Shown in Figure 1 is a diagram of a distribution network.

As shown in the Figure 1, the network contains multiple buses and nodes. The PV cells and loads are connected into the distribution network in different nodes. The output power of the PV is affected by a variety of climate factors, as shown below (Wang and Yang, 2017): U O C = A 0 K T q ln I s c I D O + 1 (1) where A ₀ is a constant; μ is the Boltzmann constant; q is the electronic charge; I _SC is the short circuit current varied with irradiation intensity; I _DO is the equivalent saturation current of the diode; T is the environment temperature.

It can be seen from the above formula that the climate factors are the main factors to result in the strong uncertainty of PV power.

The PV system can be equivalent to a PQ node if the PV system is connected into a power system. With the fluctuation of grid-connected power, the power quality problem may be generated, such as voltage fluctuations, flicker, and other problems.

Uncertainty modeling is helpful to quantitatively understand the uncertainty of PV power. In this paper, a Markov chain is used to model the power state of PV cells.

The Markov chain is one of the typical algorithms used to describe the uncertainty of a random variable process, in which the present state can be used to describe the future state of the variable; the state of the variable at different times can present the variable development uncertainty. For a discrete power sequence {p ₁, p ₂ … … p _t , p _t+1}, the state in t+1 can be described by the following probabilities (He, 2008; Tabone and Callaway, 2015; Zhang et al., 2021): P x t + 1 = p t + 1 | x t = p t , x t − 1 = p t − 1 , ⋅ ⋅ ⋅ x 1 = p 1 = P x t + 1 = p t + 1 | x t = p t (2) where p is the probability of each state.

Figure 2 shows the process transfer between various random states:

As can be seen from Figure 2, each power state can be transferred to its own initial state or other states, and the transition uncertainty can be described by the transition probability. All the state transition processes can be represented as a state transition matrix: A = A 11 A 12 ⋯ A 1 n A 21 A 22 ⋯ A 2 m ⋮ ⋮ ⋱ ⋮ A m 1 A m 2 ⋯ A m n (3) where A _mn is the transition probability. Namely, in the state transition matrix, each element corresponds to the transition probability of a variable state.

When modeling the power uncertainty of a PV cell, the historic data of the PV cell’s power are classified first to obtain different states. And then, the transfer probability matrix of the Markov chain is trained to obtain the uncertain Markov chain model for the target PV system.

However, in the case where multiple PV systems are connected into the distribution network in different nodes, the impact on the network from the PV cells is global and comprehensive, in other words, the node voltage of the distribution network is affected by all the grid-connected PV cells, to present the mutual coupling, which increases the difficulty of the relational analysis.

Therefore, how to intuitively and accurately characterize the effect caused by the grid-connected PV is one of the key aspects addressed in this paper.

The power flow of a traditional distribution network is in the form of a feeder line; the voltage of node i is related to the power and the line impedance, as in the following (Demazy et al., 2020; Wan, 2022; Yang et al., 2023): U i ≈ U i − 1 − PR − Q X U i (4) where P is the active power; Q is the reactive power; R and X are the equivalent line impedance of the node i-1 and node i, respectively.

In traditional distribution networks, most of time, the load R _L is connected into a node. However, with the development of new energy, more and more DGs are connected into distribution networks. The connected node is usually called PCC (point of common coupling). The modeling for the PCC voltage can be shown as the following: U P C C = U i − 1 − P P V − P L R − Q P V − Q C X U P C C (5) where P _PV and Q _PV are the active power and reactive power of the PV system into the PCC, respectively; Q _C is the reactive power of the local compensation.

It is assumed that the reactive power demand in PV systems and the load can be compensated completely by the local compensation device. Thus, the above formula can be simplified as the following: U P C C = U i − 1 − P P V − P L R i − 1 P C C − Q X i − 1 P C C U P C C (6) where Q is the reactive power flow in the bus.

For the next node i+1, its active power can be obtained as the following: P P C C i + 1 = p i − 1 P C C − P L P C C + P P V P C C (7)

The above discussion is about a distribution network connected a single PV. In cases where multiple PVs are connected into the distribution network, the modeling for node voltage is as follows: U 1 = U N − U e 01 = U N − P 01 R 01 − Q 01 X 01 U 1 (8) where P ₀₁ and Q ₀₁ are the active power and reactive power of the initial node and its successive node, respectively; R ₀₁ and X ₀₁ are the equivalent line impedance between the initial node and its successive node, respectively.

Similarly, the voltage of node two and node three of the system is as follows: U 2 = U 1 − U e 12 = U 1 − P 12 R 12 − Q 12 X 12 U 2 = U N − P 01 R 01 − Q 01 X 01 U 1 − P 12 R 12 − Q 12 X 12 U 2 (9) U 3 = U 2 − U e 23 = U 2 − P 23 R 23 − Q 23 X 23 U 3 = U N − U e 01 − U e 12 − U e 23 (10)

And then, the voltage of node j can be obtained as the following: U j = U N − ∑ U e i j = U N − P i j R i j − Q i j X i j U j (11)

If there are multiple PV cells connected into the node of a distribution network, the power between the node i and node j is as follows: P i j = p i − 1 i − P L i + P P V i = P 01 − ∑ P L i + ∑ P P V i (12) Q i j = Q i − 1 i − Q L i − Q P V i − Q C i (13) where P _(i-1)i is the active power; Q _(i-1)i is the reactive power; P _Li , Q _Li , and Q _Ci are the real power, reactive power, and the compensation reactive power of node i, respectively. P _PVi is the real power of the PV cells in node i. Assuming that the reactive power of the node can be compensated completely and locally, the above, Formula (13), can be simplified as follows: Q i j ≈ Q 01 (14)

Taking Formula (14) and Formula (12) into Formula (11), the voltage model for node j is as follows: U j = U N − ∑ P 01 − ∑ P L i + ∑ P P V i R i j − Q 01 X i j U j (15)

As mentioned above, the output power of a PV can be seen as a discrete process. Therefore, the Markov chain can be used to describe the uncertainty in power transition. For the PV system in node i of a network, the PV power can be presented as a time series: P P V j = P P V 0 , P P V 1 , P P V 2 , ⋅ ⋅ ⋅ , P P V n (16) where n is the time number; P _PVj is the output power in each time.

