fitness interval is compared, the grade system of the chicken flock is reestablished, the chicken flock is divided into a plurality of chicken flocks, and the unique cock, the hen, the mama hen and the corresponding chick of each chicken flock are determined;
step 303: giving a voltage and phase angle initial interval, solving an interval power flow equation (2) by using an interval Newton method, calculating each individual fitness interval of the chicken flock after a converged voltage amplitude and a converged phase angle interval are obtained, and determining pbest and gbest of an initial individual;
step 304: for each individual chicken, comparing the fitness interval with the best position, if the fitness interval is better, updating the best position of the individual by using the current position, and taking the best position as the optimal interval pbest of the individual chicken;
step 305: comparing the fitness interval of the cock of each sub-chicken flock with the gbest, if the fitness interval is better, updating the global best position by using the position of the cock, and taking the global best position as the gbest of the chicken flock optimal interval;
step 306: the positions of the cock, the hen and the chick are respectively updated according to the following methods:
in order to ensure that a global optimal solution is obtained, a new cock position is generated according to formulas (9) to (10):
in the formula:
is the position of the cock i after the kth iteration, and t is the individual of all cocks except the cock i;
the adaptive intervals of the cocks i and t are respectively sigma 2 Is an intermediate variable;
secondly, if
Then the new location is accepted
Otherwise, selecting whether to accept the new position according to the formula (11);
in the formula: t is the simulated annealing temperature; lamda is the attenuation coefficient; k is a radical of B Boltzmann constant;
updating the fitness interval of the cock i position;
the fitness interval before updating the i position of the cock, s is a random number generated between 0 and 1, and p T Expressed as a probability of choosing to update the current location;
the hen position updating method is updated according to formulas (12) to (14):
in the formula: r is a radical of hydrogen 1 The chicken is a cock of a chick group where the ith hen is located; r is 2 Randomly selected from cock and hen in the whole chicken group 1 ≠r 2 (ii) a rand is [0,1 ]]The random number of (2);
the individual chicken i, r1 and r2 correspond to fitness intervals respectively, and L1 and L2 are hen position updating factors;
the chicken position updating method updates according to a formula (15):
wherein w is determined according to equation (16)
In the formula:
the chicken individuals i and r correspond to fitness intervals; r is the cock of the chicken group in which the chicken is located; m is a mother hen corresponding to the chicken; w is the inertia coefficient; u is a random weight value, u max Take 0.7, u min Taking 0.4; FM is [0,2 ]]The random number of (2).
Step 307: judging whether the maximum iteration number is reached; if yes, go to step 308; otherwise, let the iteration counter k be k +1, and go to step 302;
step 308: and outputting the optimal solution, and ending.
The invention has the beneficial effects that: the invention provides an interval reactive power optimization model, and solves the interval reactive power optimization model by using a reactive power optimization method of a power distribution network containing a distributed power supply based on a simulated annealing chicken swarm algorithm, and the analysis of an example shows that the method can obtain a better optimization effect on the reactive power optimization of the power distribution network considering the uncertainty of the output of the distributed power supply, has a optimistic convergence rate, and has good engineering application prospect by comparing and finding that an objective function obtained by the method is a more reasonable interval and contains a deterministic optimization result.
Detailed Description
The following examples are described in detail.
The reactive power optimization method of the power distribution network with the distributed power supply based on the simulated annealing chicken swarm algorithm comprises the following steps:
step 1, providing an interval reactive power optimization model.
An objective function:
in the formula: n is the number of the nodes,
are respectively nodesi. The voltage interval value of j; g ij ，B ij Respectively is a conductance value and a susceptance value between the nodes i and j;
the phase angle difference interval value of the nodes i and j is obtained;
is the expected value of the voltage at node i; u shape imin 、U imax Respectively an upper limit value and a lower limit value of the voltage at the node i;
considering the reactive interval value at the reactive out-of-limit node j; q jmax Considering the maximum allowable reactive value at the reactive out-of-limit node j; mu.s 1 ,μ 2 Is an out-of-limit penalty factor.
The equation constrains:
in the formula: delta P i Is the value of the active power change, Δ Q, at node i i Is the value of the reactive power change at node i,
the output interval of the active power and the reactive power of the generator set at the node i is shown;
the output interval of the active power and the reactive power of the distributed power supply at the node i is set;
an active load and reactive load fluctuation interval at a node i is set;
voltage interval values of the nodes i and j are respectively;
is the phase angle difference interval value of the node i, j.
The inequality constrains:
in the formula: p Di 、Q Di Respectively an active power output and a reactive power output of the distributed power supply; p Dimin 、P Dimax Respectively representing the upper limit value and the lower limit value of the active power output of the distributed power supply; q Dimin 、Q Dimax Respectively are the upper limit value and the lower limit value of reactive power output of the distributed power supply; q Ci Is a reactive compensation capacity; q Cimin 、Q Cimax The reactive compensation capacity is an upper limit value and a lower limit value; t is k Is a tap position of the on-load tap changing transformer; t is kmin 、T kmax The upper and lower limit values of the tap position of the on-load tap changing transformer; u shape i Is the voltage at node i; u shape imin 、U imax The upper and lower limit values of the voltage at the node i.
Step 2, solving method of interval power flow model
The interval power flow model (2) is a nonlinear equation on an interval, an interval Newton method is adopted for solving, and the detailed description process is as follows:
for convenience of expression, the model (2) is expressed as: and h (x) ═ 0(4), and the dot values in the intervals are marked by mid (where represents the number of the intervals.
Given an initial voltage, the phase angle interval is:
in the formula:
in the voltage amplitude interval, U min 、U max Upper and lower voltage amplitude limits;
is a voltage phase angle interval, theta min 、θ max Upper and lower voltage phase angle limits; let y be mid (X), and denote the midpoint value of the interval variable X.
The monotonicity-containing interval of the derivative with respect to the variable U, θ in equation set (5) extends as:
this results in the interval newton operator N:
N＝y-[H'(X)I- 1 h(y)(7)
iteration is carried out by using an interval Newton operator:
X (k+1) ＝X (k) ∩N (k) ，(k＝1，2，3...)(8)
and obtaining the voltage and phase angle interval solution of the interval power flow equation (2).
Step 3, solving step of interval reactive power optimization model
Step 301: the chicken flock individuals are initialized, and the chicken flock individuals are initialized,
in the formula:
voltage regulation interval for the voltage of the generator terminal;
compensating a capacity interval for reactive compensation equipment;
is a tap gear range of the on-load tap changing transformer; and determining the number N of the chicken flocks; the number of cocks, hens, chicks and mama hens is NR, NH, NC and NM respectively; maximum number of iterations k max The initialization is to make k equal to 0.
Step 302: if the k% G is 1, wherein G is a reconstruction rank system parameter, the fitness interval is compared, the rank system of the chicken flocks is re-established, the chicken flocks are divided into a plurality of sub-chicken flocks, and the unique cock of each sub-chicken flock, the hen, the mother hen and the corresponding chick are determined.
Step 303: giving a voltage and phase angle initial interval, solving an interval power flow model (2) by using an interval Newton method, calculating each individual fitness interval of the chicken flock after a converged voltage amplitude and a converged phase angle interval are obtained, and determining pbest and gbest of an initial individual.
Step 304: and comparing the fitness interval of each individual chicken with the best position experienced, if the fitness interval is better, updating the best position of the individual by using the current position, and taking the best position as the optimal interval pbest of the individual chicken.
Step 305: and comparing the fitness interval of the cock of each sub-chicken flock with the gbest, if the fitness interval is better, updating the global best position by using the position of the cock, and taking the global best position as the optimal interval gbest of the chicken flock.
Step 306: the positions of the cock, the hen and the chick are respectively updated according to the following methods:
in order to ensure that a global optimal solution is obtained, a new cock position is generated according to formulas (9) to (10):
in the formula: t is an individual of all cocks except the cock i;
the cock i and the cock i correspond to the fitness intervals respectively.
Secondly, if
Then the new location is accepted
Otherwise, whether to accept the new location is selected according to equation (11).
In the formula: t is the simulated annealing temperature; lamda is the attenuation coefficient; k is a radical of B Boltzmann constant;
updating the fitness interval of the cock i position;
the fitness interval before updating the i position of the cock, s is a random number generated between 0 and 1, and p T Represented as the probability of choosing to update the current location.
The hen position updating method is updated according to formulas (12) to (14):
in the formula: r is 1 The chicken is a cock of a chick group where the ith hen is located; r is a radical of hydrogen 2 Randomly selected from cock and hen in the whole chicken group 1 ≠r 2 (ii) a rand is [0,1 ]]The random number of (2);
the chicken individuals i, r1 and r2 correspond to fitness intervals respectively, and L1 and L2 are hen position updating factors.
The chicken position updating method updates according to a formula (15):
wherein w is determined according to the formula (16)
In the formula:
the chicken individuals i and r correspond to fitness intervals; r is the cock of the chicken group in which the chicken is located; m is a mother hen corresponding to the chicken; w is the inertia coefficient; u is a random weight value, u max Take 0.7, u min Taking 0.4; FM is [0,2 ]]The random number of (2).
Step 307: judging whether the maximum iteration number is reached; if yes, go to step 308; otherwise, let the iteration counter k be k +1, and go to step 302;
step 308: and outputting the optimal solution, and ending.