The output power P _PVj can be converted to the power state as follows: M P V j = M P V 0 , M P V 1 , M P V 2 , ⋅ ⋅ ⋅ , M P V k (17) where k is the number of state; M _PVj is the power state.

In this case, a T function can be defined to achieve the conversion between the PV power and the power state: M P V j k = T P P V j n − 1 (18)

As mentioned above, the state transition processes of the PV power state can be represented as a state transition matrix A in Formula (3). Therefore, the PV power state conversion can be presented as follows: M P V j r = M P V j k A (19)

And the PV power in the nth time can be obtained by the inverse transformation, as follows: P P V j n = T − 1 M P V j r + σ (20) where σ is the correction factor. In the above process, the power error may be appear, therefore this correction factor is used to correct the PV power. While the Markov chain is constructed for a PV system, the transition error distribution can be extracted and it is used to generate the correction factor, namely, the correction factor can be obtained using the historical error statistics.

As can be seen from the above formulas, the Markov chain is used to describe the PV power uncertainty transfer process; the power uncertainty is represented as a probability function.

Taking Formula (20) into Formula (15), the voltage of node j can be obtained as follows: U j = U N − ∑ U e i j (21)

The voltage error is as follows: U e i j = ∑ P 01 − ∑ P L i + ∑ ( T − 1 ( ( T P P V j n − 1 A + σ ) R i j − Q 01 X i j U j (22)

As can be seen from the above formulas, the PV power uncertainty is represented by the Markov chain, and then, Formula (22) shows the global voltage effect of the distribution network resulting from the PV power uncertainty.

Generally, In this paper, the Markov chain is used to present the uncertainty of PV power, in which the PV power in time t+1 can be obtained by the PV power in time t. Furthermore, the Markov model is combined with the above mentioned voltage model of distribution network, to describe influence of the voltage on the distribution network nodes resulted from the uncertainty of PV power.

A sensitivity function is usually used to describe the relationship among the variables quantificationally, as in the following (Xiong et al., 2021; Zhou and Zhang, 2021): S = d y d x (23) where y is dependent variable; x is the independent variable.

In this paper, in order to describe the relationship between the node voltage and the grid-tied PV power, the sensitivity function is developed as follows: S e P P V j U e = ∂ U e i j ∂ P P V j = ∂ ∑ P 01 − ∑ P L i + ∑ ( T − 1 ( ( T P P V j n − 1 A + σ ) R i j − Q 01 X i j U j ∂ P P V j (24)

The above function is called the Markov global sensitivity, in which the power uncertainty of all the PV systems in a distribution network can be represented by the Markov chain.

As seen from the above mentioned, the proposed model is combined with the Markov chain, the voltage model of distribution network, and the sensitively function. By using the proposed model, the influence of the distribution network resulting from the PV power uncertainty can be characterized.

In order to test the feasibility of the proposed method, the voltage distribution of network is calculated by the proposed method and the traditional method, as shown in Figure 3.

It can be seen that the voltage distribution is basically consistent by using the proposed method and the traditional method, this verify the feasibility of the proposed method. Further, comparing with the traditional method, the proposed method combines the uncertainly analysis and the sensitivity analysis in the following test expediently.

The PV systems are connected into node nine of the distribution network. In this case, the power flow of the network will change, resulting in changing of the voltage distribution of the nodes; the voltage distribution and the voltage deviation are shown in Figure 4.

It can be seen that the effect on the voltage is finite because the capacity of a single PV system is not large compared with the distribution network. It can be seen that though the voltage of the grid-tied node is mainly affected, the voltage of the other nodes are also effected by a grid-tied PV system.

In this case, the three PVs are connected into the distribution network in node 4, node 7, and node 13, respectively. The voltage distribution of the nodes and the voltage deviation is shown in Figure 5.

It can be seen that, in terms of the connection of the PV system, the power flow of system is changed, leading the voltage distribution changing. And the effect of the node voltage is increasing, especially for the nodes which are grid-tied in the PV system.

From the sensitivity analysis of the node voltage, it can be seen that the node voltage will be affected by the all grid-tied PV system; the degree of influence can be estimated by the proposed Markov global sensitivity in this paper. For example, the voltage of node five is effected by the grid-tied PV cells, though it is not a node that is connected to a PV.

In order to analyze the voltage of node 5, in the experiment, the grid-tied node of the PV cell is changed to observe the voltage change, as shown in Figure 6.

As can be seen from the figures, the voltage distributed of node is different as the PV system are connected into the system. The changing trend of the node voltage is different, which can be described by the voltage sensitivity by using the proposed Markov global sensitivity, as shown in Figure 7 from the sensitivity analysis, it can be seen that the node voltage will effected by all the grid-tied PV cells, and the degree of influence is different.