For a better understanding of the invention and to show the subject matter thereof, those skilled in the art will recognize that the invention may be practiced with other embodiments that depart from these specific details.
And setting a reactive power optimization method for the power distribution network with the distributed power supply based on the simulated annealing chicken swarm algorithm by using an IEEE standard system. Experiments modifications to the IEEE-14 system were made: a distributed power supply is connected to the node 3, and the maximum value of active output is 40MW, and the maximum value of reactive output is 40 MW; each of the nodes 6 and 8 is provided with 10 groups of 2MW reactive power compensation devices; the load output interval is +/-5% of the rated load; the node voltage (pu) is 0.95-1.05; the chicken flock scale is 30, the cock proportion is 20%, the hen proportion is 60%, the mama hen proportion is 10%, the chick proportion is 10%, the rebuilding grade system parameter G is 10, the maximum iteration times are 30, and the simulated annealing temperature T is 200; the attenuation coefficient lamda is 0.7, and the penalty coefficients μ 1, μ 2 are both 0.1. The test environment is a PC, the CPU is Intel (R) core (TM) i3M370, the main frequency is 2.40GHz, and the memory is 2.00 GB.
The method is compared with a deterministic reactive power optimization method (namely that the output of the distributed power supply is a determined value, namely active power 40MW, reactive power 17MW, and the load is a rated load value) based on a simulated annealing chicken flock algorithm.
Simulation calculation was performed using matlab, and the obtained simulation calculation results are shown in tables 1 and 2. TABLE 1 deterministic optimized and Interval optimized node Voltage case
TABLE 2 deterministic optimization and Interval optimization the first 20 iteration values of the objective function
As can be seen from table 1, when the distributed power output is a determined value, the interval optimization model degenerates into a deterministic optimization model, and the voltage of each node can obtain a determined value; when the distributed power output and the load are interval values, the result obtained by the method is the interval value, and the result obtained by deterministic optimization is included in the interval obtained by interval optimization through comparison.
As can be seen from Table 2, when the distributed power supply output is a determined value, a deterministic reactive power optimization method based on a simulated annealing chicken flock algorithm can obtain a better optimization effect, and the convergence rate is optimistic.
In conclusion, the reasonability and the feasibility of the method can be verified, the method is suitable for practical application, and a reasonable method is provided for power system reactive power optimization considering uncertain factors of distributed power output.
The above embodiments are only preferred embodiments of the present invention, but the scope of the present invention is not limited thereto, and any changes or substitutions that can be easily conceived by those skilled in the art within the technical scope of the present invention are also within the scope of the present invention. Therefore, the protection scope of the present invention shall be subject to the protection scope of the claims.
Claims (4)
1. A reactive power optimization method for a power distribution network with distributed power supplies based on a simulated annealing chicken flock algorithm is characterized by comprising the following steps:
step 1, establishing an interval reactive power optimization model which comprises the constraints that the power grid loss is minimized as a target function, an interval power flow equation is an equality constraint and the output and reactive power compensation capacity of a distributed power supply are inequality constraints;
step 2, solving the interval power flow equation by adopting an interval Newton method, giving an initial voltage, expanding a monotonicity-containing interval of a variable derivative to obtain an interval Newton operator, and then performing iteration to obtain a voltage and phase angle interval solution of the interval power flow equation;
step 3, solving an interval reactive power optimization model by using a simulated annealing chicken swarm algorithm;
the equation constraints are:
in the formula: delta P i Is the value of active power change, Δ Q, at node i i Is the value of the reactive power change at node i,
the output interval of the active power and the reactive power of the generator set at the node i is shown;
the output interval of the active power and the reactive power of the distributed power supply at the node i is set;
the fluctuation interval of the active load and the reactive load at the node i is set;
voltage interval values of the nodes i and j are respectively;
is the phase angle difference interval value of the node i, j; g ij ，B ij Respectively is a conductance value and a susceptance value between the nodes i and j;
the step 3 comprises the following steps:
step 301: the chicken flock individuals are initialized, and the chicken flock individuals are initialized,
in the formula:
voltage regulating interval for the voltage of the generator terminal;
compensating a capacity interval for reactive compensation equipment;
is a tap gear range of the on-load tap changing transformer; determining the number N of chicken flocks; the number of cocks, hens, chicks and mama hens is NR, NH, NC and NM respectively; maximum number of iterations k max Initializing to enable k to be 0;
step 302: if the k% G is 1, wherein G is a reconstruction grade system parameter, the fitness interval is compared, the grade system of the chicken flock is reestablished, the chicken flock is divided into a plurality of chicken flocks, and the unique cock, the hen, the mama hen and the corresponding chick of each chicken flock are determined;
step 303: giving a voltage and phase angle initial interval, solving a formula (2) by using an interval Newton method, calculating each individual fitness interval of the chicken flock after a converged voltage amplitude and a converged phase angle interval are obtained, and determining pbest and gbest of an initial individual;
step 304: for each individual chicken, comparing the fitness interval with the best position, if the fitness interval is better, updating the best position of the individual by using the current position, and taking the best position as the optimal interval pbest of the individual chicken;
step 305: comparing the fitness interval of the cock of each sub-chicken flock with the gbest, if the fitness interval is better, updating the global best position by using the position of the cock, and taking the global best position as the gbest of the chicken flock optimal interval;
step 306: the positions of the cock, the hen and the chick are respectively updated according to the following methods:
in order to ensure that a global optimal solution is obtained, a new cock position is generated according to formulas (9) to (10):
in the formula:
the position of the cock i after the k iteration is shown, and t is an individual of all cocks except the cock i;
the fitness interval, sigma, is corresponding to the cock i and t respectively 2 Is an intermediate variable;
secondly, if
Then the new location is accepted
Otherwise, selecting whether to accept the new position according to the formula (11);
in the formula: t is the simulated annealing temperature; lamda is the attenuation coefficient; k is a radical of B Boltzmann constant;
updating the fitness interval of the cock i position;
the fitness interval before the i position of the cock is updated, s is a random number generated between 0 and 1, and p T Updating the probability of the current position for selection;
the hen position updating method is updated according to formulas (12) to (14):
in the formula: r is 1 Is a cock of a chick group of the ith hen; r is 2 Randomly selected from cock and hen in whole chicken group 1 ≠r 2 (ii) a rand is [0,1 ]]The random number of (2);
the adaptive range is respectively a cock r1 and a cock r2, L1 is a hen position updating factor L1, and L2 is a hen position updating factor L2;
the chicken position updating method updates according to a formula (15):
wherein w is determined according to the formula (16)
In the formula:
the individual i and r of the cock correspond to the fitness interval; r is the cock of the chicken group in which the chicken is located; m is a mother hen corresponding to the chicken; w is the inertia coefficient; u is a random weight value max Take 0.7, u min Taking 0.4; FM is [0,2 ]]The random number of (2); marking the midpoint value of the interval as mid (mid), wherein mid represents an interval number, and p is an intermediate variable;
step 307: judging whether the maximum iteration times is reached; if yes, go to step 308; otherwise, let the iteration counter k be k +1, and go to step 302;
step 308: and outputting the optimal solution, and ending.
2. The method of claim 1, wherein the objective function is:
in the formula: n is the number of the nodes,
voltage interval values of the nodes i and j are respectively; g ij ，B ij Respectively is a conductance value and a susceptance value between the nodes i and j;
the phase angle difference interval value of the nodes i and j is obtained;
is the expected value of the voltage at node i; u shape imin 、U imax Respectively an upper limit value and a lower limit value of the voltage at the node i;
considering the reactive interval value at the reactive out-of-limit node j; q jmax Considering the maximum allowable reactive value at the reactive out-of-limit node j; mu.s 1 ,μ 2 Is an out-of-limit penalty factor.
3. The method of claim 1, wherein the inequality constraint is:
in the formula: p Di 、Q Di Respectively an active power output and a reactive power output of the distributed power supply; p Dimin 、P Dimax Respectively representing the upper limit value and the lower limit value of the active power output of the distributed power supply; q Dimin 、Q Dimax Respectively are the upper limit value and the lower limit value of reactive power output of the distributed power supply; q Ci Is a reactive compensation capacity; q Cimin 、Q Cimax The reactive compensation capacity is an upper limit value and a lower limit value; t is k Is a tap position of the on-load tap changing transformer; t is a unit of kmin 、T kmax The upper and lower limit values of the tap position of the on-load tap changing transformer; u shape i Is the voltage at node i; u shape imin 、U imax The upper and lower limit values of the voltage at the node i.
4. The method of claim 1, wherein the step 2 comprises:
expression (2) is: h (x) is 0(4),
meanwhile, the mark value of the middle point value of the interval is mid, wherein, the mid represents the number of the interval,
given an initial voltage, the phase angle interval is:
in the formula:
in the voltage amplitude interval, U min 、U max Upper and lower voltage amplitude limits;
is a voltage phase angle interval, theta min 、θ max Upper and lower voltage phase angle limits; let intermediate variable y ═ mid (X), denote the midpoint value of interval variable X;
the inclusive monotonicity interval extension of the derivative of the voltage phase angle θ with respect to the voltage magnitude U in equation set (5) is:
this results in the interval newton operator N:
N＝y-[H′(X)] -1 h(y) (7)
iteration is carried out by using an interval Newton operator:
X (k+1) ＝X (k) ∩N (k) ，(k＝1,2,3…) (8)
the voltage and phase angle interval solution of formula (2) can be obtained.
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